# Approaches to designing the revenue sharing contract under asymmetric cost information

Approaches to designing the revenue sharing contract under asymmetric cost information Abstract We investigate different approaches to designing price-only and revenue sharing (RS) contracts between a manufacturer and a newsvendor retailer in the case of asymmetric cost information. The manufacturer, i.e. contract designer, has $$N$$ estimations about the retailer’s cost with the corresponding probabilities. We address two general approaches: in Approach 1, the manufacturer designs the contracts using the average of cost estimations, while in Approach 2, the manufacturer designs the contracts using all cost estimations. In Approach 2, we first model a reservation profit for each retailer type, explicitly based on the price-only contract (Approach 2-1). Then, we investigate how to design a unique RS contract (Approach 2-2) and a menu of coordinating RS contracts (Approaches 2-3 and 2-4). Approaches 2-2, 2-3 and 2-4 are designed based on a cut-off policy in which the manufacturer offers a price-only contract to the retailer if the retailer’s cost is greater than a specific level. We find that in Approach 1, the price-only contract is more profitable than the RS contract for the manufacturer. Both firms’ expected profits are increased considerably in Approach 2, as compared to Approach 1. Approach 2-4 results in the highest expected profit for both firms. However, if this approach leads to high administrative costs for the manufacturer, Approach 2-2 can be an appropriate substitute. Numerical analysis and a number of managerial and analytical insights are presented. 1. Introduction In this paper, we focus on the revenue sharing (RS) contract as one of the most important types of coordinating contracts. We investigate different approaches to designing this contract under cost information asymmetry between a manufacturer and a retailer. According to the regular parameters of an RS contract, i.e. $$(w,\Phi )$$, the contract designer reduces its wholesale price $$(w)$$ in return for receiving a portion of its partner’s revenue $$(1-\Phi )$$. Video rental industries are the potential users of this type of contract (Cachon & Lariviere, 2005). In practice, the RS contract designer is not usually aware of its contract partner’s cost information, which is often private. Thus, in this research we seek to investigate how a manufacturer should design this contract in the presence of information asymmetry about the retailer’s cost. We assume that the manufacturer knows $$N$$ different estimations about the retailer’s cost. We define a retailer type for each cost estimation. According to the literature on the mechanisms for contract design in the case of information asymmetry (see, for instance, Baron et al., 1982), a ‘menu of contracts’ can maximize the contract designer’s expected profit. A menu of contracts consists of several contracts designed so that each retailer type selects a specific contract that is appropriately designed for its type. In this paper, in addition to the design of a menu of contracts, we investigate other approaches to contract design in the case of information asymmetry. In reality, contract partners expect a minimum level of profit as a reservation profit from each trade. The reservation profit can be the profit that each entity may earn by investing in an alternative market. In this paper, we model a specific reservation profit for the retailer based on the price-only contract. This is the simplest contract imaginable between a manufacturer and a retailer in which the manufacturer offers only a wholesale price to the retailer and the retailer decides on order quantities. The reservation profit considered for the retailer depends on the retailer type, i.e. the retailer’s cost level. Çakanyildirim et al. (2012) have emphasized the importance of considering type-dependent reservation profit, suggesting that an important issue for future research is the analysis of a model assuming type-dependent reservation profit and $$N$$ different states for asymmetric information. In this paper, we consider these assumptions. We include an outside option for the manufacturer as well as the retailer, based on the price-only contract. Along this line, we investigate some approaches based on the cut-off level policy. Based on this, the manufacturer may not want to motivate every retailer type with any cost level to participate in the RS schemes. The manufacturer can offer a price-only contract to the retailer if the retailer’s cost is greater than a specific level. We address three research questions, considering both theoretical and managerial concerns: (1) How can we simplify the design of a menu of contracts, under information asymmetry? For example, whenever the number of contracts in a menu is increased, the menu becomes too complicated to use, thus we should look for other simpler approaches. (2) How will the contract designer’s expected profit change under a simpler approach, as compared to designing a menu of contracts? Suppose a menu of contracts as the most sophisticated approach for contract design is used. The last research question is: (3) How should the manufacturer design a menu of RS contracts so that it leads to a truthful retailer behaviour as well as system coordination? This study, by investigating the research questions brought above, proposes two parent approaches. The approaches are illustrated in Fig. 1: Approach 1 (containing two child approaches) and Approach 2 (containing four child approaches). Fig. 1. View largeDownload slide Introduction of the investigated approaches. Fig. 1. View largeDownload slide Introduction of the investigated approaches. Regarding research question (1), the first simplification for menu design is considered based on the information the manufacturer uses for contract design. In Approach 1, we create a major simplification for contract design by utilizing the average of cost estimations instead of all ($$N)$$ estimations. Approach 1-1 corresponds to the design of the price-only contract, and Approach 1-2 corresponds to the design of RS contract. We introduced a slightly modified RS (SMRS) contract in Approach 1-2 as well. In Approach 2, as opposed to Approach 1, we investigate contract design based on all ($$N)$$ cost estimations. In Approach 2-1, we introduce the outside option and type-dependent reservation profit in which the manufacturer designs a price-only contract based on all cost estimations. By considering research question (1), the second simplification for menu design is proposed in Approach 2, so we consider the option of designing a unique contract instead of designing several contracts for a menu of contracts. Along this line, in Approach 2, in addition to models based on the menu design (Approaches 2-3 and 2-4), we investigate a model as Approach 2-2 that considers the design of a unique RS contract. We believe that when a menu of contracts is not usable for the manufacturer, it can design a unique RS contract using Approach 2-2. Notice that here as opposed to the first simplification, all the estimations are used for contract design. Regarding research question (2), we conduct an extensive numerical analysis to compare expected profits between sophisticated and simplified approaches. Regarding research question (3), first in Approach 2-3, we investigate the design of a menu of coordinating RS contracts based on the regular parameters of the RS contract $$(w,\Phi )$$. Then, we move onto a tailored version of Approach 2-3, as Approach 2-4, in which the manufacturer offers coordinated order quantities in addition to the regular RS parameters. It should be mentioned that our main focus is on the RS contract. However, in different approaches, we consider the price-only contract as well. In Approach 1, we investigate the design of the price-only contract (Approach 1-1) so that we can compare the expected profits in Approaches 1-1 and 2-1. In Approach 2, we use the price-only contract as an outside option that determines the reservation profit of contract partners. Seven interesting theoretical and managerial insights are derived from this paper. First, we find that in Approach 1, the design of a price-only contract is more profitable for the manufacturer than the design of an RS contract. This is against our usual expectation in the case of information symmetry, where a coordinating contract can improve the contract designer’s expected profit. Second, we find that the manufacturer’s expected profit under Approach 2 is considerably more than that under Approach 1. This finding highlights the value of considering all estimations. Third, we show that a menu of coordinating RS contracts, according to Approach 2-3, does not motivate the retailer to behave truthfully. Fourth, the most sophisticated approach (Approach 2-4) yields the highest expected profit for the manufacturer. However, our numerical analysis shows that the preference for Approach 2-4 over Approach 2-2 is insignificant. The key managerial insight here is that if the implementation of Approach 2-4 in practice is difficult, Approach 2-2 could be an appropriate substitute. Fifth, the optimal features of the model in Approach 2-2 are investigated when demand distribution has an increasing generalized failure rate (IGFR). IGFR distributions, such as normal distribution, usually occur in practice. Sixth, all of our findings and observations hold true for other contracts that are equivalent to an RS contract, such as a buy-back contract. Seventh, based on Approach 2-4, we obtain each retailer type’s optimal profit by considering type-independent reservation profit. This paper is organized as follows: first we review the related literature in Section 2 and present the model statements, assumptions and notation in Section 3. Then, in Sections 4 and 5, we investigate Approach 1 and Approach 2, respectively. In Section 6, we present numerical analysis. Section 7 investigates two other cases of the paper. Finally, the conclusion is presented in Section 8. 2. Literature review In this section, we shed light on the similarities and differences between our paper and the related literature on contract design under cost information asymmetry, from different perspectives. 2.1. The contract designer: upstream or downstream member A contract designer might be an upstream or downstream member in a supply chain. A change in the contract designer’s position leads to changes in the leadership situation and managerial aspects of the supply chain. Some previous studies have considered the issue of cost information asymmetry when a downstream supply chain member designs a contract for its upstream member. For instance, Lau et al. (2007) and Wang et al. (2009) studied how a dominant retailer who did not have a perfect knowledge from the manufacturer’s cost should design a contract. Lau et al. (2007) examined a volume discount scheme from the retailer’s perspective, while Wang et al. (2009) investigated whether the dominated manufacturer could be motivated by the retailer to improve the quality of the retailer’s knowledge. In both studies, market demand was deterministic. In contrast, in our paper demand is stochastic and the upstream member, i.e. the manufacturer, is the contract designer. Xu et al. (2010) considered a supply chain with overseas suppliers, local suppliers and one manufacturer in which the manufacturer did not know the local suppliers’ production cost. They investigated designing a contract menu consisting of a transfer payment and a lead-time quotation, while in this paper we focus on designing the RS contract with the price-only contract as an outside option. Xu et al. (2013) extended their study in Xu et al. (2010) by investigating the effects of time-sensitive characteristics on decisions. Çakanyildirim et al. (2012) analysed a supply chain in which a downstream member (a retailer) did not know the production cost of its upstream member (a supplier). They showed that information asymmetry alone did not induce loss in channel efficiency. Similar to Ha (2001), Corbett et al. (2004), Mukhopadhyay et al. (2008) and Lutze & Özer (2008), we investigate contract design under information asymmetry from the perspective of an upstream member (the manufacturer) for a downstream member (the retailer). 2.2. The implemented contracts: price-only and RS contracts Different contracts are used in the literature on contract design under cost information asymmetry. Several studies, like Ha (2001), Corbett et al. (2004), Mukhopadhyay et al. (2008) and Özer & Raz (2011), have investigated two part linear and non-linear pricing contracts. Cachon & Zhang (2006), Chakravarty & Zhang et al. (2007) and Lutze & Özer (2008), respectively, investigated late-fee mechanisms, Bayesian capacity investment games and promised lead-time contracts. We present some models under specific approaches considering the RS and price-only contracts. These contracts have been used before in several studies under information asymmetry. Studies like Corbett et al. (2004) and Cao et al. (2013) implemented wholesale price contract under information asymmetry, while designing a menu of contracts. In this paper, the price-only contract is considered as an outside option for the RS contract partners. In the literature dealing with the RS contract (see, for instance, Giannoccaro & Pontrandolfo, 2004; Cachon & Lariviere, 2005; Qin, 2008; Wang et al., 2012; Xu et al., 2014; Peth & Thonemann, 2016), by assuming access to all information, the major emphasis has been placed on showing how the contract leads to supply chain coordination and a win–win situation for supply chain members. A few studies investigated the RS contract under information asymmetry. Çakanyildirim et al. (2012) investigated a profit sharing contract as a scheme equivalent to the RS contract, between a retailer and a supplier. The retailer knew two low and high cost states from the supplier’s cost. They analysed the effects of outside opportunities on the retailer-supplier relationship. Different from Çakanyildirim et al. (2012), our study considers $$N$$ different states for the asymmetric information. Zhou et al. (2012) investigated the impact of customers learning about product value on the marketing efforts of a supplier and a retailer. The supplier offered an RS contract to the retailer. Their model considered asymmetric information about the supplier’s and the retailer’s marketing efforts. Our study is similar to their work in that we study the RS contract design by an upstream member in a supply chain in the case of information asymmetry. However in this study, we focus on the modelling of cost information asymmetry under specific approaches. Shen & Willems (2012) studied the coordination of a manufacturer-retailer system using a buy-back contract as a scheme equivalent to the RS contract. Our research, like theirs, considers the retailer’s cost information as private information. However, we consider discrete cost estimations and type dependent reservation profit for the retailer. 2.3. The approach to model information asymmetry: discrete or continuous Several studies, like Xu et al. (2010), Özer & Raz (2011), Çakanyildirim et al. (2012), Xu et al. (2013) and Cao et al. (2013), have modelled information asymmetry by considering just two low and high states for the asymmetric information. Our paper extends these studies by considering, in general, $$N$$ different states. To our knowledge, Lutze & Özer (2008) is the only study that has considered, in general, $$N$$ different states for the asymmetric information. They investigated a promised lead-time contract between a supplier and a retailer in which the supplier knew a finite set of estimations from the retailer’s service level. The investigated contracts and approaches in our paper are essentially different from their work. Some other studies, like Ha (2001), Corbett et al. (2004), Cachon & Zhang (2006), Chakravarty & Zhang et al. (2007), Mukhopadhyay et al. (2008) and Zhang et al. (2014), have modelled information asymmetry by considering a prior continuous probability distribution for the asymmetric information, e.g. cost. 2.4. The menu design approach The usual approach to contract design in the case of information asymmetry is to design a menu of contracts; see, e.g. Ha (2001), Corbett et al. (2004), Cachon & Zhang (2006), Chakravarty & Zhang et al. (2007), Lutze & Özer (2008) and Fang et al. (2014). Our paper differs from these studies by taking into account and comparing other approaches based on administrative considerations, in addition to menu design. Zhang et al. (2014) analysed an information asymmetry model in a supply chain that included a dominant supplier and a manufacturer using a different approach. Their approach was based on a principle-agent model as Pontryagin’s maximum principle. 2.5. The reservation profit: type-independent or type-dependent Another specific feature that distinguishes our paper from the aforementioned studies is that we consider type-dependent reservation profit for the retailer in some of the approaches. The issue of type-dependent reservation profit has not received due consideration in operations management literature (Çakanyildirim et al., 2012). A few studies, like Laffont & Martimort (2002), Chakravarty & Zhang et al. (2007), Xu et al. (2010) and Çakanyildirim et al. (2012), have considered type-dependent reservation profit for the contract partner in their works. However, in these studies, the reservation profit is a given parameter whose origin might not be clear. In this paper, we model the reservation profit explicitly based on a widely used contract, i.e. the price-only contract. In addition, in Section 7.2, we examine Approach 2-4 under the assumption of type-independent reservation profit in order to get some analytical results. Therefore, our paper has a number of unique features compared to the literature reviewed. As far as we know, there are few studies addressing the implementation of the RS contract under information asymmetry. Thus, in this research, we design our models by assuming the RS contract between the manufacturer and the retailer. The investigated approaches are different from those in the studies mentioned. In this research, in addition to designing a coordinating menu of RS contracts, we investigate simplified approaches that might be relatively easier to use. Also, we consider discrete cost estimations and type-dependent reservation profit for the retailer. 3. Model statements, assumptions and notation Consider a two-echelon supply chain with a dominant manufacturer and a retailer. The retailer, operating in a newsvendor-type environment, sources its products from the manufacturer. The manufacturer’s and the retailer’s unit costs are $$c_{m}$$ and $$c_{r}$$, respectively. These costs might include handling, processing, assembling or packaging costs. The manufacturer’s wholesale price is $$w>c_{m}$$ and the retailer’s exogenous sale price is $$p>w+c_{r}$$. $$X$$ represents the random market demand with a probability distribution function of $$f_{X} (x)$$. $$F$$(.) is a continuous and differentiable distribution function, where $$\bar{F}$$(.)$$=1-F$$(.). We assume that there is no shortage penalty and no salvage value for inventory leftover after the selling season. Similar to Giannoccaro & Pontrandolfo (2004) and Cachon & Lariviere (2005), suppose that the manufacturer offers an RS contract with parameters $$(w,\Phi )$$ to the retailer. According to the analysis presented by Cachon & Lariviere (2005), in order to coordinate the retailer’s optimal order quantity the RS contract parameters have to be adjusted to answer: $$w=\Phi \left( {c_{m} +c_{r} } \right)-c_{r}$$. Therefore, the manufacturer needs to know the retailer’s cost in order to optimally adjust the RS contract parameters. But such information is usually the retailer’s private property, so the manufacturer might not be aware of its exact value. Assume the manufacturer has a set of $$N$$ positive prior estimations from the retailer’s cost structure $$\{c_{r,1} <c_{r,2} <c_{r,3} <...<c_{r,N} \}$$ representing retailer type $$i=1,2,3,\ldots ,N$$, with probabilities $$\lambda_{1} ,\lambda_{2} ,\lambda_{3} ,\ldots ,\lambda_{N} (\sum_{i=1}^{N} \lambda_{i} =1)$$. The estimations only include those that allow for beneficial business, denoting that: $$p>c_{m} +c_{r,N}$$. Assume that the retailer’s true cost belongs to this set and this discrete distribution is public information. These assumptions are similar to Lutze & Özer (2008) as they assumed, in general, $$N$$ different states for the asymmetric information with the corresponding probabilities as the public information. 4. Approach 1: contract design based on the average of cost estimations Suppose the manufacturer wants to design a contract only based on the average value of cost estimations $$\bar{c}_{r} = \sum_{i=1}^{N} \lambda_{i} c_{r,i}$$. 4.1. Approach 1-1: price-only contract based on $$\bar{c}_{r}$$ Suppose the manufacturer offers a wholesale price $$(\bar{w}_{p} )$$ to the retailer, assuming the retailer’s cost is $$\bar{c}_{r}$$. In this situation, the manufacturer’s optimization problem is as follows:   Maxw¯p>cmΠmPC¯=(w¯p−cm)qPC¯(w¯p), (1)  qPC¯(w¯p)={F−1[p−w¯p−c¯rp] if w¯p<p−c¯r0 Otherwise . (2) $$\Pi_{m}^{\overline{\rm PC}}$$ represents the manufacturer’s expected profit under the price-only contract. Constraint (2) represents dependency of a retailer’s order quantity with a cost of $$\bar{c}_{r}$$ to $$\bar{w}_{p}$$. According to Lariviere & Porteus (2001), by assuming an IGFR demand distribution, the unique optimal $$\bar{w}_{p}$$ ($$\bar{w}_{p}^{\ast})$$ satisfies the following equation:   dΠmPC¯dw¯p=F−1[p−w¯p∗−c¯rp]−(w¯p∗−cm)pf(F−1[p−w¯p∗−c¯rp])=0. (3) It should be mentioned that a continuous random variable $$X$$ with density function $$f(x)$$ and distribution function $$F(x)$$ is IGFR if its generalized failure rate (GFR) $$g\left( x \right)=\frac{xf(x)}{1-F(x)}$$ increases in $$x.$$ IGFR distribution is a mild restriction because it captures most common distributions with increasing failure rates e.g. Normal, Uniform, Gamma and Weibull distributions. In reality, however, there may be various retailer types with different costs. A retailer’s order quantity with the actual cost of $$c_{r,i}$$ can be obtained from Relation (4):   qiPC¯(w¯p∗)={F−1[p−w¯p∗−cr,ip] if w¯p∗<p−cr,i0 Otherwise . (4) Consider $$S\left( q \right)=q-\smallint_{0}^{q} F\left( x \right)dx$$ as expected sales when the retailer places an order of $$q$$ units. Therefore, each retailer type’s expected profit in this price-only contract is as follows:   Πr,iPC¯=pS(qiPC¯(w¯p∗))−qiPC¯(w¯p∗)(w¯p∗+cr,i)∀i=1,2,3,…,N. (5) In this approach, the manufacturer does not consider all the cost estimations. In reality, different costs of different retailer types lead to an expected profit for the manufacturer, which differs from $$\Pi_{m}^{\overline{\rm PC}}$$ in (1). In this situation, the manufacturer’s actual expected profit is as follows:   ΠmaPC¯=∑i=1Nλi(w¯p∗−cm)qiPC¯(w¯p∗). (6) 4.2. Approach 1-2: RS contract based on $$\bar{c}_{r}$$ Suppose the manufacturer wants to offer an RS contract $$(\bar{w},\bar{\Phi})$$ to the retailer, assuming the retailer’s cost is $$\bar{c}_{r}$$. In this situation, the manufacturer’s optimization problem is as follows:   Max(w¯,Φ¯)ΠmRS¯=(w¯−cm)qRS¯+(1−Φ¯)pS(qRS¯), (7)  S.T.qRS¯={F−1[Φ¯p−w¯−c¯rΦ¯p] if w¯<Φ¯p−c¯r0 Otherwise , (8)  0<Φ¯⩽1,−c¯r<w¯. (9) $$\Pi_{m}^{\overline{\rm RS}}$$ represents the manufacturer’s expected profit under the RS contract. Constraint (8) represents the dependency of a retailer’s order quantity with the cost of $$\bar{c}_{r}$$ to $$\bar{w}$$ and $$\bar{\Phi}$$. Constraint (9) represents a feasible range for the decision variables. The lower bound of $$\bar{w}$$ prevents infinite orders from a retailer with the cost of $$\bar{c}_{r}$$. Since in the RS contract the manufacturer can obtain some portion of its partner’s revenue $$(1-\bar{\Phi})$$, $$\bar{w}$$ can become negative. According to Cachon & Lariviere (2005), the interpretation for a negative $$\bar{w}$$ is as follows: whenever the retailer’s cost is high, the manufacturer should subsidize the retailer in a low-margin business in order to claim a specific portion of the retailer’s revenue. The manufacturer can adjust the RS contract parameters to maximize its expected profit, while allocating zero profit to a retailer with the cost of $$\bar{c}_{r}$$. The maximum profit will be available for the manufacturer if the retailer’s order quantity equals the optimal order quantity in the system. According to the classic newsvendor model, the system’s optimal order quantity satisfies $$F^{-1}\left[ {\frac{p-c_{m} -\bar{c}_{r} }{p}} \right]$$. Therefore, an optimal relation between $$\bar{w}$$ and $$\bar{\Phi}$$ can be obtained through the following equation:   F−1[Φ¯p−w¯−c¯rΦ¯p]=F−1[p−cm−c¯rp]⟹w¯=Φ¯(cm+c¯r)−c¯r. (10) According to the analysis of Cachon & Lariviere (2005), in this situation $$\bar{\Phi}$$ is the retailer’s share of the supply chain’s profit. Thus, in order to allocate the least profit share to the retailer, the optimal value of $$\bar{\Phi}$$ ($$\bar{\Phi}^{\ast})$$ moves toward zero. In this situation, the optimal value of $$\bar{w}$$ ($$\bar{w}^{\ast})$$ moves toward $$-\bar{c}_{r}$$. However, the fact is that there are different retailer types each with a specific cost. A retailer’s order quantity with the actual cost of $$c_{r,i}$$ can be obtained from Relation (11):   qiRS¯={F−1[Φ¯∗p−w¯∗−cr,iΦ¯∗p] if w¯∗<Φ¯∗p−cr,i0 Otherwise . (11) Therefore, each retailer type’s actual expected profit in the case of this RS contract is as follows:   Πr,iRS¯=Φ¯∗pS(qiRS¯)−qiRS¯(w¯∗+cr,i) ∀i=1,2,3,…,N. (12) Putting it differently, the manufacturer’s actual expected profit, which differs from $$\Pi_{m}^{\overline{\rm RS}}$$ in (7), is as follows:   ΠmaRS¯=∑i=1N⁡λi[(w¯∗−cm)qiRS¯+(1−Φ¯∗)pS(qiRS¯)]. (13) Proposition 1 In Approach 1, the manufacturer’s actual expected profit under the price-only contract is higher than that under the RS contract. All proofs are presented in Appendix. According to Proposition 1, the manufacturer always prefers the price-only contract to the RS contract, when it designs a contract only based on $$\bar{c}_{r}$$. We can improve the manufacturer’s actual expected profit in the case of RS contract by implementing a minor modification. If we increase the lower bound of $$\bar{w}$$ from $$-\bar{c}_{r}$$ to $$-c_{r,1}$$, then the retailer’s order quantity with any cost level will not move toward infinity. In this situation, by considering the manufacturer’s expected profit in Relation (13), the positive profit share from the retailer’s revenue, i.e. $$\left( {1-\bar{\Phi}^{\ast}}\right)pS\left( {q_{i}^{\overline{\rm RS}}} \right),$$ might compensate the contingent negative income from selling to the retailer, i.e., $$\left( {\bar{w} ^{\ast}-c_{m}} \right)q_{i}^{\overline{\rm RS}}$$ if $$\bar{w}^{\ast}<0$$. As a result of this modification, the manufacturer finds a chance for a positive expected profit. We call this a SMRS contract and use it later in the numerical analysis. 5. Approach 2: contract design based on all cost estimations In this section, we investigate more sophisticated approaches in which the manufacturer considers all cost estimations for contract design in the case of cost information asymmetry. Approach 2-1 addresses the price-only contract as an outside option that establishes each firm’s reservation profit in Approaches 2-2, 2-3 and 2-4. In Approaches 2-2, 2-3 and 2-4, we focus on the design of the RS contract. 5.1. Approach 2-1: price-only contract as an outside option Suppose the manufacturer offers a wholesale price ($$w_{p})$$ to the retailer, by considering all types of the retailer. The manufacturer’s problem is:   Maxwp>cmΠmPC1=∑i=1N⁡λi(wp−cm)qiPC1(wp), (14)  S.T.qiPC1(wp)={F−1[p−wp−cr,ip]  if wp<p−cr,i0 Otherwise . (15) Relation (15) represents the retailer’s optimal order for each $$w_{p}$$, based on the newsvendor model. According to Lariviere & Porteus (2001), by assuming an IGFR demand distribution, the unique optimal $$w_{p}$$ satisfies:   dΠmPC1dwp=∑i=1Nλi(F−1[p−wp∗−cr,ip]−(wp∗−cm)pf(F−1[p−wp∗−cr,ip]))=0. (16) Notice that if the IGFR assumption does not hold, we can find the optimal value of $$w_{p}$$ through a one-dimensional search procedure. Consider $$w_{p}^{\ast}$$ as the optimal value of $$w_{p}$$. We assume each retailer type’s expected profit, in the case of this price-only contract, as its reservation profit to participate in any other contract offered by the manufacturer:   RPr,i=pS(qiPC1(wp∗))−qiPC1(wp∗)(wp∗+cr,i) ∀i=1,2,3,…,N. (17) Note that $$\mbox{RP}_{r,i}$$ depends on the retailer type, and regarding (15), it might be zero for some types. We also consider the price-only scheme as an outside option for the manufacturer. It provides the manufacturer with the right to sign a price-only contract with some retailer types if they are not chosen for the RS contract. 5.2. Approach 2-2: design of an RS contract and a price-only contract Regarding the manufacturer’s outside option based on the price only contract, it may not accept to enter the RS scheme with every retailer type. We considered this issue in the following models through a cut-off policy (Ha, 2001; Corbett et al., 2004). We define $$c_{r,U} \in \{c_{r,1} ,c_{r,2} ,c_{r,3} ,\ldots ,c_{r,N} \}$$ as a cut-off point. The manufacturer designs a single RS contract for a retailer with a cost equal to or less than $$c_{r,U}$$ and it will trade with the other types of the retailer $$\left( {i=U+1,\ldots ,N} \right)$$ according to a price-only contract. The single RS contract is identical for different retailer types. The following steps determine the sequence of decisions in Approach 2-2. Step 1: The manufacturer offers an RS contract with parameters $$(w,\Phi )$$ to the retailer. The RS contract is designed so that if the retailer’s cost is equal to or less than $$c_{r,U}$$, its participation constraint is satisfied and therefore, the retailer will accept the RS contract. Otherwise, the retailer’s cost is greater than $$c_{r,U}$$ and it waits for the next offer in Step 2. Step 2: The manufacturer offers a price-only contract to the retailer. The price-only contract satisfies the retailer’s participation constraint as well. Therefore, the manufacturer’s optimization problem in Approach 2-2 is:   Max(U,w,Φ,wp ′)ΠmRS1=∑i=1Uλi[(w−cm)qiRS1+(1−Φ)pS(qiRS1)]+∑i=U+1Nλi(wp ′−cm)qi ′PC1(wp ′), (18)  S.T.ΦpS(qiRS1)−qiRS1(w+cr,i)⩾RPr,i∀i=1,2,3,…,U, (19)  ΦpS(qiRS1)−qiRS1(w+cr,i)⩽RPr,i∀i=U+1, …, N, (20)  qiRS1={F−1[Φp−w−cr,iΦp]  if w<Φp−cr,i0 Otherwise ∀i=1,2,3,…,N, (21)  pS(qi ′PC1)−qi ′PC1(wp ′+cr,i)⩾RPr,i∀i=U+1, …, N, (22)  qi ′PC1(wp ′)={F−1[p−wp ′−cr,ip] if wp ′<p−cr,i0 Otherwise  ∀i=U+1, …, N, (23)  U∈{0,1,2,3,…,N},0<Φ⩽1,−cr,1<w,cm⩽wp ′. (24) The first summation of the objective function is the manufacturer’s expected profit based on the RS contract, with retailer types being no more than $$U$$. Since it is decreasing in the retailer’s cost, the cut-off policy will be optimal (Ha, 2001). The second summation of the objective function represents the manufacturer’s expected profit based on the price-only contract with retailer types more than $$U$$. Notice that the wholesale price in the second summation of the objective function ($$w_{p}^{'})$$ differs from that in Relation (14), i.e. $$w_{p}$$, unless we have $$U=0$$. Constraints (19)–(21) belong to Step 1 of the sequence of decisions. Constraint (19) represents the participation constraint for retailer types no more than $$U$$ in the RS scheme. $$\mbox{RP}_{r,i}$$ in the constraints comes from Relation (17). Constraint (20) prevents participation of a retailer with a cost more than $$c_{r,U}$$ in the RS scheme. Relation (21) represents the retailer’s order quantity under the RS scheme. Constraints (22) and (23) belong to Step 2. Constraint (22) represents the participation constraint for retailer types more than $$U$$ in the price-only contract. Relation (23) represents the retailer’s order quantity under the price-only contract. Relation (24) represents the feasible range of variables. In order to prevent the retailer’s tendency toward infinite orders, $$w$$ has to be greater than $$-c_{r,1}$$. 5.2.1. General solution methodology for Approach 2-2 It should be mentioned that the objective function in Approach 2-1 is the simplest case of the objective function in Approach 2-2, and also Approaches 2-3 and 2-4, for $$U=0.$$Lariviere & Porteus (2001) have established that all we need to guarantee the concavity of this type of objective function, with respect to the retailer’s order quantity, is an IGFR demand distribution. We can then find the corresponding optimal wholesale price as well. However, in general without an IGFR assumption, we cannot be sure about the concavity of this type of objective function in Approach 2-1 and the other more sophisticated approaches (Approaches 2-2, 2-3 and 2-4). Thus, here we first propose a general solution methodology. Then in Section 5.2.2, we present some analytical results considering IGFR distribution for demand. Step 1: Set $$U=0$$. Step 2: Find the optimal RS contract for a retailer type $$i=1,2,3,\ldots ,U,$$ as shown below: Step 2.1: By considering Constraints (19)–(21) and (24), generate feasible values for $$(w,\Phi )$$ from their entire feasible range. Step 2.2: Find the optimal value for $$(w,\Phi )$$, i.e. the values that optimize the first summation of (18). Step 3: Find the optimal price-only contract for a retailer type $$i=U+1,\ldots ,N$$, as shown below: Step 3.1: By considering Constraints (22)–(24), generate feasible values for $$w_{p}^{'}$$ from its entire feasible range. Step 3.2: Find the optimal value for $$w_{p}^{'}$$, i.e. the value that optimizes the second summation of (18). Step 4: If $$U<N$$, compute the whole Relation (18), set $$U=U+1$$ and go to Step 2. Otherwise, select the optimal value for $$U$$, i.e. the one that optimizes (18). The optimal values for $$(w,\Phi )$$ and $$w_{p}^{'}$$, corresponding to the optimal $$U$$, are determined in Steps 2 and 3. According to the steps mentioned above, we find the optimal solution based on a complete enumeration method by searching the entire space of decision variables with a specific precision. This way, we ensure that without losing any solution space, the search procedure is converged to the optimal solution. 5.2.2. A special case in Approach 2-2: IGFR distribution When the demand distribution is IGFR, we can make sure that the first and second summations in the objective function are unimodal in terms of $$q_{i}^{\rm RS1}$$ and $$q_{i}^{'{\rm PC}1}$$, respectively. Proposition 2 Suppose the demand distribution is IGFR. I. The first summation of (18) is concave with respect to $$q_{i}^{\rm RS1}$$, for each feasible value of $$\Phi$$. II. The second summation of (18) is concave with respect to $$q_{i}^{'{\rm PC}1}$$. We can characterize optimal orders from the manufacturer’s perspective based on the analysis provided by Lariviere & Porteus (2001). Suppose that the support of $$F$$ is $$[a,b)$$ and define $$\bar{q}_{i}^{\rm RS1}$$ as the largest value of $$q_{i}^{\rm RS1}$$ that satisfies $$g(q_{i}^{\rm RS1} )\leqslant \frac{1}{\Phi}$$. In this situation, any solution such as $$q_{i}^{\ast {\rm RS}1}$$ from equalizing Relation (A.6) to zero is unique and must lie in the interval $$[a,\bar{q}_{i}^{{\rm RS}1}]$$. The optimal order quantity for the manufacturer will be either $$q_{i}^{\ast RS1}$$ or $$a$$. The story is the same for $$q_{i}^{\ast {\rm PC}1}$$, by considering $$\bar{q}_{i}^{{\rm PC}1}$$ as the largest value of $$q_{i}^{{\rm PC}1}$$ that satisfies $$g(q_{i}^{{\rm PC}1} )\leqslant 1$$. Note that by finding the optimal value of $$q_{i}^{{\rm RS}1}$$ and $$q_{i}^{'{\rm PC}1}$$, we can find the corresponding optimal value for $$w$$ and $$w_{p}^{'}$$ through inverse demand curves, e.g. Relation (A.4). Proposition 3 Suppose the demand distribution is IGFR. Let $$w^{\ast }=\Phi p \bar{F} \left( {q_{i}^{\ast {\rm RS}1} } \right)-c_{r,i}$$ for each feasible $$\Phi$$ and $$w_{p}^{\ast'} =p \bar{F} \left( {q_{i}^{\ast '{\rm PC}1} } \right)-c_{r,i}$$ for one of the values of ‘$$i$$’. I. If $$w^{\ast }$$ satisfies Relations (19), (20) and (24), it will be the optimal value. If $$w^{\ast }$$ violates one of the constraints in Relation (19), then the optimal value of $$w$$ will be $$w_{U}$$, where $$w_{U}$$ is the largest $$w$$ that satisfies all the constraints in Relation (19). $$w_{U}$$ can be found by considering the constraints in Relation (19) as equations. If $$w^{\ast}$$ violates one of the Constraints (20) or (24), then the optimal value of $$w$$ will be $$w_{L}$$, where $$w_{L}$$ is the smallest $$w$$ that satisfies Constraints (20) and (24). $$w_{L}$$ is determined by considering the Constraints (20) and (24) as equations. II. If $$w_{p}^{\ast'}$$ satisfies Relation (22), it will be the optimal value. If $$w_{p}^{\ast '}$$ violates one of the constraints in Relation (22), then the optimal value of $$w_{p}^{'}$$ will be $$w_{pU}^{'}$$, where $$w_{pU}^{'}$$ is the largest $$w_{p}^{'}$$ that satisfies all the constraints in Relation (22). $$w_{pU}^{'}$$ can be found by considering the constraints in Relation (22) as equations. The optimal value of $$w_{p}^{'}$$, i.e. $$\mbox{Min}\{w_{p}^{\ast '} ,w_{pU}^{'} \}$$, satisfies Relation (24) as well. Proposition 3 helps us find $$w^{\ast }$$ for each feasible $$\Phi$$. We can find the optimal value for $$\Phi$$ based on the general solution methodology through a one-dimensional search procedure. 5.3. Approach 2-3: designing a menu of coordinating RS contracts and a price-only contract The sequence of decisions in Approach 2-3 (and Approach 2-4) is similar to that in Approach 2-2, except that in Step 1, the manufacturer designs a menu of coordinating RS contracts, instead of a single RS contract. Suppose the manufacturer designs a menu of coordinating RS contracts so that each contract in the menu is particular for each retailer type $$i=1,2,3,\ldots ,N$$. The retailer has to choose one contract from the menu. According to the revelation principle, the manufacturer can limit its search for an optimal menu to the class of ‘truth-telling’ contracts (Baron et al., 1982). In this approach, we define $$q_{j,i}^{{\rm RS}2}$$ as the order quantity of a retailer whose actual cost is $$c_{r,i}$$ and it selects the contract designed for a retailer with the cost of $$c_{r,j}$$. The manufacturer’s problem in Approach 2-3 is:   Max(U,wi,Φi,wp ′)ΠmRS2=∑i=1U⁡λi[(wi−cm)qi,iRS2+(1−Φi)pS(qi,iRS2)]+∑i=U+1N⁡λi(wp ′−cm)qi ′PC1(wp ′), (25)  S.T.ΦipS(qi,iRS2)−qi,iRS2(wi+cr,i)⩾RPr,i∀i=1,2,3,…,U, (26)  ΦipS(qi,iRS2)−qi,iRS2(wi+cr,i)⩾ΦjpS(qj,iRS2)−qj,iRS2(wj+cr,i) (27)  ∀i=1,2,3,…,U,∀j=1,2,3,…,U, i≠j,ΦipS(qi,fRS2)−qi,fRS2(wi+cr,f)⩽RPr,f∀i=1,2,3,…,U, f=U+1,…,N, (28)  wi=Φi(cm+cr,i)−cr,i∀i=1,2,3,…,U, (29)  qj,iRS2={F−1[Φjp−wj−cr,iΦjp] if Φjp−wj−cr,i>00 Otherwise ∀i,j=1,2,3,…,N. (30) In addition to Constraints (22) and (23),   U∈{0,1,2,3,…,N},0<Φi⩽1,−cr,1<wi,cm⩽wp ′,∀i=1,2,3,…,U. (31) The first summation in the objective function represents the manufacturer’s expected profit based on the menu of RS contracts. The second summation in the objective function represents the manufacturer’s expected profit based on the price-only contract with retailer types more than $$U$$. Constraints (26)–(30) belong to Step 1 of the decision sequences, as introduced in Section 5.2. Constraints (26) and (27) are ‘individual rationality’ and ‘incentive compatibility’ constraints, respectively (Corbett et al., 2004). Constraint (28) prevents a retailer’s tendency with a cost more than $$c_{r,U}$$ from using the RS menu. Relation (29) determines the coordinating relation between parameters $$(w_{i} ,\Phi_{\mbox{i}} )$$. Relation (30) determines the optimal value of $$q_{j,i}^{\rm RS2}$$. Constraints (22) and (23) for Step 2 recur in this model as well. Constraint (31) determines the feasible value of the decision variables. The lower bound of $$w_{i}$$ ensures that $$F(q_{j,i}^{\rm RS2} )$$ will not be more than one. In order to solve this model, by fixing the value of $$U$$, we have to find the optimal menu of coordinating RS contracts. According to our investigations, this model is analytically intractable. Consequently, we turned to numerical analysis. We conducted extensive numerical investigations with various model parameters and observed in all of them that the menu of contracts was not capable of motivating the retailer to behave truthfully. The following example illustrates this point. 5.3.1. An example for Approach 2-3 Consider two estimations from the retailer’s cost as $$c_{r,l} <c_{r,h}$$. Assume that the parameters of the model are estimated as: $$p=150,\,c_{m} =40,c_{r,l} =10,c_{r,h} =30, \ X\sim \mbox{Normal}(\mu =100,\sigma =20).$$ Suppose the manufacturer wants to design a menu including two coordinating RS contracts. Depending on the retailer’s actual cost and its decision, four cases may occur: Case 1: The retailer’s actual cost is $$c_{r,l}$$ and it selects the contract that is designed for type $$l$$ (coloured in blue). Case 2: The retailer’s actual cost is $$c_{r,l}$$ and it selects the contract that is designed for type $$h$$ (coloured in red). Case 3: The retailer’s actual cost is $$c_{r,h}$$ and it selects the contract that is designed for type $$h$$ (coloured in green). Case 4: The retailer’s actual cost is $$c_{r,h}$$ and it selects the contract that is designed for type $$l$$ (coloured in yellow). In Cases 1 and 3, the retailer reacts truthfully. Figure 2(a) depicts the retailer’s profit under different cases for all feasible values of $$\Phi _{l}$$ and $$\Phi_{h}$$. In order to clearly show the retailer’s behaviour, in Fig. 2(a), we only depict the regions in which each retailer type benefits from not selecting its particular contract, i.e. where the retailer is not motivated for truth-telling. Figure 2(b) shows Fig. 2(a) from the top view. Cases 2 and 4 occur, respectively, in the red and yellow regions. We observe that there is no region in Fig. 2(b) where both retailer types react truthfully. Fig. 2. View largeDownload slide (a) Retailer’s profit in Cases 1-4. (b) Top view of (a). Fig. 2. View largeDownload slide (a) Retailer’s profit in Cases 1-4. (b) Top view of (a). Therefore, Constraint (27) cannot be satisfied and the manufacturer, even when facing just two estimations, cannot motivate the retailer towards truth-telling. Note that the only constraint we can relax from the menu design issue in Approach 2-3 is (29). If we can somehow relax this constraint, while preserving system coordination, by generating different values for $$(w_{i} ,\Phi_{\mbox{i}} )$$ a truth-telling menu may be obtained. We try to create such conditions in Approach 2-4. 5.4. Approach 2-4: designing a tailored menu of coordinating RS contracts and a price-only contract In this approach, we relax Constraint (29); however, in order to keep coordinated order quantities, we use contracts with quantity commitments as used, for instance, by Li et al. (2009) and Gan et al. (2010). Thus, the manufacturer offers a menu of RS contracts as: $$(w_{i} ,\Phi_{\mbox{i}} ,q_{i}^{\rm RS3} )$$ designed for each retailer type $$i=1,2,3,\ldots ,U$$. The manufacturer’s optimization problem is:   Max(U,wi,Φi,wp ′)ΠmRS3=∑i=1U⁡λi[(wi−cm)qiRS3+(1−Φi)pS(qiRS3)]+∑i=U+1N⁡λi(wp ′−cm)qi ′PC1(wp ′), (32)  S.T.ΦipS(qiRS3)−qiRS3(wi+cr,i)⩾RPr,i∀i=1,2,3,…,U, (33)  ΦipS(qiRS3)−qiRS3(wi+cr,i)⩾ΦjpS(qjRS3)−qjRS3(wj+cr,i) (34)  ∀i=1,2,3,…,U,∀j=1,2,3,…,U, i≠j,ΦipS(qiRS3)−qiRS3(wi+cr,f)⩽RPr,f∀i=1,2,3,…,U, f=U+1,…,N, (35)  qiRS3=F−1[p−cm−cr,ip]∀i=1,2,3,…,N. (36) In addition to Constraints (22) and (23),   U∈{0,1,2,3,…,N},0⩽Φi⩽1,wi:free in sign,cm⩽wp ′,∀i=1,2,3,…,U. (37) Description of the objective function is similar to the previous models. The role of the constraints is similar to the model in Approach 2-3. In Relation (36), $$q_{i}^{\rm RS3}$$ represents the coordinated order quantity for a retailer with the cost of $$c_{r,i}$$. Regarding the objective function for each $$U$$, the manufacturer tries to determine $$w_{i}$$ and $$\Phi_{i}$$ so that it owns as much coordinated system profit as possible. Since in this model the order quantity is offered in the contract, there is no concern about the sign of $$w_{i}$$. The objective function determines its optimal level. While Approach 2-4 is a coordinating approach, it does not require the restrictive relation between $$(w_{i} ,\Phi_{\mbox{i}} )$$, i.e. Relation (29). In fact, even if the model in Approach 2-3 has a feasible solution space, its solution can be generated by Approach 2-4 as well, because Approach 2-3 is a special case of Approach 2-4. 5.4.1. General solution methodology for Approach 2-4 Step 1: Set $$U=0$$. Step 2: Find the optimal tailored menu of RS contracts for retailer types $$i=1,2,3,\ldots ,U_{\mathrm{\thinspace }}$$ Note that the set of variables and Constraints (33)–(37), which are related to the issue of menu design, constitute a linear programming problem, the optimal solution of which can be found by the CPLEX algorithm. Step 3: Find the optimal price-only contract for a retailer type $$i=U+1,\ldots ,N.$$ Step 3.1: By considering Constraints (22), (23) and (37), generate feasible values of $$w_{p}^{'}$$ from its entire feasible range. Step 3.2: Find the optimal value of $$w_{p}^{'}$$, i.e. the value that optimizes the second summation of (32). Step 4: If $$U<N$$, compute the whole Relation (32), set $$U=U+1$$ and go to Step 2. Otherwise, select the optimal value of $$U$$, i.e. the one that optimizes (32). The optimal values of $$(w_{i} ,\Phi_{\mbox{i}} )$$ and $$w_{p}^{'}$$ corresponding to the optimal value of $$U$$ are determined in Steps 2 and 3. We can find the optimal value of $$w_{p}^{'}$$ in Step 3, under the IGFR assumption, through Propositions 2 and 3. Notice that a specific case of the tailored menu may occur when $$\Phi_{\mbox{i}} =1, \ \forall i=1,2,3,\ldots ,U$$. In this case, the tailored menu consists of order quantities and wholesale prices. In this specific case, there will not be a collaborative relationship like the one in the RS contract between the manufacturer and the retailer. According to our investigations, a truth-telling menu can be designed by Approach 2-4. We provide some analytical results for this approach in Section 7.2. 6. Numerical analysis We first present an example in Section 6.1 to clarify how the proposed models in Approaches 2-2 and 2-4 work. Since the other investigated approaches are specific cases of Approaches 2-2 and 2-4, in Section 6.1, we only concentrate on these two approaches. Then, in Section 6.2, we conduct a sensitivity analysis and compare the manufacturer’s, the retailer’s and the system’s expected profits under different approaches, with different parameter settings. Finally, in Section 6.3, we present some managerial insights from the numerical analysis. 6.1. An example for Approaches 2-2 and 2-4 Assume that the parameters of the model are estimated as follows:   p =150, cm=40, cr,i={10,20,30,40,50}, λ1=λ2=...=λ5 =0.2, X∼Normal(μ=100,σ=20). In order to find the optimal value of variables in Approaches 2-2 and 2-4, e.g. $$(w,\Phi)$$, we wrote a computer program in MATLAB software on the basis of the general solution methodologies. We generated feasible values for the decision variables in the increments of 0.01. Table 1 presents the results of Approach 2-2. We computed the manufacturer’s, the retailer’s and the system’s expected profits for each value of $$U$$ and in terms of $$_{\mathrm{\thinspace }}$$ estimations from the retailer’s cost. The column ‘Exp. Prof.’ represents the expected profits. Table 1 Results of Approach 2-2      Note that the results for $$U=0$$ represent the reservation profit case in which the manufacturer, based on Approach 2-1, designs a price-only contract for all retailer types. The highlighted cells represent the cases in which the manufacturer offers a price-only contract to the corresponding retailer types. For instance, for $$U=2$$ an RS contract is designed for a retailer whose cost is 10 or 20 and a price-only contract is designed for a retailer whose cost is > 20. In the column ‘Exp. Prof.’, we observe that the manufacturer’s expected profit is increased as $$U$$ is increased up to 4. This means the optimal value of $$U$$ is 4 and the manufacturer should design an RS contract for a retailer whose cost is < 50, and a price-only contract for a retailer whose cost is 50. The optimal parameters of the RS contract are $$(w=7.99,\Phi =0.39)$$ and the optimal wholesale price is $$w_{p}^{'} =92.42$$. Table 2 presents the results of Approach 2-4. From the column ‘Exp. Prof.’ in Table 2, we realize that the optimal value of $$U$$ is 5 and the manufacturer should design a tailored menu of RS contracts for all retailer types. The optimal tailored menu is: $$\{(w_{1} =44.77,\Phi_{1} =0.7, q_{1}^{\rm RS3} =108.61)$$, $$\left( {w_{2} =45.32,\Phi_{2} =0.7,q_{2}^{\rm RS3} =105.07} \right), \ \left( {w_{3} =45.98,\Phi_{3} =0.7,q_{3}^{\rm RS3} =101.67} \right), ( w_{4} =46.78$$, $$\Phi_{4}=0.7, q_{4}^{\rm RS3} =98.33),\left( {w_{5} =47.75,\Phi_{5} =0.7,q_{5}^{\rm RS3} =94.93} \right)$$ Table 2 Results of Approach 2-4      6.2. Sensitivity analysis: comparison of different approaches In this subsection, we want to compare the effects of each approach studied on the expected profits. In Section 6.2.1, we compare expected profits based on $$\bar{c}_{r}$$ (Approach 1) with that based on all cost estimations (Approach 2). Along this line, we compare expected profits under the price-only contract in Approach 1-1 with those under Approach 2-1. We compare the expected profits under the RS contract in Approach 1-2 with those under Approaches 2-2 and 2-4 as well. Focusing on Approach 2 in Section 6.2.2, we then compare the expected profits under Approaches 2-2 and 2-4. Suppose the manufacturer has three estimations of the retailer’s cost with the same probabilities, corresponding to optimistic, moderate and pessimistic estimations. Assume that the parameters of the model in the base case are estimated as: $$p=150, \ c_{m} =40, \ c_{r,i} =\left\{ {10,30,50} \right\}, \ X\sim \mbox{Normal}(\mu =100,\sigma =20)$$. In the following subsections, different tables with different values for $$p, \ c_{m} , \ \mu , \ \sigma$$ and $$c_{r,i}$$ are presented. Only one parameter at a time is varied while the remaining parameters are fixed as above. In all the tables, through the following subsections, the optimal expected profit and an index are represented in columns ‘Man.’ (manufacturer), ‘Ret.’ (retailer) and ‘System’. 6.2.1. Comparison of Approaches 1 and 2 In order to make easier comparisons, hereafter we define $$VP$$ as an index that compares expected profit under Approaches 1 and 2 and returns variations as a percentage of the expected profit in Approach 1:   VP=(Expected Profit in Approach 2−Expected Profit in Approach 1)|Expected Profit in Approach 1|∗100%. (38) For example, assume the expected profit is 10 and 30, respectively, in Approaches 1 and 2. In this situation, $$VP$$ becomes 200%. Putting it differently, the expected profit in Approach 2 is the expected profit in Approach 1 plus 200% of the expected profit in Approach 1. $$VP$$ may be positive or negative and it can be calculated based on either the price-only or the RS contract, for the manufacturer, the retailer or the system. 6.2.2. Comparison of Approaches 1-1 and 2-1 Tables 3–7 present expected profits as well as $$VP$$. These tables are for the price-only contract in Approaches 1 and 2. Table 3 Comparison of the expected profits in Approaches 1-1 and 2-1, under different values for ‘$$p$$’    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{p}$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  100  1313.63  655.59  1969.22  1.96  -14.02  -3.36  150  3568.50  939.40  4507.90  21.00  76.09  32.48  200  5824.13  1237.13  7061.26  33.99  49.92  36.78  250  11029.75  1604.09  12633.84  1.87  33.59  5.90  300  14573.87  2046.79  16620.66  0.47  21.57  3.07    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{p}$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  100  1313.63  655.59  1969.22  1.96  -14.02  -3.36  150  3568.50  939.40  4507.90  21.00  76.09  32.48  200  5824.13  1237.13  7061.26  33.99  49.92  36.78  250  11029.75  1604.09  12633.84  1.87  33.59  5.90  300  14573.87  2046.79  16620.66  0.47  21.57  3.07  Table 4 Comparison of the expected profits in Approaches 1-1 and 2-1, under different values for ‘$$c_{m}$$’    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{c}_{\mathbf{m}}$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  10  5067.1  1162.21  6229.31  30.56  56.70  35.44  20  4560.61  1085.65  5646.26  28.09  62.12  34.64  30  4061.08  1011.03  5072.11  24.99  68.56  33.67  40  3568.50  939.40  4507.90  21.00  76.09  32.48  80  1680.31  674.48  2354.79  1.71  -16.42  -3.49    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{c}_{\mathbf{m}}$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  10  5067.1  1162.21  6229.31  30.56  56.70  35.44  20  4560.61  1085.65  5646.26  28.09  62.12  34.64  30  4061.08  1011.03  5072.11  24.99  68.56  33.67  40  3568.50  939.40  4507.90  21.00  76.09  32.48  80  1680.31  674.48  2354.79  1.71  -16.42  -3.49  Table 5 Comparison of the expected profits in Approaches 1-1 and 2-1, under different values for ‘$$\mu$$’    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{\mu }$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  50  1513.22  558.97  2072.19  4.12  29.52  10.97  100  3568.50  939.40  4507.90  21.00  76.09  32.48  200  8646.83  1570.78  10217.61  17.56  129.86  34.82  300  13872.18  2209.29  16081.47  16.03  153.41  34.91  500  24425.64  3512.22  27937.86  14.72  172.96  34.61    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{\mu }$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  50  1513.22  558.97  2072.19  4.12  29.52  10.97  100  3568.50  939.40  4507.90  21.00  76.09  32.48  200  8646.83  1570.78  10217.61  17.56  129.86  34.82  300  13872.18  2209.29  16081.47  16.03  153.41  34.91  500  24425.64  3512.22  27937.86  14.72  172.96  34.61  Table 6 Comparison of the expected profits in Approaches 1-1 and 2-1, under different values for ‘$$\sigma$$’    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{\sigma }$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  5  4784.87  714.46  5499.33  15.22  165.66  34.77  10  4323.41  785.39  5108.8  17.56  129.86  34.82  20  3568.50  939.40  4507.90  21.00  76.09  32.48  30  3001.71  1057.25  4058.96  22.39  46.11  28.57  40  3026.44  1117.95  4144.39  4.12  29.52  10.97    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{\sigma }$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  5  4784.87  714.46  5499.33  15.22  165.66  34.77  10  4323.41  785.39  5108.8  17.56  129.86  34.82  20  3568.50  939.40  4507.90  21.00  76.09  32.48  30  3001.71  1057.25  4058.96  22.39  46.11  28.57  40  3026.44  1117.95  4144.39  4.12  29.52  10.97  Table 7 Comparison of the expected profits in Approaches 1-1 and 2-1, under different values for ‘$$c_{r,i}$$’    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{c}_{\mathbf{r},\mathbf{i}}$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  0,30,60  3669.64  1224.55  4894.19  2.64  102.33  27.58  10,30,50  3568.50  939.40  4507.90  21.00  76.09  32.48  15,30,45  3510.44  802.6  4313.04  29.54  59.91  35.19  20,30,40  4430.37  673.88  5104.25  6.82  42.43  11.52  25,30,35  4849.43  641.78  5491.21  0.12  12.58  1.58    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{c}_{\mathbf{r},\mathbf{i}}$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  0,30,60  3669.64  1224.55  4894.19  2.64  102.33  27.58  10,30,50  3568.50  939.40  4507.90  21.00  76.09  32.48  15,30,45  3510.44  802.6  4313.04  29.54  59.91  35.19  20,30,40  4430.37  673.88  5104.25  6.82  42.43  11.52  25,30,35  4849.43  641.78  5491.21  0.12  12.58  1.58  In all Tables 3–7, we observe that the manufacturer’s expected profit under Approach 2-1 is more than that under Approach 1-1, because $$VP$$ is always positive. This improvement can be as high as nearly 34% for $$p=200$$ in Table 3. We observe that for $$p=100, \ c_{m} =80$$ the retailer’s share from the system’s profit under Approach 2-1 is so low that $$VP$$ becomes negative. However, depending on the model parameters the retailer’s and the system’s expected profits, as well as the manufacturer’s, are most often improved in Approach 2-1. This improvement for the retailer might be as high as 172% for $$\mu =500$$ in Table 5. Therefore, we observe that a manufacturer should always design the price-only contract considering all estimations of the retailer’s cost. This option is most often profitable for the retailer as well. 6.2.3. Comparison of Approach 1-2 with Approaches 2-2 and 2-4 Here we compare the expected profits in Approaches 1 and 2 by considering the RS contract. We take into account the expected profits in Approach 1-2 based on the SMRS contract introduced in Section 4.2. In the SMRS contract, we increase the lower bound of $$\bar{w}$$ from $$-\bar{c}_{r}$$ to $$-c_{r,1}$$, so that $$F(q_{i}^{\overline{RS}})<1 \ \forall i=1,2,3,...,N$$ holds. It should be mentioned that depending on the model parameters even with this small modification in the RS contract, the manufacturer’s expected profit might become negative; however, according to Proposition 1, it will be definitely negative without this modification. Tables 8–12 present expected profits as well as $$VP$$, which is calculated based on Approach 2-2 or Approach 2-4. Table 8 Comparison of the expected profits in Approach 1-2 with those in Approaches 2-2 and 2-4, under different values for ‘$$p$$’    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$p$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  100  779.90  818.73  1598.63  100.12  -20.59  38.30  114.10  -16.97  46.97  150  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  200  5473.02  1260.61  6733.63  60.14  76.19  63.15  66.54  93.53  71.60  250  8511.07  1417.20  9928.27  53.62  82.99  57.81  58.98  98.83  64.67  300  11538.11  1638.52  13176.63  51.17  82.70  55.09  55.74  96.73  60.84    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$p$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  100  779.90  818.73  1598.63  100.12  -20.59  38.30  114.10  -16.97  46.97  150  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  200  5473.02  1260.61  6733.63  60.14  76.19  63.15  66.54  93.53  71.60  250  8511.07  1417.20  9928.27  53.62  82.99  57.81  58.98  98.83  64.67  300  11538.11  1638.52  13176.63  51.17  82.70  55.09  55.74  96.73  60.84  Table 9 Comparison of the expected profits in Approach 1-2 with those in Approaches 2-2 and 2-4, under different values for ‘$$c_{m}$$’    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$\mathbf{c}_{\mathbf{m}}$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  10  5483.36  1167.45  6650.81  33.79  87.09  43.15  39.47  107.48  51.41  20  4465.96  1167.45  5633.41  43.61  74.78  50.07  49.06  97.19  59.03  30  3506.04  1075.03  4581.07  57.42  81.25  63.01  62.52  103.66  72.18  40  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  80  -1301.27  804.92  -496.35  265.86  -15.37  672.08  275.36  -11.46  703.32    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$\mathbf{c}_{\mathbf{m}}$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  10  5483.36  1167.45  6650.81  33.79  87.09  43.15  39.47  107.48  51.41  20  4465.96  1167.45  5633.41  43.61  74.78  50.07  49.06  97.19  59.03  30  3506.04  1075.03  4581.07  57.42  81.25  63.01  62.52  103.66  72.18  40  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  80  -1301.27  804.92  -496.35  265.86  -15.37  672.08  275.36  -11.46  703.32  Table 10 Comparison of the expected profits in Approach 1-2 with those in Approaches 2-2 and 2-4, under different values for ‘$$\mu$$’    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$\mathbf{\mu }$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  50  23.27  593.95  617.22  7606.32  47.04  332.04  7754.14  49.49  339.97  100  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  200  7648.21  1826.69  9474.90  37.63  109.52  51.49  40.93  123.03  56.76  300  12906.74  2471.59  15378.33  27.54  135.18  44.84  30.02  145.62  48.60  500  24772.37  3813.25  28585.62  14.58  156.95  33.57  16.21  163.97  35.92    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$\mathbf{\mu }$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  50  23.27  593.95  617.22  7606.32  47.04  332.04  7754.14  49.49  339.97  100  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  200  7648.21  1826.69  9474.90  37.63  109.52  51.49  40.93  123.03  56.76  300  12906.74  2471.59  15378.33  27.54  135.18  44.84  30.02  145.62  48.60  500  24772.37  3813.25  28585.62  14.58  156.95  33.57  16.21  163.97  35.92  Table 11 Comparison of the expected profits in Approach 1-2 with those in Approaches 2-2 and 2-4, under different values for ‘$$\sigma$$’    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$\sigma$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  5  4851.51  789.95  5641.46  15.52  146.80  33.90  17.41  155.35  36.72  10  3824.11  913.35  4737.46  37.62  109.28  51.44  40.93  123.03  56.76  20  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  30  1263.39  1134.50  2397.89  223.77  60.57  146.56  229.40  86.62  161.85  40  46.54  1187.89  1234.43  7608.49  47.54  332.60  7754.12  49.50  339.97    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$\sigma$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  5  4851.51  789.95  5641.46  15.52  146.80  33.90  17.41  155.35  36.72  10  3824.11  913.35  4737.46  37.62  109.28  51.44  40.93  123.03  56.76  20  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  30  1263.39  1134.50  2397.89  223.77  60.57  146.56  229.40  86.62  161.85  40  46.54  1187.89  1234.43  7608.49  47.54  332.60  7754.12  49.50  339.97  Table 12 Comparison of the expected profits in Approach 1-2 with those in Approaches 2-2 and 2-4, under different values for ‘$$c_{r,i}$$’    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$c_{r,i}$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  0,30,60  2786.44  1430.01  4216.45  41.97  87.14  57.28  43.47  87.56  58.43  10,30,50  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  15,30,45  2396.58  852.41  3248.99  107.64  73.97  98.81  116.16  93.82  110.30  20,30,40  2313.92  630.31  2944.23  133.12  82.27  122.23  141.24  96.18  131.59  25,30,35  2324.89  315.15  2640.04  150.66  166.86  152.60  155.34  177.22  157.96    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$c_{r,i}$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  0,30,60  2786.44  1430.01  4216.45  41.97  87.14  57.28  43.47  87.56  58.43  10,30,50  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  15,30,45  2396.58  852.41  3248.99  107.64  73.97  98.81  116.16  93.82  110.30  20,30,40  2313.92  630.31  2944.23  133.12  82.27  122.23  141.24  96.18  131.59  25,30,35  2324.89  315.15  2640.04  150.66  166.86  152.60  155.34  177.22  157.96  In Tables 8–12, we notice that the manufacturer’s expected profit under Approaches 2-2 and 2-4 is always greater than that under Approach 1-2, because $$VP$$ is always positive. This improvement can be as high as almost 7600% for $$\mu =50$$ or $$\sigma =40$$, as shown in Tables 10 and 11, respectively. We observe that the retailer’s and the system’s expected profits are usually improved considerably under Approaches 2-2 and 2-4 as well. 6.2.4. Comparison of Approaches 2-2 and 2-4 In order to save space, we compare the expected profits in Approaches 2-2 and 2-4, based on their relative preference over Approach 1-2, i.e. based on the values of $$VP$$ in Tables 8–12. By comparing $$VP$$ based on Approach 2-2 and $$VP$$ based on Approach 2-4, we observe that the manufacturer’s, the retailer’s and the system’s expected profits are increased when moving from Approach 2-2 to Approach 2-4. However, from the values of $$VP$$ in Tables 8–12 we realize that the preference for Approach 2-4 over Approach 2-2 is not substantial. This can be an appropriate motivation for the manufacturer to implement Approach 2-2 if there are some administrative drawbacks for designing a menu of contracts according to Approach 2-4. 6.3. Managerial insights from sensitivity analysis In Section 6.2.1 (Tables 3–12), we compared the expected profit in Approach 1 with that of Approach 2. As an important managerial insight from this subsection, we observe that the manufacturer’s expected profit is improved considerably in Approach 2 by taking into account all cost estimations at the time of contract design. It is clear that the manufacturer’s expected profit in Approaches 2-2 and 2-4 is always higher than that under Approach 2-1. That is because Approach 2-1 is a specific case of Approaches 2-2 and 2-4 which is generated when we set $$U=0$$ in Approaches 2-2 and 2-4. Therefore, another point here is that the manufacturer should choose Approaches 2-2 and 2-4 over Approach 2-1. The manufacturer is now left with the choice between Approaches 2-2 and 2-4. By comparing the values of $$VP$$ in Tables 8–12, we observe improvement in both the manufacturer’s and the retailer’s expected profits, by moving from Approach 2-2 to Approach 2-4. Since Approach 2-4 leads to system coordination, more profit is generated in the system according to this case as compared to Approach 2-2. Both manufacturer and retailer benefit from this additional profit in the coordinated system. Therefore, our first suggestion for the manufacturer is to use Approach 2-4. However, if executing Approach 2-4 and designing a menu can cause some difficulties in practice, e.g. whenever there is a high number of contracts in the menu, Approach 2-2 will be an appropriate substitute. By comparing the manufacturer’s expected profit under Approaches 2-2 and 2-4, we realize that on average, Approach 2-4 can improve Approach 2-2 by almost 3%, which may not be a considerable value. In short, by comparing the manufacturer’s expected profit in different approaches, we can place the approaches in the order of preference for the manufacturer as follows:   Approach 2−4>Approach 2−2>Approach 2−1>Approach 1−1>Approach 1−2. 7. Other cases 7.1. Other contracts In this paper, we assumed the RS contract between a manufacturer and a retailer. However, our results hold for any other contract that is equivalent to the RS contract. The RS contract and the buy-back contract are equivalent in a fixed-price newsvendor model (Cachon & Lariviere, 2005). This means that for any RS contract with parameters $$(w,{\Phi })$$ there exists a unique buyback contract with parameters $$(b,w_{b} )$$ that leads to the same profit for the manufacturer and the retailer, for any realization of demand. This equivalence takes place by setting $$b=\left( {1-{\Phi }} \right)p$$ and $$w_{b} =w+\left( {1-{\Phi }} \right)p$$, where $$b$$ and $$w_{b}$$ are wholesale and buyback prices. In addition, since the buy-back contract can be interpreted as a kind of advanced procurement or option contract, we can consider the option contract as another equivalent of the RS contract. In an option contract, the retailer in the first stage buys some options at price $$w_{o}$$ to use later. In the second stage after demand realization, it commits to pay an additional payment as $$w_{e}$$ to exercise the options (Wang & Tsao, 2006). This pricing mechanism creates the same profit allocation between the manufacturer and the retailer as in the RS and buy-back contracts, by setting: $$w_{o} =w_{b} -b=w, \ w_{e} =b=\left( {1-{\Phi }} \right)p$$. 7.2. Type-independent reservation profit: the case of Approach 2-4 We tried to develop some comprehensive models in Approach 2 by considering different significant conditions, such as $$N$$ different estimations for the cost information, cut-off policy and type-dependent reservation profit. Ignoring some or all of these conditions leads to easier cases of the models that might be used by a manager. We introduce one such case in this subsection. So far, by considering type-dependent reservation profit, the retailer’s reservation profit and the manufacturer’s outside option are modelled according to the price-only contract. Several studies, such as Several studies, such as Ha (2001), Corbett et al. (2004) and Lutze & Özer (2008), have considered a situation in which the reservation profit is not type-dependent. In this situation, the reservation profit can be interpreted as an outside opportunity or a minimum profit level that a contract partner demands regardless of its cost level. For the manufacturer, it is a minimum profit level that it intends to achieve for a trade to take place. In this subsection, we study the menu design problem in Approach 2-4 by considering a given reservation profit. Other approaches may be studied in a similar way. Assume $$\mbox{RP}_{m}$$ as the manufacturer’s reservation profit and $$\mbox{RP}_{r}$$ as the retailer’s reservation profit. Type-independent reservation profit is equivalent to: $$\mbox{RP}_{r,1} =\mbox{RP}_{r,2} =...=\mbox{RP}_{r,i} =...=\mbox{RP}_{r,N} =\mbox{RP}_{r}$$. If the contract is rejected, each manufacturer and retailer can pursue other investment opportunities, thereby ensuring profits of $$\mbox{RP}_{m}$$ and $$\mbox{RP}_{r}$$ respectively. Notice that in this situation, based on the cut-off policy, there might be no trade with a retailer whose cost is higher than the cut-off point. This is while both firms were continuing to trade based on a price-only contract in Approaches 2-2, 2-3 and 2-4 when the retailer’s cost was higher than the cut-off point. The manufacturer’s problem in this case is:   Max(U,wi,Φi)ΠmRS3=∑i=1U⁡λi[(wi−cm)qiRS3+(1−Φi)pS(qiRS3)]+∑i=U+1N⁡λiRPm, (39)  S.T.ΦipS(qiRS3)−qiRS3(wi+cr,i)⩾RPr ∀i=1,2,3,...,U, (40)  ΦipS(qiRS3)−qiRS3(wi+cr,i)⩾ΦjpS(qjRS3)−qjRS3(wj+cr,i) (41)  ∀i=1,2,3,...,U, ∀j=1,2,3,...,U, i≠j,ΦipS(qiRS3)−qiRS3(wi+cr,f)⩽RPr ∀i=1,2,3,...,U, f=U+1,...,N, (42)  qiRS3=F−1[p−cm−cr,ip] ∀i=1,2,3,...,N, (43)  U∈{0,1,2,3,...,N},0⩽Φi⩽1,wi:free in sign, ∀i=1,2,3,...,U. (44) In this model, the manufacturer does not offer a price-only contract to a retailer with a cost more than $$c_{r,U}$$. Therefore, there is no need to Constraints (22) and (23). Proposition 4 Consider $$c_{r,U} \in \{c_{r,1} ,c_{r,2},c_{r,3} ,...,c_{r,N} \}$$ as a cut-off point. In an optimal solution, a retailer’s expected profit with the cost of $$c_{r,i} , \ i=1,2,...,U-1$$, i.e. $${\Pi }_{r,i}^{\rm RS3}$$, is more than $$\mbox{RP}_{r}$$ and a retailer’s expected profit with the highest cost, i.e. $${\Pi }_{r,U}^{\rm RS3}$$, is $$\mbox{RP}_{r}$$. Proposition 5 Consider $$c_{r,U} \in \{c_{r,1} ,c_{r,2},c_{r,3} ,...,c_{r,N} \}$$ as a cut-off point. In the optimal solution, a retailer’s expected profit of type $$i=1,2,3,...,U-1$$ is: $${\Pi }_{r,i}^{\rm RS3} =\mbox{RP}_{r} + \sum_{j=i}^{U-1} q_{j+1}^{\rm RS3} (c_{r,j+1} -c_{r,j})$$. From Propositions 4 and 5, we find that in the optimal solution, a retailer’s expected profit with a lower cost is more than a retailer’s expected profit with a higher cost. For each value of $$U$$, through Proposition 5, we can find optimal values of $$w_{i}$$ and $$\Phi_{\mbox{i}}$$ so that they satisfy the optimal profit allocation to each retailer type. Finally, we can select the optimal value for $$U$$, according to the general solution methodology in Section 5.4.1. 8. Conclusion In this paper, we investigate different approaches to designing the RS contract by a manufacturer under asymmetric cost information about a retailer that is the contract partner. The retailer, operating in a newsvendor-type environment, sources its products from the manufacturer. The manufacturer knows several discrete estimations from the retailer’s cost structure. We consider two general approaches. In Approach 1, the manufacturer designs the RS and price-only contracts based on the average of cost estimations, which is a substantial simplification. In Approach 2, the manufacturer uses all cost estimations for contract design. We show that in Approach 1 the price-only contract creates a larger profit for the manufacturer, as compared to the RS contract. In Approach 2, we consider type-dependent reservation profit for the retailer explicitly based on the price-only contract, as introduced in Approach 2-1. We investigate Approaches 2-2, 2-3 and 2-4, for the design of the RS contract based on the cut-off policy. According to the cut-off policy in the mentioned approaches, we allow the manufacturer to refrain from signing an RS scheme if the retailer’s cost is higher than a specific level; thus, we include the price-only contract as an outside option for the manufacturer. Specifically, in Approach 2-2 a single RS contract is designed for different types of the retailer, while in Approaches 2-3 and 2-4 the manufacturer designs a menu of coordinating RS contracts. Thus, Approach 2-2 is a simplified version of Approaches 2-3 and 2-4. We address the optimal features of the model developed in Approach 2-2 when demand has an IGFR. We illustrate that Approach 2-3 cannot motivate the retailer to behave truthfully. According to the revelation principle, Approach 2-3 cannot be the optimal menu of RS contracts. Thus, we work with a tailored menu of coordinating RS contracts in Approach 2-4, which includes the coordinated order commitment as well as the regular RS parameters. We present a general solution methodology to find the optimal contract parameters in Approaches 2-2 and 2-4. We find from numerical analysis that contract design, according to each model in Approach 2, can increase both the manufacturer’s and the retailer’s profits considerably, as compared to Approach 1. This highlights the value of utilizing all cost estimations for contract design. Approach 2-1 is a specific case of Approaches 2-2 and 2-4. Thus, it is an inferior option, as compared to Approaches 2-2 and 2-4. By comparing Approaches 2-2 and 2-4, we show that both the manufacturer and the retailer obtain more expected profit in Approach 2-4, under different parameter settings. However, the important managerial insight is that if the execution of Approach 2-4 (designing a menu) creates unreasonable administrative costs for the manufacturer, Approach 2-2 can be an appropriate substitute. Our results and comparisons among different approaches hold under any other pricing mechanism that is equivalent to the RS contract, such as the buy-back contract. We also present some analytical results by considering type-independent reservation profit based on Approach 2-4. In this case, we determine the optimal profit allocation to each retailer type. One direction for future research is to investigate other approaches, such as the case in which the manufacturer designs a menu of RS contracts without specifying order quantities and coordinating the system. Approach 2-2 is a special case of this approach in practice. Taking our investigations as an indication, we anticipate that finding a solution for such a scenario will not be an easy task, because of the non-linearity of such a model. References Baron, D. B. & Myerson, R. B. ( 1982) Regulating a monopolist with unknown costs. Econometrica,  50, 911– 930. Google Scholar CrossRef Search ADS   Cachon, G. P. & Lariviere, M. A. ( 2005) Supply chain coordination with revenue-sharing contracts: strengths and limitations. Manage. Sci.,  51, 30– 44. Google Scholar CrossRef Search ADS   Cachon, G. P. & Zhang, F. ( 2006) Procuring fast delivery: sole sourcing with information asymmetry. Manage. Sci.,  52, 881– 896. Google Scholar CrossRef Search ADS   Çakanyildirim, M., Feng, Q., Gan, X. & Sethi, S. P. ( 2012) Contracting and coordination under asymmetric production cost information. Prod. Oper. Manage.,  21, 345– 360. Google Scholar CrossRef Search ADS   Cao, E., Ma, Y., Wan, C. & Lai, M. 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# Approaches to designing the revenue sharing contract under asymmetric cost information

, Volume 29 (1) – Jan 1, 2018
29 pages

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Oxford University Press
© The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
ISSN
1471-678X
eISSN
1471-6798
D.O.I.
10.1093/imaman/dpw013
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### Abstract

Abstract We investigate different approaches to designing price-only and revenue sharing (RS) contracts between a manufacturer and a newsvendor retailer in the case of asymmetric cost information. The manufacturer, i.e. contract designer, has $$N$$ estimations about the retailer’s cost with the corresponding probabilities. We address two general approaches: in Approach 1, the manufacturer designs the contracts using the average of cost estimations, while in Approach 2, the manufacturer designs the contracts using all cost estimations. In Approach 2, we first model a reservation profit for each retailer type, explicitly based on the price-only contract (Approach 2-1). Then, we investigate how to design a unique RS contract (Approach 2-2) and a menu of coordinating RS contracts (Approaches 2-3 and 2-4). Approaches 2-2, 2-3 and 2-4 are designed based on a cut-off policy in which the manufacturer offers a price-only contract to the retailer if the retailer’s cost is greater than a specific level. We find that in Approach 1, the price-only contract is more profitable than the RS contract for the manufacturer. Both firms’ expected profits are increased considerably in Approach 2, as compared to Approach 1. Approach 2-4 results in the highest expected profit for both firms. However, if this approach leads to high administrative costs for the manufacturer, Approach 2-2 can be an appropriate substitute. Numerical analysis and a number of managerial and analytical insights are presented. 1. Introduction In this paper, we focus on the revenue sharing (RS) contract as one of the most important types of coordinating contracts. We investigate different approaches to designing this contract under cost information asymmetry between a manufacturer and a retailer. According to the regular parameters of an RS contract, i.e. $$(w,\Phi )$$, the contract designer reduces its wholesale price $$(w)$$ in return for receiving a portion of its partner’s revenue $$(1-\Phi )$$. Video rental industries are the potential users of this type of contract (Cachon & Lariviere, 2005). In practice, the RS contract designer is not usually aware of its contract partner’s cost information, which is often private. Thus, in this research we seek to investigate how a manufacturer should design this contract in the presence of information asymmetry about the retailer’s cost. We assume that the manufacturer knows $$N$$ different estimations about the retailer’s cost. We define a retailer type for each cost estimation. According to the literature on the mechanisms for contract design in the case of information asymmetry (see, for instance, Baron et al., 1982), a ‘menu of contracts’ can maximize the contract designer’s expected profit. A menu of contracts consists of several contracts designed so that each retailer type selects a specific contract that is appropriately designed for its type. In this paper, in addition to the design of a menu of contracts, we investigate other approaches to contract design in the case of information asymmetry. In reality, contract partners expect a minimum level of profit as a reservation profit from each trade. The reservation profit can be the profit that each entity may earn by investing in an alternative market. In this paper, we model a specific reservation profit for the retailer based on the price-only contract. This is the simplest contract imaginable between a manufacturer and a retailer in which the manufacturer offers only a wholesale price to the retailer and the retailer decides on order quantities. The reservation profit considered for the retailer depends on the retailer type, i.e. the retailer’s cost level. Çakanyildirim et al. (2012) have emphasized the importance of considering type-dependent reservation profit, suggesting that an important issue for future research is the analysis of a model assuming type-dependent reservation profit and $$N$$ different states for asymmetric information. In this paper, we consider these assumptions. We include an outside option for the manufacturer as well as the retailer, based on the price-only contract. Along this line, we investigate some approaches based on the cut-off level policy. Based on this, the manufacturer may not want to motivate every retailer type with any cost level to participate in the RS schemes. The manufacturer can offer a price-only contract to the retailer if the retailer’s cost is greater than a specific level. We address three research questions, considering both theoretical and managerial concerns: (1) How can we simplify the design of a menu of contracts, under information asymmetry? For example, whenever the number of contracts in a menu is increased, the menu becomes too complicated to use, thus we should look for other simpler approaches. (2) How will the contract designer’s expected profit change under a simpler approach, as compared to designing a menu of contracts? Suppose a menu of contracts as the most sophisticated approach for contract design is used. The last research question is: (3) How should the manufacturer design a menu of RS contracts so that it leads to a truthful retailer behaviour as well as system coordination? This study, by investigating the research questions brought above, proposes two parent approaches. The approaches are illustrated in Fig. 1: Approach 1 (containing two child approaches) and Approach 2 (containing four child approaches). Fig. 1. View largeDownload slide Introduction of the investigated approaches. Fig. 1. View largeDownload slide Introduction of the investigated approaches. Regarding research question (1), the first simplification for menu design is considered based on the information the manufacturer uses for contract design. In Approach 1, we create a major simplification for contract design by utilizing the average of cost estimations instead of all ($$N)$$ estimations. Approach 1-1 corresponds to the design of the price-only contract, and Approach 1-2 corresponds to the design of RS contract. We introduced a slightly modified RS (SMRS) contract in Approach 1-2 as well. In Approach 2, as opposed to Approach 1, we investigate contract design based on all ($$N)$$ cost estimations. In Approach 2-1, we introduce the outside option and type-dependent reservation profit in which the manufacturer designs a price-only contract based on all cost estimations. By considering research question (1), the second simplification for menu design is proposed in Approach 2, so we consider the option of designing a unique contract instead of designing several contracts for a menu of contracts. Along this line, in Approach 2, in addition to models based on the menu design (Approaches 2-3 and 2-4), we investigate a model as Approach 2-2 that considers the design of a unique RS contract. We believe that when a menu of contracts is not usable for the manufacturer, it can design a unique RS contract using Approach 2-2. Notice that here as opposed to the first simplification, all the estimations are used for contract design. Regarding research question (2), we conduct an extensive numerical analysis to compare expected profits between sophisticated and simplified approaches. Regarding research question (3), first in Approach 2-3, we investigate the design of a menu of coordinating RS contracts based on the regular parameters of the RS contract $$(w,\Phi )$$. Then, we move onto a tailored version of Approach 2-3, as Approach 2-4, in which the manufacturer offers coordinated order quantities in addition to the regular RS parameters. It should be mentioned that our main focus is on the RS contract. However, in different approaches, we consider the price-only contract as well. In Approach 1, we investigate the design of the price-only contract (Approach 1-1) so that we can compare the expected profits in Approaches 1-1 and 2-1. In Approach 2, we use the price-only contract as an outside option that determines the reservation profit of contract partners. Seven interesting theoretical and managerial insights are derived from this paper. First, we find that in Approach 1, the design of a price-only contract is more profitable for the manufacturer than the design of an RS contract. This is against our usual expectation in the case of information symmetry, where a coordinating contract can improve the contract designer’s expected profit. Second, we find that the manufacturer’s expected profit under Approach 2 is considerably more than that under Approach 1. This finding highlights the value of considering all estimations. Third, we show that a menu of coordinating RS contracts, according to Approach 2-3, does not motivate the retailer to behave truthfully. Fourth, the most sophisticated approach (Approach 2-4) yields the highest expected profit for the manufacturer. However, our numerical analysis shows that the preference for Approach 2-4 over Approach 2-2 is insignificant. The key managerial insight here is that if the implementation of Approach 2-4 in practice is difficult, Approach 2-2 could be an appropriate substitute. Fifth, the optimal features of the model in Approach 2-2 are investigated when demand distribution has an increasing generalized failure rate (IGFR). IGFR distributions, such as normal distribution, usually occur in practice. Sixth, all of our findings and observations hold true for other contracts that are equivalent to an RS contract, such as a buy-back contract. Seventh, based on Approach 2-4, we obtain each retailer type’s optimal profit by considering type-independent reservation profit. This paper is organized as follows: first we review the related literature in Section 2 and present the model statements, assumptions and notation in Section 3. Then, in Sections 4 and 5, we investigate Approach 1 and Approach 2, respectively. In Section 6, we present numerical analysis. Section 7 investigates two other cases of the paper. Finally, the conclusion is presented in Section 8. 2. Literature review In this section, we shed light on the similarities and differences between our paper and the related literature on contract design under cost information asymmetry, from different perspectives. 2.1. The contract designer: upstream or downstream member A contract designer might be an upstream or downstream member in a supply chain. A change in the contract designer’s position leads to changes in the leadership situation and managerial aspects of the supply chain. Some previous studies have considered the issue of cost information asymmetry when a downstream supply chain member designs a contract for its upstream member. For instance, Lau et al. (2007) and Wang et al. (2009) studied how a dominant retailer who did not have a perfect knowledge from the manufacturer’s cost should design a contract. Lau et al. (2007) examined a volume discount scheme from the retailer’s perspective, while Wang et al. (2009) investigated whether the dominated manufacturer could be motivated by the retailer to improve the quality of the retailer’s knowledge. In both studies, market demand was deterministic. In contrast, in our paper demand is stochastic and the upstream member, i.e. the manufacturer, is the contract designer. Xu et al. (2010) considered a supply chain with overseas suppliers, local suppliers and one manufacturer in which the manufacturer did not know the local suppliers’ production cost. They investigated designing a contract menu consisting of a transfer payment and a lead-time quotation, while in this paper we focus on designing the RS contract with the price-only contract as an outside option. Xu et al. (2013) extended their study in Xu et al. (2010) by investigating the effects of time-sensitive characteristics on decisions. Çakanyildirim et al. (2012) analysed a supply chain in which a downstream member (a retailer) did not know the production cost of its upstream member (a supplier). They showed that information asymmetry alone did not induce loss in channel efficiency. Similar to Ha (2001), Corbett et al. (2004), Mukhopadhyay et al. (2008) and Lutze & Özer (2008), we investigate contract design under information asymmetry from the perspective of an upstream member (the manufacturer) for a downstream member (the retailer). 2.2. The implemented contracts: price-only and RS contracts Different contracts are used in the literature on contract design under cost information asymmetry. Several studies, like Ha (2001), Corbett et al. (2004), Mukhopadhyay et al. (2008) and Özer & Raz (2011), have investigated two part linear and non-linear pricing contracts. Cachon & Zhang (2006), Chakravarty & Zhang et al. (2007) and Lutze & Özer (2008), respectively, investigated late-fee mechanisms, Bayesian capacity investment games and promised lead-time contracts. We present some models under specific approaches considering the RS and price-only contracts. These contracts have been used before in several studies under information asymmetry. Studies like Corbett et al. (2004) and Cao et al. (2013) implemented wholesale price contract under information asymmetry, while designing a menu of contracts. In this paper, the price-only contract is considered as an outside option for the RS contract partners. In the literature dealing with the RS contract (see, for instance, Giannoccaro & Pontrandolfo, 2004; Cachon & Lariviere, 2005; Qin, 2008; Wang et al., 2012; Xu et al., 2014; Peth & Thonemann, 2016), by assuming access to all information, the major emphasis has been placed on showing how the contract leads to supply chain coordination and a win–win situation for supply chain members. A few studies investigated the RS contract under information asymmetry. Çakanyildirim et al. (2012) investigated a profit sharing contract as a scheme equivalent to the RS contract, between a retailer and a supplier. The retailer knew two low and high cost states from the supplier’s cost. They analysed the effects of outside opportunities on the retailer-supplier relationship. Different from Çakanyildirim et al. (2012), our study considers $$N$$ different states for the asymmetric information. Zhou et al. (2012) investigated the impact of customers learning about product value on the marketing efforts of a supplier and a retailer. The supplier offered an RS contract to the retailer. Their model considered asymmetric information about the supplier’s and the retailer’s marketing efforts. Our study is similar to their work in that we study the RS contract design by an upstream member in a supply chain in the case of information asymmetry. However in this study, we focus on the modelling of cost information asymmetry under specific approaches. Shen & Willems (2012) studied the coordination of a manufacturer-retailer system using a buy-back contract as a scheme equivalent to the RS contract. Our research, like theirs, considers the retailer’s cost information as private information. However, we consider discrete cost estimations and type dependent reservation profit for the retailer. 2.3. The approach to model information asymmetry: discrete or continuous Several studies, like Xu et al. (2010), Özer & Raz (2011), Çakanyildirim et al. (2012), Xu et al. (2013) and Cao et al. (2013), have modelled information asymmetry by considering just two low and high states for the asymmetric information. Our paper extends these studies by considering, in general, $$N$$ different states. To our knowledge, Lutze & Özer (2008) is the only study that has considered, in general, $$N$$ different states for the asymmetric information. They investigated a promised lead-time contract between a supplier and a retailer in which the supplier knew a finite set of estimations from the retailer’s service level. The investigated contracts and approaches in our paper are essentially different from their work. Some other studies, like Ha (2001), Corbett et al. (2004), Cachon & Zhang (2006), Chakravarty & Zhang et al. (2007), Mukhopadhyay et al. (2008) and Zhang et al. (2014), have modelled information asymmetry by considering a prior continuous probability distribution for the asymmetric information, e.g. cost. 2.4. The menu design approach The usual approach to contract design in the case of information asymmetry is to design a menu of contracts; see, e.g. Ha (2001), Corbett et al. (2004), Cachon & Zhang (2006), Chakravarty & Zhang et al. (2007), Lutze & Özer (2008) and Fang et al. (2014). Our paper differs from these studies by taking into account and comparing other approaches based on administrative considerations, in addition to menu design. Zhang et al. (2014) analysed an information asymmetry model in a supply chain that included a dominant supplier and a manufacturer using a different approach. Their approach was based on a principle-agent model as Pontryagin’s maximum principle. 2.5. The reservation profit: type-independent or type-dependent Another specific feature that distinguishes our paper from the aforementioned studies is that we consider type-dependent reservation profit for the retailer in some of the approaches. The issue of type-dependent reservation profit has not received due consideration in operations management literature (Çakanyildirim et al., 2012). A few studies, like Laffont & Martimort (2002), Chakravarty & Zhang et al. (2007), Xu et al. (2010) and Çakanyildirim et al. (2012), have considered type-dependent reservation profit for the contract partner in their works. However, in these studies, the reservation profit is a given parameter whose origin might not be clear. In this paper, we model the reservation profit explicitly based on a widely used contract, i.e. the price-only contract. In addition, in Section 7.2, we examine Approach 2-4 under the assumption of type-independent reservation profit in order to get some analytical results. Therefore, our paper has a number of unique features compared to the literature reviewed. As far as we know, there are few studies addressing the implementation of the RS contract under information asymmetry. Thus, in this research, we design our models by assuming the RS contract between the manufacturer and the retailer. The investigated approaches are different from those in the studies mentioned. In this research, in addition to designing a coordinating menu of RS contracts, we investigate simplified approaches that might be relatively easier to use. Also, we consider discrete cost estimations and type-dependent reservation profit for the retailer. 3. Model statements, assumptions and notation Consider a two-echelon supply chain with a dominant manufacturer and a retailer. The retailer, operating in a newsvendor-type environment, sources its products from the manufacturer. The manufacturer’s and the retailer’s unit costs are $$c_{m}$$ and $$c_{r}$$, respectively. These costs might include handling, processing, assembling or packaging costs. The manufacturer’s wholesale price is $$w>c_{m}$$ and the retailer’s exogenous sale price is $$p>w+c_{r}$$. $$X$$ represents the random market demand with a probability distribution function of $$f_{X} (x)$$. $$F$$(.) is a continuous and differentiable distribution function, where $$\bar{F}$$(.)$$=1-F$$(.). We assume that there is no shortage penalty and no salvage value for inventory leftover after the selling season. Similar to Giannoccaro & Pontrandolfo (2004) and Cachon & Lariviere (2005), suppose that the manufacturer offers an RS contract with parameters $$(w,\Phi )$$ to the retailer. According to the analysis presented by Cachon & Lariviere (2005), in order to coordinate the retailer’s optimal order quantity the RS contract parameters have to be adjusted to answer: $$w=\Phi \left( {c_{m} +c_{r} } \right)-c_{r}$$. Therefore, the manufacturer needs to know the retailer’s cost in order to optimally adjust the RS contract parameters. But such information is usually the retailer’s private property, so the manufacturer might not be aware of its exact value. Assume the manufacturer has a set of $$N$$ positive prior estimations from the retailer’s cost structure $$\{c_{r,1} <c_{r,2} <c_{r,3} <...<c_{r,N} \}$$ representing retailer type $$i=1,2,3,\ldots ,N$$, with probabilities $$\lambda_{1} ,\lambda_{2} ,\lambda_{3} ,\ldots ,\lambda_{N} (\sum_{i=1}^{N} \lambda_{i} =1)$$. The estimations only include those that allow for beneficial business, denoting that: $$p>c_{m} +c_{r,N}$$. Assume that the retailer’s true cost belongs to this set and this discrete distribution is public information. These assumptions are similar to Lutze & Özer (2008) as they assumed, in general, $$N$$ different states for the asymmetric information with the corresponding probabilities as the public information. 4. Approach 1: contract design based on the average of cost estimations Suppose the manufacturer wants to design a contract only based on the average value of cost estimations $$\bar{c}_{r} = \sum_{i=1}^{N} \lambda_{i} c_{r,i}$$. 4.1. Approach 1-1: price-only contract based on $$\bar{c}_{r}$$ Suppose the manufacturer offers a wholesale price $$(\bar{w}_{p} )$$ to the retailer, assuming the retailer’s cost is $$\bar{c}_{r}$$. In this situation, the manufacturer’s optimization problem is as follows:   Maxw¯p>cmΠmPC¯=(w¯p−cm)qPC¯(w¯p), (1)  qPC¯(w¯p)={F−1[p−w¯p−c¯rp] if w¯p<p−c¯r0 Otherwise . (2) $$\Pi_{m}^{\overline{\rm PC}}$$ represents the manufacturer’s expected profit under the price-only contract. Constraint (2) represents dependency of a retailer’s order quantity with a cost of $$\bar{c}_{r}$$ to $$\bar{w}_{p}$$. According to Lariviere & Porteus (2001), by assuming an IGFR demand distribution, the unique optimal $$\bar{w}_{p}$$ ($$\bar{w}_{p}^{\ast})$$ satisfies the following equation:   dΠmPC¯dw¯p=F−1[p−w¯p∗−c¯rp]−(w¯p∗−cm)pf(F−1[p−w¯p∗−c¯rp])=0. (3) It should be mentioned that a continuous random variable $$X$$ with density function $$f(x)$$ and distribution function $$F(x)$$ is IGFR if its generalized failure rate (GFR) $$g\left( x \right)=\frac{xf(x)}{1-F(x)}$$ increases in $$x.$$ IGFR distribution is a mild restriction because it captures most common distributions with increasing failure rates e.g. Normal, Uniform, Gamma and Weibull distributions. In reality, however, there may be various retailer types with different costs. A retailer’s order quantity with the actual cost of $$c_{r,i}$$ can be obtained from Relation (4):   qiPC¯(w¯p∗)={F−1[p−w¯p∗−cr,ip] if w¯p∗<p−cr,i0 Otherwise . (4) Consider $$S\left( q \right)=q-\smallint_{0}^{q} F\left( x \right)dx$$ as expected sales when the retailer places an order of $$q$$ units. Therefore, each retailer type’s expected profit in this price-only contract is as follows:   Πr,iPC¯=pS(qiPC¯(w¯p∗))−qiPC¯(w¯p∗)(w¯p∗+cr,i)∀i=1,2,3,…,N. (5) In this approach, the manufacturer does not consider all the cost estimations. In reality, different costs of different retailer types lead to an expected profit for the manufacturer, which differs from $$\Pi_{m}^{\overline{\rm PC}}$$ in (1). In this situation, the manufacturer’s actual expected profit is as follows:   ΠmaPC¯=∑i=1Nλi(w¯p∗−cm)qiPC¯(w¯p∗). (6) 4.2. Approach 1-2: RS contract based on $$\bar{c}_{r}$$ Suppose the manufacturer wants to offer an RS contract $$(\bar{w},\bar{\Phi})$$ to the retailer, assuming the retailer’s cost is $$\bar{c}_{r}$$. In this situation, the manufacturer’s optimization problem is as follows:   Max(w¯,Φ¯)ΠmRS¯=(w¯−cm)qRS¯+(1−Φ¯)pS(qRS¯), (7)  S.T.qRS¯={F−1[Φ¯p−w¯−c¯rΦ¯p] if w¯<Φ¯p−c¯r0 Otherwise , (8)  0<Φ¯⩽1,−c¯r<w¯. (9) $$\Pi_{m}^{\overline{\rm RS}}$$ represents the manufacturer’s expected profit under the RS contract. Constraint (8) represents the dependency of a retailer’s order quantity with the cost of $$\bar{c}_{r}$$ to $$\bar{w}$$ and $$\bar{\Phi}$$. Constraint (9) represents a feasible range for the decision variables. The lower bound of $$\bar{w}$$ prevents infinite orders from a retailer with the cost of $$\bar{c}_{r}$$. Since in the RS contract the manufacturer can obtain some portion of its partner’s revenue $$(1-\bar{\Phi})$$, $$\bar{w}$$ can become negative. According to Cachon & Lariviere (2005), the interpretation for a negative $$\bar{w}$$ is as follows: whenever the retailer’s cost is high, the manufacturer should subsidize the retailer in a low-margin business in order to claim a specific portion of the retailer’s revenue. The manufacturer can adjust the RS contract parameters to maximize its expected profit, while allocating zero profit to a retailer with the cost of $$\bar{c}_{r}$$. The maximum profit will be available for the manufacturer if the retailer’s order quantity equals the optimal order quantity in the system. According to the classic newsvendor model, the system’s optimal order quantity satisfies $$F^{-1}\left[ {\frac{p-c_{m} -\bar{c}_{r} }{p}} \right]$$. Therefore, an optimal relation between $$\bar{w}$$ and $$\bar{\Phi}$$ can be obtained through the following equation:   F−1[Φ¯p−w¯−c¯rΦ¯p]=F−1[p−cm−c¯rp]⟹w¯=Φ¯(cm+c¯r)−c¯r. (10) According to the analysis of Cachon & Lariviere (2005), in this situation $$\bar{\Phi}$$ is the retailer’s share of the supply chain’s profit. Thus, in order to allocate the least profit share to the retailer, the optimal value of $$\bar{\Phi}$$ ($$\bar{\Phi}^{\ast})$$ moves toward zero. In this situation, the optimal value of $$\bar{w}$$ ($$\bar{w}^{\ast})$$ moves toward $$-\bar{c}_{r}$$. However, the fact is that there are different retailer types each with a specific cost. A retailer’s order quantity with the actual cost of $$c_{r,i}$$ can be obtained from Relation (11):   qiRS¯={F−1[Φ¯∗p−w¯∗−cr,iΦ¯∗p] if w¯∗<Φ¯∗p−cr,i0 Otherwise . (11) Therefore, each retailer type’s actual expected profit in the case of this RS contract is as follows:   Πr,iRS¯=Φ¯∗pS(qiRS¯)−qiRS¯(w¯∗+cr,i) ∀i=1,2,3,…,N. (12) Putting it differently, the manufacturer’s actual expected profit, which differs from $$\Pi_{m}^{\overline{\rm RS}}$$ in (7), is as follows:   ΠmaRS¯=∑i=1N⁡λi[(w¯∗−cm)qiRS¯+(1−Φ¯∗)pS(qiRS¯)]. (13) Proposition 1 In Approach 1, the manufacturer’s actual expected profit under the price-only contract is higher than that under the RS contract. All proofs are presented in Appendix. According to Proposition 1, the manufacturer always prefers the price-only contract to the RS contract, when it designs a contract only based on $$\bar{c}_{r}$$. We can improve the manufacturer’s actual expected profit in the case of RS contract by implementing a minor modification. If we increase the lower bound of $$\bar{w}$$ from $$-\bar{c}_{r}$$ to $$-c_{r,1}$$, then the retailer’s order quantity with any cost level will not move toward infinity. In this situation, by considering the manufacturer’s expected profit in Relation (13), the positive profit share from the retailer’s revenue, i.e. $$\left( {1-\bar{\Phi}^{\ast}}\right)pS\left( {q_{i}^{\overline{\rm RS}}} \right),$$ might compensate the contingent negative income from selling to the retailer, i.e., $$\left( {\bar{w} ^{\ast}-c_{m}} \right)q_{i}^{\overline{\rm RS}}$$ if $$\bar{w}^{\ast}<0$$. As a result of this modification, the manufacturer finds a chance for a positive expected profit. We call this a SMRS contract and use it later in the numerical analysis. 5. Approach 2: contract design based on all cost estimations In this section, we investigate more sophisticated approaches in which the manufacturer considers all cost estimations for contract design in the case of cost information asymmetry. Approach 2-1 addresses the price-only contract as an outside option that establishes each firm’s reservation profit in Approaches 2-2, 2-3 and 2-4. In Approaches 2-2, 2-3 and 2-4, we focus on the design of the RS contract. 5.1. Approach 2-1: price-only contract as an outside option Suppose the manufacturer offers a wholesale price ($$w_{p})$$ to the retailer, by considering all types of the retailer. The manufacturer’s problem is:   Maxwp>cmΠmPC1=∑i=1N⁡λi(wp−cm)qiPC1(wp), (14)  S.T.qiPC1(wp)={F−1[p−wp−cr,ip]  if wp<p−cr,i0 Otherwise . (15) Relation (15) represents the retailer’s optimal order for each $$w_{p}$$, based on the newsvendor model. According to Lariviere & Porteus (2001), by assuming an IGFR demand distribution, the unique optimal $$w_{p}$$ satisfies:   dΠmPC1dwp=∑i=1Nλi(F−1[p−wp∗−cr,ip]−(wp∗−cm)pf(F−1[p−wp∗−cr,ip]))=0. (16) Notice that if the IGFR assumption does not hold, we can find the optimal value of $$w_{p}$$ through a one-dimensional search procedure. Consider $$w_{p}^{\ast}$$ as the optimal value of $$w_{p}$$. We assume each retailer type’s expected profit, in the case of this price-only contract, as its reservation profit to participate in any other contract offered by the manufacturer:   RPr,i=pS(qiPC1(wp∗))−qiPC1(wp∗)(wp∗+cr,i) ∀i=1,2,3,…,N. (17) Note that $$\mbox{RP}_{r,i}$$ depends on the retailer type, and regarding (15), it might be zero for some types. We also consider the price-only scheme as an outside option for the manufacturer. It provides the manufacturer with the right to sign a price-only contract with some retailer types if they are not chosen for the RS contract. 5.2. Approach 2-2: design of an RS contract and a price-only contract Regarding the manufacturer’s outside option based on the price only contract, it may not accept to enter the RS scheme with every retailer type. We considered this issue in the following models through a cut-off policy (Ha, 2001; Corbett et al., 2004). We define $$c_{r,U} \in \{c_{r,1} ,c_{r,2} ,c_{r,3} ,\ldots ,c_{r,N} \}$$ as a cut-off point. The manufacturer designs a single RS contract for a retailer with a cost equal to or less than $$c_{r,U}$$ and it will trade with the other types of the retailer $$\left( {i=U+1,\ldots ,N} \right)$$ according to a price-only contract. The single RS contract is identical for different retailer types. The following steps determine the sequence of decisions in Approach 2-2. Step 1: The manufacturer offers an RS contract with parameters $$(w,\Phi )$$ to the retailer. The RS contract is designed so that if the retailer’s cost is equal to or less than $$c_{r,U}$$, its participation constraint is satisfied and therefore, the retailer will accept the RS contract. Otherwise, the retailer’s cost is greater than $$c_{r,U}$$ and it waits for the next offer in Step 2. Step 2: The manufacturer offers a price-only contract to the retailer. The price-only contract satisfies the retailer’s participation constraint as well. Therefore, the manufacturer’s optimization problem in Approach 2-2 is:   Max(U,w,Φ,wp ′)ΠmRS1=∑i=1Uλi[(w−cm)qiRS1+(1−Φ)pS(qiRS1)]+∑i=U+1Nλi(wp ′−cm)qi ′PC1(wp ′), (18)  S.T.ΦpS(qiRS1)−qiRS1(w+cr,i)⩾RPr,i∀i=1,2,3,…,U, (19)  ΦpS(qiRS1)−qiRS1(w+cr,i)⩽RPr,i∀i=U+1, …, N, (20)  qiRS1={F−1[Φp−w−cr,iΦp]  if w<Φp−cr,i0 Otherwise ∀i=1,2,3,…,N, (21)  pS(qi ′PC1)−qi ′PC1(wp ′+cr,i)⩾RPr,i∀i=U+1, …, N, (22)  qi ′PC1(wp ′)={F−1[p−wp ′−cr,ip] if wp ′<p−cr,i0 Otherwise  ∀i=U+1, …, N, (23)  U∈{0,1,2,3,…,N},0<Φ⩽1,−cr,1<w,cm⩽wp ′. (24) The first summation of the objective function is the manufacturer’s expected profit based on the RS contract, with retailer types being no more than $$U$$. Since it is decreasing in the retailer’s cost, the cut-off policy will be optimal (Ha, 2001). The second summation of the objective function represents the manufacturer’s expected profit based on the price-only contract with retailer types more than $$U$$. Notice that the wholesale price in the second summation of the objective function ($$w_{p}^{'})$$ differs from that in Relation (14), i.e. $$w_{p}$$, unless we have $$U=0$$. Constraints (19)–(21) belong to Step 1 of the sequence of decisions. Constraint (19) represents the participation constraint for retailer types no more than $$U$$ in the RS scheme. $$\mbox{RP}_{r,i}$$ in the constraints comes from Relation (17). Constraint (20) prevents participation of a retailer with a cost more than $$c_{r,U}$$ in the RS scheme. Relation (21) represents the retailer’s order quantity under the RS scheme. Constraints (22) and (23) belong to Step 2. Constraint (22) represents the participation constraint for retailer types more than $$U$$ in the price-only contract. Relation (23) represents the retailer’s order quantity under the price-only contract. Relation (24) represents the feasible range of variables. In order to prevent the retailer’s tendency toward infinite orders, $$w$$ has to be greater than $$-c_{r,1}$$. 5.2.1. General solution methodology for Approach 2-2 It should be mentioned that the objective function in Approach 2-1 is the simplest case of the objective function in Approach 2-2, and also Approaches 2-3 and 2-4, for $$U=0.$$Lariviere & Porteus (2001) have established that all we need to guarantee the concavity of this type of objective function, with respect to the retailer’s order quantity, is an IGFR demand distribution. We can then find the corresponding optimal wholesale price as well. However, in general without an IGFR assumption, we cannot be sure about the concavity of this type of objective function in Approach 2-1 and the other more sophisticated approaches (Approaches 2-2, 2-3 and 2-4). Thus, here we first propose a general solution methodology. Then in Section 5.2.2, we present some analytical results considering IGFR distribution for demand. Step 1: Set $$U=0$$. Step 2: Find the optimal RS contract for a retailer type $$i=1,2,3,\ldots ,U,$$ as shown below: Step 2.1: By considering Constraints (19)–(21) and (24), generate feasible values for $$(w,\Phi )$$ from their entire feasible range. Step 2.2: Find the optimal value for $$(w,\Phi )$$, i.e. the values that optimize the first summation of (18). Step 3: Find the optimal price-only contract for a retailer type $$i=U+1,\ldots ,N$$, as shown below: Step 3.1: By considering Constraints (22)–(24), generate feasible values for $$w_{p}^{'}$$ from its entire feasible range. Step 3.2: Find the optimal value for $$w_{p}^{'}$$, i.e. the value that optimizes the second summation of (18). Step 4: If $$U<N$$, compute the whole Relation (18), set $$U=U+1$$ and go to Step 2. Otherwise, select the optimal value for $$U$$, i.e. the one that optimizes (18). The optimal values for $$(w,\Phi )$$ and $$w_{p}^{'}$$, corresponding to the optimal $$U$$, are determined in Steps 2 and 3. According to the steps mentioned above, we find the optimal solution based on a complete enumeration method by searching the entire space of decision variables with a specific precision. This way, we ensure that without losing any solution space, the search procedure is converged to the optimal solution. 5.2.2. A special case in Approach 2-2: IGFR distribution When the demand distribution is IGFR, we can make sure that the first and second summations in the objective function are unimodal in terms of $$q_{i}^{\rm RS1}$$ and $$q_{i}^{'{\rm PC}1}$$, respectively. Proposition 2 Suppose the demand distribution is IGFR. I. The first summation of (18) is concave with respect to $$q_{i}^{\rm RS1}$$, for each feasible value of $$\Phi$$. II. The second summation of (18) is concave with respect to $$q_{i}^{'{\rm PC}1}$$. We can characterize optimal orders from the manufacturer’s perspective based on the analysis provided by Lariviere & Porteus (2001). Suppose that the support of $$F$$ is $$[a,b)$$ and define $$\bar{q}_{i}^{\rm RS1}$$ as the largest value of $$q_{i}^{\rm RS1}$$ that satisfies $$g(q_{i}^{\rm RS1} )\leqslant \frac{1}{\Phi}$$. In this situation, any solution such as $$q_{i}^{\ast {\rm RS}1}$$ from equalizing Relation (A.6) to zero is unique and must lie in the interval $$[a,\bar{q}_{i}^{{\rm RS}1}]$$. The optimal order quantity for the manufacturer will be either $$q_{i}^{\ast RS1}$$ or $$a$$. The story is the same for $$q_{i}^{\ast {\rm PC}1}$$, by considering $$\bar{q}_{i}^{{\rm PC}1}$$ as the largest value of $$q_{i}^{{\rm PC}1}$$ that satisfies $$g(q_{i}^{{\rm PC}1} )\leqslant 1$$. Note that by finding the optimal value of $$q_{i}^{{\rm RS}1}$$ and $$q_{i}^{'{\rm PC}1}$$, we can find the corresponding optimal value for $$w$$ and $$w_{p}^{'}$$ through inverse demand curves, e.g. Relation (A.4). Proposition 3 Suppose the demand distribution is IGFR. Let $$w^{\ast }=\Phi p \bar{F} \left( {q_{i}^{\ast {\rm RS}1} } \right)-c_{r,i}$$ for each feasible $$\Phi$$ and $$w_{p}^{\ast'} =p \bar{F} \left( {q_{i}^{\ast '{\rm PC}1} } \right)-c_{r,i}$$ for one of the values of ‘$$i$$’. I. If $$w^{\ast }$$ satisfies Relations (19), (20) and (24), it will be the optimal value. If $$w^{\ast }$$ violates one of the constraints in Relation (19), then the optimal value of $$w$$ will be $$w_{U}$$, where $$w_{U}$$ is the largest $$w$$ that satisfies all the constraints in Relation (19). $$w_{U}$$ can be found by considering the constraints in Relation (19) as equations. If $$w^{\ast}$$ violates one of the Constraints (20) or (24), then the optimal value of $$w$$ will be $$w_{L}$$, where $$w_{L}$$ is the smallest $$w$$ that satisfies Constraints (20) and (24). $$w_{L}$$ is determined by considering the Constraints (20) and (24) as equations. II. If $$w_{p}^{\ast'}$$ satisfies Relation (22), it will be the optimal value. If $$w_{p}^{\ast '}$$ violates one of the constraints in Relation (22), then the optimal value of $$w_{p}^{'}$$ will be $$w_{pU}^{'}$$, where $$w_{pU}^{'}$$ is the largest $$w_{p}^{'}$$ that satisfies all the constraints in Relation (22). $$w_{pU}^{'}$$ can be found by considering the constraints in Relation (22) as equations. The optimal value of $$w_{p}^{'}$$, i.e. $$\mbox{Min}\{w_{p}^{\ast '} ,w_{pU}^{'} \}$$, satisfies Relation (24) as well. Proposition 3 helps us find $$w^{\ast }$$ for each feasible $$\Phi$$. We can find the optimal value for $$\Phi$$ based on the general solution methodology through a one-dimensional search procedure. 5.3. Approach 2-3: designing a menu of coordinating RS contracts and a price-only contract The sequence of decisions in Approach 2-3 (and Approach 2-4) is similar to that in Approach 2-2, except that in Step 1, the manufacturer designs a menu of coordinating RS contracts, instead of a single RS contract. Suppose the manufacturer designs a menu of coordinating RS contracts so that each contract in the menu is particular for each retailer type $$i=1,2,3,\ldots ,N$$. The retailer has to choose one contract from the menu. According to the revelation principle, the manufacturer can limit its search for an optimal menu to the class of ‘truth-telling’ contracts (Baron et al., 1982). In this approach, we define $$q_{j,i}^{{\rm RS}2}$$ as the order quantity of a retailer whose actual cost is $$c_{r,i}$$ and it selects the contract designed for a retailer with the cost of $$c_{r,j}$$. The manufacturer’s problem in Approach 2-3 is:   Max(U,wi,Φi,wp ′)ΠmRS2=∑i=1U⁡λi[(wi−cm)qi,iRS2+(1−Φi)pS(qi,iRS2)]+∑i=U+1N⁡λi(wp ′−cm)qi ′PC1(wp ′), (25)  S.T.ΦipS(qi,iRS2)−qi,iRS2(wi+cr,i)⩾RPr,i∀i=1,2,3,…,U, (26)  ΦipS(qi,iRS2)−qi,iRS2(wi+cr,i)⩾ΦjpS(qj,iRS2)−qj,iRS2(wj+cr,i) (27)  ∀i=1,2,3,…,U,∀j=1,2,3,…,U, i≠j,ΦipS(qi,fRS2)−qi,fRS2(wi+cr,f)⩽RPr,f∀i=1,2,3,…,U, f=U+1,…,N, (28)  wi=Φi(cm+cr,i)−cr,i∀i=1,2,3,…,U, (29)  qj,iRS2={F−1[Φjp−wj−cr,iΦjp] if Φjp−wj−cr,i>00 Otherwise ∀i,j=1,2,3,…,N. (30) In addition to Constraints (22) and (23),   U∈{0,1,2,3,…,N},0<Φi⩽1,−cr,1<wi,cm⩽wp ′,∀i=1,2,3,…,U. (31) The first summation in the objective function represents the manufacturer’s expected profit based on the menu of RS contracts. The second summation in the objective function represents the manufacturer’s expected profit based on the price-only contract with retailer types more than $$U$$. Constraints (26)–(30) belong to Step 1 of the decision sequences, as introduced in Section 5.2. Constraints (26) and (27) are ‘individual rationality’ and ‘incentive compatibility’ constraints, respectively (Corbett et al., 2004). Constraint (28) prevents a retailer’s tendency with a cost more than $$c_{r,U}$$ from using the RS menu. Relation (29) determines the coordinating relation between parameters $$(w_{i} ,\Phi_{\mbox{i}} )$$. Relation (30) determines the optimal value of $$q_{j,i}^{\rm RS2}$$. Constraints (22) and (23) for Step 2 recur in this model as well. Constraint (31) determines the feasible value of the decision variables. The lower bound of $$w_{i}$$ ensures that $$F(q_{j,i}^{\rm RS2} )$$ will not be more than one. In order to solve this model, by fixing the value of $$U$$, we have to find the optimal menu of coordinating RS contracts. According to our investigations, this model is analytically intractable. Consequently, we turned to numerical analysis. We conducted extensive numerical investigations with various model parameters and observed in all of them that the menu of contracts was not capable of motivating the retailer to behave truthfully. The following example illustrates this point. 5.3.1. An example for Approach 2-3 Consider two estimations from the retailer’s cost as $$c_{r,l} <c_{r,h}$$. Assume that the parameters of the model are estimated as: $$p=150,\,c_{m} =40,c_{r,l} =10,c_{r,h} =30, \ X\sim \mbox{Normal}(\mu =100,\sigma =20).$$ Suppose the manufacturer wants to design a menu including two coordinating RS contracts. Depending on the retailer’s actual cost and its decision, four cases may occur: Case 1: The retailer’s actual cost is $$c_{r,l}$$ and it selects the contract that is designed for type $$l$$ (coloured in blue). Case 2: The retailer’s actual cost is $$c_{r,l}$$ and it selects the contract that is designed for type $$h$$ (coloured in red). Case 3: The retailer’s actual cost is $$c_{r,h}$$ and it selects the contract that is designed for type $$h$$ (coloured in green). Case 4: The retailer’s actual cost is $$c_{r,h}$$ and it selects the contract that is designed for type $$l$$ (coloured in yellow). In Cases 1 and 3, the retailer reacts truthfully. Figure 2(a) depicts the retailer’s profit under different cases for all feasible values of $$\Phi _{l}$$ and $$\Phi_{h}$$. In order to clearly show the retailer’s behaviour, in Fig. 2(a), we only depict the regions in which each retailer type benefits from not selecting its particular contract, i.e. where the retailer is not motivated for truth-telling. Figure 2(b) shows Fig. 2(a) from the top view. Cases 2 and 4 occur, respectively, in the red and yellow regions. We observe that there is no region in Fig. 2(b) where both retailer types react truthfully. Fig. 2. View largeDownload slide (a) Retailer’s profit in Cases 1-4. (b) Top view of (a). Fig. 2. View largeDownload slide (a) Retailer’s profit in Cases 1-4. (b) Top view of (a). Therefore, Constraint (27) cannot be satisfied and the manufacturer, even when facing just two estimations, cannot motivate the retailer towards truth-telling. Note that the only constraint we can relax from the menu design issue in Approach 2-3 is (29). If we can somehow relax this constraint, while preserving system coordination, by generating different values for $$(w_{i} ,\Phi_{\mbox{i}} )$$ a truth-telling menu may be obtained. We try to create such conditions in Approach 2-4. 5.4. Approach 2-4: designing a tailored menu of coordinating RS contracts and a price-only contract In this approach, we relax Constraint (29); however, in order to keep coordinated order quantities, we use contracts with quantity commitments as used, for instance, by Li et al. (2009) and Gan et al. (2010). Thus, the manufacturer offers a menu of RS contracts as: $$(w_{i} ,\Phi_{\mbox{i}} ,q_{i}^{\rm RS3} )$$ designed for each retailer type $$i=1,2,3,\ldots ,U$$. The manufacturer’s optimization problem is:   Max(U,wi,Φi,wp ′)ΠmRS3=∑i=1U⁡λi[(wi−cm)qiRS3+(1−Φi)pS(qiRS3)]+∑i=U+1N⁡λi(wp ′−cm)qi ′PC1(wp ′), (32)  S.T.ΦipS(qiRS3)−qiRS3(wi+cr,i)⩾RPr,i∀i=1,2,3,…,U, (33)  ΦipS(qiRS3)−qiRS3(wi+cr,i)⩾ΦjpS(qjRS3)−qjRS3(wj+cr,i) (34)  ∀i=1,2,3,…,U,∀j=1,2,3,…,U, i≠j,ΦipS(qiRS3)−qiRS3(wi+cr,f)⩽RPr,f∀i=1,2,3,…,U, f=U+1,…,N, (35)  qiRS3=F−1[p−cm−cr,ip]∀i=1,2,3,…,N. (36) In addition to Constraints (22) and (23),   U∈{0,1,2,3,…,N},0⩽Φi⩽1,wi:free in sign,cm⩽wp ′,∀i=1,2,3,…,U. (37) Description of the objective function is similar to the previous models. The role of the constraints is similar to the model in Approach 2-3. In Relation (36), $$q_{i}^{\rm RS3}$$ represents the coordinated order quantity for a retailer with the cost of $$c_{r,i}$$. Regarding the objective function for each $$U$$, the manufacturer tries to determine $$w_{i}$$ and $$\Phi_{i}$$ so that it owns as much coordinated system profit as possible. Since in this model the order quantity is offered in the contract, there is no concern about the sign of $$w_{i}$$. The objective function determines its optimal level. While Approach 2-4 is a coordinating approach, it does not require the restrictive relation between $$(w_{i} ,\Phi_{\mbox{i}} )$$, i.e. Relation (29). In fact, even if the model in Approach 2-3 has a feasible solution space, its solution can be generated by Approach 2-4 as well, because Approach 2-3 is a special case of Approach 2-4. 5.4.1. General solution methodology for Approach 2-4 Step 1: Set $$U=0$$. Step 2: Find the optimal tailored menu of RS contracts for retailer types $$i=1,2,3,\ldots ,U_{\mathrm{\thinspace }}$$ Note that the set of variables and Constraints (33)–(37), which are related to the issue of menu design, constitute a linear programming problem, the optimal solution of which can be found by the CPLEX algorithm. Step 3: Find the optimal price-only contract for a retailer type $$i=U+1,\ldots ,N.$$ Step 3.1: By considering Constraints (22), (23) and (37), generate feasible values of $$w_{p}^{'}$$ from its entire feasible range. Step 3.2: Find the optimal value of $$w_{p}^{'}$$, i.e. the value that optimizes the second summation of (32). Step 4: If $$U<N$$, compute the whole Relation (32), set $$U=U+1$$ and go to Step 2. Otherwise, select the optimal value of $$U$$, i.e. the one that optimizes (32). The optimal values of $$(w_{i} ,\Phi_{\mbox{i}} )$$ and $$w_{p}^{'}$$ corresponding to the optimal value of $$U$$ are determined in Steps 2 and 3. We can find the optimal value of $$w_{p}^{'}$$ in Step 3, under the IGFR assumption, through Propositions 2 and 3. Notice that a specific case of the tailored menu may occur when $$\Phi_{\mbox{i}} =1, \ \forall i=1,2,3,\ldots ,U$$. In this case, the tailored menu consists of order quantities and wholesale prices. In this specific case, there will not be a collaborative relationship like the one in the RS contract between the manufacturer and the retailer. According to our investigations, a truth-telling menu can be designed by Approach 2-4. We provide some analytical results for this approach in Section 7.2. 6. Numerical analysis We first present an example in Section 6.1 to clarify how the proposed models in Approaches 2-2 and 2-4 work. Since the other investigated approaches are specific cases of Approaches 2-2 and 2-4, in Section 6.1, we only concentrate on these two approaches. Then, in Section 6.2, we conduct a sensitivity analysis and compare the manufacturer’s, the retailer’s and the system’s expected profits under different approaches, with different parameter settings. Finally, in Section 6.3, we present some managerial insights from the numerical analysis. 6.1. An example for Approaches 2-2 and 2-4 Assume that the parameters of the model are estimated as follows:   p =150, cm=40, cr,i={10,20,30,40,50}, λ1=λ2=...=λ5 =0.2, X∼Normal(μ=100,σ=20). In order to find the optimal value of variables in Approaches 2-2 and 2-4, e.g. $$(w,\Phi)$$, we wrote a computer program in MATLAB software on the basis of the general solution methodologies. We generated feasible values for the decision variables in the increments of 0.01. Table 1 presents the results of Approach 2-2. We computed the manufacturer’s, the retailer’s and the system’s expected profits for each value of $$U$$ and in terms of $$_{\mathrm{\thinspace }}$$ estimations from the retailer’s cost. The column ‘Exp. Prof.’ represents the expected profits. Table 1 Results of Approach 2-2      Note that the results for $$U=0$$ represent the reservation profit case in which the manufacturer, based on Approach 2-1, designs a price-only contract for all retailer types. The highlighted cells represent the cases in which the manufacturer offers a price-only contract to the corresponding retailer types. For instance, for $$U=2$$ an RS contract is designed for a retailer whose cost is 10 or 20 and a price-only contract is designed for a retailer whose cost is > 20. In the column ‘Exp. Prof.’, we observe that the manufacturer’s expected profit is increased as $$U$$ is increased up to 4. This means the optimal value of $$U$$ is 4 and the manufacturer should design an RS contract for a retailer whose cost is < 50, and a price-only contract for a retailer whose cost is 50. The optimal parameters of the RS contract are $$(w=7.99,\Phi =0.39)$$ and the optimal wholesale price is $$w_{p}^{'} =92.42$$. Table 2 presents the results of Approach 2-4. From the column ‘Exp. Prof.’ in Table 2, we realize that the optimal value of $$U$$ is 5 and the manufacturer should design a tailored menu of RS contracts for all retailer types. The optimal tailored menu is: $$\{(w_{1} =44.77,\Phi_{1} =0.7, q_{1}^{\rm RS3} =108.61)$$, $$\left( {w_{2} =45.32,\Phi_{2} =0.7,q_{2}^{\rm RS3} =105.07} \right), \ \left( {w_{3} =45.98,\Phi_{3} =0.7,q_{3}^{\rm RS3} =101.67} \right), ( w_{4} =46.78$$, $$\Phi_{4}=0.7, q_{4}^{\rm RS3} =98.33),\left( {w_{5} =47.75,\Phi_{5} =0.7,q_{5}^{\rm RS3} =94.93} \right)$$ Table 2 Results of Approach 2-4      6.2. Sensitivity analysis: comparison of different approaches In this subsection, we want to compare the effects of each approach studied on the expected profits. In Section 6.2.1, we compare expected profits based on $$\bar{c}_{r}$$ (Approach 1) with that based on all cost estimations (Approach 2). Along this line, we compare expected profits under the price-only contract in Approach 1-1 with those under Approach 2-1. We compare the expected profits under the RS contract in Approach 1-2 with those under Approaches 2-2 and 2-4 as well. Focusing on Approach 2 in Section 6.2.2, we then compare the expected profits under Approaches 2-2 and 2-4. Suppose the manufacturer has three estimations of the retailer’s cost with the same probabilities, corresponding to optimistic, moderate and pessimistic estimations. Assume that the parameters of the model in the base case are estimated as: $$p=150, \ c_{m} =40, \ c_{r,i} =\left\{ {10,30,50} \right\}, \ X\sim \mbox{Normal}(\mu =100,\sigma =20)$$. In the following subsections, different tables with different values for $$p, \ c_{m} , \ \mu , \ \sigma$$ and $$c_{r,i}$$ are presented. Only one parameter at a time is varied while the remaining parameters are fixed as above. In all the tables, through the following subsections, the optimal expected profit and an index are represented in columns ‘Man.’ (manufacturer), ‘Ret.’ (retailer) and ‘System’. 6.2.1. Comparison of Approaches 1 and 2 In order to make easier comparisons, hereafter we define $$VP$$ as an index that compares expected profit under Approaches 1 and 2 and returns variations as a percentage of the expected profit in Approach 1:   VP=(Expected Profit in Approach 2−Expected Profit in Approach 1)|Expected Profit in Approach 1|∗100%. (38) For example, assume the expected profit is 10 and 30, respectively, in Approaches 1 and 2. In this situation, $$VP$$ becomes 200%. Putting it differently, the expected profit in Approach 2 is the expected profit in Approach 1 plus 200% of the expected profit in Approach 1. $$VP$$ may be positive or negative and it can be calculated based on either the price-only or the RS contract, for the manufacturer, the retailer or the system. 6.2.2. Comparison of Approaches 1-1 and 2-1 Tables 3–7 present expected profits as well as $$VP$$. These tables are for the price-only contract in Approaches 1 and 2. Table 3 Comparison of the expected profits in Approaches 1-1 and 2-1, under different values for ‘$$p$$’    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{p}$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  100  1313.63  655.59  1969.22  1.96  -14.02  -3.36  150  3568.50  939.40  4507.90  21.00  76.09  32.48  200  5824.13  1237.13  7061.26  33.99  49.92  36.78  250  11029.75  1604.09  12633.84  1.87  33.59  5.90  300  14573.87  2046.79  16620.66  0.47  21.57  3.07    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{p}$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  100  1313.63  655.59  1969.22  1.96  -14.02  -3.36  150  3568.50  939.40  4507.90  21.00  76.09  32.48  200  5824.13  1237.13  7061.26  33.99  49.92  36.78  250  11029.75  1604.09  12633.84  1.87  33.59  5.90  300  14573.87  2046.79  16620.66  0.47  21.57  3.07  Table 4 Comparison of the expected profits in Approaches 1-1 and 2-1, under different values for ‘$$c_{m}$$’    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{c}_{\mathbf{m}}$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  10  5067.1  1162.21  6229.31  30.56  56.70  35.44  20  4560.61  1085.65  5646.26  28.09  62.12  34.64  30  4061.08  1011.03  5072.11  24.99  68.56  33.67  40  3568.50  939.40  4507.90  21.00  76.09  32.48  80  1680.31  674.48  2354.79  1.71  -16.42  -3.49    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{c}_{\mathbf{m}}$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  10  5067.1  1162.21  6229.31  30.56  56.70  35.44  20  4560.61  1085.65  5646.26  28.09  62.12  34.64  30  4061.08  1011.03  5072.11  24.99  68.56  33.67  40  3568.50  939.40  4507.90  21.00  76.09  32.48  80  1680.31  674.48  2354.79  1.71  -16.42  -3.49  Table 5 Comparison of the expected profits in Approaches 1-1 and 2-1, under different values for ‘$$\mu$$’    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{\mu }$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  50  1513.22  558.97  2072.19  4.12  29.52  10.97  100  3568.50  939.40  4507.90  21.00  76.09  32.48  200  8646.83  1570.78  10217.61  17.56  129.86  34.82  300  13872.18  2209.29  16081.47  16.03  153.41  34.91  500  24425.64  3512.22  27937.86  14.72  172.96  34.61    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{\mu }$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  50  1513.22  558.97  2072.19  4.12  29.52  10.97  100  3568.50  939.40  4507.90  21.00  76.09  32.48  200  8646.83  1570.78  10217.61  17.56  129.86  34.82  300  13872.18  2209.29  16081.47  16.03  153.41  34.91  500  24425.64  3512.22  27937.86  14.72  172.96  34.61  Table 6 Comparison of the expected profits in Approaches 1-1 and 2-1, under different values for ‘$$\sigma$$’    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{\sigma }$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  5  4784.87  714.46  5499.33  15.22  165.66  34.77  10  4323.41  785.39  5108.8  17.56  129.86  34.82  20  3568.50  939.40  4507.90  21.00  76.09  32.48  30  3001.71  1057.25  4058.96  22.39  46.11  28.57  40  3026.44  1117.95  4144.39  4.12  29.52  10.97    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{\sigma }$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  5  4784.87  714.46  5499.33  15.22  165.66  34.77  10  4323.41  785.39  5108.8  17.56  129.86  34.82  20  3568.50  939.40  4507.90  21.00  76.09  32.48  30  3001.71  1057.25  4058.96  22.39  46.11  28.57  40  3026.44  1117.95  4144.39  4.12  29.52  10.97  Table 7 Comparison of the expected profits in Approaches 1-1 and 2-1, under different values for ‘$$c_{r,i}$$’    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{c}_{\mathbf{r},\mathbf{i}}$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  0,30,60  3669.64  1224.55  4894.19  2.64  102.33  27.58  10,30,50  3568.50  939.40  4507.90  21.00  76.09  32.48  15,30,45  3510.44  802.6  4313.04  29.54  59.91  35.19  20,30,40  4430.37  673.88  5104.25  6.82  42.43  11.52  25,30,35  4849.43  641.78  5491.21  0.12  12.58  1.58    Expected profits in Approach 1-1  $$VP$$ (%)  $$\mathbf{c}_{\mathbf{r},\mathbf{i}}$$  (Price-only contract)  (Based on Approach 2-1)    Man.  Ret.  System  Man.  Ret.  System  0,30,60  3669.64  1224.55  4894.19  2.64  102.33  27.58  10,30,50  3568.50  939.40  4507.90  21.00  76.09  32.48  15,30,45  3510.44  802.6  4313.04  29.54  59.91  35.19  20,30,40  4430.37  673.88  5104.25  6.82  42.43  11.52  25,30,35  4849.43  641.78  5491.21  0.12  12.58  1.58  In all Tables 3–7, we observe that the manufacturer’s expected profit under Approach 2-1 is more than that under Approach 1-1, because $$VP$$ is always positive. This improvement can be as high as nearly 34% for $$p=200$$ in Table 3. We observe that for $$p=100, \ c_{m} =80$$ the retailer’s share from the system’s profit under Approach 2-1 is so low that $$VP$$ becomes negative. However, depending on the model parameters the retailer’s and the system’s expected profits, as well as the manufacturer’s, are most often improved in Approach 2-1. This improvement for the retailer might be as high as 172% for $$\mu =500$$ in Table 5. Therefore, we observe that a manufacturer should always design the price-only contract considering all estimations of the retailer’s cost. This option is most often profitable for the retailer as well. 6.2.3. Comparison of Approach 1-2 with Approaches 2-2 and 2-4 Here we compare the expected profits in Approaches 1 and 2 by considering the RS contract. We take into account the expected profits in Approach 1-2 based on the SMRS contract introduced in Section 4.2. In the SMRS contract, we increase the lower bound of $$\bar{w}$$ from $$-\bar{c}_{r}$$ to $$-c_{r,1}$$, so that $$F(q_{i}^{\overline{RS}})<1 \ \forall i=1,2,3,...,N$$ holds. It should be mentioned that depending on the model parameters even with this small modification in the RS contract, the manufacturer’s expected profit might become negative; however, according to Proposition 1, it will be definitely negative without this modification. Tables 8–12 present expected profits as well as $$VP$$, which is calculated based on Approach 2-2 or Approach 2-4. Table 8 Comparison of the expected profits in Approach 1-2 with those in Approaches 2-2 and 2-4, under different values for ‘$$p$$’    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$p$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  100  779.90  818.73  1598.63  100.12  -20.59  38.30  114.10  -16.97  46.97  150  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  200  5473.02  1260.61  6733.63  60.14  76.19  63.15  66.54  93.53  71.60  250  8511.07  1417.20  9928.27  53.62  82.99  57.81  58.98  98.83  64.67  300  11538.11  1638.52  13176.63  51.17  82.70  55.09  55.74  96.73  60.84    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$p$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  100  779.90  818.73  1598.63  100.12  -20.59  38.30  114.10  -16.97  46.97  150  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  200  5473.02  1260.61  6733.63  60.14  76.19  63.15  66.54  93.53  71.60  250  8511.07  1417.20  9928.27  53.62  82.99  57.81  58.98  98.83  64.67  300  11538.11  1638.52  13176.63  51.17  82.70  55.09  55.74  96.73  60.84  Table 9 Comparison of the expected profits in Approach 1-2 with those in Approaches 2-2 and 2-4, under different values for ‘$$c_{m}$$’    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$\mathbf{c}_{\mathbf{m}}$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  10  5483.36  1167.45  6650.81  33.79  87.09  43.15  39.47  107.48  51.41  20  4465.96  1167.45  5633.41  43.61  74.78  50.07  49.06  97.19  59.03  30  3506.04  1075.03  4581.07  57.42  81.25  63.01  62.52  103.66  72.18  40  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  80  -1301.27  804.92  -496.35  265.86  -15.37  672.08  275.36  -11.46  703.32    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$\mathbf{c}_{\mathbf{m}}$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  10  5483.36  1167.45  6650.81  33.79  87.09  43.15  39.47  107.48  51.41  20  4465.96  1167.45  5633.41  43.61  74.78  50.07  49.06  97.19  59.03  30  3506.04  1075.03  4581.07  57.42  81.25  63.01  62.52  103.66  72.18  40  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  80  -1301.27  804.92  -496.35  265.86  -15.37  672.08  275.36  -11.46  703.32  Table 10 Comparison of the expected profits in Approach 1-2 with those in Approaches 2-2 and 2-4, under different values for ‘$$\mu$$’    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$\mathbf{\mu }$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  50  23.27  593.95  617.22  7606.32  47.04  332.04  7754.14  49.49  339.97  100  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  200  7648.21  1826.69  9474.90  37.63  109.52  51.49  40.93  123.03  56.76  300  12906.74  2471.59  15378.33  27.54  135.18  44.84  30.02  145.62  48.60  500  24772.37  3813.25  28585.62  14.58  156.95  33.57  16.21  163.97  35.92    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$\mathbf{\mu }$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  50  23.27  593.95  617.22  7606.32  47.04  332.04  7754.14  49.49  339.97  100  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  200  7648.21  1826.69  9474.90  37.63  109.52  51.49  40.93  123.03  56.76  300  12906.74  2471.59  15378.33  27.54  135.18  44.84  30.02  145.62  48.60  500  24772.37  3813.25  28585.62  14.58  156.95  33.57  16.21  163.97  35.92  Table 11 Comparison of the expected profits in Approach 1-2 with those in Approaches 2-2 and 2-4, under different values for ‘$$\sigma$$’    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$\sigma$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  5  4851.51  789.95  5641.46  15.52  146.80  33.90  17.41  155.35  36.72  10  3824.11  913.35  4737.46  37.62  109.28  51.44  40.93  123.03  56.76  20  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  30  1263.39  1134.50  2397.89  223.77  60.57  146.56  229.40  86.62  161.85  40  46.54  1187.89  1234.43  7608.49  47.54  332.60  7754.12  49.50  339.97    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$\sigma$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  5  4851.51  789.95  5641.46  15.52  146.80  33.90  17.41  155.35  36.72  10  3824.11  913.35  4737.46  37.62  109.28  51.44  40.93  123.03  56.76  20  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  30  1263.39  1134.50  2397.89  223.77  60.57  146.56  229.40  86.62  161.85  40  46.54  1187.89  1234.43  7608.49  47.54  332.60  7754.12  49.50  339.97  Table 12 Comparison of the expected profits in Approach 1-2 with those in Approaches 2-2 and 2-4, under different values for ‘$$c_{r,i}$$’    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$c_{r,i}$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  0,30,60  2786.44  1430.01  4216.45  41.97  87.14  57.28  43.47  87.56  58.43  10,30,50  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  15,30,45  2396.58  852.41  3248.99  107.64  73.97  98.81  116.16  93.82  110.30  20,30,40  2313.92  630.31  2944.23  133.12  82.27  122.23  141.24  96.18  131.59  25,30,35  2324.89  315.15  2640.04  150.66  166.86  152.60  155.34  177.22  157.96    Expected profits in Approach 1-2  $$VP$$ (%)  $$VP$$ (%)  $$c_{r,i}$$  (SMRS contract)  (Based on Approach 2-2)  (Based on Approach 2-4)    Man.  Ret.  System  Man.  Ret.  System  Man.  Ret.  System  0,30,60  2786.44  1430.01  4216.45  41.97  87.14  57.28  43.47  87.56  58.43  10,30,50  2496.99  1075.03  3572.02  86.15  73.21  82.26  91.03  93.72  91.84  15,30,45  2396.58  852.41  3248.99  107.64  73.97  98.81  116.16  93.82  110.30  20,30,40  2313.92  630.31  2944.23  133.12  82.27  122.23  141.24  96.18  131.59  25,30,35  2324.89  315.15  2640.04  150.66  166.86  152.60  155.34  177.22  157.96  In Tables 8–12, we notice that the manufacturer’s expected profit under Approaches 2-2 and 2-4 is always greater than that under Approach 1-2, because $$VP$$ is always positive. This improvement can be as high as almost 7600% for $$\mu =50$$ or $$\sigma =40$$, as shown in Tables 10 and 11, respectively. We observe that the retailer’s and the system’s expected profits are usually improved considerably under Approaches 2-2 and 2-4 as well. 6.2.4. Comparison of Approaches 2-2 and 2-4 In order to save space, we compare the expected profits in Approaches 2-2 and 2-4, based on their relative preference over Approach 1-2, i.e. based on the values of $$VP$$ in Tables 8–12. By comparing $$VP$$ based on Approach 2-2 and $$VP$$ based on Approach 2-4, we observe that the manufacturer’s, the retailer’s and the system’s expected profits are increased when moving from Approach 2-2 to Approach 2-4. However, from the values of $$VP$$ in Tables 8–12 we realize that the preference for Approach 2-4 over Approach 2-2 is not substantial. This can be an appropriate motivation for the manufacturer to implement Approach 2-2 if there are some administrative drawbacks for designing a menu of contracts according to Approach 2-4. 6.3. Managerial insights from sensitivity analysis In Section 6.2.1 (Tables 3–12), we compared the expected profit in Approach 1 with that of Approach 2. As an important managerial insight from this subsection, we observe that the manufacturer’s expected profit is improved considerably in Approach 2 by taking into account all cost estimations at the time of contract design. It is clear that the manufacturer’s expected profit in Approaches 2-2 and 2-4 is always higher than that under Approach 2-1. That is because Approach 2-1 is a specific case of Approaches 2-2 and 2-4 which is generated when we set $$U=0$$ in Approaches 2-2 and 2-4. Therefore, another point here is that the manufacturer should choose Approaches 2-2 and 2-4 over Approach 2-1. The manufacturer is now left with the choice between Approaches 2-2 and 2-4. By comparing the values of $$VP$$ in Tables 8–12, we observe improvement in both the manufacturer’s and the retailer’s expected profits, by moving from Approach 2-2 to Approach 2-4. Since Approach 2-4 leads to system coordination, more profit is generated in the system according to this case as compared to Approach 2-2. Both manufacturer and retailer benefit from this additional profit in the coordinated system. Therefore, our first suggestion for the manufacturer is to use Approach 2-4. However, if executing Approach 2-4 and designing a menu can cause some difficulties in practice, e.g. whenever there is a high number of contracts in the menu, Approach 2-2 will be an appropriate substitute. By comparing the manufacturer’s expected profit under Approaches 2-2 and 2-4, we realize that on average, Approach 2-4 can improve Approach 2-2 by almost 3%, which may not be a considerable value. In short, by comparing the manufacturer’s expected profit in different approaches, we can place the approaches in the order of preference for the manufacturer as follows:   Approach 2−4>Approach 2−2>Approach 2−1>Approach 1−1>Approach 1−2. 7. Other cases 7.1. Other contracts In this paper, we assumed the RS contract between a manufacturer and a retailer. However, our results hold for any other contract that is equivalent to the RS contract. The RS contract and the buy-back contract are equivalent in a fixed-price newsvendor model (Cachon & Lariviere, 2005). This means that for any RS contract with parameters $$(w,{\Phi })$$ there exists a unique buyback contract with parameters $$(b,w_{b} )$$ that leads to the same profit for the manufacturer and the retailer, for any realization of demand. This equivalence takes place by setting $$b=\left( {1-{\Phi }} \right)p$$ and $$w_{b} =w+\left( {1-{\Phi }} \right)p$$, where $$b$$ and $$w_{b}$$ are wholesale and buyback prices. In addition, since the buy-back contract can be interpreted as a kind of advanced procurement or option contract, we can consider the option contract as another equivalent of the RS contract. In an option contract, the retailer in the first stage buys some options at price $$w_{o}$$ to use later. In the second stage after demand realization, it commits to pay an additional payment as $$w_{e}$$ to exercise the options (Wang & Tsao, 2006). This pricing mechanism creates the same profit allocation between the manufacturer and the retailer as in the RS and buy-back contracts, by setting: $$w_{o} =w_{b} -b=w, \ w_{e} =b=\left( {1-{\Phi }} \right)p$$. 7.2. Type-independent reservation profit: the case of Approach 2-4 We tried to develop some comprehensive models in Approach 2 by considering different significant conditions, such as $$N$$ different estimations for the cost information, cut-off policy and type-dependent reservation profit. Ignoring some or all of these conditions leads to easier cases of the models that might be used by a manager. We introduce one such case in this subsection. So far, by considering type-dependent reservation profit, the retailer’s reservation profit and the manufacturer’s outside option are modelled according to the price-only contract. Several studies, such as Several studies, such as Ha (2001), Corbett et al. (2004) and Lutze & Özer (2008), have considered a situation in which the reservation profit is not type-dependent. In this situation, the reservation profit can be interpreted as an outside opportunity or a minimum profit level that a contract partner demands regardless of its cost level. For the manufacturer, it is a minimum profit level that it intends to achieve for a trade to take place. In this subsection, we study the menu design problem in Approach 2-4 by considering a given reservation profit. Other approaches may be studied in a similar way. Assume $$\mbox{RP}_{m}$$ as the manufacturer’s reservation profit and $$\mbox{RP}_{r}$$ as the retailer’s reservation profit. Type-independent reservation profit is equivalent to: $$\mbox{RP}_{r,1} =\mbox{RP}_{r,2} =...=\mbox{RP}_{r,i} =...=\mbox{RP}_{r,N} =\mbox{RP}_{r}$$. If the contract is rejected, each manufacturer and retailer can pursue other investment opportunities, thereby ensuring profits of $$\mbox{RP}_{m}$$ and $$\mbox{RP}_{r}$$ respectively. Notice that in this situation, based on the cut-off policy, there might be no trade with a retailer whose cost is higher than the cut-off point. This is while both firms were continuing to trade based on a price-only contract in Approaches 2-2, 2-3 and 2-4 when the retailer’s cost was higher than the cut-off point. The manufacturer’s problem in this case is:   Max(U,wi,Φi)ΠmRS3=∑i=1U⁡λi[(wi−cm)qiRS3+(1−Φi)pS(qiRS3)]+∑i=U+1N⁡λiRPm, (39)  S.T.ΦipS(qiRS3)−qiRS3(wi+cr,i)⩾RPr ∀i=1,2,3,...,U, (40)  ΦipS(qiRS3)−qiRS3(wi+cr,i)⩾ΦjpS(qjRS3)−qjRS3(wj+cr,i) (41)  ∀i=1,2,3,...,U, ∀j=1,2,3,...,U, i≠j,ΦipS(qiRS3)−qiRS3(wi+cr,f)⩽RPr ∀i=1,2,3,...,U, f=U+1,...,N, (42)  qiRS3=F−1[p−cm−cr,ip] ∀i=1,2,3,...,N, (43)  U∈{0,1,2,3,...,N},0⩽Φi⩽1,wi:free in sign, ∀i=1,2,3,...,U. (44) In this model, the manufacturer does not offer a price-only contract to a retailer with a cost more than $$c_{r,U}$$. Therefore, there is no need to Constraints (22) and (23). Proposition 4 Consider $$c_{r,U} \in \{c_{r,1} ,c_{r,2},c_{r,3} ,...,c_{r,N} \}$$ as a cut-off point. In an optimal solution, a retailer’s expected profit with the cost of $$c_{r,i} , \ i=1,2,...,U-1$$, i.e. $${\Pi }_{r,i}^{\rm RS3}$$, is more than $$\mbox{RP}_{r}$$ and a retailer’s expected profit with the highest cost, i.e. $${\Pi }_{r,U}^{\rm RS3}$$, is $$\mbox{RP}_{r}$$. Proposition 5 Consider $$c_{r,U} \in \{c_{r,1} ,c_{r,2},c_{r,3} ,...,c_{r,N} \}$$ as a cut-off point. In the optimal solution, a retailer’s expected profit of type $$i=1,2,3,...,U-1$$ is: $${\Pi }_{r,i}^{\rm RS3} =\mbox{RP}_{r} + \sum_{j=i}^{U-1} q_{j+1}^{\rm RS3} (c_{r,j+1} -c_{r,j})$$. From Propositions 4 and 5, we find that in the optimal solution, a retailer’s expected profit with a lower cost is more than a retailer’s expected profit with a higher cost. For each value of $$U$$, through Proposition 5, we can find optimal values of $$w_{i}$$ and $$\Phi_{\mbox{i}}$$ so that they satisfy the optimal profit allocation to each retailer type. Finally, we can select the optimal value for $$U$$, according to the general solution methodology in Section 5.4.1. 8. Conclusion In this paper, we investigate different approaches to designing the RS contract by a manufacturer under asymmetric cost information about a retailer that is the contract partner. The retailer, operating in a newsvendor-type environment, sources its products from the manufacturer. The manufacturer knows several discrete estimations from the retailer’s cost structure. We consider two general approaches. In Approach 1, the manufacturer designs the RS and price-only contracts based on the average of cost estimations, which is a substantial simplification. In Approach 2, the manufacturer uses all cost estimations for contract design. We show that in Approach 1 the price-only contract creates a larger profit for the manufacturer, as compared to the RS contract. In Approach 2, we consider type-dependent reservation profit for the retailer explicitly based on the price-only contract, as introduced in Approach 2-1. We investigate Approaches 2-2, 2-3 and 2-4, for the design of the RS contract based on the cut-off policy. According to the cut-off policy in the mentioned approaches, we allow the manufacturer to refrain from signing an RS scheme if the retailer’s cost is higher than a specific level; thus, we include the price-only contract as an outside option for the manufacturer. Specifically, in Approach 2-2 a single RS contract is designed for different types of the retailer, while in Approaches 2-3 and 2-4 the manufacturer designs a menu of coordinating RS contracts. Thus, Approach 2-2 is a simplified version of Approaches 2-3 and 2-4. We address the optimal features of the model developed in Approach 2-2 when demand has an IGFR. We illustrate that Approach 2-3 cannot motivate the retailer to behave truthfully. According to the revelation principle, Approach 2-3 cannot be the optimal menu of RS contracts. Thus, we work with a tailored menu of coordinating RS contracts in Approach 2-4, which includes the coordinated order commitment as well as the regular RS parameters. We present a general solution methodology to find the optimal contract parameters in Approaches 2-2 and 2-4. We find from numerical analysis that contract design, according to each model in Approach 2, can increase both the manufacturer’s and the retailer’s profits considerably, as compared to Approach 1. This highlights the value of utilizing all cost estimations for contract design. Approach 2-1 is a specific case of Approaches 2-2 and 2-4. Thus, it is an inferior option, as compared to Approaches 2-2 and 2-4. By comparing Approaches 2-2 and 2-4, we show that both the manufacturer and the retailer obtain more expected profit in Approach 2-4, under different parameter settings. However, the important managerial insight is that if the execution of Approach 2-4 (designing a menu) creates unreasonable administrative costs for the manufacturer, Approach 2-2 can be an appropriate substitute. Our results and comparisons among different approaches hold under any other pricing mechanism that is equivalent to the RS contract, such as the buy-back contract. We also present some analytical results by considering type-independent reservation profit based on Approach 2-4. In this case, we determine the optimal profit allocation to each retailer type. One direction for future research is to investigate other approaches, such as the case in which the manufacturer designs a menu of RS contracts without specifying order quantities and coordinating the system. Approach 2-2 is a special case of this approach in practice. 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IMA Journal of Management MathematicsOxford University Press

Published: Jan 1, 2018

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