No-wait resource allocation flowshop scheduling with learning effect under limited cost availability

No-wait resource allocation flowshop scheduling with learning effect under limited cost availability Abstract This article considers the no-wait flowshop scheduling with learning effect and resource allocation on two-machine. Our goal is to find the optimal resource allocations and job sequence that minimize the scheduling criterion (the total weighted resource consumption) subject to the constraint that the total weighted resource consumption (the scheduling criterion) is less than or equal to a given constant, where the schedule criteria include weighted makespan, total completion (waiting) time and total absolute differences in completion (waiting) times. We show that these two problems are polynomially solvable respectively. 1. INTRODUCTION In traditional scheduling models and problems, the job processing times are generally treated as constant values. However, we often encounter production systems, job processing times may be dependent on additional resource (such as energy, money and manpower) and their position in a sequence (i.e. learning effect). Extensive surveys of scheduling problems with learning effects and resource allocation (controllable processing times) can be found in Biskup [1] and Shabtay and Steiner [2]. Recently Kuo [3] and Wang and Wang [4] considered single machine scheduling problems with learning effects. Wang et al. [5], Shiau et al. [6] and Wang et al. [7] considered flowshop scheduling problems with learning effects. Wu et al. [8] considered a multi-machine order scheduling with learning effects. Mor and Mosheiov [9] considered batch scheduling problems with controllable processing times. For identical jobs and linear compression cost function, they gave some results. Yang et al. [10] and Liu et al. [11] considered single machine due-window scheduling problems with controllable processing times. Hsieh et al. [12] considered scheduling problem with discrete controllable processing times on unrelated parallel machine. Zhao et al. [13] and Wang et al. [14] considered single machine scheduling with a controllable rate modifying activity (i.e. a resource-dependent maintenance activity). Wang et al. [15], Zhu et al. [16] and Wang et al. [17] considered single machine scheduling problems with learning effects and resource allocation. Lu et al. [18], Wang and Wang [19] and Wang et al. [20] considered single machine due date assignment scheduling problems with learning effects and resource allocation. Wang and Wang [21] and Li et al. [22] considered single machine due-window assignment scheduling problems with learning effects and resource allocation. Liu and Feng [23] considered flowshop scheduling with learning effect and resource allocation. In this paper, we consider the no-wait flowshop scheduling problems with learning effect and resource allocation. ‘The ‘no-wait’ constraint means that each job, once started, has to be processed without interruption until all of its operations are completed. In practice, this requirement may arise out of certain job characteristics (such as in the ‘hot ingot’ problem, where metal has to be processed at a continuously high temperature) or out of the unavailability of intermediate storage in between machines’ (Wang and Xia [24]). However, to the best of our knowledge, there exist only a result concerning no-wait flowshop scheduling problems with learning effects and resource allocation simultaneously. Liu and Feng [23] considered two-machine no-wait flowshop scheduling with simultaneous considerations of learning effect and resource allocation. For the weighted makespan, total completion (waiting) time, total absolute differences in completion (waiting) times and total resource cost, they proved that the problem can be solved in polynomial time. In the real production process, the resources are limited, one might wang to control the total resource consumption (Janiak and Li [25]). Hence, in this study, we continue the work of Liu and Feng [23], i.e. we consider resource allocation scheduling problem with learning effects in the two-machine no-wait permutation flowshop environment. The first type objective is to minimize the schedule criterion (i.e. the weighted makespan, total completion (waiting) time and total absolute differences in completion (waiting) times) subject to the constraint that the total resource cost is less than or equal to an given constant. The second type objective is to minimize the total resource cost subject to the constraint that the schedule criterion is less than or equal to an given constant. We show that these two problems are polynomially solvable, respectively. The remainder of this article is organized as follows: in Section 2, we introduce and formulate the problem. In Sections 3 and 4 , we provide some properties to optimally solve these two problems, respectively. In Section 5, we consider a special case. We conclude the paper in Section 6. 2. PROBLEM FORMULATION Consider a two-machine no-wait permutation flowshop setting. There are n independent jobs J={J1,J2,…,Jn} and two machines M1 and M2 available at time zero, and job preemption and machine interruption are not allowed. Job Jj must be processed first on machine M1 and then on machine M2 (i.e. operations Oji, i=1,2, j=1,2,…,n), and the jobs are not allowed to wait between the two machines (i.e. p[j]2=p[j+1]1, there is a no-wait restriction, where [j] denotes the job scheduled in the jth position in a sequence). As in Liu and Feng [23], we assume that the actual processing time of job Jj on machine Mi is pji=p¯jiraujik, (1) where p¯ji is the basic (normal) processing time of operation Oji (i.e. the processing time without any learning effects and resource allocation), uji is the amount of resource allocated to operation Oji and k is a positive constant, r is the position of job Jj which is scheduled in a sequence. Let Cmax=max{Cj∣j=1,2,…,n} be the maximum completion time (makespan), TC=∑j=1nCj be the total completion times, TW=∑j=1nWj be the total waiting times, TADC=∑k=1n∑j=kn∣Ck−Cj∣ be the total absolute differences in completion times and ∑k=1n∑j=kn∣Wk−Wj∣ be the total absolute differences in waiting times, where Cj=Cj(π) and Wj=Cj−pj1−pj2 are the completion time and the waiting time for job Jj on machine M2, respectively. 3. THE PROBLEM F2con,NW,LE,∑∑Gjiuji≤Uγ In this section, we aim to determine the set of optimal resource allocations u=(u11,u21,…,un1; u12,u22,…,un2) and the job sequence π such that the following cost function: Z1(π,u)=δ1Cmax+δ2TC+δ3TADC, (2) and Z2(π,u)=δ1Cmax+δ2TW+δ3TADW, (3) is minimized, respectively, subject to the constraint that the total resource cost ∑i=12∑j=1nGjiuji≤U, where weights δ1≥0,δ2≥0,δ3≥0 are given constants, and Gji is the cost associated with the resource allocation per time unit. Similar to Graham et al. [26] and Liu and Feng [23], the problem can be denoted as F2con,NW,LE,∑∑Gjiuji≤Uγ, where γ∈{δ1Cmax+δ2TC+δ3TADC,δ1Cmax+δ2TW+δ3TADW}. Liu and Feng [23] considered the problems F2∣con,NW,LE∣ δ1Cmax+δ2TC+δ3TADC+δ4∑∑Gjiuji and F2∣con,NW,LE∣ δ1Cmax+δ2TW+δ3TADW+δ4∑∑Gjiuji, where δ4 is a given constant. 3.1. The problem F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TC+δ3TADC Similar to Liu and Feng [23], for the problem F2|con,NW,LE,∑∑Gjiuji≤U| δ1Cmax+δ2TC+δ3TADC, we have Z1(π,u)=δ1Cmax+δ2TC+δ3TADC=∑i=12∑j=1nωjip¯[j]ijau[j]ik, (4) where ωj1=δ1+δ2n,ifj=1,δ1+δ2(n−j+1)+δ3(j−2)(n−j+2),ifj=2,…,n, (5) and ωj2=δ2,ifj=1,…,n−1,δ1+δ2+δ3(n−1),ifj=n. (6) Obviously, in an optimal solution for the problem F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TC+δ3TADC, the constraint will be satisfied as equality, i.e. ∑∑Gjiuji=U. Lemma 1 For a given job sequence, the optimal resource allocation policy can be obtained by the following expression, which minimizes δ1Cmax+δ2TC+δ3TADC: u[j]i*=(ωji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1×U∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1,j=1,2,…,n, (7)where ωjiare given by Equations (5) and (6). Proof For a given sequence π=(J[1],J[2],…,J[n]), where [k] denotes the job in the kth position, the Lagrange function is L(u,λ)=δ1Cmax+δ2TC+δ3TADC+λ∑i=12∑j=1nG[j]iu[j]i−U=∑i=12∑j=1nωjip¯[j]ijau[j]ik+λ∑i=12∑j=1nG[j]iu[j]i−U (8) where λ is the Lagrangian multiplier. Deriving (8) with respect to u[j]i and λ, we have ∂L(u,λ)∂u[j]i=λG[j]i−kωji×(p¯[j]ija)k(u[j]i)k+1=0,∀j=1,2,…,n. (9) ∂L(u,λ)∂λ=∑i=12∑j=1nG[j]iu[j]i−U=0 (10) It follows that u[j]i=(kωji(p¯[j]ija)k)1k+1(λG[j]i)1k+1 (11) and λ1k+1=∑i=12∑j=1n(kωji)1k+1(p¯[j]iG[j]ija)kk+1U. (12) Finally, inserting (12) into (11), we have u[j]i*=(ωji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1×U∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1. □ Lemma 2 Given that an optimal resource allocation is chosen, δ1Cmax+δ2TC+δ3TADCis given by the following expression: Z1(π,u*(π))=U−k∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1k+1. (13)where ωjiare given by Equations (5) and (6). Proof By substituting the values u[j]i* given in (7) into the expression given in (4), we obtain Z1(π,u*(π))=∑i=12∑j=1nωjip¯[j]ijau[j]ik=∑i=12∑j=1nωjip¯[j]ija∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1(ωji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1×Uk=∑i=12∑j=1nωjip¯[j]ija(ωji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1×Uk·∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1k=U−k∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1k+1. □ Obviously, minimizing the term U−k(∑i=12∑j=1n(ωji)1k+1·(p¯[j]iG[j]ija)kk+1)k+1 is equal to minimizing the term ∑i=12·∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1. In order to obtain the optimal job sequence, we formulate the F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TC+δ3TADC problem as an assignment problem. Let Xjr be a 0/1 variable such that Xjr=1 if job Jj ( j=1,2,…,n) is scheduled at position r ( r=1,2,…,n), and Xjr=0, otherwise. Then, the problem F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TC+δ3TADC can be solved by the following assignment problem: Min∑r=1n∑j=1nΩjrXjr (14) St ∑r=1nXjr=1,j=1,2,…,n, (15) ∑j=1nXjr=1,r=1,2,…,n, (16) Xjr=0or1,j=1,2,…,n,r=1,2,…,n, (17) where Ωjr=(ωr1)1k+1(Gj1p¯j1ra)kk+1+(ωr2)1k+1(Gj2p¯j2ra)kk+1, (18) ωr1 and ωr2 are given by Equations (5) and (6) Algorithm 1. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 1. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 1. Algorithm 1. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 1. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 1. Based on the above analysis, the problem F2|con,NW,LE,∑∑Gjiuji≤U|δ1Cmax+δ2TC+δ3TADC can be optimally solved by the following algorithm: Theorem 1 The F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TC+δ3TADCproblem can be solved by Algorithm1in O(n3)time. Proof The correctness of Algorithm 1 follows from Lemmas 1 and 2 and the assignment problem. The time complexity of Step 1 is O(n2), Step 2 is O(n3) time and Step 3 is O(n) time. Thus the overall computational complexity of Algorithm 1 is O(n3). Algorithm 2. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 3. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 3. Algorithm 2. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 3. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 3. □ 3.2. The problem F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TW+δ3TADW Similar to Liu and Feng [23] and Section 3.1, for the problem F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TW+δ3TADW, we have Z2(π,u)=δ1Cmax+δ2TW+δ3TADW=∑i=12∑j=1nψjip¯[j]ijau[j]ik, (19) where ψj1=δ1+δ2(n−1)+δ3(n−1),ifj=1,δ1+δ2(n−j)+δ3(j−1)(n−j+1),ifj=2,…,n, (20) and ψj2=δ2,ifj=1,…,n−1,δ1,ifj=n. (21) Lemma 3 For a given job sequence, the optimal resource allocation policy can be obtained by the following expression, which minimizes δ1Cmax+δ2TW+δ3TADW: u[j]i*=(ψji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1×U∑i=12∑j=1n(ψji)1k+1(p¯[j]iG[j]ija)kk+1,j=1,2,…,n, (22)where ψjiare given by Equations (20) and (21). Lemma 4 Given that an optimal resource allocation is chosen, δ1Cmax+δ2TW+δ3TADWis given by the following expression: Z2(π,u*(π))=U−k∑i=12∑j=1n(ψji)1k+1(p¯[j]iG[j]ija)kk+1k+1. (23)where ψjiare given by Equations (20) and (21). The problem F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TW+δ3TADW can be solved by the following assignment problem: Min∑r=1n∑j=1nΨjrXjr (24) St(15),(16),(17) where Ψjr=(ψr1)1k+1(Gj1p¯j1ra)kk+1+(ψr2)1k+1(Gj2p¯j2ra)kk+1. (25) The optimal solution for F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TC+δ3TADC can be obtained by the following algorithm: Theorem 2 The F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TW+δ3TADWproblem can be solved by Algorithm2in O(n3)time. 4. THE PROBLEM F2∣con,NW,LE,γ≤V∣∑∑Gjiuji In this section, we will deal with the ‘inverse version’ of the F2con,NW,LE,∑∑Gjiuji≤Uγ problem, where γ∈{δ1Cmax+δ2TC+δ3TADC,δ1Cmax+δ2TW+δ3TADW}. That is, the problem of minimizing the total cost of resource consumed subject to limited schedule criterion, i.e. the problem F2∣con,NW,LE,γ≤V∣∑∑Gjiuji, where γ∈{δ1Cmax+δ2TC+δ3TADC,δ1Cmax+δ2TW+δ3TADW} and V is a real number. 4.1. The problem F2∣con,NW,LE,δ1Cmax+δ2TC+δ3TADC≤V∣∑∑Gjiuji Similar to Section 3.1, for the F2∣con,NW,LE,δ1Cmax+δ2TC+δ3TADC≤V∣∑∑Gjiuji problem, the constraint will be satisfied as equality, i.e. δ1Cmax+δ2TC+δ3TADC=∑i=12∑j=1nωjip¯[j]ijau[j]ik=V. Lemma 5 For a given job sequence, the optimal resource allocation policy can be obtained by the following expression, which minimizes ∑i=12∑j=1nGjiuji: u[j]i*=V−1k(ωji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1·∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+11k,j=1,2,…,n, (26)where ωjiare given by Equations (5) and (6). Proof For a given sequence π=(J[1],J[2],…,J[n]), the Lagrange function is L(u,λ)=∑i=12∑j=1nG[j]iu[j]i+λ(δ1Cmax+δ2TC+δ3TADC−V)=∑i=12∑j=1nG[j]iu[j]i+λ(∑i=12∑j=1nωji(p¯[j]ijau[j]i)k−V), (27) where λ is the Lagrangian multiplier. Deriving (27) with respect to u[j]i and λ, we have ∂L(u,λ)∂u[j]i=G[j]i−λkωji×(p¯[j]ija)k(u[j]i)k+1=0,∀j=1,2,…,n. (28) ∂L(u,λ)∂λ=∑i=12∑j=1nωji(p¯[j]ijau[j]i)k−V=0. (29) It follows that u[j]i=(λkωji(p¯[j]ija)k)1k+1(G[j]i)1k+1 (30) and (λk)kk+1=∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1V. (31) Finally, inserting (31) into (30), we have u[j]i*=V−1k(ωji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1·∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+11k. □ Lemma 6 Given that an optimal resource allocation is chosen, ∑i=12∑j=1nGjiujiis given by the following expression: ∑i=12∑j=1nGjiuji(π,u*(π))=V−1k∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+11k+1. (32)where ωjiare given by Equations (5) and (6). Algorithm 3. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 5. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 5. Algorithm 3. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 5. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 5. Algorithm 4. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 7. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 7. Algorithm 4. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 7. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 7. Proof By substituting the values u[j]i* given in (26) into ∑i=12∑j=1nGjiuji, the result can be easily obtained.□ Similar to Section 3.1, the F2∣con,NW,LE,δ1Cmax+δ2TC+δ3TADC≤V∣∑∑Gjiuji problem can be solved by the following algorithm: Theorem 3 The F2∣con,NW,LE,δ1Cmax+δ2TC+δ3TADC≤V∣∑∑Gjiujiproblem can be solved by Algorithm3in O(n3)time. 4.2. The problem F2∣con,NW,LE,δ1Cmax+δ2TW+δ3TADW≤V∣∑∑Gjiuji Similar to Sections 3.2 and 4.1 , for F2∣con,NW,LE,δ1Cmax+δ2TW+δ3TADW≤V∣∑∑Gjiuji problem, we have Lemma 7 For a given job sequence, the optimal resource allocation policy can be obtained by the following expression, which minimizes ∑i=12∑j=1nGjiuji: u[j]i*=V−1k(ψji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1·∑i=12∑j=1n(ψji)1k+1(p¯[j]iG[j]ija)kk+11k,j=1,2,…,n, (33)where ψjiare given by Equations. (20) and (21). Lemma 8 Given that an optimal resource allocation is chosen, ∑i=12∑j=1nGjiujiis given by the following expression: ∑i=12∑j=1nGjiuji(π,u*(π))=V−1k∑i=12∑j=1n(ψji)1k+1(p¯[j]iG[j]ija)kk+11k+1. (34)where ψjiare given by Equations (20) and (21). The optimal solution for F2∣con,NW,LE,δ1Cmax+δ2TW+δ3TADW≤V∣∑∑Gjiuji can be obtained by the following algorithm: Theorem 4 The F2∣con,NW,LE,δ1Cmax+δ2TW+δ3TADW≤V∣∑∑Gjiujiproblem can be solved by Algorithm4in O(n3)time. Remark Obviously, the optimal sequence for the F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TC+δ3TADC ( F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TW+δ3TADW) problem is the same as the optimal sequence for the F2∣con,NW,LE,δ1Cmax+δ2TC+δ3TADC≤V∣∑∑Gjiuji ( F2∣con,NW,LE,δ1Cmax+δ2TW+δ3TADW≤V∣∑∑Gjiuji) problem. 5. A SPECIAL CASE If Gj1p¯j1=Gj2p¯j2=Gjp¯j for j=1,2,…,n, then the objective function (13) can be expressed as Z1(π,u*(π))=U−k∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1k+1=U−k∑j=1nΘjP[j]k+1, (35) where Θj=(ωj1)1k+1+(ωj2)1m+1jkak+1 and P[j]=(G[j]p¯[j])kk+1. Obviously, the term ∑j=1nΘjP[j] can be minimized by the HLP rule (Hardy et al. [27]) in O(nlogn) time. Hence, for the special case (i.e. Gj1p¯j1=Gj2p¯j2=Gjp¯j for j=1,2,…,n), we have: Theorem 5 For the special case (i.e. Gj1p¯j1=Gj2p¯j2=Gjp¯j, j=1,2,…,n), the problems F2∣con,NW,LE,∑∑Gjiuji≤U∣γand F2∣con,NW,LE,γ≤V∣∑∑Gjiuji ( γ∈{δ1Cmax+δ2TC+δ3TADC,δ1Cmax+δ2TW+δ3TADW}) can be solved in O(nlogn)time. 6. CONCLUSIONS This article considered the no-wait permutation scheduling problem with the learning effect and resource allocation at the same time. For the two-machine scheduling, we provided a bicriteria analysis where the first criteria is to minimize the schedule criterion and the second criteria is to minimize the total weighted resource cost. The objective is to minimize one of the two criteria subject to the constraint that the other criteria is less than or equal to an given constant. We showed that these two problems are polynomially solvable respectively. Further research may consider job shop scheduling, two-agent scheduling, no-idle flowshop scheduling, due-window assignment scheduling, and optimize other performance measures. FUNDING This research was supported by the National Natural Science Foundation of China [Grant no. 71471120], the Support Program for Innovative Talents in Liaoning University [LR2016017] and the Liaoning BaiQianWan Talents Program. 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Ann. Discrete Math. , 5 , 287 – 326 . Google Scholar CrossRef Search ADS 27 Hardy , G.H. , Littlewood , J.E. and Polya , G. ( 1967 ) Inequalities . Cambridge University Press , Cambridge . Author notes Handling editor: Antonio Fernandez Anta © The British Computer Society 2018. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Computer Journal Oxford University Press

No-wait resource allocation flowshop scheduling with learning effect under limited cost availability

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Abstract

Abstract This article considers the no-wait flowshop scheduling with learning effect and resource allocation on two-machine. Our goal is to find the optimal resource allocations and job sequence that minimize the scheduling criterion (the total weighted resource consumption) subject to the constraint that the total weighted resource consumption (the scheduling criterion) is less than or equal to a given constant, where the schedule criteria include weighted makespan, total completion (waiting) time and total absolute differences in completion (waiting) times. We show that these two problems are polynomially solvable respectively. 1. INTRODUCTION In traditional scheduling models and problems, the job processing times are generally treated as constant values. However, we often encounter production systems, job processing times may be dependent on additional resource (such as energy, money and manpower) and their position in a sequence (i.e. learning effect). Extensive surveys of scheduling problems with learning effects and resource allocation (controllable processing times) can be found in Biskup [1] and Shabtay and Steiner [2]. Recently Kuo [3] and Wang and Wang [4] considered single machine scheduling problems with learning effects. Wang et al. [5], Shiau et al. [6] and Wang et al. [7] considered flowshop scheduling problems with learning effects. Wu et al. [8] considered a multi-machine order scheduling with learning effects. Mor and Mosheiov [9] considered batch scheduling problems with controllable processing times. For identical jobs and linear compression cost function, they gave some results. Yang et al. [10] and Liu et al. [11] considered single machine due-window scheduling problems with controllable processing times. Hsieh et al. [12] considered scheduling problem with discrete controllable processing times on unrelated parallel machine. Zhao et al. [13] and Wang et al. [14] considered single machine scheduling with a controllable rate modifying activity (i.e. a resource-dependent maintenance activity). Wang et al. [15], Zhu et al. [16] and Wang et al. [17] considered single machine scheduling problems with learning effects and resource allocation. Lu et al. [18], Wang and Wang [19] and Wang et al. [20] considered single machine due date assignment scheduling problems with learning effects and resource allocation. Wang and Wang [21] and Li et al. [22] considered single machine due-window assignment scheduling problems with learning effects and resource allocation. Liu and Feng [23] considered flowshop scheduling with learning effect and resource allocation. In this paper, we consider the no-wait flowshop scheduling problems with learning effect and resource allocation. ‘The ‘no-wait’ constraint means that each job, once started, has to be processed without interruption until all of its operations are completed. In practice, this requirement may arise out of certain job characteristics (such as in the ‘hot ingot’ problem, where metal has to be processed at a continuously high temperature) or out of the unavailability of intermediate storage in between machines’ (Wang and Xia [24]). However, to the best of our knowledge, there exist only a result concerning no-wait flowshop scheduling problems with learning effects and resource allocation simultaneously. Liu and Feng [23] considered two-machine no-wait flowshop scheduling with simultaneous considerations of learning effect and resource allocation. For the weighted makespan, total completion (waiting) time, total absolute differences in completion (waiting) times and total resource cost, they proved that the problem can be solved in polynomial time. In the real production process, the resources are limited, one might wang to control the total resource consumption (Janiak and Li [25]). Hence, in this study, we continue the work of Liu and Feng [23], i.e. we consider resource allocation scheduling problem with learning effects in the two-machine no-wait permutation flowshop environment. The first type objective is to minimize the schedule criterion (i.e. the weighted makespan, total completion (waiting) time and total absolute differences in completion (waiting) times) subject to the constraint that the total resource cost is less than or equal to an given constant. The second type objective is to minimize the total resource cost subject to the constraint that the schedule criterion is less than or equal to an given constant. We show that these two problems are polynomially solvable, respectively. The remainder of this article is organized as follows: in Section 2, we introduce and formulate the problem. In Sections 3 and 4 , we provide some properties to optimally solve these two problems, respectively. In Section 5, we consider a special case. We conclude the paper in Section 6. 2. PROBLEM FORMULATION Consider a two-machine no-wait permutation flowshop setting. There are n independent jobs J={J1,J2,…,Jn} and two machines M1 and M2 available at time zero, and job preemption and machine interruption are not allowed. Job Jj must be processed first on machine M1 and then on machine M2 (i.e. operations Oji, i=1,2, j=1,2,…,n), and the jobs are not allowed to wait between the two machines (i.e. p[j]2=p[j+1]1, there is a no-wait restriction, where [j] denotes the job scheduled in the jth position in a sequence). As in Liu and Feng [23], we assume that the actual processing time of job Jj on machine Mi is pji=p¯jiraujik, (1) where p¯ji is the basic (normal) processing time of operation Oji (i.e. the processing time without any learning effects and resource allocation), uji is the amount of resource allocated to operation Oji and k is a positive constant, r is the position of job Jj which is scheduled in a sequence. Let Cmax=max{Cj∣j=1,2,…,n} be the maximum completion time (makespan), TC=∑j=1nCj be the total completion times, TW=∑j=1nWj be the total waiting times, TADC=∑k=1n∑j=kn∣Ck−Cj∣ be the total absolute differences in completion times and ∑k=1n∑j=kn∣Wk−Wj∣ be the total absolute differences in waiting times, where Cj=Cj(π) and Wj=Cj−pj1−pj2 are the completion time and the waiting time for job Jj on machine M2, respectively. 3. THE PROBLEM F2con,NW,LE,∑∑Gjiuji≤Uγ In this section, we aim to determine the set of optimal resource allocations u=(u11,u21,…,un1; u12,u22,…,un2) and the job sequence π such that the following cost function: Z1(π,u)=δ1Cmax+δ2TC+δ3TADC, (2) and Z2(π,u)=δ1Cmax+δ2TW+δ3TADW, (3) is minimized, respectively, subject to the constraint that the total resource cost ∑i=12∑j=1nGjiuji≤U, where weights δ1≥0,δ2≥0,δ3≥0 are given constants, and Gji is the cost associated with the resource allocation per time unit. Similar to Graham et al. [26] and Liu and Feng [23], the problem can be denoted as F2con,NW,LE,∑∑Gjiuji≤Uγ, where γ∈{δ1Cmax+δ2TC+δ3TADC,δ1Cmax+δ2TW+δ3TADW}. Liu and Feng [23] considered the problems F2∣con,NW,LE∣ δ1Cmax+δ2TC+δ3TADC+δ4∑∑Gjiuji and F2∣con,NW,LE∣ δ1Cmax+δ2TW+δ3TADW+δ4∑∑Gjiuji, where δ4 is a given constant. 3.1. The problem F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TC+δ3TADC Similar to Liu and Feng [23], for the problem F2|con,NW,LE,∑∑Gjiuji≤U| δ1Cmax+δ2TC+δ3TADC, we have Z1(π,u)=δ1Cmax+δ2TC+δ3TADC=∑i=12∑j=1nωjip¯[j]ijau[j]ik, (4) where ωj1=δ1+δ2n,ifj=1,δ1+δ2(n−j+1)+δ3(j−2)(n−j+2),ifj=2,…,n, (5) and ωj2=δ2,ifj=1,…,n−1,δ1+δ2+δ3(n−1),ifj=n. (6) Obviously, in an optimal solution for the problem F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TC+δ3TADC, the constraint will be satisfied as equality, i.e. ∑∑Gjiuji=U. Lemma 1 For a given job sequence, the optimal resource allocation policy can be obtained by the following expression, which minimizes δ1Cmax+δ2TC+δ3TADC: u[j]i*=(ωji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1×U∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1,j=1,2,…,n, (7)where ωjiare given by Equations (5) and (6). Proof For a given sequence π=(J[1],J[2],…,J[n]), where [k] denotes the job in the kth position, the Lagrange function is L(u,λ)=δ1Cmax+δ2TC+δ3TADC+λ∑i=12∑j=1nG[j]iu[j]i−U=∑i=12∑j=1nωjip¯[j]ijau[j]ik+λ∑i=12∑j=1nG[j]iu[j]i−U (8) where λ is the Lagrangian multiplier. Deriving (8) with respect to u[j]i and λ, we have ∂L(u,λ)∂u[j]i=λG[j]i−kωji×(p¯[j]ija)k(u[j]i)k+1=0,∀j=1,2,…,n. (9) ∂L(u,λ)∂λ=∑i=12∑j=1nG[j]iu[j]i−U=0 (10) It follows that u[j]i=(kωji(p¯[j]ija)k)1k+1(λG[j]i)1k+1 (11) and λ1k+1=∑i=12∑j=1n(kωji)1k+1(p¯[j]iG[j]ija)kk+1U. (12) Finally, inserting (12) into (11), we have u[j]i*=(ωji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1×U∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1. □ Lemma 2 Given that an optimal resource allocation is chosen, δ1Cmax+δ2TC+δ3TADCis given by the following expression: Z1(π,u*(π))=U−k∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1k+1. (13)where ωjiare given by Equations (5) and (6). Proof By substituting the values u[j]i* given in (7) into the expression given in (4), we obtain Z1(π,u*(π))=∑i=12∑j=1nωjip¯[j]ijau[j]ik=∑i=12∑j=1nωjip¯[j]ija∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1(ωji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1×Uk=∑i=12∑j=1nωjip¯[j]ija(ωji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1×Uk·∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1k=U−k∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1k+1. □ Obviously, minimizing the term U−k(∑i=12∑j=1n(ωji)1k+1·(p¯[j]iG[j]ija)kk+1)k+1 is equal to minimizing the term ∑i=12·∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1. In order to obtain the optimal job sequence, we formulate the F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TC+δ3TADC problem as an assignment problem. Let Xjr be a 0/1 variable such that Xjr=1 if job Jj ( j=1,2,…,n) is scheduled at position r ( r=1,2,…,n), and Xjr=0, otherwise. Then, the problem F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TC+δ3TADC can be solved by the following assignment problem: Min∑r=1n∑j=1nΩjrXjr (14) St ∑r=1nXjr=1,j=1,2,…,n, (15) ∑j=1nXjr=1,r=1,2,…,n, (16) Xjr=0or1,j=1,2,…,n,r=1,2,…,n, (17) where Ωjr=(ωr1)1k+1(Gj1p¯j1ra)kk+1+(ωr2)1k+1(Gj2p¯j2ra)kk+1, (18) ωr1 and ωr2 are given by Equations (5) and (6) Algorithm 1. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 1. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 1. Algorithm 1. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 1. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 1. Based on the above analysis, the problem F2|con,NW,LE,∑∑Gjiuji≤U|δ1Cmax+δ2TC+δ3TADC can be optimally solved by the following algorithm: Theorem 1 The F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TC+δ3TADCproblem can be solved by Algorithm1in O(n3)time. Proof The correctness of Algorithm 1 follows from Lemmas 1 and 2 and the assignment problem. The time complexity of Step 1 is O(n2), Step 2 is O(n3) time and Step 3 is O(n) time. Thus the overall computational complexity of Algorithm 1 is O(n3). Algorithm 2. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 3. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 3. Algorithm 2. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 3. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 3. □ 3.2. The problem F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TW+δ3TADW Similar to Liu and Feng [23] and Section 3.1, for the problem F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TW+δ3TADW, we have Z2(π,u)=δ1Cmax+δ2TW+δ3TADW=∑i=12∑j=1nψjip¯[j]ijau[j]ik, (19) where ψj1=δ1+δ2(n−1)+δ3(n−1),ifj=1,δ1+δ2(n−j)+δ3(j−1)(n−j+1),ifj=2,…,n, (20) and ψj2=δ2,ifj=1,…,n−1,δ1,ifj=n. (21) Lemma 3 For a given job sequence, the optimal resource allocation policy can be obtained by the following expression, which minimizes δ1Cmax+δ2TW+δ3TADW: u[j]i*=(ψji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1×U∑i=12∑j=1n(ψji)1k+1(p¯[j]iG[j]ija)kk+1,j=1,2,…,n, (22)where ψjiare given by Equations (20) and (21). Lemma 4 Given that an optimal resource allocation is chosen, δ1Cmax+δ2TW+δ3TADWis given by the following expression: Z2(π,u*(π))=U−k∑i=12∑j=1n(ψji)1k+1(p¯[j]iG[j]ija)kk+1k+1. (23)where ψjiare given by Equations (20) and (21). The problem F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TW+δ3TADW can be solved by the following assignment problem: Min∑r=1n∑j=1nΨjrXjr (24) St(15),(16),(17) where Ψjr=(ψr1)1k+1(Gj1p¯j1ra)kk+1+(ψr2)1k+1(Gj2p¯j2ra)kk+1. (25) The optimal solution for F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TC+δ3TADC can be obtained by the following algorithm: Theorem 2 The F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TW+δ3TADWproblem can be solved by Algorithm2in O(n3)time. 4. THE PROBLEM F2∣con,NW,LE,γ≤V∣∑∑Gjiuji In this section, we will deal with the ‘inverse version’ of the F2con,NW,LE,∑∑Gjiuji≤Uγ problem, where γ∈{δ1Cmax+δ2TC+δ3TADC,δ1Cmax+δ2TW+δ3TADW}. That is, the problem of minimizing the total cost of resource consumed subject to limited schedule criterion, i.e. the problem F2∣con,NW,LE,γ≤V∣∑∑Gjiuji, where γ∈{δ1Cmax+δ2TC+δ3TADC,δ1Cmax+δ2TW+δ3TADW} and V is a real number. 4.1. The problem F2∣con,NW,LE,δ1Cmax+δ2TC+δ3TADC≤V∣∑∑Gjiuji Similar to Section 3.1, for the F2∣con,NW,LE,δ1Cmax+δ2TC+δ3TADC≤V∣∑∑Gjiuji problem, the constraint will be satisfied as equality, i.e. δ1Cmax+δ2TC+δ3TADC=∑i=12∑j=1nωjip¯[j]ijau[j]ik=V. Lemma 5 For a given job sequence, the optimal resource allocation policy can be obtained by the following expression, which minimizes ∑i=12∑j=1nGjiuji: u[j]i*=V−1k(ωji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1·∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+11k,j=1,2,…,n, (26)where ωjiare given by Equations (5) and (6). Proof For a given sequence π=(J[1],J[2],…,J[n]), the Lagrange function is L(u,λ)=∑i=12∑j=1nG[j]iu[j]i+λ(δ1Cmax+δ2TC+δ3TADC−V)=∑i=12∑j=1nG[j]iu[j]i+λ(∑i=12∑j=1nωji(p¯[j]ijau[j]i)k−V), (27) where λ is the Lagrangian multiplier. Deriving (27) with respect to u[j]i and λ, we have ∂L(u,λ)∂u[j]i=G[j]i−λkωji×(p¯[j]ija)k(u[j]i)k+1=0,∀j=1,2,…,n. (28) ∂L(u,λ)∂λ=∑i=12∑j=1nωji(p¯[j]ijau[j]i)k−V=0. (29) It follows that u[j]i=(λkωji(p¯[j]ija)k)1k+1(G[j]i)1k+1 (30) and (λk)kk+1=∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1V. (31) Finally, inserting (31) into (30), we have u[j]i*=V−1k(ωji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1·∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+11k. □ Lemma 6 Given that an optimal resource allocation is chosen, ∑i=12∑j=1nGjiujiis given by the following expression: ∑i=12∑j=1nGjiuji(π,u*(π))=V−1k∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+11k+1. (32)where ωjiare given by Equations (5) and (6). Algorithm 3. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 5. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 5. Algorithm 3. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 5. Step 1. Calculate the values Ωjr by using (18). Step 2. Solve the assignment problem (14–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 5. Algorithm 4. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 7. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 7. Algorithm 4. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 7. Step 1. Calculate the values Ψjr by using (25). Step 2. Solve the assignment problem (24), (15–17) to determine the optimal job sequence. Step 3. Calculate the optimal resource allocation by using Lemma 7. Proof By substituting the values u[j]i* given in (26) into ∑i=12∑j=1nGjiuji, the result can be easily obtained.□ Similar to Section 3.1, the F2∣con,NW,LE,δ1Cmax+δ2TC+δ3TADC≤V∣∑∑Gjiuji problem can be solved by the following algorithm: Theorem 3 The F2∣con,NW,LE,δ1Cmax+δ2TC+δ3TADC≤V∣∑∑Gjiujiproblem can be solved by Algorithm3in O(n3)time. 4.2. The problem F2∣con,NW,LE,δ1Cmax+δ2TW+δ3TADW≤V∣∑∑Gjiuji Similar to Sections 3.2 and 4.1 , for F2∣con,NW,LE,δ1Cmax+δ2TW+δ3TADW≤V∣∑∑Gjiuji problem, we have Lemma 7 For a given job sequence, the optimal resource allocation policy can be obtained by the following expression, which minimizes ∑i=12∑j=1nGjiuji: u[j]i*=V−1k(ψji)1k+1(G[j]i)−1k+1(p¯[j]ija)kk+1·∑i=12∑j=1n(ψji)1k+1(p¯[j]iG[j]ija)kk+11k,j=1,2,…,n, (33)where ψjiare given by Equations. (20) and (21). Lemma 8 Given that an optimal resource allocation is chosen, ∑i=12∑j=1nGjiujiis given by the following expression: ∑i=12∑j=1nGjiuji(π,u*(π))=V−1k∑i=12∑j=1n(ψji)1k+1(p¯[j]iG[j]ija)kk+11k+1. (34)where ψjiare given by Equations (20) and (21). The optimal solution for F2∣con,NW,LE,δ1Cmax+δ2TW+δ3TADW≤V∣∑∑Gjiuji can be obtained by the following algorithm: Theorem 4 The F2∣con,NW,LE,δ1Cmax+δ2TW+δ3TADW≤V∣∑∑Gjiujiproblem can be solved by Algorithm4in O(n3)time. Remark Obviously, the optimal sequence for the F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TC+δ3TADC ( F2con,NW,LE,∑∑Gjiuji≤Uδ1Cmax+δ2TW+δ3TADW) problem is the same as the optimal sequence for the F2∣con,NW,LE,δ1Cmax+δ2TC+δ3TADC≤V∣∑∑Gjiuji ( F2∣con,NW,LE,δ1Cmax+δ2TW+δ3TADW≤V∣∑∑Gjiuji) problem. 5. A SPECIAL CASE If Gj1p¯j1=Gj2p¯j2=Gjp¯j for j=1,2,…,n, then the objective function (13) can be expressed as Z1(π,u*(π))=U−k∑i=12∑j=1n(ωji)1k+1(p¯[j]iG[j]ija)kk+1k+1=U−k∑j=1nΘjP[j]k+1, (35) where Θj=(ωj1)1k+1+(ωj2)1m+1jkak+1 and P[j]=(G[j]p¯[j])kk+1. Obviously, the term ∑j=1nΘjP[j] can be minimized by the HLP rule (Hardy et al. [27]) in O(nlogn) time. Hence, for the special case (i.e. Gj1p¯j1=Gj2p¯j2=Gjp¯j for j=1,2,…,n), we have: Theorem 5 For the special case (i.e. Gj1p¯j1=Gj2p¯j2=Gjp¯j, j=1,2,…,n), the problems F2∣con,NW,LE,∑∑Gjiuji≤U∣γand F2∣con,NW,LE,γ≤V∣∑∑Gjiuji ( γ∈{δ1Cmax+δ2TC+δ3TADC,δ1Cmax+δ2TW+δ3TADW}) can be solved in O(nlogn)time. 6. CONCLUSIONS This article considered the no-wait permutation scheduling problem with the learning effect and resource allocation at the same time. For the two-machine scheduling, we provided a bicriteria analysis where the first criteria is to minimize the schedule criterion and the second criteria is to minimize the total weighted resource cost. The objective is to minimize one of the two criteria subject to the constraint that the other criteria is less than or equal to an given constant. We showed that these two problems are polynomially solvable respectively. Further research may consider job shop scheduling, two-agent scheduling, no-idle flowshop scheduling, due-window assignment scheduling, and optimize other performance measures. FUNDING This research was supported by the National Natural Science Foundation of China [Grant no. 71471120], the Support Program for Innovative Talents in Liaoning University [LR2016017] and the Liaoning BaiQianWan Talents Program. REFERENCES 1 Biskup , D. ( 2008 ) A state-of-the-art review on scheduling with learning effects . Eur. J. Oper. Res. , 188 , 315 – 329 . Google Scholar CrossRef Search ADS 2 Shabtay , D. and Steiner , G. ( 2007 ) A survey of scheduling with controllable processing times . Discrete Appl. Math. , 155 , 1643 – 1666 . Google Scholar CrossRef Search ADS 3 Kuo , W.-H. ( 2012 ) Single-machine group scheduling with time-dependent learning effect and position-based setup time learning effect . Ann. Oper. Res. , 196 , 349 – 359 . 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For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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The Computer JournalOxford University Press

Published: Apr 3, 2018

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