A study of a robust multi-objective supplier-material selection problem

A study of a robust multi-objective supplier-material selection problem Abstract This paper develops a multi-objective mathematical formulation for supplier-material selection in cardboard box manufacturing. The aim is to minimize three different scaled objectives: wastage of raw material, raw material cost and product surplus. To model industry reality more closely, the formulation is extended to an environment that includes uncertain costs and demands. In this sense, the model is reformulated as a robust optimization problem. To solve this problem, a weighted global criterion approach is developed and applied to find Pareto optimal solutions. A case study from a cardboard box manufacturer is used to test the efficiency of the proposed robust formulation and the proposed solution approach. The sensitivity of the Pareto optimal solutions to different levels of uncertainty in the parameters and different weights of the objective functions is also studied. 1. Introduction The selection of suppliers and materials in a manufacturing supply chain is an important and challenging problem (see Ayhan, 2013; Batuhan & Selcuk, 2015; Tan & Alp, 2009; Asgari et al., 2016). The problem is important for a manufacturer as supplier selection affects price of the raw material, due date of orders, quality of raw materials etc. On the other hand, the problem is challenging as there is a tradeoff between criteria such as quality, price, due date etc. Material purchase cost is an important criteria in supplier selection. It can account for as much as 60% of total sales (Krajewski & Ritzman, 2001). This cost may vary from 70% of total revenue in automotive industries to 80% in high technology industries (Weber et al., 1991). Additionally, since good supplier selection can significantly contribute to cost reduction, enterprizes must consider their supply chains (SCs) more carefully (Ozgen & Gulsun (2014)). Following the same aim, Cakravastia et al. (2002) develop an analytical model for the process of supplier selection while an SC network is being established. The objective of their study was to minimize the level of customer dissatisfaction, through two factors—price and delivery lead time. Alp & Tan (2008) and Tan & Alp (2009) investigated a problem with two supply options, in a multi-period, having fixed procurement cost. Later on, Alp et al. (2013) considered a linear cost function with identical suppliers in an infinite horizon version of the problem with fixed components. This study considers a typical multi-objective supplier and material selection problem from a real cardboard box production company. This problem is important because it combines the problem of cutting raw paper sheets (Russo et al., 2014) with two key issues that contribute to profitability: (1) choice of material dimensions and (2) supplier selection. To the best of our knowledge this study is the first to consider these issues through simultaneous minimization of (1) wastage of raw paper sheets, (2) product surplus relative to demand and (3) procurement costs. The problem is modelled as a multi-objective mixed integer linear programme in an uncertain environment (Fullér & Majlender, 2004; Fullér et al., 2012; Bouzembrak et al., 2013; Almeida et al., 2017) where some parameters e.g. demand and raw material price are uncertain. These uncertainties is reflected using robust optimization which is a suitable uncertain approach for formulating real-world problems. The crisp form of the uncertain multi-objective formulation is solved by a weighted global criterion approach for the real data obtained from the cardboard box production company. The paper is structured as follows. The next section describes the problem of the cardboard box production company. Section 3 proposes a deterministic multi-objective mathematical model and its robust form for the problem. Section 4 focuses on a solution approach for the proposed robust formulation. Section 5 performs some computational experiments on the robust formulation using the real data obtained from the company. The paper ends with conclusion. 2. Problem description We consider a real life problem in a cardboard box production company. The company produces cardboard boxes in different sizes (where each size has its specified dimensions) in response to customer demands. A cardboard box is packaging used for shipping goods, typically between manufacturer and retailer. In the rest of this paper we shall refer to a ‘cardboard box’ simply as ‘box’. These boxes are produced from raw sheets of cardboard, which are sold to the company in various dimensions. Each supplier of the company is able to provide several dimensions of the raw sheets. Another component of the problem is the ordering style, where the orders take place by the customers in specific horizon and quantity according to their need. In the rest of this paper we refer to different sizes of box by ‘box types’ or ‘types of box’, where raw sheets with different dimensions are referred by ‘raw sheet types’ or ‘types of raw sheet’. More details about the problem is as follows: The customers order a specific number of each box type in a planning horizon. There are several types of raw sheets available in all suppliers. The suppliers offer competitive prices. Each type of raw sheet produces a specified number of each box type that is known for the company. One or more than one type of the raw sheets may be used to produce a type of box. Employing a specific type of a raw sheet for producing a box type corresponds to a known amount of wastage. This wastage varies depending on the box type. For each type of the raw sheet, there exist several alternative suppliers, the supplier who offers a lower price gains a higher priority. In order to be more competitive, the suppliers offer a discount based on the raw sheet type. Their discount policy is based on the ordered quantity. They determine a price break point for the ordered quantity of each sheet type. If the order quantity of a raw sheet is above the determined break point quantity of the supplier, there will be a price discount for the demanded quantity. By using the same raw sheet for more than one box type, the company can maximize its benefit from the supplier discount policy. Considering the given descriptions, the aim of the company is to determine the sizes and the order quantity of the raw sheet for producing each box type as well as selecting the suitable supplier for the required raw sheet such that the below-mentioned three objectives are considered. Wastage of raw paper sheets. The wastage of the raw material is minimized. For this purpose, the types of the ordered raw sheet and the quantity of each type, which is used for producing a box type, should be determined such that the total amount of wastage is minimized (wastage is scaled by units of area such as meters square, etc.). Raw material cost. Ordering the required raw sheets has a certain purchasing cost for the company. Therefore, the company should decide to order the raw sheets such that the total payment per purchase is minimized over the planning horizon. In this respect, the break points offered by the suppliers play a critical role in minimizing total purchasing cost of the raw sheets. Product surplus. The total amount of the boxes produced from the ordered raw sheets must be greater than or equal to the demand of the boxes. When this condition is met, the extra products should be stored in the inventory to be used on the next planning horizon. These extra boxes are considered as a typical surplus. However, the company requires minimizing the total surpluses of all types of boxes. The parameters like the price of the sheets, break points of the discount, which is determined by the suppliers, and demand of the boxes have high degree of uncertainty. However, the previous experiences and historical data of the company provide a suitable base for calculating an estimate for them. In this study, intervals of values for these uncertain parameters are considered, which are obtained from the historical data of the previous planning horizons. 3. Mathematical formulation In this section, first a deterministic mathematical formulation is proposed for the problem described in previous section and then a robust formulation of the deterministic model is proposed to satisfy the uncertainty of the parameters of the problem. 3.1. Deterministic formulation The following notations are used in the deterministic formulation of the problem: $$\begin{array}{lrl} \textit{i} & (\textrm{index}): & \textrm{index used for types of box,} \\ \textit{j} & (\textrm{index}): & \textrm{index used for types of raw sheet,} \\ \textit{k} & (\textrm{index}): & \textrm{index used for suppliers,} \\ \textit{I} & (\textrm{parameter}): & \textrm{number of box types to be produced in a planning horizon,}\\ \textit{J} & (\textrm{parameter}): & \textrm{number of types of raw sheet presented by suppliers,} \\ \textit{K} & (\textrm{parameter}): & \textrm{number of suppliers,} \\ \textit{M} & (\textrm{parameter}): & \textrm{a large positive value,} \\ d_{i} & (\textrm{parameter}): & \textrm{demand of box type}\ \textit{i}\ \textrm{shown by unit of quantity,} \\ a_{ij} & (\textrm{parameter}): & \textrm{number of box type}\ \textit{i}\ \textrm{that can be cut from raw sheet type}\ \textit{j}, \\ w_{ij} & (\textrm{parameter}): & \textrm{waste amount remained after cutting box type}\ \textit{i}\ \textrm{from raw sheet}\ \textit{j}, \\ g_{jk} & (\textrm{parameter}): & \textrm{break point for ordering raw sheet type}\ \textit{j}\ \textrm{offered by supplier}\ \textit{k}.\\ & & \textrm{For orders more than this amount discounted price will be applied for the order}, \\ c_{jk}^{1},c_{jk}^{2} & (\textrm{parameter}): & \textrm{normal and discounted unit prices for raw sheet type}\ \textit{j}\ \textrm{by supplier}\ \textit{k}, (c_{jk}^{1} \ge c_{jk}^{2}), \\ T_{jk}^{1},T_{jk}^{2} & (\textrm{variable}): & \textrm{binary variables indicating that whether or not discount is applied for sheet type}\ \textit{j}\\ && \textrm{by supplier}\ \textit{k}. \textrm{Normal price is applied if}\ T_{jk}^{1}=1\ \textrm{and discounted price is applied if}\ T_{jk}^{2} =1. \\ Y_{ijk} & (\textrm{variable}): & \textrm{number of raw sheets type}\ \textit{j}\ \textrm{which is ordered to supplier}\ \textit{k}\\ && \textrm{for producing box type}\ \textit{i}\textrm{,} \\ Z_{jk}^{1},Z_{jk}^{2} & (\textrm{variable}): & \textrm{non-negative continuous values to be used instead of non-linear terms}\\ && T_{jk}^{1} \left(\sum_{i=1}^{I}Y_{ijk} \right),T_{jk}^{2} \left(\sum_{i=1}^{I}Y_{ijk} \right) \textrm{respectively}. \end{array}$$ Based on the above-mentioned notations, the following non-linear model is proposed for the supplier-material selection problem described in Section 2: \begin{equation} {\hskip-90pt}\textrm{Objective function 1}: OF_{1} =\min \; \sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{K}w_{ij} Y_{ijk} \end{equation} (1) \begin{equation} \textrm{Objective function 2}: OF_{2} =\min \; \sum_{j=1}^{J}\sum_{k=1}^{K}\left(c_{jk}^{1} T_{jk}^{1} \sum_{i=1}^{I}Y_{ijk} +c_{jk}^{2} T_{jk}^{2} \sum_{i=1}^{I}Y_{ijk} \right) \end{equation} (2) \begin{equation} \textrm{Objective function 3}: OF_{3} =\min \; \sum_{i=1}^{I}\left(\left(\sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \right)-d_{i} \right)\qquad\qquad \end{equation} (3) \begin{align}\textrm{subject to}\nonumber\\ & \sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \ge d_{i}\qquad{\forall i} {\hskip160pt}\end{align} (4) \begin{equation} {\hskip-130pt}T_{jk}^{1} +T_{jk}^{2} \le 1 \qquad{\forall j,k} \end{equation} (5) \begin{equation} {\hskip-60pt}\sum_{i=1}^{I}Y_{ijk} \ge T_{jk}^{1} \qquad{\forall j,k}\qquad\qquad\qquad\ \ \ \end{equation} (6) \begin{equation} {\hskip-65pt}\sum_{i=1}^{I}Y_{ijk} \le g_{jk} T_{jk}^{1} +MT_{jk}^{2} \qquad{\forall j,k}\qquad\ \ \end{equation} (7) \begin{equation} {\hskip-85pt}\sum_{i=1}^{I}Y_{ijk} \ge \left(g_{jk} +1\right)T_{jk}^{2} \qquad{\forall j,k}\qquad\quad \end{equation} (8) \begin{equation} {\hskip-85pt}T_{jk}^{1},T_{jk}^{2} \in \left\{0,1\right\} \qquad{\forall j,k}\; \qquad\qquad\quad \end{equation} (9) \begin{equation} {\hskip-115pt}Y_{ijk} \ge 0\ \textrm{and integer} \qquad{\forall i,j,k.} \end{equation} (10) The objective function (1) minimizes the total wastage remained by all raw sheets bought to produce all box types. Objective function (2) calculates and minimizes the total raw material cost due to price of raw sheets. The discount policy described in Section 2 is considered in this objective function such that at most one of the non-linear terms $$c_{jk}^{1} T_{jk}^{1} \sum _{i=1}^{I}Y_{ijk} $$ or $$c_{jk}^{2} T_{jk}^{2} \sum _{i=1}^{I}Y_{ijk} $$ can take positive value. Objective function (3) minimizes the surplus. (For example, if the demand of a box type is 95 and 10 numbers of a type of raw sheet is selected and each raw sheet can produce 10 number of the box, the surplus in this case is $$10\left (10\right )-95=5$$). Constraint set (4) satisfies the demand of the box types. Constraint sets (5)–(8) together guarantee that for buying type j of raw sheet from supplier k, only one of the following cases may happen: $$\sum _{i=1}^{I}Y_{ijk} \le g_{jk} $$ and $$T_{jk}^{1} =1$$, then the normal price in objective function (2) is considered by supplier k, $$\sum _{i=1}^{I}Y_{ijk} \ge g_{jk} +1 $$ and $$T_{jk}^{2} =1$$, therefore, the discounted price in objective function (2) is considered by supplier k, $$\sum _{i=1}^{I}Y_{ijk} =0 $$, $$T_{jk}^{1} =0$$ and$$T_{jk}^{2} =0$$, thus the raw sheet j is not bought from supplier k. Finally, constraint sets (9) and (10) define the nature of the variables. The non-linear term $$T_{jk}^{1} \left (\sum _{i=1}^{I}Y_{ijk} \right )$$ (as well as$$T_{jk}^{2} \left (\sum _{i=1}^{I}Y_{ijk} \right )$$), can be linearized by a linearization technique which is discussed here. If a new continuous non-negative variable $$Z_{jk}^{1} =T_{jk}^{1} \left (\sum _{i=1}^{I}Y_{ijk} \right )$$ is defined, the following set of constraints guarantee that the value of $$Z_{jk}^{1} $$ can be either zero (if $$T_{jk}^{1} =0$$) or $$\sum _{i=1}^{I}Y_{ijk} $$ (if $$T_{jk}^{1} =1$$): \begin{equation} {\hskip-83pt}Z_{jk}^{1} \le MT_{jk}^{1} \qquad\forall j,k \quad\end{equation} (11) \begin{equation} {\hskip-83pt}Z_{jk}^{1} \le \sum_{i=1}^{I}Y_{ijk} \qquad\forall j,k \end{equation} (12) \begin{equation} Z_{jk}^{1} \ge \left(\sum_{i=1}^{I}Y_{ijk} \right)-M\left(1-T_{jk}^{1} \right) \qquad\forall j,k \end{equation} (13) \begin{equation} Z_{jk}^{1} \ge 0 \qquad\forall j,k. \qquad\qquad\qquad\qquad\qquad\end{equation} (14) Similarly, $$Z_{jk}^{2} =T_{jk}^{2} \left (\sum _{i=1}^{I}Y_{ijk} \right )$$ is defined in a similar way. Therefore, the non-linear models (1)–(10) is linearized as the mixed integer linear model (MILP) given below, \begin{equation} \textrm{Objective function 1}: OF_{1} =\min \; \sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{K}w_{ij} Y_{ijk} \qquad\qquad\quad \end{equation} (15) \begin{equation} \textrm{Objective function 2}: OF_{2} =\min \; \sum_{j=1}^{J}\sum_{k=1}^{K}\left(c_{jk}^{1} Z_{jk}^{1} +c_{jk}^{2} Z_{jk}^{2} \right) \qquad\ \end{equation} (16) \begin{equation} \textrm{Objective function 3}: OF_{3} =\min \; \sum_{i=1}^{I}\left(\left(\sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \right)-d_{i} \right) \end{equation} (17) \begin{align}\textrm{subject to}&\nonumber\\& \sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \ge d_{i}\qquad{\forall i} {\hskip115pt}\end{align} (18) \begin{equation} {\hskip-30pt}T_{jk}^{1} +T_{jk}^{2} \le 1 \qquad{\forall j,k} \qquad\qquad\qquad\ \end{equation} (19) \begin{equation} {\hskip-40pt}\sum_{i=1}^{I}Y_{ijk} \ge T_{jk}^{1}\qquad{\forall j,k} \qquad\qquad\quad \ \end{equation} (20) \begin{equation} {\hskip-40pt}\sum_{i=1}^{I}Y_{ijk} \le g_{jk} T_{jk}^{1} +MT_{jk}^{2}\qquad{\forall j,k} \quad\end{equation} (21) \begin{equation} {\hskip-40pt}\sum_{i=1}^{I}Y_{ijk} \ge g_{jk} T_{jk}^{2}\qquad{\forall j,k} \qquad\qquad\end{equation} (22) \begin{equation} {\hskip-70pt}Z_{jk}^{1} \le MT_{jk}^{1}\qquad{\forall j,k} \qquad\ \ \end{equation} (23) \begin{equation} {\hskip-10pt}Z_{jk}^{1} \le \sum_{i=1}^{I}Y_{ijk}\qquad{\forall j,k} \qquad\qquad\qquad\quad\ \ \ \end{equation} (24) \begin{equation}{\hskip-5pt} Z_{jk}^{1} \ge \left(\sum_{i=1}^{I}Y_{ijk} \right)-M\left(1-T_{jk}^{1} \right)\qquad{\forall j,k} \end{equation} (25) \begin{equation}\!\!\!\! Z_{jk}^{2} \le MT_{jk}^{2}\qquad{\forall j,k} \qquad\qquad\qquad\qquad\ \ \ \end{equation} (26) \begin{equation} Z_{jk}^{2} \le \sum_{i=1}^{I}Y_{ijk}\qquad{\forall j,k} \qquad\qquad\qquad\quad\ \ \ \end{equation} (27) \begin{equation} Z_{jk}^{2} \ge \left(\sum_{i=1}^{I}Y_{ijk} \right)-M\left(1-T_{jk}^{2} \right)\qquad{\forall j,k} \end{equation} (28) \begin{equation} T_{jk}^{1},T_{jk}^{2} \in \left\{0,1\right\}\qquad{\forall j,k} \qquad\qquad\qquad\quad\end{equation} (29) \begin{equation} Y_{ijk} \ge 0\ \textrm{and integer} \qquad{\forall i,j,k} \qquad\qquad\quad\end{equation} (30) \begin{equation} Z_{jk}^{1},Z_{jk}^{2} \ge 0 \qquad{\forall j,k.}\quad\qquad\qquad\qquad\quad\ \ \end{equation} (31) 3.2. A robust approach for the introduced deterministic formulation As mentioned in Section 2, there exists a high degree of uncertainty on the parameters of this problem. In this section, the deterministic formulation of the problem using the concepts of robust optimization approach is reformulated to cover the uncertain nature of the parameters. Robust approach for a problem has the following properties: (1) The uncertain parameters can take any value from a given interval while it does not assume any known distribution on the value of the uncertain parameters. (2) The optimization is done with respect to the uncertainty of the parameters. The obtained solution is a robust solution to the problem such that it considers the whole range of possible values for the uncertain variables. Consequently, the feasible solutions of the problem hold for all possible values of the uncertain parameters. The concepts of robust optimization methods can be found in the study of Bertsimas & Sim (2004). Moreover, robust optimization has been applied to model the uncertainty of many optimization problems in the field of SC network design and other combinatorial optimization problems. For more detailed information the studies of Thiele (2008), Klibi et al. (2010), Pishvaee et al. (2011), Ramezani et al. (2013), Recchia & Scutellé (2014), Hosseini et al. (2014) etc. can be referred. Here, the robust optimization method introduced by Bertsimas & Sim (2004) is briefly explained. Consider the following optimization problem: \begin{align}\min& \quad c\, x \nonumber\\ \textrm{subject to}& \nonumber\\ &\quad Ax\le b \\ &\quad x\in S, \nonumber \end{align} (32) where c is an n dimensional cost vector, A is a m × n matrix, b is an m dimensional vector and S is a polyhedron. We assume that data uncertainty affects the cost vector c and the elements of the matrix A.Each entry aij of matrix Acan be modeled as an independent, symmetric and bounded parameter, which can take values in interval $$\left [a_{ij} -\hat{a}_{ij},a_{ij} +\hat{a}_{ij} \right ]\, $$. Notations aij and $$\hat{a}_{ij} $$ denote the nominal value and the maximum deviation from the nominal value, respectively. Same as elements of matrix A, each entry cj can take values from$$\left [c_{j},c_{j} +\hat{c}_{j} \right ]$$. So, cj and $$\hat{c}_{j} $$ denote the nominal value and the maximum deviation from the nominal value, respectively. Let J0 be the sets of the uncertain parameters in objective function and Ji be the set of the uncertain parameters in i-th constraint. Now, problem (32) can be reformulated as follows: \begin{align} \min&\quad \sum_{j=1}^{n}\tilde{c}_{j} x_{j} \nonumber\\ \textrm{subject to}&\quad \nonumber\\ &\quad\sum_{j=1}^{n}\tilde{a}_{ij} x_{j} \le b_{i} \qquad\forall i \\ \nonumber &\quad x\in S, \end{align} (33) where $$\tilde{c}_{j} |j\in J_{0} $$ and $$\tilde{a}_{ij} |j\in J_{i} $$. The scaled deviation of $$\tilde{c}_{j} $$ and $$\tilde{a}_{ij} $$ from their nominal values are defined as follows: \begin{align} \mu_{j} &=\frac{\tilde{c}_{j} -c_{j}}{\hat{c}_{j}} \nonumber \\ \eta_{ij} &=\frac{\tilde{a}_{ij} -a_{ij} }{\hat{a}_{ij}} \end{align} (34) where $$\tilde{c}_{j} $$ and $$\tilde{a}_{ij} $$ are uncertain data, cj and aij are nominal values and $$\hat{c}_{j} $$ and $$\hat{a}_{ij} $$ are maximum deviations from the nominal values. Furthermore, μj and ηij are random variables, which take a value in $$\, \left [0,1\right ]$$ and $$\, \left [-1,1\right ]$$ respectively. Bertsimas & Sim (2004) introduced parameters Γ0 and Γi which can vary in continuous intervals $$\left [0,|J_{0} |\, \right ]$$ and $$\left [0,|J_{i} |\, \right ]$$, respectively. The notations |J0| and |Ji| denote the number of elements of sets Γ0 and Γi. The parameters Γ0 and Γi adjust the robustness of the method against the level of conservatism of the solution. Although the aggregated scaled deviation for objective function can vary in the interval $$\left [-|J_{0} |\,,|J_{0} |\, \right ]$$, it is limited to the following: \begin{equation} \sum_{j=1}^{n}\mu_{j} \le \Gamma_{0}. \end{equation} (35) The parameter Γ0 controls the level of robustness in the objective function. By the same manner, the aggregated scaled deviation for constraint i can vary in the interval $$\left [-|J_{i} |\,,|J_{i} |\, \right ]$$, but it is limited to the following: \begin{equation} \sum_{j=1}^{n}\eta_{ij} \le \Gamma_{i} \qquad \forall i. \end{equation} (36) Based on the above definitions, the setsJ0 and Ji can be written as follows: \begin{equation} J_{0} =\left\{j|\tilde{c}_{j} =c_{j} +\mu_{j} \hat{c}_{j},\, 0\le \mu_{j} \, \le 1,\, \sum_{j=1}^{n}\mu_{j} \le \Gamma_{0},\quad \forall j\right\} \quad\end{equation} (37) \begin{equation} J_{i} =\left\{j|\tilde{a}_{ij} =a_{ij} +\hat{a}_{ij} \eta_{ij},\, \, |\eta_{ij} |\, \le 1,\, \sum_{j=1}^{n}\eta_{ij} \le \Gamma_{i}, \quad\forall i,j\right\}\!. \end{equation} (38) Finally, the tractable form of the robust formulation when uncertainty affects the cost values and coefficient of constraints can be written as follows: \begin{align} \min&\quad c\, x+z_{0} \Gamma_{0} +\sum_{j\in J_{o} }p_{oj}\nonumber \\ \textrm{subject to}&\nonumber\\ &\quad\sum_{j=1}^{n}a_{ij} x_{j} +z_{i} \Gamma_{i} +\sum_{j\in J_{i} }p_{ij} \le b_{i} \qquad\forall i \nonumber\\ &\quad z_{0} +p_{0j} \ge \hat{c}_{j} \, y_{j}\qquad\forall j\in J_{0} \nonumber\\ &\quad z_{i} +p_{ij} \ge \hat{a}_{ij} \, y_{j}\qquad\forall i,\ j\in J_{i} \nonumber\\ &\nonumber\quad -y_{j} \le x_{j} \le y_{j}\qquad\forall j \\ \nonumber &\quad p_{0j},p_{ij} \ge 0\qquad\forall i,\ j\in J_{i} \\ \nonumber &\quad z_{0},z_{i} \ge 0 \qquad\forall i \nonumber\\ &\quad y_{j} \ge 0 \qquad\forall j \nonumber\\ &\quad x\in S.\end{align} (39) The proof of equivalency of problems (33) and (39) is detailed by Bertsimas & Sim (2004). In the above problem variables z0, zi, p0j, pij and yj is the dual variables which help to convert problem (33) to problem (39) (see Bertsimas & Sim (2004)). It is important to note that, in the cases that right-hand side vector b,in problem (32) is uncertain, we can introduce a new variable xn+1, and write Ax − bxn+1 ≤ 0, 1 ≤ xn+1 ≤ 1 (Bertsimas & Sim, 2003). In the next sub-sections, the symmetric interval as above is considered for the previously mentioned uncertain parameters (price of the raw sheets, break point for ordering raw sheet and demand of box types) and using the above-mentioned concepts of robust optimization approach, the deterministic model of (15)–(31) is reformulated to a robust model (covering the uncertainty of the parameters). 3.2.1. Robust form of the constraints related to uncertain demand The demand of the types of boxes in a planning horizon may vary in an expected range of values. If a certain demand value of previous planning horizon for box type i isdi, then it is claimed that in the presence of an uncertain demand, the symmetric and bounded random demand value for the current planning horizon is noted by $$\tilde{d}_{i} $$ on interval $$\left [d_{i} -\hat{d}_{i},\, d_{i} +\hat{d}_{i} \right ]$$. Now, to convert the constraint set (18) to its robust form, they must be changed to a form that satisfies all the possible demand values of the interval $$\left [d_{i} -\hat{d}_{i},\, d_{i} +\hat{d}_{i} \right ]$$. As the highest demand value in the planning horizon is $$d_{i} +\hat{d}_{i} $$, if the constraint set (18) is satisfied for this value, then it is satisfied for all other demand values of the interval $$\left [d_{i} -\hat{d}_{i},\, d_{i} +\hat{d}_{i} \right ]$$. Therefore, the robust form of the constraint set (18) is defined by the following constraint set (Bertsimas & Sim, 2003): \begin{equation} \sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \ge d_{i} +\xi_{i} \hat{d}_{i} \qquad\forall i, \end{equation} (40) where ξi denotes the budget of uncertainty associated with the demand uncertainty, and takes a value in $$\, \left [0,1\right ]$$. 3.2.2. Robust form of the objective function related to uncertain demand Considering the above-mentioned type of uncertainty for the demand values, the objective function (17) should be changed to its robust form. As the objective function is of minimization type, its worst value (largest value) occurs when the value of parameter di turns to its lowest value as $$d_{i} -\hat{d}_{i} $$. Therefore, this change in the value of di covers the uncertainty of the demand values. Consequently, the robust version of the objective function (17) is introduced as Bertsimas & Sim, 2003, \begin{equation} \textrm{Objective function 3}: OF_{3} =\min \; \sum_{i=1}^{I}\left(\left(\sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \right)-\left(d_{i} -\xi_{i} \hat{d}_{i} \right)\right)\!, \end{equation} (41) where ξi denotes the budget of uncertainty associated with the demand uncertainty, and takes a value in $$\, \left [0,1\right ]$$. 3.2.3. Robust form of the constraints related to discount break point The discount break point determined by each supplier for each raw sheet type, has an uncertain characteristic similar to the uncertain demand. It means that for each discount break point gjk, the symmetric and bounded random discount break point on current planning horizon is noted by $$\tilde{g}_{jk} $$ from interval $$\left [g_{jk} -\hat{g}_{jk},\, g_{jk} +\hat{g}_{jk} \right ]$$. As the constraint sets (21) and (22) contain the discount break point values, they are changed to the following robust versions (with the same logic as the logic applied in Section 3.2.1): \begin{equation} \sum_{i=1}^{I}Y_{ijk} \le \left(g_{jk} -\bar{g}_{jk} \right)T_{jk}^{1} +MT_{jk}^{2} \qquad\forall j,k \end{equation} (42) \begin{equation} \sum_{i=1}^{I}Y_{ijk} \ge \left(g_{jk} +\bar{g}_{jk} \right)T_{jk}^{2}\qquad\forall j,k. \qquad\quad\end{equation} (43) 3.2.4. Robust form of the objective function related to price of raw sheet The same uncertainty as the previous cases exists in relevance to the prices of the raw sheets. The symmetric and bounded random sheet prices (due to the discount points) for the current planning horizon are noted by $$\tilde{c}_{jk}^{1} $$ and $$\tilde{c}_{jk}^{2} $$ on the intervals $$\left [c_{jk}^{1} -\hat{c}_{jk}^{1},\, c_{jk}^{1} +\hat{c}_{jk}^{1} \right ]$$ and $$\left [c_{jk}^{2} -\hat{c}_{jk}^{2},\, c_{jk}^{2} +\hat{c}_{jk}^{2} \right ]$$, respectively. Since only the second objective function of the model (15)–(31) contains sheet price values, its robust form is obtained using the following lemma: Lemma 1 The following MILP obtains the robust form of the objective function 2 that is presented by Equation (16), note that constraints (18)–(31) still are respected. \begin{equation} {\min \; \sum_{j=1}^{J}\sum_{k=1}^{K}\left(c_{jk}^{1} Z_{jk}^{1} +c_{jk}^{2} Z_{jk}^{2} \right) +A^{1} \delta^{1} +A^{2} \delta^{2} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{1} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{2}} \end{equation} (44) \begin{align} \textrm{subject to}& \nonumber\\& \delta^{1} +\gamma_{jk}^{1} \ge \hat{c}_{jk}^{1} +Z_{jk}^{1}\qquad{\forall j,k} \end{align} (45) \begin{equation} \delta^{2} +\gamma_{jk}^{2} \ge \hat{c}_{jk}^{2} +Z_{jk}^{2}\qquad{\forall j,k} \end{equation} (46) \begin{equation} \gamma_{jk}^{1},\gamma_{jk}^{2} \ge 0 \qquad{\forall j,k} \end{equation} (47) \begin{equation} \delta^{1},\delta^{2} \ge 0. \end{equation} (48) Proof. By applying the uncertainness of the sheet prices in the objective function of Equation (16), its worst case (largest possible value) occurs at the sheet prices of $$c_{jk}^{1} +\hat{c}_{jk}^{1} $$ and $$c_{jk}^{2} +\hat{c}_{jk}^{2} $$. Thus, the robust form of the objective function is introduced as follows, where variable $$\alpha _{jk}^{1} $$ ($$-1\le \alpha _{jk}^{1} \le 1$$) determines the variation of the sheet price from its mean value ($$c_{jk}^{1} $$) ($$\alpha _{jk}^{2} $$ is defined in the same way): \begin{equation} \min \; \sum_{j=1}^{J}\sum_{k=1}^{K}\left(c_{jk}^{1} +\alpha_{jk}^{1} \hat{c}_{jk}^{1} \right)Z_{jk}^{1} +\left(c_{jk}^{2} +\alpha_{jk}^{2} \hat{c}_{jk}^{2} \right)Z_{jk}^{2} \end{equation} (49) The robust form is extended as, \begin{equation} \min \; \sum_{j=1}^{J}\sum_{k=1}^{K}\left(c_{jk}^{1} Z_{jk}^{1} +c_{jk}^{2} Z_{jk}^{2} \right) +\sum_{j=1}^{J}\sum_{k=1}^{K}\left(\alpha_{jk}^{1} \hat{c}_{jk}^{1} Z_{jk}^{1} +\alpha_{jk}^{2} \hat{c}_{jk}^{2} Z_{jk}^{2} \right). \end{equation} (50) Since the worst case of the objective function value needs to be considered in the robust form, the second term, which is related to the uncertainty of the sheet prices, should be maximized as the following where A1 and A2 determine the total allowed variations of the sheet prices from their mean values. The decision maker as the output of the model determines these values. \begin{equation} {\hskip-35pt}{\max \sum_{j=1}^{J}\sum_{k=1}^{K}\left(\alpha_{jk}^{1} \hat{c}_{jk}^{1} Z_{jk}^{1} +\alpha_{jk}^{2} \hat{c}_{jk}^{2} Z_{jk}^{2} \right)} \end{equation} (51) \begin{align} \textrm{subject to}&\nonumber\\ &\quad\sum_{j=1}^{J}\sum_{k=1}^{K}\left|\alpha_{jk}^{1} \right|\le A^{1} \end{align} (52) \begin{equation} \sum_{j=1}^{J}\sum_{k=1}^{K}\left|\alpha_{jk}^{2} \right|\le A^{2} \end{equation} (53) \begin{equation} \left|\alpha_{jk}^{1} \right|,\left|\alpha_{jk}^{2} \right|\le 1\qquad{\forall j,k.} \end{equation} (54) As the formulation (51)–(54) is of maximization type, its optimization direction is different from the optimization direction of the model (15)–(31). To overcome this contradiction, the model (51)–(54) is dualized as the following: \begin{equation} {\hskip40pt}{\min \quad A^{1} \delta^{1} +A^{2} \delta^{2} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{1} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{2}} \end{equation} (55) \begin{align} \textrm{subject to}& \nonumber\\&\quad \delta^{1} +\gamma_{jk}^{1} \ge \hat{c}_{jk}^{1} +Z_{jk}^{1}\qquad{\forall j,k} \end{align} (56) \begin{equation} {\hskip5pt}\delta^{2} +\gamma_{jk}^{2} \ge \hat{c}_{jk}^{2} +Z_{jk}^{2}\qquad{\forall j,k}\end{equation} (57) \begin{equation} {\hskip-12pt}\gamma_{jk}^{1},\gamma_{jk}^{2} \ge 0 \qquad{\forall j,k} \end{equation} (58) \begin{equation} {\hskip-22pt}\delta^{1},\delta^{2} \ge 0.\end{equation} (59) Hence, the lemma is proved. 3.2.5. Final robust formulation of the deterministic model Based on the formulations introduced in the Sections 3.2.1 to 3.2.4, the robust formulation of the model (15)–(31) is finalized in this section. For this aim, the deterministic constraints (18), (21) and (22) are replaced by the robust constraints (40), (42) and (43), respectively. Similarly, the deterministic objective functions (16) and (17) are replaced by the formulation (55)–(59) and the robust objective function (41), respectively. Finally, the robust formulation of the model (15)–(31) is represented as: \begin{equation} \textrm{Objective function 1}: OF_{1} =\min \; \sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{K}w_{ij} Y_{ijk} \end{equation} (60) \begin{align} \textrm{Objective function 2}: OF_{2} &=\min \; \sum_{j=1}^{J}\sum_{k=1}^{K}\left(c_{jk}^{1} Z_{jk}^{1} +c_{jk}^{2} Z_{jk}^{2} \right) +A^{1} \delta^{1}\nonumber\\ &\quad + A^{2} \delta^{2} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{1} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{2} \end{align} (61) \begin{equation} \textrm{Objective function 3}: OF_{3} =\min \; \sum_{i=1}^{I}\left(\left(\sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \right)-\left(d_{i} -\xi_{i} \hat{d}_{i} \right)\right) \end{equation} (62) \begin{align} &\textrm{subject to}\nonumber\\ &\qquad\qquad\!\!\sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \ge d_{i} +\xi_{i} \hat{d}_{i} \qquad\forall i \end{align} (63) \begin{equation} T_{jk}^{1} +T_{jk}^{2} \le 1 \qquad\forall j,k \end{equation} (64) \begin{equation}\sum_{i=1}^{I}Y_{ijk} \ge T_{jk}^{1} \qquad\forall j,k \end{equation} (65) \begin{equation} \sum_{i=1}^{I}Y_{ijk} \le \left(g_{jk} -\bar{g}_{jk} \right)T_{jk}^{1} +MT_{jk}^{2} \qquad\forall j,k \end{equation} (66) \begin{align} \sum_{i=1}^{I}Y_{ijk} \ge \left(g_{jk} +\bar{g}_{jk} \right)T_{jk}^{2} \qquad\forall j,k \end{align} (67) \begin{equation} Z_{jk}^{1} \le MT_{jk}^{1} \qquad\forall j,k \end{equation} (68) \begin{equation} Z_{jk}^{1} \le \sum_{i=1}^{I}Y_{ijk} \qquad\forall j,k \end{equation} (69) \begin{equation} Z_{jk}^{1} \ge \left(\sum_{i=1}^{I}Y_{ijk}\right)-M\left(1-T_{jk}^{1} \right) \quad\forall j,k \end{equation} (70) \begin{equation} Z_{jk}^{2} \le MT_{jk}^{2}\qquad\forall j,k \end{equation} (71) \begin{equation} Z_{jk}^{2} \le \sum_{i=1}^{I}Y_{ijk} \qquad\forall j,k \end{equation} (72) \begin{align} Z_{jk}^{2} \ge \left(\sum_{i=1}^{I}Y_{ijk} \right)-M\left(1-T_{jk}^{2} \right) \qquad\forall j,k \end{align} (73) \begin{align} \delta^{1} +\gamma_{jk}^{1} \ge \hat{c}_{jk}^{1} +Z_{jk}^{1} \qquad\forall j,k \end{align} (74) \begin{align} \delta^{2} +\gamma_{jk}^{2} \ge \hat{c}_{jk}^{2} +Z_{jk}^{2} \qquad\forall j,k \end{align} (75) \begin{align} T_{jk}^{1},T_{jk}^{2} \in \left\{0,1\right\} \qquad\forall j,k \end{align} (76) \begin{align} Y_{ijk} \ge 0\;\textrm{and integer} \qquad\forall i,j,k \end{align} (77) \begin{align} Z_{jk}^{1},Z_{jk}^{2},\gamma_{jk}^{1},\gamma_{jk}^{2} \ge 0 \qquad\forall j,k \end{align} (78) \begin{equation} \delta^{1},\delta^{2} \ge 0. \end{equation} (79) The above-introduced multi-objective robust formulation is solvable with any multi-objective optimization method in order to obtain an acceptable Pareto optimal solution. The next section focuses on a method to solve this multi-objective model. 4. Solution approach The proposed robust formulation (60)–(79) is a multi-objective optimization model, for this type of models instead of the optimal solution, the concept of efficient solution is defined (see Steuer, 1986; Kovécs & Marian, 2002; Jablonsky, 2007, 2014; Kahraman, 2008; Franco et al., 2009; ; Hadi-Vencheh et al., 2014; Kar, 2015; Hadi-Vencheh & Mohamadghasemi, 2015; Zhou et al., 2015). The following definitions introduces this concept: Definition 1 Let S* be a feasible solution of the problem (60)–(79). Then, S* is called an efficient or a Pareto optimal solution, if there is no other feasible solution S such that $$O{F_{i}^{S}} \le OF_{i}^{S^{*} },\, \, \, i=1,2,3$$ and at least one strict inequality. If S* is efficient, the point $$(OF_{1}^{S^{*} },OF_{2}^{S^{*} },OF_{3}^{S^{*} } )$$ is then called non-dominated. Definition 2 Let S* be a feasible solution of the problem (60)–(79). Then, S* is called a weakly efficient solution, if there is no other feasible solution S such that $$O{F_{i}^{S}}\mathrm{<}OF_{i}^{S^{*} },\, \, \, i=1,2,3$$. If S* is weakly efficient, the point $$(OF_{1}^{S^{*} },OF_{2}^{S^{*} },OF_{3}^{S^{*} } )$$ is then called weakly non-dominated. An efficient solution is always weakly efficient while the inverse is not true. It should be noted that, generally, the efficient solution is not unique, and from the geometrical point of view, efficient solutions are points of the feasible space, but non-dominated points as the image of the efficient solutions are located in the objectives’ space. In this study, a modified version of the Global criterion method (GCM) is applied to find Pareto optimal solutions of the model (60)–(79). 4.1. A weighted Global criterion method The Global criterion method GCM (see Gomes et al., 2012; Saraj & Safaei, 2012) is modified as a weighted Global criterion method (WGCM) for solving the model (60)–(79). The WGCM is constructed based on the following steps: Step 1: Solve the following models separately to obtain the values $$OF_{1}^{*},OF_{2}^{*} $$ and $$OF_{3}^{*} $$. Meaning that each objective function of the model (60)–(79) is solved individually respecting to all constraints of the model. \begin{equation} OF_{1}^{*} =\min \; \sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{K}w_{ij} Y_{ijk}\end{equation} (80) subject to Constraints (63)–(79). \begin{equation} OF_{2}^{*} =\min \; \sum_{j=1}^{J}\sum_{k=1}^{K}\left(c_{jk}^{1} Z_{jk}^{1} +c_{jk}^{2} Z_{jk}^{2} \right) +A^{1} \delta^{1} +A^{2} \delta^{2} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{1} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{2}\end{equation} (81) subject to Constraints (63)–(79). \begin{equation}OF_{3}^{*} =\min \; \sum_{i=1}^{I}\left(\left(\sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij}Y_{ijk} \right)-\left(d_{i} -\xi_{i} \hat{d}_{i} \right)\right) \end{equation} (82) subject to Constraints (63)–(79). Step 2: Define the Global criterion of the method from the optimal objective function values of the models (80), (81) and (82) as follows: \begin{align} OF=\psi_{1} \left(\frac{OF_{1} -OF_{1}^{*} }{OF_{1}^{*} } \right)+\psi_{2} \left(\frac{OF_{2} -OF_{2}^{*} }{OF_{2}^{*} } \right)+\psi_{3} \left(\frac{OF_{3} -OF_{3}^{*} }{OF_{3}^{*} } \right) \end{align} (83) The above term is a typical Global criterion which converts the objective functions of the model to a single objective function. To give more flexibility to decision maker to determine the importance (weight) of each objective function of the model, the values ψ1, ψ2 and ψ3 (ψ1 + ψ2 + ψ3 = 3) are defined as the weights of the objective functions (60), (61) and (62), respectively. These values are determined by decision maker as parameter. The case with all the weights equal to 1, means that all the objective functions have equal importance for decision maker. Otherwise, the objective function with higher weights is more important for decision maker. The value 3 for the sum of the weights is selected as there are three objective functions. Taking a value other than 3 for this sum will not affect the obtained solution and objective function values if and only if the percentage of the sum value for the weights remain the same. For example, in the case of changing the sum value of 3 to 1, the ψ values should be divided by 3. Step 3: Solve the following model instead of the model (60)–(79) to obtain Pareto optimal solution for the model: \begin{equation}\min OF=\min \; \psi_{1} \left(\frac{OF_{1} -OF_{1}^{*} }{OF_{1}^{*} } \right)+\psi_{2} \left(\frac{OF_{2} -OF_{2}^{*} }{OF_{2}^{*} } \right)+\psi_{3} \left(\frac{OF_{3} -OF_{3}^{*} }{OF_{3}^{*} } \right) \end{equation} (84) subject to Constraints (63)–(79). Table 1. Demand of different types of paper box for one planning horizon given by the company of the case study Box code Demand Box code Demand Box code Demand Box code Demand Box code Demand EPA0111 19975 EPA0027 17313 EPA0015 23776 EPA0004 47915 EPA0153 54195 EPA0031 21930 EPA0010 51508 EPA0016 60578 EPA0005 63872 EPA0119 71252 EPA0029 35814 EPA0138 46582 EPA0056 49693 EPA0140 33588 EPA0014 74498 EPA0030 55257 EPA0007 20762 EPA0057 35836 EPA0141 34823 EPA0055 18882 EPA0154 17707 EPA0008 16161 EPA0059 6968 EPA0152 73472 EPA0062 80607 Box code Demand Box code Demand Box code Demand Box code Demand Box code Demand EPA0111 19975 EPA0027 17313 EPA0015 23776 EPA0004 47915 EPA0153 54195 EPA0031 21930 EPA0010 51508 EPA0016 60578 EPA0005 63872 EPA0119 71252 EPA0029 35814 EPA0138 46582 EPA0056 49693 EPA0140 33588 EPA0014 74498 EPA0030 55257 EPA0007 20762 EPA0057 35836 EPA0141 34823 EPA0055 18882 EPA0154 17707 EPA0008 16161 EPA0059 6968 EPA0152 73472 EPA0062 80607 Table 1. Demand of different types of paper box for one planning horizon given by the company of the case study Box code Demand Box code Demand Box code Demand Box code Demand Box code Demand EPA0111 19975 EPA0027 17313 EPA0015 23776 EPA0004 47915 EPA0153 54195 EPA0031 21930 EPA0010 51508 EPA0016 60578 EPA0005 63872 EPA0119 71252 EPA0029 35814 EPA0138 46582 EPA0056 49693 EPA0140 33588 EPA0014 74498 EPA0030 55257 EPA0007 20762 EPA0057 35836 EPA0141 34823 EPA0055 18882 EPA0154 17707 EPA0008 16161 EPA0059 6968 EPA0152 73472 EPA0062 80607 Box code Demand Box code Demand Box code Demand Box code Demand Box code Demand EPA0111 19975 EPA0027 17313 EPA0015 23776 EPA0004 47915 EPA0153 54195 EPA0031 21930 EPA0010 51508 EPA0016 60578 EPA0005 63872 EPA0119 71252 EPA0029 35814 EPA0138 46582 EPA0056 49693 EPA0140 33588 EPA0014 74498 EPA0030 55257 EPA0007 20762 EPA0057 35836 EPA0141 34823 EPA0055 18882 EPA0154 17707 EPA0008 16161 EPA0059 6968 EPA0152 73472 EPA0062 80607 Table 2. Some data for one planning horizon given by the company of the case study Raw sheet Dimension (cm) a1j w1j gj1 $$\hat{g}_{j1} $$ $$c_{j1}^{1} $$ $$\hat{c}_{j1}^{1} $$ $$c_{j1}^{2} $$ $$\hat{c}_{j1}^{2} $$ ( j) Length Width 1 125 90 65 804.3 2651 400 1092 200 928 50 2 125 100 75 808.8 2293 400 1178 200 1095 100 3 125 110 80 966.0 2001 400 1254 200 1148 100 4 125 120 90 970.5 1684 300 1454 200 1350 100 5 125 140 105 1132.3 1713 300 1608 200 1474 100 6 125 150 115 1136.8 1015 200 1834 200 1661 100 7 125 160 120 1294.0 1147 200 1850 200 1636 100 8 125 180 135 1455.8 1296 200 2102 300 1924 100 9 150 90 78 939.3 1747 300 1250 200 1092 100 10 150 100 90 958.8 1795 300 1491 200 1233 100 11 150 110 96 1131.0 1630 300 1511 200 1403 100 12 150 120 108 1150.5 953 100 1733 200 1566 100 13 150 140 126 1342.3 795 100 2069 300 1735 100 14 150 150 138 1361.8 1066 200 2205 300 1982 100 15 150 160 144 1534.0 758 100 2342 300 1937 100 16 150 180 162 1725.8 754 100 2627 300 2268 200 17 175 90 91 1074.3 1408 200 1544 200 1383 100 18 175 100 105 1108.8 907 100 1580 200 1453 100 19 175 110 112 1296.0 940 100 1815 200 1623 100 20 175 120 126 1330.5 934 100 2018 300 1884 100 21 175 140 147 1552.3 1076 200 2350 300 2198 200 22 175 150 161 1586.8 878 100 2376 300 2292 200 23 175 160 168 1774.0 911 100 2587 300 2461 200 24 175 180 189 1995.8 653 100 3128 400 2734 200 25 200 90 104 1209.3 851 100 1769 200 1579 100 26 200 100 120 1258.8 1154 200 1846 200 1726 100 27 200 110 128 1461.0 1209 200 1991 200 1797 100 28 200 120 144 1510.5 878 100 2273 300 2081 200 29 200 140 168 1762.3 748 100 2654 300 2503 200 30 200 150 184 1811.8 699 100 2790 300 2538 200 31 200 160 192 2014.0 603 100 3123 400 2614 200 32 200 180 216 2265.8 637 100 3341 400 2909 200 Raw sheet Dimension (cm) a1j w1j gj1 $$\hat{g}_{j1} $$ $$c_{j1}^{1} $$ $$\hat{c}_{j1}^{1} $$ $$c_{j1}^{2} $$ $$\hat{c}_{j1}^{2} $$ ( j) Length Width 1 125 90 65 804.3 2651 400 1092 200 928 50 2 125 100 75 808.8 2293 400 1178 200 1095 100 3 125 110 80 966.0 2001 400 1254 200 1148 100 4 125 120 90 970.5 1684 300 1454 200 1350 100 5 125 140 105 1132.3 1713 300 1608 200 1474 100 6 125 150 115 1136.8 1015 200 1834 200 1661 100 7 125 160 120 1294.0 1147 200 1850 200 1636 100 8 125 180 135 1455.8 1296 200 2102 300 1924 100 9 150 90 78 939.3 1747 300 1250 200 1092 100 10 150 100 90 958.8 1795 300 1491 200 1233 100 11 150 110 96 1131.0 1630 300 1511 200 1403 100 12 150 120 108 1150.5 953 100 1733 200 1566 100 13 150 140 126 1342.3 795 100 2069 300 1735 100 14 150 150 138 1361.8 1066 200 2205 300 1982 100 15 150 160 144 1534.0 758 100 2342 300 1937 100 16 150 180 162 1725.8 754 100 2627 300 2268 200 17 175 90 91 1074.3 1408 200 1544 200 1383 100 18 175 100 105 1108.8 907 100 1580 200 1453 100 19 175 110 112 1296.0 940 100 1815 200 1623 100 20 175 120 126 1330.5 934 100 2018 300 1884 100 21 175 140 147 1552.3 1076 200 2350 300 2198 200 22 175 150 161 1586.8 878 100 2376 300 2292 200 23 175 160 168 1774.0 911 100 2587 300 2461 200 24 175 180 189 1995.8 653 100 3128 400 2734 200 25 200 90 104 1209.3 851 100 1769 200 1579 100 26 200 100 120 1258.8 1154 200 1846 200 1726 100 27 200 110 128 1461.0 1209 200 1991 200 1797 100 28 200 120 144 1510.5 878 100 2273 300 2081 200 29 200 140 168 1762.3 748 100 2654 300 2503 200 30 200 150 184 1811.8 699 100 2790 300 2538 200 31 200 160 192 2014.0 603 100 3123 400 2614 200 32 200 180 216 2265.8 637 100 3341 400 2909 200 Table 2. Some data for one planning horizon given by the company of the case study Raw sheet Dimension (cm) a1j w1j gj1 $$\hat{g}_{j1} $$ $$c_{j1}^{1} $$ $$\hat{c}_{j1}^{1} $$ $$c_{j1}^{2} $$ $$\hat{c}_{j1}^{2} $$ ( j) Length Width 1 125 90 65 804.3 2651 400 1092 200 928 50 2 125 100 75 808.8 2293 400 1178 200 1095 100 3 125 110 80 966.0 2001 400 1254 200 1148 100 4 125 120 90 970.5 1684 300 1454 200 1350 100 5 125 140 105 1132.3 1713 300 1608 200 1474 100 6 125 150 115 1136.8 1015 200 1834 200 1661 100 7 125 160 120 1294.0 1147 200 1850 200 1636 100 8 125 180 135 1455.8 1296 200 2102 300 1924 100 9 150 90 78 939.3 1747 300 1250 200 1092 100 10 150 100 90 958.8 1795 300 1491 200 1233 100 11 150 110 96 1131.0 1630 300 1511 200 1403 100 12 150 120 108 1150.5 953 100 1733 200 1566 100 13 150 140 126 1342.3 795 100 2069 300 1735 100 14 150 150 138 1361.8 1066 200 2205 300 1982 100 15 150 160 144 1534.0 758 100 2342 300 1937 100 16 150 180 162 1725.8 754 100 2627 300 2268 200 17 175 90 91 1074.3 1408 200 1544 200 1383 100 18 175 100 105 1108.8 907 100 1580 200 1453 100 19 175 110 112 1296.0 940 100 1815 200 1623 100 20 175 120 126 1330.5 934 100 2018 300 1884 100 21 175 140 147 1552.3 1076 200 2350 300 2198 200 22 175 150 161 1586.8 878 100 2376 300 2292 200 23 175 160 168 1774.0 911 100 2587 300 2461 200 24 175 180 189 1995.8 653 100 3128 400 2734 200 25 200 90 104 1209.3 851 100 1769 200 1579 100 26 200 100 120 1258.8 1154 200 1846 200 1726 100 27 200 110 128 1461.0 1209 200 1991 200 1797 100 28 200 120 144 1510.5 878 100 2273 300 2081 200 29 200 140 168 1762.3 748 100 2654 300 2503 200 30 200 150 184 1811.8 699 100 2790 300 2538 200 31 200 160 192 2014.0 603 100 3123 400 2614 200 32 200 180 216 2265.8 637 100 3341 400 2909 200 Raw sheet Dimension (cm) a1j w1j gj1 $$\hat{g}_{j1} $$ $$c_{j1}^{1} $$ $$\hat{c}_{j1}^{1} $$ $$c_{j1}^{2} $$ $$\hat{c}_{j1}^{2} $$ ( j) Length Width 1 125 90 65 804.3 2651 400 1092 200 928 50 2 125 100 75 808.8 2293 400 1178 200 1095 100 3 125 110 80 966.0 2001 400 1254 200 1148 100 4 125 120 90 970.5 1684 300 1454 200 1350 100 5 125 140 105 1132.3 1713 300 1608 200 1474 100 6 125 150 115 1136.8 1015 200 1834 200 1661 100 7 125 160 120 1294.0 1147 200 1850 200 1636 100 8 125 180 135 1455.8 1296 200 2102 300 1924 100 9 150 90 78 939.3 1747 300 1250 200 1092 100 10 150 100 90 958.8 1795 300 1491 200 1233 100 11 150 110 96 1131.0 1630 300 1511 200 1403 100 12 150 120 108 1150.5 953 100 1733 200 1566 100 13 150 140 126 1342.3 795 100 2069 300 1735 100 14 150 150 138 1361.8 1066 200 2205 300 1982 100 15 150 160 144 1534.0 758 100 2342 300 1937 100 16 150 180 162 1725.8 754 100 2627 300 2268 200 17 175 90 91 1074.3 1408 200 1544 200 1383 100 18 175 100 105 1108.8 907 100 1580 200 1453 100 19 175 110 112 1296.0 940 100 1815 200 1623 100 20 175 120 126 1330.5 934 100 2018 300 1884 100 21 175 140 147 1552.3 1076 200 2350 300 2198 200 22 175 150 161 1586.8 878 100 2376 300 2292 200 23 175 160 168 1774.0 911 100 2587 300 2461 200 24 175 180 189 1995.8 653 100 3128 400 2734 200 25 200 90 104 1209.3 851 100 1769 200 1579 100 26 200 100 120 1258.8 1154 200 1846 200 1726 100 27 200 110 128 1461.0 1209 200 1991 200 1797 100 28 200 120 144 1510.5 878 100 2273 300 2081 200 29 200 140 168 1762.3 748 100 2654 300 2503 200 30 200 150 184 1811.8 699 100 2790 300 2538 200 31 200 160 192 2014.0 603 100 3123 400 2614 200 32 200 180 216 2265.8 637 100 3341 400 2909 200 Table 3. The results of the WGCM for different levels of A1 and A2 and setting ψ1 = ψ2 = ψ3 = 1 Set by Pareto optimal decision Obtained by Step solution obtained by Exp. maker 1 of the WGCM Step 3 of the WGCM $$OF_{1}^{*} $$ $$OF_{2}^{*} $$ OF1 OF2 A1 A2 (millions) (millions) $$OF_{3}^{*} $$ (millions) (millions) OF3 1 0 0 554.68 543.5332 332010 554.6896 547.2582 332060 2 0 50 554.68 577.1843 332010 554.7023 579.4770 332023 3 0 100 554.68 577.1839 332010 554.7016 578.4340 332072 4 0 150 554.68 577.1848 332010 554.7793 577.6672 332018 5 0 192 554.68 577.1841 332010 554.6920 579.7137 332017 6 50 0 554.68 543.5343 332010 554.8681 545.4129 332040 7 50 50 554.68 577.1863 332010 554.7563 579.3057 332054 8 50 100 554.68 577.1848 332010 554.6898 578.9136 332024 9 50 150 554.68 577.1863 332010 554.7028 578.1839 332010 10 50 192 554.68 578.5893 332010 554.8427 580.9199 332075 11 100 0 554.68 543.5329 332010 554.6910 545.5737 332028 12 100 50 554.68 577.1896 332010 554.6896 578.0054 332010 13 100 100 554.68 577.1866 332010 554.7388 578.3775 332010 14 100 150 554.68 577.1920 332010 554.7615 577.9540 332010 15 100 192 554.68 577.1847 332010 554.7300 578.7020 332026 16 150 0 554.68 543.5349 332010 554.7686 544.6990 332071 17 150 50 554.68 577.1868 332010 554.7843 578.4177 332075 18 150 100 554.68 578.5269 332010 555.0523 580.2028 332050 19 150 150 554.68 577.1861 332010 554.8248 581.4120 332010 20 150 192 554.68 577.1865 332010 554.7349 578.5261 332056 21 192 0 554.68 543.5334 332010 554.7204 544.0228 332020 22 192 50 554.68 577.1863 332010 554.6900 578.0897 332010 23 192 100 554.68 577.1867 332010 554.8805 581.7697 332096 24 192 150 554.68 577.1862 332010 554.7489 579.1361 332012 25 192 192 554.68 577.1861 332010 554.6907 577.4787 332010 Set by Pareto optimal decision Obtained by Step solution obtained by Exp. maker 1 of the WGCM Step 3 of the WGCM $$OF_{1}^{*} $$ $$OF_{2}^{*} $$ OF1 OF2 A1 A2 (millions) (millions) $$OF_{3}^{*} $$ (millions) (millions) OF3 1 0 0 554.68 543.5332 332010 554.6896 547.2582 332060 2 0 50 554.68 577.1843 332010 554.7023 579.4770 332023 3 0 100 554.68 577.1839 332010 554.7016 578.4340 332072 4 0 150 554.68 577.1848 332010 554.7793 577.6672 332018 5 0 192 554.68 577.1841 332010 554.6920 579.7137 332017 6 50 0 554.68 543.5343 332010 554.8681 545.4129 332040 7 50 50 554.68 577.1863 332010 554.7563 579.3057 332054 8 50 100 554.68 577.1848 332010 554.6898 578.9136 332024 9 50 150 554.68 577.1863 332010 554.7028 578.1839 332010 10 50 192 554.68 578.5893 332010 554.8427 580.9199 332075 11 100 0 554.68 543.5329 332010 554.6910 545.5737 332028 12 100 50 554.68 577.1896 332010 554.6896 578.0054 332010 13 100 100 554.68 577.1866 332010 554.7388 578.3775 332010 14 100 150 554.68 577.1920 332010 554.7615 577.9540 332010 15 100 192 554.68 577.1847 332010 554.7300 578.7020 332026 16 150 0 554.68 543.5349 332010 554.7686 544.6990 332071 17 150 50 554.68 577.1868 332010 554.7843 578.4177 332075 18 150 100 554.68 578.5269 332010 555.0523 580.2028 332050 19 150 150 554.68 577.1861 332010 554.8248 581.4120 332010 20 150 192 554.68 577.1865 332010 554.7349 578.5261 332056 21 192 0 554.68 543.5334 332010 554.7204 544.0228 332020 22 192 50 554.68 577.1863 332010 554.6900 578.0897 332010 23 192 100 554.68 577.1867 332010 554.8805 581.7697 332096 24 192 150 554.68 577.1862 332010 554.7489 579.1361 332012 25 192 192 554.68 577.1861 332010 554.6907 577.4787 332010 Table 3. The results of the WGCM for different levels of A1 and A2 and setting ψ1 = ψ2 = ψ3 = 1 Set by Pareto optimal decision Obtained by Step solution obtained by Exp. maker 1 of the WGCM Step 3 of the WGCM $$OF_{1}^{*} $$ $$OF_{2}^{*} $$ OF1 OF2 A1 A2 (millions) (millions) $$OF_{3}^{*} $$ (millions) (millions) OF3 1 0 0 554.68 543.5332 332010 554.6896 547.2582 332060 2 0 50 554.68 577.1843 332010 554.7023 579.4770 332023 3 0 100 554.68 577.1839 332010 554.7016 578.4340 332072 4 0 150 554.68 577.1848 332010 554.7793 577.6672 332018 5 0 192 554.68 577.1841 332010 554.6920 579.7137 332017 6 50 0 554.68 543.5343 332010 554.8681 545.4129 332040 7 50 50 554.68 577.1863 332010 554.7563 579.3057 332054 8 50 100 554.68 577.1848 332010 554.6898 578.9136 332024 9 50 150 554.68 577.1863 332010 554.7028 578.1839 332010 10 50 192 554.68 578.5893 332010 554.8427 580.9199 332075 11 100 0 554.68 543.5329 332010 554.6910 545.5737 332028 12 100 50 554.68 577.1896 332010 554.6896 578.0054 332010 13 100 100 554.68 577.1866 332010 554.7388 578.3775 332010 14 100 150 554.68 577.1920 332010 554.7615 577.9540 332010 15 100 192 554.68 577.1847 332010 554.7300 578.7020 332026 16 150 0 554.68 543.5349 332010 554.7686 544.6990 332071 17 150 50 554.68 577.1868 332010 554.7843 578.4177 332075 18 150 100 554.68 578.5269 332010 555.0523 580.2028 332050 19 150 150 554.68 577.1861 332010 554.8248 581.4120 332010 20 150 192 554.68 577.1865 332010 554.7349 578.5261 332056 21 192 0 554.68 543.5334 332010 554.7204 544.0228 332020 22 192 50 554.68 577.1863 332010 554.6900 578.0897 332010 23 192 100 554.68 577.1867 332010 554.8805 581.7697 332096 24 192 150 554.68 577.1862 332010 554.7489 579.1361 332012 25 192 192 554.68 577.1861 332010 554.6907 577.4787 332010 Set by Pareto optimal decision Obtained by Step solution obtained by Exp. maker 1 of the WGCM Step 3 of the WGCM $$OF_{1}^{*} $$ $$OF_{2}^{*} $$ OF1 OF2 A1 A2 (millions) (millions) $$OF_{3}^{*} $$ (millions) (millions) OF3 1 0 0 554.68 543.5332 332010 554.6896 547.2582 332060 2 0 50 554.68 577.1843 332010 554.7023 579.4770 332023 3 0 100 554.68 577.1839 332010 554.7016 578.4340 332072 4 0 150 554.68 577.1848 332010 554.7793 577.6672 332018 5 0 192 554.68 577.1841 332010 554.6920 579.7137 332017 6 50 0 554.68 543.5343 332010 554.8681 545.4129 332040 7 50 50 554.68 577.1863 332010 554.7563 579.3057 332054 8 50 100 554.68 577.1848 332010 554.6898 578.9136 332024 9 50 150 554.68 577.1863 332010 554.7028 578.1839 332010 10 50 192 554.68 578.5893 332010 554.8427 580.9199 332075 11 100 0 554.68 543.5329 332010 554.6910 545.5737 332028 12 100 50 554.68 577.1896 332010 554.6896 578.0054 332010 13 100 100 554.68 577.1866 332010 554.7388 578.3775 332010 14 100 150 554.68 577.1920 332010 554.7615 577.9540 332010 15 100 192 554.68 577.1847 332010 554.7300 578.7020 332026 16 150 0 554.68 543.5349 332010 554.7686 544.6990 332071 17 150 50 554.68 577.1868 332010 554.7843 578.4177 332075 18 150 100 554.68 578.5269 332010 555.0523 580.2028 332050 19 150 150 554.68 577.1861 332010 554.8248 581.4120 332010 20 150 192 554.68 577.1865 332010 554.7349 578.5261 332056 21 192 0 554.68 543.5334 332010 554.7204 544.0228 332020 22 192 50 554.68 577.1863 332010 554.6900 578.0897 332010 23 192 100 554.68 577.1867 332010 554.8805 581.7697 332096 24 192 150 554.68 577.1862 332010 554.7489 579.1361 332012 25 192 192 554.68 577.1861 332010 554.6907 577.4787 332010 5. Computational experiments on the case study The robust formulation (60)–(79) and the introduced solution approach of Section 4 is experimented in this section. For this aim a set of real data from the paper box manufacturing company explained in Section 2 is used. The data and the obtained results are explained in the following sub-sections. Notably, the robust formulation (60)–(79) and the introduced solution approach are coded in GAMS solver and run on a computer with an Intel Core 2 Duo 2.53 GHz processor and 4.00 GB RAM. 5.1. The real data set The real data of one planning horizon of the paper box manufacturing company, which was detailed in Section 2, is obtained to be used for computational study. The data obtained from the company include the following: there are 25 different types of boxes; demand of each type of box; there are 6 potential suppliers; there are 32 different types of raw sheet, which can be supplied by all suppliers; the wastage amount remained from each raw sheet when it is used for each type of paper box; the discount break point for each raw sheet type given by each supplier; and normal and discounted price of each raw sheet given by each supplier. Some information of each type of box is represented in Table 1, while Table 2 contains different types of raw sheets. As the full data of the case study need large matrixes to be represented, only the data of paper box type 1 and supplier 1 (as an instance) for all raw sheets are presented in Table 2. 5.2. Results and sensitivity analysis The proposed WGCM of Section 4.1 is used to solve the model (60)–(79) for the data of the cardboard box production company. As there are six suppliers supplying all 32 types of raw sheets, the total allowed variations of the sheet prices from their mean values (A1 and A2) can take minimum value of zero (meaning there is no uncertainty (variation) in the prices), and maximum value of 1 (meaning all prices are in their highest possible level). Therefore, the values of set $$\left \{0,50,100,150,192\right \}$$ is used for A1 and A2. On the other hand, the budget of uncertainty associated with the demand uncertainty (ξi) is fixed to 1 in all experiments. All steps of the proposed WGCM of Section 4.1 is applied for any combination of the A1 and A2 values where all the objective function weights are equally determined (ψ1 = ψ2 = ψ3 = 1).The obtained results are represented in Table 3. Fig. 1. View largeDownload slide Graph of variations of $$OF_{2}^{*} $$ over different levels of A1 and A2. Fig. 1. View largeDownload slide Graph of variations of $$OF_{2}^{*} $$ over different levels of A1 and A2. Fig. 2. View largeDownload slide Graph of variations of OF1 (obtained by the second step of the WGCM, where ψ1 = ψ2 = ψ3 = 1) over different levels of A1 and A2. Fig. 2. View largeDownload slide Graph of variations of OF1 (obtained by the second step of the WGCM, where ψ1 = ψ2 = ψ3 = 1) over different levels of A1 and A2. Fig. 3. View largeDownload slide Graph of variations of OF2 (obtained by the second step of the WGCM, where ψ1 = ψ2 = ψ3 = 1) over different levels of A1 and A2. Fig. 3. View largeDownload slide Graph of variations of OF2 (obtained by the second step of the WGCM, where ψ1 = ψ2 = ψ3 = 1) over different levels of A1 and A2. Fig. 4. View largeDownload slide Graph of variations of OF3 (obtained by the second step of the WGCM, where ψ1 = ψ2 = ψ3 = 1) over different levels of A1 and A2. Fig. 4. View largeDownload slide Graph of variations of OF3 (obtained by the second step of the WGCM, where ψ1 = ψ2 = ψ3 = 1) over different levels of A1 and A2. Table 4. The results of the WGCM for two combinations of the levels of A1 and A2 considering different weights Set by Set by Pareto optimal solution decision decision obtained by Step 3 Exp. maker Run maker of the WGCM OF1 OF2 A1 A1 ψ1 ψ2 ψ3 (millions) (millions) OF3 1 1 1 1 554.6896 547.2582 332060 2 1.5 1 0.5 554.6916 545.1571 332043 3 1.5 0.5 1 554.6895 547.3694 332032 2 150 50 4 1 1.5 0.5 554.8713 544.7098 332115 5 1 0.5 1.5 554.7470 549.3233 332067 6 0.5 1 1.5 555.1553 544.5439 332028 7 0.5 1.5 1 554.8128 545.6115 332076 8 1 1 1 554.7843 578.4177 332075 9 1.5 1 0.5 554.7028 581.7285 332030 10 1.5 0.5 1 554.6899 579.7565 332010 2 150 50 11 1 1.5 0.5 554.7966 579.0578 332031 12 1 0.5 1.5 554.8111 581.8725 332010 13 0.5 1 1.5 554.9912 583.3101 332020 104 0.5 1.5 1 555.6172 577.9442 332106 Set by Set by Pareto optimal solution decision decision obtained by Step 3 Exp. maker Run maker of the WGCM OF1 OF2 A1 A1 ψ1 ψ2 ψ3 (millions) (millions) OF3 1 1 1 1 554.6896 547.2582 332060 2 1.5 1 0.5 554.6916 545.1571 332043 3 1.5 0.5 1 554.6895 547.3694 332032 2 150 50 4 1 1.5 0.5 554.8713 544.7098 332115 5 1 0.5 1.5 554.7470 549.3233 332067 6 0.5 1 1.5 555.1553 544.5439 332028 7 0.5 1.5 1 554.8128 545.6115 332076 8 1 1 1 554.7843 578.4177 332075 9 1.5 1 0.5 554.7028 581.7285 332030 10 1.5 0.5 1 554.6899 579.7565 332010 2 150 50 11 1 1.5 0.5 554.7966 579.0578 332031 12 1 0.5 1.5 554.8111 581.8725 332010 13 0.5 1 1.5 554.9912 583.3101 332020 104 0.5 1.5 1 555.6172 577.9442 332106 View Large Table 4. The results of the WGCM for two combinations of the levels of A1 and A2 considering different weights Set by Set by Pareto optimal solution decision decision obtained by Step 3 Exp. maker Run maker of the WGCM OF1 OF2 A1 A1 ψ1 ψ2 ψ3 (millions) (millions) OF3 1 1 1 1 554.6896 547.2582 332060 2 1.5 1 0.5 554.6916 545.1571 332043 3 1.5 0.5 1 554.6895 547.3694 332032 2 150 50 4 1 1.5 0.5 554.8713 544.7098 332115 5 1 0.5 1.5 554.7470 549.3233 332067 6 0.5 1 1.5 555.1553 544.5439 332028 7 0.5 1.5 1 554.8128 545.6115 332076 8 1 1 1 554.7843 578.4177 332075 9 1.5 1 0.5 554.7028 581.7285 332030 10 1.5 0.5 1 554.6899 579.7565 332010 2 150 50 11 1 1.5 0.5 554.7966 579.0578 332031 12 1 0.5 1.5 554.8111 581.8725 332010 13 0.5 1 1.5 554.9912 583.3101 332020 104 0.5 1.5 1 555.6172 577.9442 332106 Set by Set by Pareto optimal solution decision decision obtained by Step 3 Exp. maker Run maker of the WGCM OF1 OF2 A1 A1 ψ1 ψ2 ψ3 (millions) (millions) OF3 1 1 1 1 554.6896 547.2582 332060 2 1.5 1 0.5 554.6916 545.1571 332043 3 1.5 0.5 1 554.6895 547.3694 332032 2 150 50 4 1 1.5 0.5 554.8713 544.7098 332115 5 1 0.5 1.5 554.7470 549.3233 332067 6 0.5 1 1.5 555.1553 544.5439 332028 7 0.5 1.5 1 554.8128 545.6115 332076 8 1 1 1 554.7843 578.4177 332075 9 1.5 1 0.5 554.7028 581.7285 332030 10 1.5 0.5 1 554.6899 579.7565 332010 2 150 50 11 1 1.5 0.5 554.7966 579.0578 332031 12 1 0.5 1.5 554.8111 581.8725 332010 13 0.5 1 1.5 554.9912 583.3101 332020 104 0.5 1.5 1 555.6172 577.9442 332106 View Large In the obtained results from Step 1 of the WGCM in Table 3, the values of $$OF_{1}^{*} $$ and $$OF_{3}^{*} $$ are not sensitive to the variations of values used for A1 and A2. This happens because in the models (80) and (82), the objective functions do not contain any value of A1 and A2. On the other hand, the values obtained for $$OF_{2}^{*} $$ varies by an unstable trend when the values of A1 and A2 are changed in the model (60)–(79). These variations are shown in the graph of Fig.1. Furthermore, the sensitivity of the objective function values of the obtained Pareto optimal solutions over different levels of A1 and A2, follow unstable trends as shown by Figs 2–4. Fig. 5. View largeDownload slide Behavior of objective function 1 for the experiments of Table 4 over various weight values. Fig. 5. View largeDownload slide Behavior of objective function 1 for the experiments of Table 4 over various weight values. Fig. 6. View largeDownload slide Behavior of objective function 2 for the experiments of Table 4 over various weight values. Fig. 6. View largeDownload slide Behavior of objective function 2 for the experiments of Table 4 over various weight values. To further analyze the behavior of the proposed robust model its sensitivity to the different levels of weights of the objective functions is studied. For this aim, for instance, two experiments from Table 3 is selected (the experiment with A1 = A2 = 0 and the experiment with A1 = 150 and A2 = 50). Seven combinations of weights of the objective functions in these experiments are selected to be of set $$\left \{0.5,1,1.5\right \}$$ (for example ψ1 = ψ2 = ψ3 = 1). In each combination of the weights the sum of weight values must be equal to 3 as assumed in Step 2 of the proposed approach. Therefore, each of the mentioned experiments results in seven runs. Applying the WGCM to the new experiments, the results is shown by Table 4. The results of Table 4 which are also plotted in the graphs of Figs 6–8, show that although the values of each objective function has an unstable trend over various values of its weight, the best value of each objective function is obtained when its weight is in the highest level (ψ = 1.5). In each of the Figs 5–7, we eliminated the runs of Table 4 with equal weights (the first run of each experiment). Excluding this run, for each objective function there is a pair of runs for each weight value. Therefore, the objective function values associated to each run of the weight values are plotted. Fig. 7. View largeDownload slide Behavior of objective function 3 for the experiments of Table 4 over various weight values. Fig. 7. View largeDownload slide Behavior of objective function 3 for the experiments of Table 4 over various weight values. Table 5. A sample detailed result for the case problem when A1 = 50, A2 = 100 and ψ1 = ψ2 = ψ3 = 1 Y Value Y Value Y Value i j k i j k i j k 1 22 1 149 8 11 2 3349 16 5 1 7 2 9 2 1 9 20 5 4294 17 32 5 9234 2 25 3 1120 10 4 3 3 18 8 5 9897 2 28 2 1 10 7 1 5038 19 32 1 737 3 10 4 1 11 7 1 7194 19 32 5 4366 3 29 3 1161 12 20 5 11763 20 6 2 1643 4 19 1 1 13 4 1 3 20 6 6 40093 4 21 2 1405 13 7 1 14171 21 30 2 15549 5 23 2 1809 14 20 5 6970 22 7 1 40626 6 3 1 1 14 23 2 2 23 8 5 42249 6 8 5 4262 15 7 1 2242 24 8 5 11441 7 11 2 3718 16 1 5 27447 25 1 5 90607 7 13 3 1 Y Value Y Value Y Value i j k i j k i j k 1 22 1 149 8 11 2 3349 16 5 1 7 2 9 2 1 9 20 5 4294 17 32 5 9234 2 25 3 1120 10 4 3 3 18 8 5 9897 2 28 2 1 10 7 1 5038 19 32 1 737 3 10 4 1 11 7 1 7194 19 32 5 4366 3 29 3 1161 12 20 5 11763 20 6 2 1643 4 19 1 1 13 4 1 3 20 6 6 40093 4 21 2 1405 13 7 1 14171 21 30 2 15549 5 23 2 1809 14 20 5 6970 22 7 1 40626 6 3 1 1 14 23 2 2 23 8 5 42249 6 8 5 4262 15 7 1 2242 24 8 5 11441 7 11 2 3718 16 1 5 27447 25 1 5 90607 7 13 3 1 View Large Table 5. A sample detailed result for the case problem when A1 = 50, A2 = 100 and ψ1 = ψ2 = ψ3 = 1 Y Value Y Value Y Value i j k i j k i j k 1 22 1 149 8 11 2 3349 16 5 1 7 2 9 2 1 9 20 5 4294 17 32 5 9234 2 25 3 1120 10 4 3 3 18 8 5 9897 2 28 2 1 10 7 1 5038 19 32 1 737 3 10 4 1 11 7 1 7194 19 32 5 4366 3 29 3 1161 12 20 5 11763 20 6 2 1643 4 19 1 1 13 4 1 3 20 6 6 40093 4 21 2 1405 13 7 1 14171 21 30 2 15549 5 23 2 1809 14 20 5 6970 22 7 1 40626 6 3 1 1 14 23 2 2 23 8 5 42249 6 8 5 4262 15 7 1 2242 24 8 5 11441 7 11 2 3718 16 1 5 27447 25 1 5 90607 7 13 3 1 Y Value Y Value Y Value i j k i j k i j k 1 22 1 149 8 11 2 3349 16 5 1 7 2 9 2 1 9 20 5 4294 17 32 5 9234 2 25 3 1120 10 4 3 3 18 8 5 9897 2 28 2 1 10 7 1 5038 19 32 1 737 3 10 4 1 11 7 1 7194 19 32 5 4366 3 29 3 1161 12 20 5 11763 20 6 2 1643 4 19 1 1 13 4 1 3 20 6 6 40093 4 21 2 1405 13 7 1 14171 21 30 2 15549 5 23 2 1809 14 20 5 6970 22 7 1 40626 6 3 1 1 14 23 2 2 23 8 5 42249 6 8 5 4262 15 7 1 2242 24 8 5 11441 7 11 2 3718 16 1 5 27447 25 1 5 90607 7 13 3 1 View Large 5.3. A sample detailed result In this subsection the detailed results of one of the Pareto optimal solutions of the model is reported. As there are 25 treatment with different combination of A1 and A2 values, the first treatment which contains values A1 = 50 and A2 = 100 is chosen for reporting the detailed results. Thus, the values of the parameters e.g. $$T_{jk}^{1} $$, $$T_{jk}^{2} $$, $$Z_{jk}^{1} $$, $$Z_{jk}^{2} $$ and Yijk obtained in the Pareto optimal solution of Table 3 is reported in Table 5–6. Note that Table 5–6 show only the variables that took positive values in the Pareto optimal solution of Table 3. For instance, paper box number 7 is produced from the sheet types 11 and 13 by 3718 and 1 sheets from each one respectively (Table 5). Table 6. A sample detailed result for the case problem when A1 = 50, A2 = 100 and ψ1 = ψ2 = ψ3 = 1 T1 Value T2 Value Z1 Value Z2 Value j k j k j k j k 3 1 1 1 5 1 3 1 1 1 5 118000 4 1 1 6 2 1 4 1 3 6 2 1643 4 3 1 6 6 1 4 3 3 6 6 40093 5 1 1 7 1 1 5 1 7 7 1 69271 9 2 1 8 5 1 9 2 1 8 5 67849 10 4 1 11 2 1 10 4 1 11 2 7067 13 3 1 20 5 1 13 3 1 20 5 23027 19 1 1 21 2 1 19 1 1 21 2 1405 22 1 1 23 2 1 22 1 149 23 2 1811 28 2 1 25 3 1 28 2 1 25 3 1120 3 29 3 29 3 1161 30 2 1 30 2 15549 32 1 1 32 1 737 32 5 1 32 5 13600 T1 Value T2 Value Z1 Value Z2 Value j k j k j k j k 3 1 1 1 5 1 3 1 1 1 5 118000 4 1 1 6 2 1 4 1 3 6 2 1643 4 3 1 6 6 1 4 3 3 6 6 40093 5 1 1 7 1 1 5 1 7 7 1 69271 9 2 1 8 5 1 9 2 1 8 5 67849 10 4 1 11 2 1 10 4 1 11 2 7067 13 3 1 20 5 1 13 3 1 20 5 23027 19 1 1 21 2 1 19 1 1 21 2 1405 22 1 1 23 2 1 22 1 149 23 2 1811 28 2 1 25 3 1 28 2 1 25 3 1120 3 29 3 29 3 1161 30 2 1 30 2 15549 32 1 1 32 1 737 32 5 1 32 5 13600 Table 6. A sample detailed result for the case problem when A1 = 50, A2 = 100 and ψ1 = ψ2 = ψ3 = 1 T1 Value T2 Value Z1 Value Z2 Value j k j k j k j k 3 1 1 1 5 1 3 1 1 1 5 118000 4 1 1 6 2 1 4 1 3 6 2 1643 4 3 1 6 6 1 4 3 3 6 6 40093 5 1 1 7 1 1 5 1 7 7 1 69271 9 2 1 8 5 1 9 2 1 8 5 67849 10 4 1 11 2 1 10 4 1 11 2 7067 13 3 1 20 5 1 13 3 1 20 5 23027 19 1 1 21 2 1 19 1 1 21 2 1405 22 1 1 23 2 1 22 1 149 23 2 1811 28 2 1 25 3 1 28 2 1 25 3 1120 3 29 3 29 3 1161 30 2 1 30 2 15549 32 1 1 32 1 737 32 5 1 32 5 13600 T1 Value T2 Value Z1 Value Z2 Value j k j k j k j k 3 1 1 1 5 1 3 1 1 1 5 118000 4 1 1 6 2 1 4 1 3 6 2 1643 4 3 1 6 6 1 4 3 3 6 6 40093 5 1 1 7 1 1 5 1 7 7 1 69271 9 2 1 8 5 1 9 2 1 8 5 67849 10 4 1 11 2 1 10 4 1 11 2 7067 13 3 1 20 5 1 13 3 1 20 5 23027 19 1 1 21 2 1 19 1 1 21 2 1405 22 1 1 23 2 1 22 1 149 23 2 1811 28 2 1 25 3 1 28 2 1 25 3 1120 3 29 3 29 3 1161 30 2 1 30 2 15549 32 1 1 32 1 737 32 5 1 32 5 13600 5.4. Discussion The optimization technique (including modelling and solution approach sections) used in this study was applied to a cardboard box manufacturing company as a case study. To overcome the big concerns of the managers of the company, as a mathematical formulation, the decisions on the amount of each raw sheet type and supplier selection had to be made in order to improve some objectives e.g. wastage amount of raw sheets, raw sheets cost and product surplus simultaneously for one planning horizon. Considering the uncertainties in the mathematical formulation, the numerical experiments done in the previous sub-sections proved that the obtained values for the objective functions are highly depended on the level of uncertainty of the parameters and also the weights used for objective functions in the solution procedure (Figs 1–7). Practically, the obtained results for a given level of uncertainty such as the results of Tables 5–6, can help the production planners (and managers) of the company to make a plan for selecting the raw sheets and their suppliers in order to improve the objectives considered in the problem of this study simultaneously. Notably, an obtained solution when considering a higher level of uncertainty in the parameters, can give a more robust decision to the managers in the unstable market of the case study. 6. Conclusion A new and real problem in cardboard box manufacturing was solved in this paper. The problem minimizes waste, surplus amount and the procurement costs through making decisions on selection of the suppliers and materials for a cardboard box manufacturer. The problem was formulated as a multi-objective mixed integer linear programme with uncertain demand and raw material price values. A typical robust optimization approach was applied to cope with the uncertainties. The crisp form of the robust multi-objective formulation was solved by a weighted global criterion approach for the real data obtained from a cardboard box manufacturer. As future study, the uncertainty of the problem can be reflected using other methods like fuzzy numbers, etc. The complexity of the deterministic form of the problem also can be studied. In the case of high complexity, meta-heuristic approaches can be used to solve the problem. Acknowledgements The authors are indebted to the editors and the referees of the journal for their helpful and constructive comments that improved the quality of this paper. References Almeida , A. T. D. , Alencar , M. H. , Garcez , T. V. & Ferreira , R. J. P. ( 2017 ) A systematic literature review of multicriteria and multi-objective models applied in risk management . IMA J. Manag. Math. , 28 , 153 -- 184 . Alp , O. & Tan , T. ( 2008 ) Tactical capacity management under capacity flexibility . IIE Trans. , 40 , 221 -- 237 . 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A study of a robust multi-objective supplier-material selection problem

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1471-678X
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Abstract

Abstract This paper develops a multi-objective mathematical formulation for supplier-material selection in cardboard box manufacturing. The aim is to minimize three different scaled objectives: wastage of raw material, raw material cost and product surplus. To model industry reality more closely, the formulation is extended to an environment that includes uncertain costs and demands. In this sense, the model is reformulated as a robust optimization problem. To solve this problem, a weighted global criterion approach is developed and applied to find Pareto optimal solutions. A case study from a cardboard box manufacturer is used to test the efficiency of the proposed robust formulation and the proposed solution approach. The sensitivity of the Pareto optimal solutions to different levels of uncertainty in the parameters and different weights of the objective functions is also studied. 1. Introduction The selection of suppliers and materials in a manufacturing supply chain is an important and challenging problem (see Ayhan, 2013; Batuhan & Selcuk, 2015; Tan & Alp, 2009; Asgari et al., 2016). The problem is important for a manufacturer as supplier selection affects price of the raw material, due date of orders, quality of raw materials etc. On the other hand, the problem is challenging as there is a tradeoff between criteria such as quality, price, due date etc. Material purchase cost is an important criteria in supplier selection. It can account for as much as 60% of total sales (Krajewski & Ritzman, 2001). This cost may vary from 70% of total revenue in automotive industries to 80% in high technology industries (Weber et al., 1991). Additionally, since good supplier selection can significantly contribute to cost reduction, enterprizes must consider their supply chains (SCs) more carefully (Ozgen & Gulsun (2014)). Following the same aim, Cakravastia et al. (2002) develop an analytical model for the process of supplier selection while an SC network is being established. The objective of their study was to minimize the level of customer dissatisfaction, through two factors—price and delivery lead time. Alp & Tan (2008) and Tan & Alp (2009) investigated a problem with two supply options, in a multi-period, having fixed procurement cost. Later on, Alp et al. (2013) considered a linear cost function with identical suppliers in an infinite horizon version of the problem with fixed components. This study considers a typical multi-objective supplier and material selection problem from a real cardboard box production company. This problem is important because it combines the problem of cutting raw paper sheets (Russo et al., 2014) with two key issues that contribute to profitability: (1) choice of material dimensions and (2) supplier selection. To the best of our knowledge this study is the first to consider these issues through simultaneous minimization of (1) wastage of raw paper sheets, (2) product surplus relative to demand and (3) procurement costs. The problem is modelled as a multi-objective mixed integer linear programme in an uncertain environment (Fullér & Majlender, 2004; Fullér et al., 2012; Bouzembrak et al., 2013; Almeida et al., 2017) where some parameters e.g. demand and raw material price are uncertain. These uncertainties is reflected using robust optimization which is a suitable uncertain approach for formulating real-world problems. The crisp form of the uncertain multi-objective formulation is solved by a weighted global criterion approach for the real data obtained from the cardboard box production company. The paper is structured as follows. The next section describes the problem of the cardboard box production company. Section 3 proposes a deterministic multi-objective mathematical model and its robust form for the problem. Section 4 focuses on a solution approach for the proposed robust formulation. Section 5 performs some computational experiments on the robust formulation using the real data obtained from the company. The paper ends with conclusion. 2. Problem description We consider a real life problem in a cardboard box production company. The company produces cardboard boxes in different sizes (where each size has its specified dimensions) in response to customer demands. A cardboard box is packaging used for shipping goods, typically between manufacturer and retailer. In the rest of this paper we shall refer to a ‘cardboard box’ simply as ‘box’. These boxes are produced from raw sheets of cardboard, which are sold to the company in various dimensions. Each supplier of the company is able to provide several dimensions of the raw sheets. Another component of the problem is the ordering style, where the orders take place by the customers in specific horizon and quantity according to their need. In the rest of this paper we refer to different sizes of box by ‘box types’ or ‘types of box’, where raw sheets with different dimensions are referred by ‘raw sheet types’ or ‘types of raw sheet’. More details about the problem is as follows: The customers order a specific number of each box type in a planning horizon. There are several types of raw sheets available in all suppliers. The suppliers offer competitive prices. Each type of raw sheet produces a specified number of each box type that is known for the company. One or more than one type of the raw sheets may be used to produce a type of box. Employing a specific type of a raw sheet for producing a box type corresponds to a known amount of wastage. This wastage varies depending on the box type. For each type of the raw sheet, there exist several alternative suppliers, the supplier who offers a lower price gains a higher priority. In order to be more competitive, the suppliers offer a discount based on the raw sheet type. Their discount policy is based on the ordered quantity. They determine a price break point for the ordered quantity of each sheet type. If the order quantity of a raw sheet is above the determined break point quantity of the supplier, there will be a price discount for the demanded quantity. By using the same raw sheet for more than one box type, the company can maximize its benefit from the supplier discount policy. Considering the given descriptions, the aim of the company is to determine the sizes and the order quantity of the raw sheet for producing each box type as well as selecting the suitable supplier for the required raw sheet such that the below-mentioned three objectives are considered. Wastage of raw paper sheets. The wastage of the raw material is minimized. For this purpose, the types of the ordered raw sheet and the quantity of each type, which is used for producing a box type, should be determined such that the total amount of wastage is minimized (wastage is scaled by units of area such as meters square, etc.). Raw material cost. Ordering the required raw sheets has a certain purchasing cost for the company. Therefore, the company should decide to order the raw sheets such that the total payment per purchase is minimized over the planning horizon. In this respect, the break points offered by the suppliers play a critical role in minimizing total purchasing cost of the raw sheets. Product surplus. The total amount of the boxes produced from the ordered raw sheets must be greater than or equal to the demand of the boxes. When this condition is met, the extra products should be stored in the inventory to be used on the next planning horizon. These extra boxes are considered as a typical surplus. However, the company requires minimizing the total surpluses of all types of boxes. The parameters like the price of the sheets, break points of the discount, which is determined by the suppliers, and demand of the boxes have high degree of uncertainty. However, the previous experiences and historical data of the company provide a suitable base for calculating an estimate for them. In this study, intervals of values for these uncertain parameters are considered, which are obtained from the historical data of the previous planning horizons. 3. Mathematical formulation In this section, first a deterministic mathematical formulation is proposed for the problem described in previous section and then a robust formulation of the deterministic model is proposed to satisfy the uncertainty of the parameters of the problem. 3.1. Deterministic formulation The following notations are used in the deterministic formulation of the problem: $$\begin{array}{lrl} \textit{i} & (\textrm{index}): & \textrm{index used for types of box,} \\ \textit{j} & (\textrm{index}): & \textrm{index used for types of raw sheet,} \\ \textit{k} & (\textrm{index}): & \textrm{index used for suppliers,} \\ \textit{I} & (\textrm{parameter}): & \textrm{number of box types to be produced in a planning horizon,}\\ \textit{J} & (\textrm{parameter}): & \textrm{number of types of raw sheet presented by suppliers,} \\ \textit{K} & (\textrm{parameter}): & \textrm{number of suppliers,} \\ \textit{M} & (\textrm{parameter}): & \textrm{a large positive value,} \\ d_{i} & (\textrm{parameter}): & \textrm{demand of box type}\ \textit{i}\ \textrm{shown by unit of quantity,} \\ a_{ij} & (\textrm{parameter}): & \textrm{number of box type}\ \textit{i}\ \textrm{that can be cut from raw sheet type}\ \textit{j}, \\ w_{ij} & (\textrm{parameter}): & \textrm{waste amount remained after cutting box type}\ \textit{i}\ \textrm{from raw sheet}\ \textit{j}, \\ g_{jk} & (\textrm{parameter}): & \textrm{break point for ordering raw sheet type}\ \textit{j}\ \textrm{offered by supplier}\ \textit{k}.\\ & & \textrm{For orders more than this amount discounted price will be applied for the order}, \\ c_{jk}^{1},c_{jk}^{2} & (\textrm{parameter}): & \textrm{normal and discounted unit prices for raw sheet type}\ \textit{j}\ \textrm{by supplier}\ \textit{k}, (c_{jk}^{1} \ge c_{jk}^{2}), \\ T_{jk}^{1},T_{jk}^{2} & (\textrm{variable}): & \textrm{binary variables indicating that whether or not discount is applied for sheet type}\ \textit{j}\\ && \textrm{by supplier}\ \textit{k}. \textrm{Normal price is applied if}\ T_{jk}^{1}=1\ \textrm{and discounted price is applied if}\ T_{jk}^{2} =1. \\ Y_{ijk} & (\textrm{variable}): & \textrm{number of raw sheets type}\ \textit{j}\ \textrm{which is ordered to supplier}\ \textit{k}\\ && \textrm{for producing box type}\ \textit{i}\textrm{,} \\ Z_{jk}^{1},Z_{jk}^{2} & (\textrm{variable}): & \textrm{non-negative continuous values to be used instead of non-linear terms}\\ && T_{jk}^{1} \left(\sum_{i=1}^{I}Y_{ijk} \right),T_{jk}^{2} \left(\sum_{i=1}^{I}Y_{ijk} \right) \textrm{respectively}. \end{array}$$ Based on the above-mentioned notations, the following non-linear model is proposed for the supplier-material selection problem described in Section 2: \begin{equation} {\hskip-90pt}\textrm{Objective function 1}: OF_{1} =\min \; \sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{K}w_{ij} Y_{ijk} \end{equation} (1) \begin{equation} \textrm{Objective function 2}: OF_{2} =\min \; \sum_{j=1}^{J}\sum_{k=1}^{K}\left(c_{jk}^{1} T_{jk}^{1} \sum_{i=1}^{I}Y_{ijk} +c_{jk}^{2} T_{jk}^{2} \sum_{i=1}^{I}Y_{ijk} \right) \end{equation} (2) \begin{equation} \textrm{Objective function 3}: OF_{3} =\min \; \sum_{i=1}^{I}\left(\left(\sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \right)-d_{i} \right)\qquad\qquad \end{equation} (3) \begin{align}\textrm{subject to}\nonumber\\ & \sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \ge d_{i}\qquad{\forall i} {\hskip160pt}\end{align} (4) \begin{equation} {\hskip-130pt}T_{jk}^{1} +T_{jk}^{2} \le 1 \qquad{\forall j,k} \end{equation} (5) \begin{equation} {\hskip-60pt}\sum_{i=1}^{I}Y_{ijk} \ge T_{jk}^{1} \qquad{\forall j,k}\qquad\qquad\qquad\ \ \ \end{equation} (6) \begin{equation} {\hskip-65pt}\sum_{i=1}^{I}Y_{ijk} \le g_{jk} T_{jk}^{1} +MT_{jk}^{2} \qquad{\forall j,k}\qquad\ \ \end{equation} (7) \begin{equation} {\hskip-85pt}\sum_{i=1}^{I}Y_{ijk} \ge \left(g_{jk} +1\right)T_{jk}^{2} \qquad{\forall j,k}\qquad\quad \end{equation} (8) \begin{equation} {\hskip-85pt}T_{jk}^{1},T_{jk}^{2} \in \left\{0,1\right\} \qquad{\forall j,k}\; \qquad\qquad\quad \end{equation} (9) \begin{equation} {\hskip-115pt}Y_{ijk} \ge 0\ \textrm{and integer} \qquad{\forall i,j,k.} \end{equation} (10) The objective function (1) minimizes the total wastage remained by all raw sheets bought to produce all box types. Objective function (2) calculates and minimizes the total raw material cost due to price of raw sheets. The discount policy described in Section 2 is considered in this objective function such that at most one of the non-linear terms $$c_{jk}^{1} T_{jk}^{1} \sum _{i=1}^{I}Y_{ijk} $$ or $$c_{jk}^{2} T_{jk}^{2} \sum _{i=1}^{I}Y_{ijk} $$ can take positive value. Objective function (3) minimizes the surplus. (For example, if the demand of a box type is 95 and 10 numbers of a type of raw sheet is selected and each raw sheet can produce 10 number of the box, the surplus in this case is $$10\left (10\right )-95=5$$). Constraint set (4) satisfies the demand of the box types. Constraint sets (5)–(8) together guarantee that for buying type j of raw sheet from supplier k, only one of the following cases may happen: $$\sum _{i=1}^{I}Y_{ijk} \le g_{jk} $$ and $$T_{jk}^{1} =1$$, then the normal price in objective function (2) is considered by supplier k, $$\sum _{i=1}^{I}Y_{ijk} \ge g_{jk} +1 $$ and $$T_{jk}^{2} =1$$, therefore, the discounted price in objective function (2) is considered by supplier k, $$\sum _{i=1}^{I}Y_{ijk} =0 $$, $$T_{jk}^{1} =0$$ and$$T_{jk}^{2} =0$$, thus the raw sheet j is not bought from supplier k. Finally, constraint sets (9) and (10) define the nature of the variables. The non-linear term $$T_{jk}^{1} \left (\sum _{i=1}^{I}Y_{ijk} \right )$$ (as well as$$T_{jk}^{2} \left (\sum _{i=1}^{I}Y_{ijk} \right )$$), can be linearized by a linearization technique which is discussed here. If a new continuous non-negative variable $$Z_{jk}^{1} =T_{jk}^{1} \left (\sum _{i=1}^{I}Y_{ijk} \right )$$ is defined, the following set of constraints guarantee that the value of $$Z_{jk}^{1} $$ can be either zero (if $$T_{jk}^{1} =0$$) or $$\sum _{i=1}^{I}Y_{ijk} $$ (if $$T_{jk}^{1} =1$$): \begin{equation} {\hskip-83pt}Z_{jk}^{1} \le MT_{jk}^{1} \qquad\forall j,k \quad\end{equation} (11) \begin{equation} {\hskip-83pt}Z_{jk}^{1} \le \sum_{i=1}^{I}Y_{ijk} \qquad\forall j,k \end{equation} (12) \begin{equation} Z_{jk}^{1} \ge \left(\sum_{i=1}^{I}Y_{ijk} \right)-M\left(1-T_{jk}^{1} \right) \qquad\forall j,k \end{equation} (13) \begin{equation} Z_{jk}^{1} \ge 0 \qquad\forall j,k. \qquad\qquad\qquad\qquad\qquad\end{equation} (14) Similarly, $$Z_{jk}^{2} =T_{jk}^{2} \left (\sum _{i=1}^{I}Y_{ijk} \right )$$ is defined in a similar way. Therefore, the non-linear models (1)–(10) is linearized as the mixed integer linear model (MILP) given below, \begin{equation} \textrm{Objective function 1}: OF_{1} =\min \; \sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{K}w_{ij} Y_{ijk} \qquad\qquad\quad \end{equation} (15) \begin{equation} \textrm{Objective function 2}: OF_{2} =\min \; \sum_{j=1}^{J}\sum_{k=1}^{K}\left(c_{jk}^{1} Z_{jk}^{1} +c_{jk}^{2} Z_{jk}^{2} \right) \qquad\ \end{equation} (16) \begin{equation} \textrm{Objective function 3}: OF_{3} =\min \; \sum_{i=1}^{I}\left(\left(\sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \right)-d_{i} \right) \end{equation} (17) \begin{align}\textrm{subject to}&\nonumber\\& \sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \ge d_{i}\qquad{\forall i} {\hskip115pt}\end{align} (18) \begin{equation} {\hskip-30pt}T_{jk}^{1} +T_{jk}^{2} \le 1 \qquad{\forall j,k} \qquad\qquad\qquad\ \end{equation} (19) \begin{equation} {\hskip-40pt}\sum_{i=1}^{I}Y_{ijk} \ge T_{jk}^{1}\qquad{\forall j,k} \qquad\qquad\quad \ \end{equation} (20) \begin{equation} {\hskip-40pt}\sum_{i=1}^{I}Y_{ijk} \le g_{jk} T_{jk}^{1} +MT_{jk}^{2}\qquad{\forall j,k} \quad\end{equation} (21) \begin{equation} {\hskip-40pt}\sum_{i=1}^{I}Y_{ijk} \ge g_{jk} T_{jk}^{2}\qquad{\forall j,k} \qquad\qquad\end{equation} (22) \begin{equation} {\hskip-70pt}Z_{jk}^{1} \le MT_{jk}^{1}\qquad{\forall j,k} \qquad\ \ \end{equation} (23) \begin{equation} {\hskip-10pt}Z_{jk}^{1} \le \sum_{i=1}^{I}Y_{ijk}\qquad{\forall j,k} \qquad\qquad\qquad\quad\ \ \ \end{equation} (24) \begin{equation}{\hskip-5pt} Z_{jk}^{1} \ge \left(\sum_{i=1}^{I}Y_{ijk} \right)-M\left(1-T_{jk}^{1} \right)\qquad{\forall j,k} \end{equation} (25) \begin{equation}\!\!\!\! Z_{jk}^{2} \le MT_{jk}^{2}\qquad{\forall j,k} \qquad\qquad\qquad\qquad\ \ \ \end{equation} (26) \begin{equation} Z_{jk}^{2} \le \sum_{i=1}^{I}Y_{ijk}\qquad{\forall j,k} \qquad\qquad\qquad\quad\ \ \ \end{equation} (27) \begin{equation} Z_{jk}^{2} \ge \left(\sum_{i=1}^{I}Y_{ijk} \right)-M\left(1-T_{jk}^{2} \right)\qquad{\forall j,k} \end{equation} (28) \begin{equation} T_{jk}^{1},T_{jk}^{2} \in \left\{0,1\right\}\qquad{\forall j,k} \qquad\qquad\qquad\quad\end{equation} (29) \begin{equation} Y_{ijk} \ge 0\ \textrm{and integer} \qquad{\forall i,j,k} \qquad\qquad\quad\end{equation} (30) \begin{equation} Z_{jk}^{1},Z_{jk}^{2} \ge 0 \qquad{\forall j,k.}\quad\qquad\qquad\qquad\quad\ \ \end{equation} (31) 3.2. A robust approach for the introduced deterministic formulation As mentioned in Section 2, there exists a high degree of uncertainty on the parameters of this problem. In this section, the deterministic formulation of the problem using the concepts of robust optimization approach is reformulated to cover the uncertain nature of the parameters. Robust approach for a problem has the following properties: (1) The uncertain parameters can take any value from a given interval while it does not assume any known distribution on the value of the uncertain parameters. (2) The optimization is done with respect to the uncertainty of the parameters. The obtained solution is a robust solution to the problem such that it considers the whole range of possible values for the uncertain variables. Consequently, the feasible solutions of the problem hold for all possible values of the uncertain parameters. The concepts of robust optimization methods can be found in the study of Bertsimas & Sim (2004). Moreover, robust optimization has been applied to model the uncertainty of many optimization problems in the field of SC network design and other combinatorial optimization problems. For more detailed information the studies of Thiele (2008), Klibi et al. (2010), Pishvaee et al. (2011), Ramezani et al. (2013), Recchia & Scutellé (2014), Hosseini et al. (2014) etc. can be referred. Here, the robust optimization method introduced by Bertsimas & Sim (2004) is briefly explained. Consider the following optimization problem: \begin{align}\min& \quad c\, x \nonumber\\ \textrm{subject to}& \nonumber\\ &\quad Ax\le b \\ &\quad x\in S, \nonumber \end{align} (32) where c is an n dimensional cost vector, A is a m × n matrix, b is an m dimensional vector and S is a polyhedron. We assume that data uncertainty affects the cost vector c and the elements of the matrix A.Each entry aij of matrix Acan be modeled as an independent, symmetric and bounded parameter, which can take values in interval $$\left [a_{ij} -\hat{a}_{ij},a_{ij} +\hat{a}_{ij} \right ]\, $$. Notations aij and $$\hat{a}_{ij} $$ denote the nominal value and the maximum deviation from the nominal value, respectively. Same as elements of matrix A, each entry cj can take values from$$\left [c_{j},c_{j} +\hat{c}_{j} \right ]$$. So, cj and $$\hat{c}_{j} $$ denote the nominal value and the maximum deviation from the nominal value, respectively. Let J0 be the sets of the uncertain parameters in objective function and Ji be the set of the uncertain parameters in i-th constraint. Now, problem (32) can be reformulated as follows: \begin{align} \min&\quad \sum_{j=1}^{n}\tilde{c}_{j} x_{j} \nonumber\\ \textrm{subject to}&\quad \nonumber\\ &\quad\sum_{j=1}^{n}\tilde{a}_{ij} x_{j} \le b_{i} \qquad\forall i \\ \nonumber &\quad x\in S, \end{align} (33) where $$\tilde{c}_{j} |j\in J_{0} $$ and $$\tilde{a}_{ij} |j\in J_{i} $$. The scaled deviation of $$\tilde{c}_{j} $$ and $$\tilde{a}_{ij} $$ from their nominal values are defined as follows: \begin{align} \mu_{j} &=\frac{\tilde{c}_{j} -c_{j}}{\hat{c}_{j}} \nonumber \\ \eta_{ij} &=\frac{\tilde{a}_{ij} -a_{ij} }{\hat{a}_{ij}} \end{align} (34) where $$\tilde{c}_{j} $$ and $$\tilde{a}_{ij} $$ are uncertain data, cj and aij are nominal values and $$\hat{c}_{j} $$ and $$\hat{a}_{ij} $$ are maximum deviations from the nominal values. Furthermore, μj and ηij are random variables, which take a value in $$\, \left [0,1\right ]$$ and $$\, \left [-1,1\right ]$$ respectively. Bertsimas & Sim (2004) introduced parameters Γ0 and Γi which can vary in continuous intervals $$\left [0,|J_{0} |\, \right ]$$ and $$\left [0,|J_{i} |\, \right ]$$, respectively. The notations |J0| and |Ji| denote the number of elements of sets Γ0 and Γi. The parameters Γ0 and Γi adjust the robustness of the method against the level of conservatism of the solution. Although the aggregated scaled deviation for objective function can vary in the interval $$\left [-|J_{0} |\,,|J_{0} |\, \right ]$$, it is limited to the following: \begin{equation} \sum_{j=1}^{n}\mu_{j} \le \Gamma_{0}. \end{equation} (35) The parameter Γ0 controls the level of robustness in the objective function. By the same manner, the aggregated scaled deviation for constraint i can vary in the interval $$\left [-|J_{i} |\,,|J_{i} |\, \right ]$$, but it is limited to the following: \begin{equation} \sum_{j=1}^{n}\eta_{ij} \le \Gamma_{i} \qquad \forall i. \end{equation} (36) Based on the above definitions, the setsJ0 and Ji can be written as follows: \begin{equation} J_{0} =\left\{j|\tilde{c}_{j} =c_{j} +\mu_{j} \hat{c}_{j},\, 0\le \mu_{j} \, \le 1,\, \sum_{j=1}^{n}\mu_{j} \le \Gamma_{0},\quad \forall j\right\} \quad\end{equation} (37) \begin{equation} J_{i} =\left\{j|\tilde{a}_{ij} =a_{ij} +\hat{a}_{ij} \eta_{ij},\, \, |\eta_{ij} |\, \le 1,\, \sum_{j=1}^{n}\eta_{ij} \le \Gamma_{i}, \quad\forall i,j\right\}\!. \end{equation} (38) Finally, the tractable form of the robust formulation when uncertainty affects the cost values and coefficient of constraints can be written as follows: \begin{align} \min&\quad c\, x+z_{0} \Gamma_{0} +\sum_{j\in J_{o} }p_{oj}\nonumber \\ \textrm{subject to}&\nonumber\\ &\quad\sum_{j=1}^{n}a_{ij} x_{j} +z_{i} \Gamma_{i} +\sum_{j\in J_{i} }p_{ij} \le b_{i} \qquad\forall i \nonumber\\ &\quad z_{0} +p_{0j} \ge \hat{c}_{j} \, y_{j}\qquad\forall j\in J_{0} \nonumber\\ &\quad z_{i} +p_{ij} \ge \hat{a}_{ij} \, y_{j}\qquad\forall i,\ j\in J_{i} \nonumber\\ &\nonumber\quad -y_{j} \le x_{j} \le y_{j}\qquad\forall j \\ \nonumber &\quad p_{0j},p_{ij} \ge 0\qquad\forall i,\ j\in J_{i} \\ \nonumber &\quad z_{0},z_{i} \ge 0 \qquad\forall i \nonumber\\ &\quad y_{j} \ge 0 \qquad\forall j \nonumber\\ &\quad x\in S.\end{align} (39) The proof of equivalency of problems (33) and (39) is detailed by Bertsimas & Sim (2004). In the above problem variables z0, zi, p0j, pij and yj is the dual variables which help to convert problem (33) to problem (39) (see Bertsimas & Sim (2004)). It is important to note that, in the cases that right-hand side vector b,in problem (32) is uncertain, we can introduce a new variable xn+1, and write Ax − bxn+1 ≤ 0, 1 ≤ xn+1 ≤ 1 (Bertsimas & Sim, 2003). In the next sub-sections, the symmetric interval as above is considered for the previously mentioned uncertain parameters (price of the raw sheets, break point for ordering raw sheet and demand of box types) and using the above-mentioned concepts of robust optimization approach, the deterministic model of (15)–(31) is reformulated to a robust model (covering the uncertainty of the parameters). 3.2.1. Robust form of the constraints related to uncertain demand The demand of the types of boxes in a planning horizon may vary in an expected range of values. If a certain demand value of previous planning horizon for box type i isdi, then it is claimed that in the presence of an uncertain demand, the symmetric and bounded random demand value for the current planning horizon is noted by $$\tilde{d}_{i} $$ on interval $$\left [d_{i} -\hat{d}_{i},\, d_{i} +\hat{d}_{i} \right ]$$. Now, to convert the constraint set (18) to its robust form, they must be changed to a form that satisfies all the possible demand values of the interval $$\left [d_{i} -\hat{d}_{i},\, d_{i} +\hat{d}_{i} \right ]$$. As the highest demand value in the planning horizon is $$d_{i} +\hat{d}_{i} $$, if the constraint set (18) is satisfied for this value, then it is satisfied for all other demand values of the interval $$\left [d_{i} -\hat{d}_{i},\, d_{i} +\hat{d}_{i} \right ]$$. Therefore, the robust form of the constraint set (18) is defined by the following constraint set (Bertsimas & Sim, 2003): \begin{equation} \sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \ge d_{i} +\xi_{i} \hat{d}_{i} \qquad\forall i, \end{equation} (40) where ξi denotes the budget of uncertainty associated with the demand uncertainty, and takes a value in $$\, \left [0,1\right ]$$. 3.2.2. Robust form of the objective function related to uncertain demand Considering the above-mentioned type of uncertainty for the demand values, the objective function (17) should be changed to its robust form. As the objective function is of minimization type, its worst value (largest value) occurs when the value of parameter di turns to its lowest value as $$d_{i} -\hat{d}_{i} $$. Therefore, this change in the value of di covers the uncertainty of the demand values. Consequently, the robust version of the objective function (17) is introduced as Bertsimas & Sim, 2003, \begin{equation} \textrm{Objective function 3}: OF_{3} =\min \; \sum_{i=1}^{I}\left(\left(\sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \right)-\left(d_{i} -\xi_{i} \hat{d}_{i} \right)\right)\!, \end{equation} (41) where ξi denotes the budget of uncertainty associated with the demand uncertainty, and takes a value in $$\, \left [0,1\right ]$$. 3.2.3. Robust form of the constraints related to discount break point The discount break point determined by each supplier for each raw sheet type, has an uncertain characteristic similar to the uncertain demand. It means that for each discount break point gjk, the symmetric and bounded random discount break point on current planning horizon is noted by $$\tilde{g}_{jk} $$ from interval $$\left [g_{jk} -\hat{g}_{jk},\, g_{jk} +\hat{g}_{jk} \right ]$$. As the constraint sets (21) and (22) contain the discount break point values, they are changed to the following robust versions (with the same logic as the logic applied in Section 3.2.1): \begin{equation} \sum_{i=1}^{I}Y_{ijk} \le \left(g_{jk} -\bar{g}_{jk} \right)T_{jk}^{1} +MT_{jk}^{2} \qquad\forall j,k \end{equation} (42) \begin{equation} \sum_{i=1}^{I}Y_{ijk} \ge \left(g_{jk} +\bar{g}_{jk} \right)T_{jk}^{2}\qquad\forall j,k. \qquad\quad\end{equation} (43) 3.2.4. Robust form of the objective function related to price of raw sheet The same uncertainty as the previous cases exists in relevance to the prices of the raw sheets. The symmetric and bounded random sheet prices (due to the discount points) for the current planning horizon are noted by $$\tilde{c}_{jk}^{1} $$ and $$\tilde{c}_{jk}^{2} $$ on the intervals $$\left [c_{jk}^{1} -\hat{c}_{jk}^{1},\, c_{jk}^{1} +\hat{c}_{jk}^{1} \right ]$$ and $$\left [c_{jk}^{2} -\hat{c}_{jk}^{2},\, c_{jk}^{2} +\hat{c}_{jk}^{2} \right ]$$, respectively. Since only the second objective function of the model (15)–(31) contains sheet price values, its robust form is obtained using the following lemma: Lemma 1 The following MILP obtains the robust form of the objective function 2 that is presented by Equation (16), note that constraints (18)–(31) still are respected. \begin{equation} {\min \; \sum_{j=1}^{J}\sum_{k=1}^{K}\left(c_{jk}^{1} Z_{jk}^{1} +c_{jk}^{2} Z_{jk}^{2} \right) +A^{1} \delta^{1} +A^{2} \delta^{2} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{1} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{2}} \end{equation} (44) \begin{align} \textrm{subject to}& \nonumber\\& \delta^{1} +\gamma_{jk}^{1} \ge \hat{c}_{jk}^{1} +Z_{jk}^{1}\qquad{\forall j,k} \end{align} (45) \begin{equation} \delta^{2} +\gamma_{jk}^{2} \ge \hat{c}_{jk}^{2} +Z_{jk}^{2}\qquad{\forall j,k} \end{equation} (46) \begin{equation} \gamma_{jk}^{1},\gamma_{jk}^{2} \ge 0 \qquad{\forall j,k} \end{equation} (47) \begin{equation} \delta^{1},\delta^{2} \ge 0. \end{equation} (48) Proof. By applying the uncertainness of the sheet prices in the objective function of Equation (16), its worst case (largest possible value) occurs at the sheet prices of $$c_{jk}^{1} +\hat{c}_{jk}^{1} $$ and $$c_{jk}^{2} +\hat{c}_{jk}^{2} $$. Thus, the robust form of the objective function is introduced as follows, where variable $$\alpha _{jk}^{1} $$ ($$-1\le \alpha _{jk}^{1} \le 1$$) determines the variation of the sheet price from its mean value ($$c_{jk}^{1} $$) ($$\alpha _{jk}^{2} $$ is defined in the same way): \begin{equation} \min \; \sum_{j=1}^{J}\sum_{k=1}^{K}\left(c_{jk}^{1} +\alpha_{jk}^{1} \hat{c}_{jk}^{1} \right)Z_{jk}^{1} +\left(c_{jk}^{2} +\alpha_{jk}^{2} \hat{c}_{jk}^{2} \right)Z_{jk}^{2} \end{equation} (49) The robust form is extended as, \begin{equation} \min \; \sum_{j=1}^{J}\sum_{k=1}^{K}\left(c_{jk}^{1} Z_{jk}^{1} +c_{jk}^{2} Z_{jk}^{2} \right) +\sum_{j=1}^{J}\sum_{k=1}^{K}\left(\alpha_{jk}^{1} \hat{c}_{jk}^{1} Z_{jk}^{1} +\alpha_{jk}^{2} \hat{c}_{jk}^{2} Z_{jk}^{2} \right). \end{equation} (50) Since the worst case of the objective function value needs to be considered in the robust form, the second term, which is related to the uncertainty of the sheet prices, should be maximized as the following where A1 and A2 determine the total allowed variations of the sheet prices from their mean values. The decision maker as the output of the model determines these values. \begin{equation} {\hskip-35pt}{\max \sum_{j=1}^{J}\sum_{k=1}^{K}\left(\alpha_{jk}^{1} \hat{c}_{jk}^{1} Z_{jk}^{1} +\alpha_{jk}^{2} \hat{c}_{jk}^{2} Z_{jk}^{2} \right)} \end{equation} (51) \begin{align} \textrm{subject to}&\nonumber\\ &\quad\sum_{j=1}^{J}\sum_{k=1}^{K}\left|\alpha_{jk}^{1} \right|\le A^{1} \end{align} (52) \begin{equation} \sum_{j=1}^{J}\sum_{k=1}^{K}\left|\alpha_{jk}^{2} \right|\le A^{2} \end{equation} (53) \begin{equation} \left|\alpha_{jk}^{1} \right|,\left|\alpha_{jk}^{2} \right|\le 1\qquad{\forall j,k.} \end{equation} (54) As the formulation (51)–(54) is of maximization type, its optimization direction is different from the optimization direction of the model (15)–(31). To overcome this contradiction, the model (51)–(54) is dualized as the following: \begin{equation} {\hskip40pt}{\min \quad A^{1} \delta^{1} +A^{2} \delta^{2} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{1} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{2}} \end{equation} (55) \begin{align} \textrm{subject to}& \nonumber\\&\quad \delta^{1} +\gamma_{jk}^{1} \ge \hat{c}_{jk}^{1} +Z_{jk}^{1}\qquad{\forall j,k} \end{align} (56) \begin{equation} {\hskip5pt}\delta^{2} +\gamma_{jk}^{2} \ge \hat{c}_{jk}^{2} +Z_{jk}^{2}\qquad{\forall j,k}\end{equation} (57) \begin{equation} {\hskip-12pt}\gamma_{jk}^{1},\gamma_{jk}^{2} \ge 0 \qquad{\forall j,k} \end{equation} (58) \begin{equation} {\hskip-22pt}\delta^{1},\delta^{2} \ge 0.\end{equation} (59) Hence, the lemma is proved. 3.2.5. Final robust formulation of the deterministic model Based on the formulations introduced in the Sections 3.2.1 to 3.2.4, the robust formulation of the model (15)–(31) is finalized in this section. For this aim, the deterministic constraints (18), (21) and (22) are replaced by the robust constraints (40), (42) and (43), respectively. Similarly, the deterministic objective functions (16) and (17) are replaced by the formulation (55)–(59) and the robust objective function (41), respectively. Finally, the robust formulation of the model (15)–(31) is represented as: \begin{equation} \textrm{Objective function 1}: OF_{1} =\min \; \sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{K}w_{ij} Y_{ijk} \end{equation} (60) \begin{align} \textrm{Objective function 2}: OF_{2} &=\min \; \sum_{j=1}^{J}\sum_{k=1}^{K}\left(c_{jk}^{1} Z_{jk}^{1} +c_{jk}^{2} Z_{jk}^{2} \right) +A^{1} \delta^{1}\nonumber\\ &\quad + A^{2} \delta^{2} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{1} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{2} \end{align} (61) \begin{equation} \textrm{Objective function 3}: OF_{3} =\min \; \sum_{i=1}^{I}\left(\left(\sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \right)-\left(d_{i} -\xi_{i} \hat{d}_{i} \right)\right) \end{equation} (62) \begin{align} &\textrm{subject to}\nonumber\\ &\qquad\qquad\!\!\sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij} Y_{ijk} \ge d_{i} +\xi_{i} \hat{d}_{i} \qquad\forall i \end{align} (63) \begin{equation} T_{jk}^{1} +T_{jk}^{2} \le 1 \qquad\forall j,k \end{equation} (64) \begin{equation}\sum_{i=1}^{I}Y_{ijk} \ge T_{jk}^{1} \qquad\forall j,k \end{equation} (65) \begin{equation} \sum_{i=1}^{I}Y_{ijk} \le \left(g_{jk} -\bar{g}_{jk} \right)T_{jk}^{1} +MT_{jk}^{2} \qquad\forall j,k \end{equation} (66) \begin{align} \sum_{i=1}^{I}Y_{ijk} \ge \left(g_{jk} +\bar{g}_{jk} \right)T_{jk}^{2} \qquad\forall j,k \end{align} (67) \begin{equation} Z_{jk}^{1} \le MT_{jk}^{1} \qquad\forall j,k \end{equation} (68) \begin{equation} Z_{jk}^{1} \le \sum_{i=1}^{I}Y_{ijk} \qquad\forall j,k \end{equation} (69) \begin{equation} Z_{jk}^{1} \ge \left(\sum_{i=1}^{I}Y_{ijk}\right)-M\left(1-T_{jk}^{1} \right) \quad\forall j,k \end{equation} (70) \begin{equation} Z_{jk}^{2} \le MT_{jk}^{2}\qquad\forall j,k \end{equation} (71) \begin{equation} Z_{jk}^{2} \le \sum_{i=1}^{I}Y_{ijk} \qquad\forall j,k \end{equation} (72) \begin{align} Z_{jk}^{2} \ge \left(\sum_{i=1}^{I}Y_{ijk} \right)-M\left(1-T_{jk}^{2} \right) \qquad\forall j,k \end{align} (73) \begin{align} \delta^{1} +\gamma_{jk}^{1} \ge \hat{c}_{jk}^{1} +Z_{jk}^{1} \qquad\forall j,k \end{align} (74) \begin{align} \delta^{2} +\gamma_{jk}^{2} \ge \hat{c}_{jk}^{2} +Z_{jk}^{2} \qquad\forall j,k \end{align} (75) \begin{align} T_{jk}^{1},T_{jk}^{2} \in \left\{0,1\right\} \qquad\forall j,k \end{align} (76) \begin{align} Y_{ijk} \ge 0\;\textrm{and integer} \qquad\forall i,j,k \end{align} (77) \begin{align} Z_{jk}^{1},Z_{jk}^{2},\gamma_{jk}^{1},\gamma_{jk}^{2} \ge 0 \qquad\forall j,k \end{align} (78) \begin{equation} \delta^{1},\delta^{2} \ge 0. \end{equation} (79) The above-introduced multi-objective robust formulation is solvable with any multi-objective optimization method in order to obtain an acceptable Pareto optimal solution. The next section focuses on a method to solve this multi-objective model. 4. Solution approach The proposed robust formulation (60)–(79) is a multi-objective optimization model, for this type of models instead of the optimal solution, the concept of efficient solution is defined (see Steuer, 1986; Kovécs & Marian, 2002; Jablonsky, 2007, 2014; Kahraman, 2008; Franco et al., 2009; ; Hadi-Vencheh et al., 2014; Kar, 2015; Hadi-Vencheh & Mohamadghasemi, 2015; Zhou et al., 2015). The following definitions introduces this concept: Definition 1 Let S* be a feasible solution of the problem (60)–(79). Then, S* is called an efficient or a Pareto optimal solution, if there is no other feasible solution S such that $$O{F_{i}^{S}} \le OF_{i}^{S^{*} },\, \, \, i=1,2,3$$ and at least one strict inequality. If S* is efficient, the point $$(OF_{1}^{S^{*} },OF_{2}^{S^{*} },OF_{3}^{S^{*} } )$$ is then called non-dominated. Definition 2 Let S* be a feasible solution of the problem (60)–(79). Then, S* is called a weakly efficient solution, if there is no other feasible solution S such that $$O{F_{i}^{S}}\mathrm{<}OF_{i}^{S^{*} },\, \, \, i=1,2,3$$. If S* is weakly efficient, the point $$(OF_{1}^{S^{*} },OF_{2}^{S^{*} },OF_{3}^{S^{*} } )$$ is then called weakly non-dominated. An efficient solution is always weakly efficient while the inverse is not true. It should be noted that, generally, the efficient solution is not unique, and from the geometrical point of view, efficient solutions are points of the feasible space, but non-dominated points as the image of the efficient solutions are located in the objectives’ space. In this study, a modified version of the Global criterion method (GCM) is applied to find Pareto optimal solutions of the model (60)–(79). 4.1. A weighted Global criterion method The Global criterion method GCM (see Gomes et al., 2012; Saraj & Safaei, 2012) is modified as a weighted Global criterion method (WGCM) for solving the model (60)–(79). The WGCM is constructed based on the following steps: Step 1: Solve the following models separately to obtain the values $$OF_{1}^{*},OF_{2}^{*} $$ and $$OF_{3}^{*} $$. Meaning that each objective function of the model (60)–(79) is solved individually respecting to all constraints of the model. \begin{equation} OF_{1}^{*} =\min \; \sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{K}w_{ij} Y_{ijk}\end{equation} (80) subject to Constraints (63)–(79). \begin{equation} OF_{2}^{*} =\min \; \sum_{j=1}^{J}\sum_{k=1}^{K}\left(c_{jk}^{1} Z_{jk}^{1} +c_{jk}^{2} Z_{jk}^{2} \right) +A^{1} \delta^{1} +A^{2} \delta^{2} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{1} +\sum_{j=1}^{J}\sum_{k=1}^{K}\gamma_{jk}^{2}\end{equation} (81) subject to Constraints (63)–(79). \begin{equation}OF_{3}^{*} =\min \; \sum_{i=1}^{I}\left(\left(\sum_{j=1}^{J}\sum_{k=1}^{K}a_{ij}Y_{ijk} \right)-\left(d_{i} -\xi_{i} \hat{d}_{i} \right)\right) \end{equation} (82) subject to Constraints (63)–(79). Step 2: Define the Global criterion of the method from the optimal objective function values of the models (80), (81) and (82) as follows: \begin{align} OF=\psi_{1} \left(\frac{OF_{1} -OF_{1}^{*} }{OF_{1}^{*} } \right)+\psi_{2} \left(\frac{OF_{2} -OF_{2}^{*} }{OF_{2}^{*} } \right)+\psi_{3} \left(\frac{OF_{3} -OF_{3}^{*} }{OF_{3}^{*} } \right) \end{align} (83) The above term is a typical Global criterion which converts the objective functions of the model to a single objective function. To give more flexibility to decision maker to determine the importance (weight) of each objective function of the model, the values ψ1, ψ2 and ψ3 (ψ1 + ψ2 + ψ3 = 3) are defined as the weights of the objective functions (60), (61) and (62), respectively. These values are determined by decision maker as parameter. The case with all the weights equal to 1, means that all the objective functions have equal importance for decision maker. Otherwise, the objective function with higher weights is more important for decision maker. The value 3 for the sum of the weights is selected as there are three objective functions. Taking a value other than 3 for this sum will not affect the obtained solution and objective function values if and only if the percentage of the sum value for the weights remain the same. For example, in the case of changing the sum value of 3 to 1, the ψ values should be divided by 3. Step 3: Solve the following model instead of the model (60)–(79) to obtain Pareto optimal solution for the model: \begin{equation}\min OF=\min \; \psi_{1} \left(\frac{OF_{1} -OF_{1}^{*} }{OF_{1}^{*} } \right)+\psi_{2} \left(\frac{OF_{2} -OF_{2}^{*} }{OF_{2}^{*} } \right)+\psi_{3} \left(\frac{OF_{3} -OF_{3}^{*} }{OF_{3}^{*} } \right) \end{equation} (84) subject to Constraints (63)–(79). Table 1. Demand of different types of paper box for one planning horizon given by the company of the case study Box code Demand Box code Demand Box code Demand Box code Demand Box code Demand EPA0111 19975 EPA0027 17313 EPA0015 23776 EPA0004 47915 EPA0153 54195 EPA0031 21930 EPA0010 51508 EPA0016 60578 EPA0005 63872 EPA0119 71252 EPA0029 35814 EPA0138 46582 EPA0056 49693 EPA0140 33588 EPA0014 74498 EPA0030 55257 EPA0007 20762 EPA0057 35836 EPA0141 34823 EPA0055 18882 EPA0154 17707 EPA0008 16161 EPA0059 6968 EPA0152 73472 EPA0062 80607 Box code Demand Box code Demand Box code Demand Box code Demand Box code Demand EPA0111 19975 EPA0027 17313 EPA0015 23776 EPA0004 47915 EPA0153 54195 EPA0031 21930 EPA0010 51508 EPA0016 60578 EPA0005 63872 EPA0119 71252 EPA0029 35814 EPA0138 46582 EPA0056 49693 EPA0140 33588 EPA0014 74498 EPA0030 55257 EPA0007 20762 EPA0057 35836 EPA0141 34823 EPA0055 18882 EPA0154 17707 EPA0008 16161 EPA0059 6968 EPA0152 73472 EPA0062 80607 Table 1. Demand of different types of paper box for one planning horizon given by the company of the case study Box code Demand Box code Demand Box code Demand Box code Demand Box code Demand EPA0111 19975 EPA0027 17313 EPA0015 23776 EPA0004 47915 EPA0153 54195 EPA0031 21930 EPA0010 51508 EPA0016 60578 EPA0005 63872 EPA0119 71252 EPA0029 35814 EPA0138 46582 EPA0056 49693 EPA0140 33588 EPA0014 74498 EPA0030 55257 EPA0007 20762 EPA0057 35836 EPA0141 34823 EPA0055 18882 EPA0154 17707 EPA0008 16161 EPA0059 6968 EPA0152 73472 EPA0062 80607 Box code Demand Box code Demand Box code Demand Box code Demand Box code Demand EPA0111 19975 EPA0027 17313 EPA0015 23776 EPA0004 47915 EPA0153 54195 EPA0031 21930 EPA0010 51508 EPA0016 60578 EPA0005 63872 EPA0119 71252 EPA0029 35814 EPA0138 46582 EPA0056 49693 EPA0140 33588 EPA0014 74498 EPA0030 55257 EPA0007 20762 EPA0057 35836 EPA0141 34823 EPA0055 18882 EPA0154 17707 EPA0008 16161 EPA0059 6968 EPA0152 73472 EPA0062 80607 Table 2. Some data for one planning horizon given by the company of the case study Raw sheet Dimension (cm) a1j w1j gj1 $$\hat{g}_{j1} $$ $$c_{j1}^{1} $$ $$\hat{c}_{j1}^{1} $$ $$c_{j1}^{2} $$ $$\hat{c}_{j1}^{2} $$ ( j) Length Width 1 125 90 65 804.3 2651 400 1092 200 928 50 2 125 100 75 808.8 2293 400 1178 200 1095 100 3 125 110 80 966.0 2001 400 1254 200 1148 100 4 125 120 90 970.5 1684 300 1454 200 1350 100 5 125 140 105 1132.3 1713 300 1608 200 1474 100 6 125 150 115 1136.8 1015 200 1834 200 1661 100 7 125 160 120 1294.0 1147 200 1850 200 1636 100 8 125 180 135 1455.8 1296 200 2102 300 1924 100 9 150 90 78 939.3 1747 300 1250 200 1092 100 10 150 100 90 958.8 1795 300 1491 200 1233 100 11 150 110 96 1131.0 1630 300 1511 200 1403 100 12 150 120 108 1150.5 953 100 1733 200 1566 100 13 150 140 126 1342.3 795 100 2069 300 1735 100 14 150 150 138 1361.8 1066 200 2205 300 1982 100 15 150 160 144 1534.0 758 100 2342 300 1937 100 16 150 180 162 1725.8 754 100 2627 300 2268 200 17 175 90 91 1074.3 1408 200 1544 200 1383 100 18 175 100 105 1108.8 907 100 1580 200 1453 100 19 175 110 112 1296.0 940 100 1815 200 1623 100 20 175 120 126 1330.5 934 100 2018 300 1884 100 21 175 140 147 1552.3 1076 200 2350 300 2198 200 22 175 150 161 1586.8 878 100 2376 300 2292 200 23 175 160 168 1774.0 911 100 2587 300 2461 200 24 175 180 189 1995.8 653 100 3128 400 2734 200 25 200 90 104 1209.3 851 100 1769 200 1579 100 26 200 100 120 1258.8 1154 200 1846 200 1726 100 27 200 110 128 1461.0 1209 200 1991 200 1797 100 28 200 120 144 1510.5 878 100 2273 300 2081 200 29 200 140 168 1762.3 748 100 2654 300 2503 200 30 200 150 184 1811.8 699 100 2790 300 2538 200 31 200 160 192 2014.0 603 100 3123 400 2614 200 32 200 180 216 2265.8 637 100 3341 400 2909 200 Raw sheet Dimension (cm) a1j w1j gj1 $$\hat{g}_{j1} $$ $$c_{j1}^{1} $$ $$\hat{c}_{j1}^{1} $$ $$c_{j1}^{2} $$ $$\hat{c}_{j1}^{2} $$ ( j) Length Width 1 125 90 65 804.3 2651 400 1092 200 928 50 2 125 100 75 808.8 2293 400 1178 200 1095 100 3 125 110 80 966.0 2001 400 1254 200 1148 100 4 125 120 90 970.5 1684 300 1454 200 1350 100 5 125 140 105 1132.3 1713 300 1608 200 1474 100 6 125 150 115 1136.8 1015 200 1834 200 1661 100 7 125 160 120 1294.0 1147 200 1850 200 1636 100 8 125 180 135 1455.8 1296 200 2102 300 1924 100 9 150 90 78 939.3 1747 300 1250 200 1092 100 10 150 100 90 958.8 1795 300 1491 200 1233 100 11 150 110 96 1131.0 1630 300 1511 200 1403 100 12 150 120 108 1150.5 953 100 1733 200 1566 100 13 150 140 126 1342.3 795 100 2069 300 1735 100 14 150 150 138 1361.8 1066 200 2205 300 1982 100 15 150 160 144 1534.0 758 100 2342 300 1937 100 16 150 180 162 1725.8 754 100 2627 300 2268 200 17 175 90 91 1074.3 1408 200 1544 200 1383 100 18 175 100 105 1108.8 907 100 1580 200 1453 100 19 175 110 112 1296.0 940 100 1815 200 1623 100 20 175 120 126 1330.5 934 100 2018 300 1884 100 21 175 140 147 1552.3 1076 200 2350 300 2198 200 22 175 150 161 1586.8 878 100 2376 300 2292 200 23 175 160 168 1774.0 911 100 2587 300 2461 200 24 175 180 189 1995.8 653 100 3128 400 2734 200 25 200 90 104 1209.3 851 100 1769 200 1579 100 26 200 100 120 1258.8 1154 200 1846 200 1726 100 27 200 110 128 1461.0 1209 200 1991 200 1797 100 28 200 120 144 1510.5 878 100 2273 300 2081 200 29 200 140 168 1762.3 748 100 2654 300 2503 200 30 200 150 184 1811.8 699 100 2790 300 2538 200 31 200 160 192 2014.0 603 100 3123 400 2614 200 32 200 180 216 2265.8 637 100 3341 400 2909 200 Table 2. Some data for one planning horizon given by the company of the case study Raw sheet Dimension (cm) a1j w1j gj1 $$\hat{g}_{j1} $$ $$c_{j1}^{1} $$ $$\hat{c}_{j1}^{1} $$ $$c_{j1}^{2} $$ $$\hat{c}_{j1}^{2} $$ ( j) Length Width 1 125 90 65 804.3 2651 400 1092 200 928 50 2 125 100 75 808.8 2293 400 1178 200 1095 100 3 125 110 80 966.0 2001 400 1254 200 1148 100 4 125 120 90 970.5 1684 300 1454 200 1350 100 5 125 140 105 1132.3 1713 300 1608 200 1474 100 6 125 150 115 1136.8 1015 200 1834 200 1661 100 7 125 160 120 1294.0 1147 200 1850 200 1636 100 8 125 180 135 1455.8 1296 200 2102 300 1924 100 9 150 90 78 939.3 1747 300 1250 200 1092 100 10 150 100 90 958.8 1795 300 1491 200 1233 100 11 150 110 96 1131.0 1630 300 1511 200 1403 100 12 150 120 108 1150.5 953 100 1733 200 1566 100 13 150 140 126 1342.3 795 100 2069 300 1735 100 14 150 150 138 1361.8 1066 200 2205 300 1982 100 15 150 160 144 1534.0 758 100 2342 300 1937 100 16 150 180 162 1725.8 754 100 2627 300 2268 200 17 175 90 91 1074.3 1408 200 1544 200 1383 100 18 175 100 105 1108.8 907 100 1580 200 1453 100 19 175 110 112 1296.0 940 100 1815 200 1623 100 20 175 120 126 1330.5 934 100 2018 300 1884 100 21 175 140 147 1552.3 1076 200 2350 300 2198 200 22 175 150 161 1586.8 878 100 2376 300 2292 200 23 175 160 168 1774.0 911 100 2587 300 2461 200 24 175 180 189 1995.8 653 100 3128 400 2734 200 25 200 90 104 1209.3 851 100 1769 200 1579 100 26 200 100 120 1258.8 1154 200 1846 200 1726 100 27 200 110 128 1461.0 1209 200 1991 200 1797 100 28 200 120 144 1510.5 878 100 2273 300 2081 200 29 200 140 168 1762.3 748 100 2654 300 2503 200 30 200 150 184 1811.8 699 100 2790 300 2538 200 31 200 160 192 2014.0 603 100 3123 400 2614 200 32 200 180 216 2265.8 637 100 3341 400 2909 200 Raw sheet Dimension (cm) a1j w1j gj1 $$\hat{g}_{j1} $$ $$c_{j1}^{1} $$ $$\hat{c}_{j1}^{1} $$ $$c_{j1}^{2} $$ $$\hat{c}_{j1}^{2} $$ ( j) Length Width 1 125 90 65 804.3 2651 400 1092 200 928 50 2 125 100 75 808.8 2293 400 1178 200 1095 100 3 125 110 80 966.0 2001 400 1254 200 1148 100 4 125 120 90 970.5 1684 300 1454 200 1350 100 5 125 140 105 1132.3 1713 300 1608 200 1474 100 6 125 150 115 1136.8 1015 200 1834 200 1661 100 7 125 160 120 1294.0 1147 200 1850 200 1636 100 8 125 180 135 1455.8 1296 200 2102 300 1924 100 9 150 90 78 939.3 1747 300 1250 200 1092 100 10 150 100 90 958.8 1795 300 1491 200 1233 100 11 150 110 96 1131.0 1630 300 1511 200 1403 100 12 150 120 108 1150.5 953 100 1733 200 1566 100 13 150 140 126 1342.3 795 100 2069 300 1735 100 14 150 150 138 1361.8 1066 200 2205 300 1982 100 15 150 160 144 1534.0 758 100 2342 300 1937 100 16 150 180 162 1725.8 754 100 2627 300 2268 200 17 175 90 91 1074.3 1408 200 1544 200 1383 100 18 175 100 105 1108.8 907 100 1580 200 1453 100 19 175 110 112 1296.0 940 100 1815 200 1623 100 20 175 120 126 1330.5 934 100 2018 300 1884 100 21 175 140 147 1552.3 1076 200 2350 300 2198 200 22 175 150 161 1586.8 878 100 2376 300 2292 200 23 175 160 168 1774.0 911 100 2587 300 2461 200 24 175 180 189 1995.8 653 100 3128 400 2734 200 25 200 90 104 1209.3 851 100 1769 200 1579 100 26 200 100 120 1258.8 1154 200 1846 200 1726 100 27 200 110 128 1461.0 1209 200 1991 200 1797 100 28 200 120 144 1510.5 878 100 2273 300 2081 200 29 200 140 168 1762.3 748 100 2654 300 2503 200 30 200 150 184 1811.8 699 100 2790 300 2538 200 31 200 160 192 2014.0 603 100 3123 400 2614 200 32 200 180 216 2265.8 637 100 3341 400 2909 200 Table 3. The results of the WGCM for different levels of A1 and A2 and setting ψ1 = ψ2 = ψ3 = 1 Set by Pareto optimal decision Obtained by Step solution obtained by Exp. maker 1 of the WGCM Step 3 of the WGCM $$OF_{1}^{*} $$ $$OF_{2}^{*} $$ OF1 OF2 A1 A2 (millions) (millions) $$OF_{3}^{*} $$ (millions) (millions) OF3 1 0 0 554.68 543.5332 332010 554.6896 547.2582 332060 2 0 50 554.68 577.1843 332010 554.7023 579.4770 332023 3 0 100 554.68 577.1839 332010 554.7016 578.4340 332072 4 0 150 554.68 577.1848 332010 554.7793 577.6672 332018 5 0 192 554.68 577.1841 332010 554.6920 579.7137 332017 6 50 0 554.68 543.5343 332010 554.8681 545.4129 332040 7 50 50 554.68 577.1863 332010 554.7563 579.3057 332054 8 50 100 554.68 577.1848 332010 554.6898 578.9136 332024 9 50 150 554.68 577.1863 332010 554.7028 578.1839 332010 10 50 192 554.68 578.5893 332010 554.8427 580.9199 332075 11 100 0 554.68 543.5329 332010 554.6910 545.5737 332028 12 100 50 554.68 577.1896 332010 554.6896 578.0054 332010 13 100 100 554.68 577.1866 332010 554.7388 578.3775 332010 14 100 150 554.68 577.1920 332010 554.7615 577.9540 332010 15 100 192 554.68 577.1847 332010 554.7300 578.7020 332026 16 150 0 554.68 543.5349 332010 554.7686 544.6990 332071 17 150 50 554.68 577.1868 332010 554.7843 578.4177 332075 18 150 100 554.68 578.5269 332010 555.0523 580.2028 332050 19 150 150 554.68 577.1861 332010 554.8248 581.4120 332010 20 150 192 554.68 577.1865 332010 554.7349 578.5261 332056 21 192 0 554.68 543.5334 332010 554.7204 544.0228 332020 22 192 50 554.68 577.1863 332010 554.6900 578.0897 332010 23 192 100 554.68 577.1867 332010 554.8805 581.7697 332096 24 192 150 554.68 577.1862 332010 554.7489 579.1361 332012 25 192 192 554.68 577.1861 332010 554.6907 577.4787 332010 Set by Pareto optimal decision Obtained by Step solution obtained by Exp. maker 1 of the WGCM Step 3 of the WGCM $$OF_{1}^{*} $$ $$OF_{2}^{*} $$ OF1 OF2 A1 A2 (millions) (millions) $$OF_{3}^{*} $$ (millions) (millions) OF3 1 0 0 554.68 543.5332 332010 554.6896 547.2582 332060 2 0 50 554.68 577.1843 332010 554.7023 579.4770 332023 3 0 100 554.68 577.1839 332010 554.7016 578.4340 332072 4 0 150 554.68 577.1848 332010 554.7793 577.6672 332018 5 0 192 554.68 577.1841 332010 554.6920 579.7137 332017 6 50 0 554.68 543.5343 332010 554.8681 545.4129 332040 7 50 50 554.68 577.1863 332010 554.7563 579.3057 332054 8 50 100 554.68 577.1848 332010 554.6898 578.9136 332024 9 50 150 554.68 577.1863 332010 554.7028 578.1839 332010 10 50 192 554.68 578.5893 332010 554.8427 580.9199 332075 11 100 0 554.68 543.5329 332010 554.6910 545.5737 332028 12 100 50 554.68 577.1896 332010 554.6896 578.0054 332010 13 100 100 554.68 577.1866 332010 554.7388 578.3775 332010 14 100 150 554.68 577.1920 332010 554.7615 577.9540 332010 15 100 192 554.68 577.1847 332010 554.7300 578.7020 332026 16 150 0 554.68 543.5349 332010 554.7686 544.6990 332071 17 150 50 554.68 577.1868 332010 554.7843 578.4177 332075 18 150 100 554.68 578.5269 332010 555.0523 580.2028 332050 19 150 150 554.68 577.1861 332010 554.8248 581.4120 332010 20 150 192 554.68 577.1865 332010 554.7349 578.5261 332056 21 192 0 554.68 543.5334 332010 554.7204 544.0228 332020 22 192 50 554.68 577.1863 332010 554.6900 578.0897 332010 23 192 100 554.68 577.1867 332010 554.8805 581.7697 332096 24 192 150 554.68 577.1862 332010 554.7489 579.1361 332012 25 192 192 554.68 577.1861 332010 554.6907 577.4787 332010 Table 3. The results of the WGCM for different levels of A1 and A2 and setting ψ1 = ψ2 = ψ3 = 1 Set by Pareto optimal decision Obtained by Step solution obtained by Exp. maker 1 of the WGCM Step 3 of the WGCM $$OF_{1}^{*} $$ $$OF_{2}^{*} $$ OF1 OF2 A1 A2 (millions) (millions) $$OF_{3}^{*} $$ (millions) (millions) OF3 1 0 0 554.68 543.5332 332010 554.6896 547.2582 332060 2 0 50 554.68 577.1843 332010 554.7023 579.4770 332023 3 0 100 554.68 577.1839 332010 554.7016 578.4340 332072 4 0 150 554.68 577.1848 332010 554.7793 577.6672 332018 5 0 192 554.68 577.1841 332010 554.6920 579.7137 332017 6 50 0 554.68 543.5343 332010 554.8681 545.4129 332040 7 50 50 554.68 577.1863 332010 554.7563 579.3057 332054 8 50 100 554.68 577.1848 332010 554.6898 578.9136 332024 9 50 150 554.68 577.1863 332010 554.7028 578.1839 332010 10 50 192 554.68 578.5893 332010 554.8427 580.9199 332075 11 100 0 554.68 543.5329 332010 554.6910 545.5737 332028 12 100 50 554.68 577.1896 332010 554.6896 578.0054 332010 13 100 100 554.68 577.1866 332010 554.7388 578.3775 332010 14 100 150 554.68 577.1920 332010 554.7615 577.9540 332010 15 100 192 554.68 577.1847 332010 554.7300 578.7020 332026 16 150 0 554.68 543.5349 332010 554.7686 544.6990 332071 17 150 50 554.68 577.1868 332010 554.7843 578.4177 332075 18 150 100 554.68 578.5269 332010 555.0523 580.2028 332050 19 150 150 554.68 577.1861 332010 554.8248 581.4120 332010 20 150 192 554.68 577.1865 332010 554.7349 578.5261 332056 21 192 0 554.68 543.5334 332010 554.7204 544.0228 332020 22 192 50 554.68 577.1863 332010 554.6900 578.0897 332010 23 192 100 554.68 577.1867 332010 554.8805 581.7697 332096 24 192 150 554.68 577.1862 332010 554.7489 579.1361 332012 25 192 192 554.68 577.1861 332010 554.6907 577.4787 332010 Set by Pareto optimal decision Obtained by Step solution obtained by Exp. maker 1 of the WGCM Step 3 of the WGCM $$OF_{1}^{*} $$ $$OF_{2}^{*} $$ OF1 OF2 A1 A2 (millions) (millions) $$OF_{3}^{*} $$ (millions) (millions) OF3 1 0 0 554.68 543.5332 332010 554.6896 547.2582 332060 2 0 50 554.68 577.1843 332010 554.7023 579.4770 332023 3 0 100 554.68 577.1839 332010 554.7016 578.4340 332072 4 0 150 554.68 577.1848 332010 554.7793 577.6672 332018 5 0 192 554.68 577.1841 332010 554.6920 579.7137 332017 6 50 0 554.68 543.5343 332010 554.8681 545.4129 332040 7 50 50 554.68 577.1863 332010 554.7563 579.3057 332054 8 50 100 554.68 577.1848 332010 554.6898 578.9136 332024 9 50 150 554.68 577.1863 332010 554.7028 578.1839 332010 10 50 192 554.68 578.5893 332010 554.8427 580.9199 332075 11 100 0 554.68 543.5329 332010 554.6910 545.5737 332028 12 100 50 554.68 577.1896 332010 554.6896 578.0054 332010 13 100 100 554.68 577.1866 332010 554.7388 578.3775 332010 14 100 150 554.68 577.1920 332010 554.7615 577.9540 332010 15 100 192 554.68 577.1847 332010 554.7300 578.7020 332026 16 150 0 554.68 543.5349 332010 554.7686 544.6990 332071 17 150 50 554.68 577.1868 332010 554.7843 578.4177 332075 18 150 100 554.68 578.5269 332010 555.0523 580.2028 332050 19 150 150 554.68 577.1861 332010 554.8248 581.4120 332010 20 150 192 554.68 577.1865 332010 554.7349 578.5261 332056 21 192 0 554.68 543.5334 332010 554.7204 544.0228 332020 22 192 50 554.68 577.1863 332010 554.6900 578.0897 332010 23 192 100 554.68 577.1867 332010 554.8805 581.7697 332096 24 192 150 554.68 577.1862 332010 554.7489 579.1361 332012 25 192 192 554.68 577.1861 332010 554.6907 577.4787 332010 5. Computational experiments on the case study The robust formulation (60)–(79) and the introduced solution approach of Section 4 is experimented in this section. For this aim a set of real data from the paper box manufacturing company explained in Section 2 is used. The data and the obtained results are explained in the following sub-sections. Notably, the robust formulation (60)–(79) and the introduced solution approach are coded in GAMS solver and run on a computer with an Intel Core 2 Duo 2.53 GHz processor and 4.00 GB RAM. 5.1. The real data set The real data of one planning horizon of the paper box manufacturing company, which was detailed in Section 2, is obtained to be used for computational study. The data obtained from the company include the following: there are 25 different types of boxes; demand of each type of box; there are 6 potential suppliers; there are 32 different types of raw sheet, which can be supplied by all suppliers; the wastage amount remained from each raw sheet when it is used for each type of paper box; the discount break point for each raw sheet type given by each supplier; and normal and discounted price of each raw sheet given by each supplier. Some information of each type of box is represented in Table 1, while Table 2 contains different types of raw sheets. As the full data of the case study need large matrixes to be represented, only the data of paper box type 1 and supplier 1 (as an instance) for all raw sheets are presented in Table 2. 5.2. Results and sensitivity analysis The proposed WGCM of Section 4.1 is used to solve the model (60)–(79) for the data of the cardboard box production company. As there are six suppliers supplying all 32 types of raw sheets, the total allowed variations of the sheet prices from their mean values (A1 and A2) can take minimum value of zero (meaning there is no uncertainty (variation) in the prices), and maximum value of 1 (meaning all prices are in their highest possible level). Therefore, the values of set $$\left \{0,50,100,150,192\right \}$$ is used for A1 and A2. On the other hand, the budget of uncertainty associated with the demand uncertainty (ξi) is fixed to 1 in all experiments. All steps of the proposed WGCM of Section 4.1 is applied for any combination of the A1 and A2 values where all the objective function weights are equally determined (ψ1 = ψ2 = ψ3 = 1).The obtained results are represented in Table 3. Fig. 1. View largeDownload slide Graph of variations of $$OF_{2}^{*} $$ over different levels of A1 and A2. Fig. 1. View largeDownload slide Graph of variations of $$OF_{2}^{*} $$ over different levels of A1 and A2. Fig. 2. View largeDownload slide Graph of variations of OF1 (obtained by the second step of the WGCM, where ψ1 = ψ2 = ψ3 = 1) over different levels of A1 and A2. Fig. 2. View largeDownload slide Graph of variations of OF1 (obtained by the second step of the WGCM, where ψ1 = ψ2 = ψ3 = 1) over different levels of A1 and A2. Fig. 3. View largeDownload slide Graph of variations of OF2 (obtained by the second step of the WGCM, where ψ1 = ψ2 = ψ3 = 1) over different levels of A1 and A2. Fig. 3. View largeDownload slide Graph of variations of OF2 (obtained by the second step of the WGCM, where ψ1 = ψ2 = ψ3 = 1) over different levels of A1 and A2. Fig. 4. View largeDownload slide Graph of variations of OF3 (obtained by the second step of the WGCM, where ψ1 = ψ2 = ψ3 = 1) over different levels of A1 and A2. Fig. 4. View largeDownload slide Graph of variations of OF3 (obtained by the second step of the WGCM, where ψ1 = ψ2 = ψ3 = 1) over different levels of A1 and A2. Table 4. The results of the WGCM for two combinations of the levels of A1 and A2 considering different weights Set by Set by Pareto optimal solution decision decision obtained by Step 3 Exp. maker Run maker of the WGCM OF1 OF2 A1 A1 ψ1 ψ2 ψ3 (millions) (millions) OF3 1 1 1 1 554.6896 547.2582 332060 2 1.5 1 0.5 554.6916 545.1571 332043 3 1.5 0.5 1 554.6895 547.3694 332032 2 150 50 4 1 1.5 0.5 554.8713 544.7098 332115 5 1 0.5 1.5 554.7470 549.3233 332067 6 0.5 1 1.5 555.1553 544.5439 332028 7 0.5 1.5 1 554.8128 545.6115 332076 8 1 1 1 554.7843 578.4177 332075 9 1.5 1 0.5 554.7028 581.7285 332030 10 1.5 0.5 1 554.6899 579.7565 332010 2 150 50 11 1 1.5 0.5 554.7966 579.0578 332031 12 1 0.5 1.5 554.8111 581.8725 332010 13 0.5 1 1.5 554.9912 583.3101 332020 104 0.5 1.5 1 555.6172 577.9442 332106 Set by Set by Pareto optimal solution decision decision obtained by Step 3 Exp. maker Run maker of the WGCM OF1 OF2 A1 A1 ψ1 ψ2 ψ3 (millions) (millions) OF3 1 1 1 1 554.6896 547.2582 332060 2 1.5 1 0.5 554.6916 545.1571 332043 3 1.5 0.5 1 554.6895 547.3694 332032 2 150 50 4 1 1.5 0.5 554.8713 544.7098 332115 5 1 0.5 1.5 554.7470 549.3233 332067 6 0.5 1 1.5 555.1553 544.5439 332028 7 0.5 1.5 1 554.8128 545.6115 332076 8 1 1 1 554.7843 578.4177 332075 9 1.5 1 0.5 554.7028 581.7285 332030 10 1.5 0.5 1 554.6899 579.7565 332010 2 150 50 11 1 1.5 0.5 554.7966 579.0578 332031 12 1 0.5 1.5 554.8111 581.8725 332010 13 0.5 1 1.5 554.9912 583.3101 332020 104 0.5 1.5 1 555.6172 577.9442 332106 View Large Table 4. The results of the WGCM for two combinations of the levels of A1 and A2 considering different weights Set by Set by Pareto optimal solution decision decision obtained by Step 3 Exp. maker Run maker of the WGCM OF1 OF2 A1 A1 ψ1 ψ2 ψ3 (millions) (millions) OF3 1 1 1 1 554.6896 547.2582 332060 2 1.5 1 0.5 554.6916 545.1571 332043 3 1.5 0.5 1 554.6895 547.3694 332032 2 150 50 4 1 1.5 0.5 554.8713 544.7098 332115 5 1 0.5 1.5 554.7470 549.3233 332067 6 0.5 1 1.5 555.1553 544.5439 332028 7 0.5 1.5 1 554.8128 545.6115 332076 8 1 1 1 554.7843 578.4177 332075 9 1.5 1 0.5 554.7028 581.7285 332030 10 1.5 0.5 1 554.6899 579.7565 332010 2 150 50 11 1 1.5 0.5 554.7966 579.0578 332031 12 1 0.5 1.5 554.8111 581.8725 332010 13 0.5 1 1.5 554.9912 583.3101 332020 104 0.5 1.5 1 555.6172 577.9442 332106 Set by Set by Pareto optimal solution decision decision obtained by Step 3 Exp. maker Run maker of the WGCM OF1 OF2 A1 A1 ψ1 ψ2 ψ3 (millions) (millions) OF3 1 1 1 1 554.6896 547.2582 332060 2 1.5 1 0.5 554.6916 545.1571 332043 3 1.5 0.5 1 554.6895 547.3694 332032 2 150 50 4 1 1.5 0.5 554.8713 544.7098 332115 5 1 0.5 1.5 554.7470 549.3233 332067 6 0.5 1 1.5 555.1553 544.5439 332028 7 0.5 1.5 1 554.8128 545.6115 332076 8 1 1 1 554.7843 578.4177 332075 9 1.5 1 0.5 554.7028 581.7285 332030 10 1.5 0.5 1 554.6899 579.7565 332010 2 150 50 11 1 1.5 0.5 554.7966 579.0578 332031 12 1 0.5 1.5 554.8111 581.8725 332010 13 0.5 1 1.5 554.9912 583.3101 332020 104 0.5 1.5 1 555.6172 577.9442 332106 View Large In the obtained results from Step 1 of the WGCM in Table 3, the values of $$OF_{1}^{*} $$ and $$OF_{3}^{*} $$ are not sensitive to the variations of values used for A1 and A2. This happens because in the models (80) and (82), the objective functions do not contain any value of A1 and A2. On the other hand, the values obtained for $$OF_{2}^{*} $$ varies by an unstable trend when the values of A1 and A2 are changed in the model (60)–(79). These variations are shown in the graph of Fig.1. Furthermore, the sensitivity of the objective function values of the obtained Pareto optimal solutions over different levels of A1 and A2, follow unstable trends as shown by Figs 2–4. Fig. 5. View largeDownload slide Behavior of objective function 1 for the experiments of Table 4 over various weight values. Fig. 5. View largeDownload slide Behavior of objective function 1 for the experiments of Table 4 over various weight values. Fig. 6. View largeDownload slide Behavior of objective function 2 for the experiments of Table 4 over various weight values. Fig. 6. View largeDownload slide Behavior of objective function 2 for the experiments of Table 4 over various weight values. To further analyze the behavior of the proposed robust model its sensitivity to the different levels of weights of the objective functions is studied. For this aim, for instance, two experiments from Table 3 is selected (the experiment with A1 = A2 = 0 and the experiment with A1 = 150 and A2 = 50). Seven combinations of weights of the objective functions in these experiments are selected to be of set $$\left \{0.5,1,1.5\right \}$$ (for example ψ1 = ψ2 = ψ3 = 1). In each combination of the weights the sum of weight values must be equal to 3 as assumed in Step 2 of the proposed approach. Therefore, each of the mentioned experiments results in seven runs. Applying the WGCM to the new experiments, the results is shown by Table 4. The results of Table 4 which are also plotted in the graphs of Figs 6–8, show that although the values of each objective function has an unstable trend over various values of its weight, the best value of each objective function is obtained when its weight is in the highest level (ψ = 1.5). In each of the Figs 5–7, we eliminated the runs of Table 4 with equal weights (the first run of each experiment). Excluding this run, for each objective function there is a pair of runs for each weight value. Therefore, the objective function values associated to each run of the weight values are plotted. Fig. 7. View largeDownload slide Behavior of objective function 3 for the experiments of Table 4 over various weight values. Fig. 7. View largeDownload slide Behavior of objective function 3 for the experiments of Table 4 over various weight values. Table 5. A sample detailed result for the case problem when A1 = 50, A2 = 100 and ψ1 = ψ2 = ψ3 = 1 Y Value Y Value Y Value i j k i j k i j k 1 22 1 149 8 11 2 3349 16 5 1 7 2 9 2 1 9 20 5 4294 17 32 5 9234 2 25 3 1120 10 4 3 3 18 8 5 9897 2 28 2 1 10 7 1 5038 19 32 1 737 3 10 4 1 11 7 1 7194 19 32 5 4366 3 29 3 1161 12 20 5 11763 20 6 2 1643 4 19 1 1 13 4 1 3 20 6 6 40093 4 21 2 1405 13 7 1 14171 21 30 2 15549 5 23 2 1809 14 20 5 6970 22 7 1 40626 6 3 1 1 14 23 2 2 23 8 5 42249 6 8 5 4262 15 7 1 2242 24 8 5 11441 7 11 2 3718 16 1 5 27447 25 1 5 90607 7 13 3 1 Y Value Y Value Y Value i j k i j k i j k 1 22 1 149 8 11 2 3349 16 5 1 7 2 9 2 1 9 20 5 4294 17 32 5 9234 2 25 3 1120 10 4 3 3 18 8 5 9897 2 28 2 1 10 7 1 5038 19 32 1 737 3 10 4 1 11 7 1 7194 19 32 5 4366 3 29 3 1161 12 20 5 11763 20 6 2 1643 4 19 1 1 13 4 1 3 20 6 6 40093 4 21 2 1405 13 7 1 14171 21 30 2 15549 5 23 2 1809 14 20 5 6970 22 7 1 40626 6 3 1 1 14 23 2 2 23 8 5 42249 6 8 5 4262 15 7 1 2242 24 8 5 11441 7 11 2 3718 16 1 5 27447 25 1 5 90607 7 13 3 1 View Large Table 5. A sample detailed result for the case problem when A1 = 50, A2 = 100 and ψ1 = ψ2 = ψ3 = 1 Y Value Y Value Y Value i j k i j k i j k 1 22 1 149 8 11 2 3349 16 5 1 7 2 9 2 1 9 20 5 4294 17 32 5 9234 2 25 3 1120 10 4 3 3 18 8 5 9897 2 28 2 1 10 7 1 5038 19 32 1 737 3 10 4 1 11 7 1 7194 19 32 5 4366 3 29 3 1161 12 20 5 11763 20 6 2 1643 4 19 1 1 13 4 1 3 20 6 6 40093 4 21 2 1405 13 7 1 14171 21 30 2 15549 5 23 2 1809 14 20 5 6970 22 7 1 40626 6 3 1 1 14 23 2 2 23 8 5 42249 6 8 5 4262 15 7 1 2242 24 8 5 11441 7 11 2 3718 16 1 5 27447 25 1 5 90607 7 13 3 1 Y Value Y Value Y Value i j k i j k i j k 1 22 1 149 8 11 2 3349 16 5 1 7 2 9 2 1 9 20 5 4294 17 32 5 9234 2 25 3 1120 10 4 3 3 18 8 5 9897 2 28 2 1 10 7 1 5038 19 32 1 737 3 10 4 1 11 7 1 7194 19 32 5 4366 3 29 3 1161 12 20 5 11763 20 6 2 1643 4 19 1 1 13 4 1 3 20 6 6 40093 4 21 2 1405 13 7 1 14171 21 30 2 15549 5 23 2 1809 14 20 5 6970 22 7 1 40626 6 3 1 1 14 23 2 2 23 8 5 42249 6 8 5 4262 15 7 1 2242 24 8 5 11441 7 11 2 3718 16 1 5 27447 25 1 5 90607 7 13 3 1 View Large 5.3. A sample detailed result In this subsection the detailed results of one of the Pareto optimal solutions of the model is reported. As there are 25 treatment with different combination of A1 and A2 values, the first treatment which contains values A1 = 50 and A2 = 100 is chosen for reporting the detailed results. Thus, the values of the parameters e.g. $$T_{jk}^{1} $$, $$T_{jk}^{2} $$, $$Z_{jk}^{1} $$, $$Z_{jk}^{2} $$ and Yijk obtained in the Pareto optimal solution of Table 3 is reported in Table 5–6. Note that Table 5–6 show only the variables that took positive values in the Pareto optimal solution of Table 3. For instance, paper box number 7 is produced from the sheet types 11 and 13 by 3718 and 1 sheets from each one respectively (Table 5). Table 6. A sample detailed result for the case problem when A1 = 50, A2 = 100 and ψ1 = ψ2 = ψ3 = 1 T1 Value T2 Value Z1 Value Z2 Value j k j k j k j k 3 1 1 1 5 1 3 1 1 1 5 118000 4 1 1 6 2 1 4 1 3 6 2 1643 4 3 1 6 6 1 4 3 3 6 6 40093 5 1 1 7 1 1 5 1 7 7 1 69271 9 2 1 8 5 1 9 2 1 8 5 67849 10 4 1 11 2 1 10 4 1 11 2 7067 13 3 1 20 5 1 13 3 1 20 5 23027 19 1 1 21 2 1 19 1 1 21 2 1405 22 1 1 23 2 1 22 1 149 23 2 1811 28 2 1 25 3 1 28 2 1 25 3 1120 3 29 3 29 3 1161 30 2 1 30 2 15549 32 1 1 32 1 737 32 5 1 32 5 13600 T1 Value T2 Value Z1 Value Z2 Value j k j k j k j k 3 1 1 1 5 1 3 1 1 1 5 118000 4 1 1 6 2 1 4 1 3 6 2 1643 4 3 1 6 6 1 4 3 3 6 6 40093 5 1 1 7 1 1 5 1 7 7 1 69271 9 2 1 8 5 1 9 2 1 8 5 67849 10 4 1 11 2 1 10 4 1 11 2 7067 13 3 1 20 5 1 13 3 1 20 5 23027 19 1 1 21 2 1 19 1 1 21 2 1405 22 1 1 23 2 1 22 1 149 23 2 1811 28 2 1 25 3 1 28 2 1 25 3 1120 3 29 3 29 3 1161 30 2 1 30 2 15549 32 1 1 32 1 737 32 5 1 32 5 13600 Table 6. A sample detailed result for the case problem when A1 = 50, A2 = 100 and ψ1 = ψ2 = ψ3 = 1 T1 Value T2 Value Z1 Value Z2 Value j k j k j k j k 3 1 1 1 5 1 3 1 1 1 5 118000 4 1 1 6 2 1 4 1 3 6 2 1643 4 3 1 6 6 1 4 3 3 6 6 40093 5 1 1 7 1 1 5 1 7 7 1 69271 9 2 1 8 5 1 9 2 1 8 5 67849 10 4 1 11 2 1 10 4 1 11 2 7067 13 3 1 20 5 1 13 3 1 20 5 23027 19 1 1 21 2 1 19 1 1 21 2 1405 22 1 1 23 2 1 22 1 149 23 2 1811 28 2 1 25 3 1 28 2 1 25 3 1120 3 29 3 29 3 1161 30 2 1 30 2 15549 32 1 1 32 1 737 32 5 1 32 5 13600 T1 Value T2 Value Z1 Value Z2 Value j k j k j k j k 3 1 1 1 5 1 3 1 1 1 5 118000 4 1 1 6 2 1 4 1 3 6 2 1643 4 3 1 6 6 1 4 3 3 6 6 40093 5 1 1 7 1 1 5 1 7 7 1 69271 9 2 1 8 5 1 9 2 1 8 5 67849 10 4 1 11 2 1 10 4 1 11 2 7067 13 3 1 20 5 1 13 3 1 20 5 23027 19 1 1 21 2 1 19 1 1 21 2 1405 22 1 1 23 2 1 22 1 149 23 2 1811 28 2 1 25 3 1 28 2 1 25 3 1120 3 29 3 29 3 1161 30 2 1 30 2 15549 32 1 1 32 1 737 32 5 1 32 5 13600 5.4. Discussion The optimization technique (including modelling and solution approach sections) used in this study was applied to a cardboard box manufacturing company as a case study. To overcome the big concerns of the managers of the company, as a mathematical formulation, the decisions on the amount of each raw sheet type and supplier selection had to be made in order to improve some objectives e.g. wastage amount of raw sheets, raw sheets cost and product surplus simultaneously for one planning horizon. Considering the uncertainties in the mathematical formulation, the numerical experiments done in the previous sub-sections proved that the obtained values for the objective functions are highly depended on the level of uncertainty of the parameters and also the weights used for objective functions in the solution procedure (Figs 1–7). Practically, the obtained results for a given level of uncertainty such as the results of Tables 5–6, can help the production planners (and managers) of the company to make a plan for selecting the raw sheets and their suppliers in order to improve the objectives considered in the problem of this study simultaneously. Notably, an obtained solution when considering a higher level of uncertainty in the parameters, can give a more robust decision to the managers in the unstable market of the case study. 6. Conclusion A new and real problem in cardboard box manufacturing was solved in this paper. The problem minimizes waste, surplus amount and the procurement costs through making decisions on selection of the suppliers and materials for a cardboard box manufacturer. The problem was formulated as a multi-objective mixed integer linear programme with uncertain demand and raw material price values. A typical robust optimization approach was applied to cope with the uncertainties. 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IMA Journal of Management MathematicsOxford University Press

Published: Feb 1, 2018

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