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The Computer Journal
, Volume Advance Article – Mar 24, 2018

9 pages

/lp/ou_press/single-machine-scheduling-with-simultaneous-considerations-of-resource-6SgIIV5Gk9

- Publisher
- Oxford University Press
- Copyright
- © The British Computer Society 2018. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
- ISSN
- 0010-4620
- eISSN
- 1460-2067
- D.O.I.
- 10.1093/comjnl/bxy030
- Publisher site
- See Article on Publisher Site

Abstract This paper considers a single-machine scheduling problem with deteriorating jobs and convex resource allocation. We assume that the actual processing time of a job is a function of its convex resource allocation and its starting time. For the multi-objective single-machine scheduling problem, we show that the problem is polynomially solvable in O(nlogn) time, where n is the total number of jobs. 1. INTRODUCTION Scheduling models and problems are widely applied to many branches of industry and logistics (Cheng et al. [1], Cheng and Janiak [2], Wang and Wang [3], Gafarov et al. [4], Janiak et al. [5], Lee et al. [6], Wu et al. [7] and Wang et al. [8]). In traditional scheduling theory, the job processing times are fixed and constant values, however, in various manufacturing settings job processing times may be subject to change due to the phenomenon of deterioration. Job deterioration appears, for example, in the steel production, steel ingots are to be heated to the required temperature before rolling can begin. Similar situations will also occur in fire fighting, where any delay in processing a job (control a fire) is penalized by incurring additional time for accomplishing the job. An extensive survey of different scheduling problems involving deteriorating jobs (time-dependent scheduling) can be found in Gawiejnowicz [9]. More recent papers that have considered scheduling problems with deteriorating jobs include Moslehi and Jafari [10], Wang et al. [11], Wang and Wang [12, 13], Wang and Wang [14], Zhao et al. [15] and Wang and Li [16]. Moslehi and Jafari [10] considered the number of tardy jobs minimization single-machine problem with piecewise-linear deterioration. To solve then problem, they proposed a branch-and bound algorithm and a heuristic algorithm. Wang et al. [11] considered a single-machine scheduling problem with simple linear deterioration. For the weighted sum of squared completion times minimization, they proved that the problem can be solved in polynomial time under the conditions of chain precedence constraints. Wang and Wang [12] considered the scheduling problems with nonlinear deterioration. They proved that the single-machine makespan minimization problem can be solved in polynomial time. For the total completion time minimization problem, they also gave some results. Wang and Wang [13] studied the makespan minimization on a three-machine permutation flow shop. The solution procedures (i.e. a branch-and-bound algorithm and two heuristic algorithms) are proposed. Wang and Wang [14] considered the single-machine makespan minimization with ready times and group technology assumption. Under the proportional linear deterioration, they proved that the problem can be solved in polynomial time. Zhao et al. [15] and Wang and Li [16] considered machine maintenance activities scheduling problems with deteriorating jobs. On the other hand, scheduling problems with resource allocation and controllable processing times have been extensively studied. A more recent survey of scheduling with controllable processing times (resource allocation) was given by Shabtay and Steiner [17]. Recently, Shabtay and Steiner [18] considered the single machine due date assignment scheduling with controllable processing times. For each combination of three different due date assignment methods and two resource allocation functions (i.e. the linear resource consumption function and the convex resource consumption function), they provided a polynomial-time algorithm to solve a regular objective function. Tseng et al. [19] and Xu et al. [20] considered single-machine total tardiness minimization scheduling with controllable processing times. Leyvand et al. [21] considered parallel machines scheduling with controllable processing times. Wang and Wang [22] considered the single-machine scheduling problem with convex resource consumption function. For the total amount of resource consumption minimization subject to a constraint on total weighted flow time, they proposed a heuristic algorithm and a branch-and-bound algorithm. Kayvanfar et al. [23] considered the single-machine earliness and tardiness problem with controllable processing times. Yin et al. [24] and Wang et al. [25] considered due date assignment single-machine scheduling with controllable processing times. Yin et al. [26] and Liu et al. [27] considered due-window assignment scheduling problems with controllable processing times. However, to the best of our knowledge, apart from the recent paper of Bachman and Janiak [28], Janiak and Iwanowski [29], Wei et al. [30], Wang and Wang [31], Wang and Wang [32], Zhao et al. [33] and Wang et al. [34], scheduling models and problems with deterioration effects and controllable processing times (resource allocation) have not been investigated at the same time. ‘The phenomena of deterioration effects and controllable processing times occurring simultaneously can be found in steel production, more precisely, in the process of preheating ingots by gas to prepare them for hot rolling on the blooming mill. Before the ingots can be hot rolled, they have to achieve the required temperature. However, the preheating time of the ingots depends on their starting temperature, i.e. the longer ingots wait for the start of the preheating process, the lower goes their temperature and therefore the longer lasts the preheating process. The preheating time can be shortened by the increase of the gas flow intensity, i.e. the more gas is consumed, the shorter lasts the preheating process. Thus, the ingot preheating time depends on the starting moment of the preheating process and the amount of gas consumed during it’ (Bachman and Janiak [28]). Bachman and Janiak [28] considered the scheduling problem with job processing times dependent on the starting time of job execution (i.e. deterioration effect) and on the amounts of resource allocation to the jobs (resource allocation). They showed that the single-machine makespan minimization subject to the constraint on the total amount of available resources is NP-hard. Janiak and Iwanowski [29] considered another scheduling model with deterioration effects and controllable processing times. Wei et al. [30] consider a scheduling model with deterioration effects and controllable processing times, i.e. the actual processing time pj is a linear function of the resource consumed uj, which is depicted by the resource consumption function pj=aj+bt−θuj,j=1,2,…,n,0≤uj≤u¯j≤ajθ, (1) where n is the number of non-preemptive jobs, aj≥0 is the normal (basic) processing time of the job Jj, uj>0 is a decision variable that represents the amount of a continuously divisible and non-renewable resource allocated to job Jj, b≥0 is the common deterioration rate for all the jobs, t≥0 is its start time, θ≥0 and u¯j is the upper bound on the amount of resource that can be allocated to job Jj. They proved that two scheduling problems can be solved in polynomial time under proposed model. Wang and Wang [31] and Zhao et al. [33] considered single-machine scheduling with convex resource-dependent processing times and deteriorating jobs, i.e. the actual processing time pj of job Jj is a convex function of the resource consumed uj, which is depicted by the resource consumption function pj=wjujk+bt,uj>0, (2) where k is a positive constant, wj≥0 is a positive parameter, which represents the workload of job Jj, b≥0 is a common deterioration rate for all the jobs, t≥0 is its start time and uj>0 is a decision variable that represents the amount of a continuously divisible and non-renewable resource allocated to job Jj. They proved that two scheduling problems can be solved in polynomial tim under proposed model. Wang and Wang [32] considered the same model with Wang and Wang [33] and a more general model than Wei et al. [30], i.e. pj=aj+bt−θjuj,j=1,2,…,n,0≤uj≤u¯j≤ajθj, (3) where θj>0. They proved that three due date assignment single-machine scheduling problems remains polynomially solvable under the proposed models. Wang et al. [34] considered resource allocation scheduling with truncated job-dependent learning and deterioration effects. In this paper, we continue the work studied by Wang and Wang [31], Wang and Wang [32] and Zhao et al. [33], but consider the model (2) and the case that minimize the completion time cost under the total resource consumption constraint, and the case that minimize the total amount of resource consumed subject to a constraint on completion time cost. The rest of the paper is organized as follows. In Section 2, we present the notation and assumption. In Sections 3 and 4 , we show that the relaxed versions of the two different problems on a single machine can be solved in polynomial time, respectively. Section 5 contains conclusions. 2. PROBLEM FORMULATION In this paper, we aim at scheduling n jobs N={J1,J2,…,Jn} on a continuously available single machine, and all the jobs are available for processing at time 0. Also it is assume that the machine is continuously available and can handle one job at a time. The actual processing time pj of job Jj is a convex function of the resource consumed uj, which is depicted by the resource consumption function (2). For a given sequence π=[J1,J2,…,Jn], Cj=Cj(π) represents the completion time for job Jj. Let [dj1,dj2] represent the due-window of job Jj such that dj1≤dj2, j=1,2,…,n, dj1 and Dj=dj2−dj1 be the starting time and the due-window size of job Jj. Let Cmax, ∑j=1nCj ( ∑j=1nWj), ∑i=1n∑j=in∣Ci−Cj∣ ( ∑i=1n∑j=in∣Wi−Wj∣) be the makespan of all jobs, the total completion (waiting) times, and the total absolute differences in completion (waiting) times, where Wj=Cj−pj be the waiting time of job Jj. We adopt the three-field notation introduced by Graham et al. [35], to specify scheduling problem. Wei et al. [30] considered the problems 1∣pj=aj+bt−θuj∣α1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣+α4∑Gjuj and 1∣pj=aj+bt−θuj∣α1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣+α4∑Gjuj, where weights α1≥0,α2≥0, α3≥0,α4≥0 are given constants, and Gj is the per time unit cost associated with the resource allocation. Wang and Wang [31] considered the problems 1pj=wjujk+btα1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣+α4∑Gjuj and 1pj=wjujk+btα1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣+α4∑Gjuj. Wang and Wang [32] considered the problems 1∣pj=aj+bt−θjuj,CON/SLK/DIF∣∑(α1Ej+α2Tj+α3dj+α4∑Gjuj) and 1pj=wjujk+bt,CON/SLK/DIF∑(α1Ej+α2Tj+α3dj+α4∑Gjuj), where CON,SLK and DIF are the most commonly used due date assignment methods, Ej,Tj and dj are the earliness, tardiness and due date of job Jj. Zhao et al. [33] considered the problem 1pj=wjujk+bt,∑uj≤Uρ, where ρ∈{Cmax,∑Cj,TADC=∑∑∣Ci−Cj∣,TADW=∑∑∣Wi−Wj∣,∑(α1Ej+α2Tj+α3dj),∑(α1Ej+α2Tj+α3d+α4D} and U is the total resource consumption upper bounded. In this paper, our objective is to determine the job schedule and the optimal resource allocation such as to (1) minimize ρ under the total resource consumption constraint, i.e. the problem 1pj=wjujk+bt,∑Gjuj≤Uρ, where ρ∈{Cmax,∑Cj,∑Wj,∑∑∣Ci−Cj∣,∑∑∣Wi−Wj∣,∑(α1Ej+α2Tj+α3dj),∑(α1Ej+α2Tj+α3dj1+α4Dj}, and U is a real number, represents the total resource cost upper bounded; 2) minimize the total resource cost with a schedule cost constraint, i.e. the 1pj=wjujk+bt,ρ≤R∑Gjuj problem, where ρ∈{Cmax,∑Cj,∑Wj,∑∑∣Ci−Cj∣,∑∑∣Wi−Wj∣,∑(α1Ej+α2Tj+α3dj),∑(α1Ej+α2Tj+α3dj1+α4Dj}, R is a real number, represents the schedule cost upper bounded. 3. PROBLEM 1pj=wjujk+bt,∑Gjuj≤Uρ Let p[r] denote the actual processing time of a job when it is scheduled in position r in a sequence, and w[r],C[r],W[r],u[r] be defined correspondingly. Then from Wang and Wang [31], we have C[j]=∑l=1j(1+b)j−lw[l]u[l]k, (4) p[j]=w[j]u[j]k+bC[j−1]=w[j]u[j]k+b∑l=1j−1(1+b)j−1−lw[l]u[l]k. (5) Three due date assignment problems: The due date assignment problem is to determine the optimal due dates d=(d1,d2,…,dn), resource allocations u=(u1,u2,…,un) and a schedule π which minimizes Z(d,u,π)=∑j=1n(α1Ej+α2Tj+α3dj), (6) where dj, Ej=max{0,dj−Cj} and Tj=max{0,Cj−dj} are the due date, the earliness and the tardiness of job Jj, j=1,2,…,n, α1,α2 and α3 be the per time unit penalties for earliness, tardiness and due date. Three more commonly used due date assignment methods are as follows: (a) The common (CON) due date assignment method, i.e. dj=d for j=1,2,…,n, where d≥0 is a decision variable (Panwalkar et al. [36]). Then Z(d,u,π)=∑j=1n(α1Ej+α2Tj+α3dj)=∑j=1n(α1Ej+α2Tj+α3d)=∑j=1nωjp[j], (7) where ωj=min{nα3+(j−1)α1,(n+1−j)α2},j=1,2,…,n. (b) The slack (SLK) due date assignment method, i.e. dj=pj+q for j=1,2,…,n, where q≥0 is a decision variable (Adamopoulos and Pappis [37]). Then Z(d,u,π)=∑j=1n(α1Ej+α2Tj+α3q)=∑j=1nωjp[j], (8) where ωj=min{nα3+jα1,(n−j)α2},j=1,2,…,n. (c) The different (DIF) due date assignment method (Seidmann et al. [38]), i.e. each job can be assigned a different due date with no restrictions. Then Z(d,u,π)=∑j=1n(α1Ej+α2Tj+α3dj)=∑j=1nωjp[j], (9) where ωj=min{α2,α3}(n+1−j),j=1,2,…,n. Three due-window assignment problems: Let [dj1,dj2] represent the due-window of job Jj such that dj1≤dj2, j=1,2,…,n, dj1 and Dj=dj2−dj1 be the starting time and the due-window size of job Jj. The due-window assignment problem is to determine the optimal starting time of due dates d1=(d11,d21,…,dn1), the due-window sizes D=(D1,D2,…,Dn), resource allocations u=(u1,u2,…,un) and a schedule π which minimizes Z(d1,D,u,π)=∑j=1n(α1Ej+α2Tj+α3dj1+α4Dj) (10) where Ej=max{0,dj1−Cj} and Tj=max{0,Cj−dj2} are the earliness and the tardiness of job Jj, j=1,2,…,n, α1,α2,α3 and α4 be the per time unit penalties for the earliness, the tardiness, the starting times of the due-windows, and the size of the due-windows. Three more commonly used due-window assignment methods are as follows: (a) The common due-window assignment method (CONW), i.e. dj1=d, D=dj2−dj1 (Liman et al. [39]). Then Z(d1,D,u,π)=∑j=1n(α1Ej+α2Tj+α3d+α4D)=∑j=1nωjp[j], (11) ωj=min{nα3+(j−1)α1,nα4,(n+1−r)α2},j=1,2,…,n. (b) The slack due-window assignment method (SLKW), i.e. dj1=pj+q1, dj2=pj+q2 (Mosheiov and Oron [40], and Mor and Mosheiov [41]). Then Z(d1,D,u,π)=∑j=1n(α1Ej+α2Tj+α3dj1+α4(q2−q1))=∑j=1nωjp[j], (12) ωj=min{(n+1)α3+α1j,nα4+α3,(n−j)α2+α3},j=1,2,…,n. (c) The different due-window assignment method (DIFW), i.e. each job can be assigned a different due-window with no restrictions (Seidmann et al. [38] and Wang et al. [42]). Then Z(d1,D,u,π)=∑j=1n(α1Ej+α2Tj+α3dj)=∑j=1nωjp[j], (13) where ωj=min{α2,α3}(n+1−j),j=1,2,…,n. Z(u,π)=∑j=1nωjp[j]=∑j=1nωjw[j]u[j]k+b∑l=1j−1(1+b)j−1−lw[l]u[l]k=∑j=1nΩjw[j]u[j]k, (14) where ωj=1 for Cmax, ωj=(n+1−j) for ∑Cj, ωj=(n−j) for ∑Wj, ωj=(j−1)(n−j+1) for ∑∑∣Ci−Cj∣, ωj=j(n−j) for ∑∑∣Wi−Wj∣, and Ω1=ω1+bω2+b(1+b)ω3+⋯+b(1+b)n−2ωnΩ2=ω2+bω3+b(1+b)ω4+⋯+b(1+b)n−3ωnΩ3=ω3+bω4+b(1+b)ω5+⋯+b(1+b)n−4ωn…Ωn−1=ωn−1+bωnΩn=ωn. (15) Lemma 1 For the problem 1pj=wjujk+bt,∑Gjuj≤Uρ, where ρ∈{Cmax,∑Cj,∑Wj, ∑∑∣Ci−Cj∣,∑∑∣Wi−Wj∣,∑(α1Ej+α2Tj+α3dj),∑(α1Ej+α2Tj+α3dj1+α4Dj}, the optimal resource allocation as a function of the job sequence π, denoted by u*(π), is u[j]*=(Ωj)1k+1(G[j])−1(w[j])kk+1×U∑j=1n(Ωj)1k+1(w[j]G[j])kk+1,j=1,2,…,n, (16) Ωjare given by Equation (15). Proof Obviously, for an optimal solution of the problem 1pj=wjujk+bt,∑Gjuj≤Uρ, ∑j=1nGjuj=U is a sufficient and necessary condition. Hence, for any given sequence, the Lagrange function is L(u,λ)=ρ+λ(∑j=1nG[j]u[j]−U)=∑j=1nΩj(w[j]u[j])k+λ(∑j=1nG[j]u[j]−U), (17) where λ is the Lagrangian multiplier. Deriving (17) with respect to the decision variables (u[j] and λ), we have ∂L(u,λ)∂u[j]=λG[j]−kΩj×(w[j])k(u[j])k+1=0,∀j=1,2,…,n. (18) ∂L(u,λ)∂λ=∑j=1nG[j]u[j]−U=0. (19) Using (18) and (19) we obtain u[j]=(kΩj(w[j])k)1k+1(λG[j])1k+1 (20) and λ1k+1=∑j=1n(kΩj)1k+1(w[j]G[j])kk+1U. (21) Finally, inserting (21) into (20), we obtain (16).□ By substituting (16) into (14), we obtain a new unified expression for the objective function ρ∈{Cmax,∑Cj,∑Wj,∑∑∣Ci−Cj∣,∑∑∣Wi−Wj∣,∑(α1Ej+α2Tj+α3dj),∑(α1Ej+α2Tj+α3dj1+α4Dj} under an optimal resource allocation and as a function of the job sequence: Z(π,u*(π))=U−k∑j=1n(Ωj)1k+1(w[j]G[j])kk+1k+1. (22) In order to find the job sequence that minimizes Z(π,u*(π)), we have to optimally match the positional penalties (Ωj)1k+1 with the job-dependent costs (w[j]G[j])kk+1. The optimal matching can be solved by sorting and matching the jth smallest wjGj to the jth largest Ωj (i.e. the HLP rule, Hardy et al. [43]). The results of our analysis are summarized in the following optimization algorithm that solves the problem 1pj=wjujk+bt,∑Gjuj≤Uρ, where ρ∈{Cmax,∑Cj,∑Wj,∑∑∣Ci−Cj∣, ∑∑∣Wi−Wj∣,∑(α1Ej+α2Tj+α3dj),∑(α1Ej+α2Tj+α3dj1+α4Dj} Algorithm 1 Calculate ωj and Ωj by Equation (15) for j=1,2,…,n. Sequence the jobs according to the HLP rule (see Equation (22)), and the resulting optimal sequence denoted by π*=[J[1],J[2],…,J[n]]. Calculate the optimal resources allocation u[j]*(π*) by using Equation (16). Calculate ωj and Ωj by Equation (15) for j=1,2,…,n. Sequence the jobs according to the HLP rule (see Equation (22)), and the resulting optimal sequence denoted by π*=[J[1],J[2],…,J[n]]. Calculate the optimal resources allocation u[j]*(π*) by using Equation (16). Algorithm 1 Calculate ωj and Ωj by Equation (15) for j=1,2,…,n. Sequence the jobs according to the HLP rule (see Equation (22)), and the resulting optimal sequence denoted by π*=[J[1],J[2],…,J[n]]. Calculate the optimal resources allocation u[j]*(π*) by using Equation (16). Calculate ωj and Ωj by Equation (15) for j=1,2,…,n. Sequence the jobs according to the HLP rule (see Equation (22)), and the resulting optimal sequence denoted by π*=[J[1],J[2],…,J[n]]. Calculate the optimal resources allocation u[j]*(π*) by using Equation (16). . Theorem 1 The problem 1pj=wjujk+bt,∑Gjuj≤Uρcan be solved by Algorithm1in O(nlogn)time, where ρ∈{Cmax,∑Cj,∑Wj,∑∑∣Ci−Cj∣,∑∑∣Wi−Wj∣,∑(α1Ej+α2Tj+α3dj),∑(α1Ej+α2Tj+α3dj1+α4Dj}. Proof The correctness of Algorithm 1 follows from Lemma 1 and the above analysis. Steps 1 and 3 require O(n) time and Step 2 requires O(nlogn) time (see Hardy et al. [43]). Thus the total computational complexity of Algorithm 1 is O(nlogn).□ Remark 3.1 Obviously, the result can be extended to any linear combination of objective function, for example the problem 1pj=wjujk+bt,∑Gjuj≤Uα1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣. For the 1pj=wjujk+bt,∑Gjuj≤Uα1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣ problem, the computational process is illustrated by the following example. Example 1 Consider n=7 jobs, w1=15,w2=27,w3=12,w4=20,w5=30,w6=25,w7=40, Gj=1 ( j=1,2,…,7), α1=α2=α3=1,k=2, b=0.1,U=30. From Algorithm 1, ω1=8,ω2=13,ω3=16,ω4=17,ω5=16,ω6=13,ω7=8, Ω1=8+0.1∗13+0.1∗1.1∗16+0.1∗1.12∗17+⋯+0.1∗1.15∗8=18.4383Ω2=13+0.1∗16+0.1∗1.1∗17+0.1∗1.12∗16+0.1∗1.13∗13+0.1∗1.14∗8=21.3076Ω3=16+0.1∗17+0.1∗1.1∗16+0.1∗1.12∗13+0.1∗1.13∗8=22.0978Ω4=17+0.1∗16+0.1∗1.1∗13+0.1∗1.12∗8=20.9980Ω5=16+0.1∗13+0.1∗1.1∗8=18.1800Ω6=13+0.1∗8=13.8000Ω7=8, the optimal schedule is [J6,J1,J3,J4,J2,J5,J7], the optimal resources are u6=(18.4383)13(25)23×30(18.4383)13(25)23+(21.3076)13(15)23+⋯+(8)13(40)23=4.6829,u1=3.4959u3=3.0494u4=4.2143u2=4.9063u5=4.8013u7=4.8498, the optimal cost Z1=α1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣= 3366.426. 4. PROBLEM 1pj=wjujk+bt,ρ≤R∑Gjuj From Section 3, the problem 1pj=wjujk+bt,ρ≤R∑Gjuj is equivalent to (P)MinZ(π,u)=∑j=1nG[j]u[j] (23) s.t.∑j=1nΩjw[j]u[j]k≤R. In the following lemma, we determine the optimal resource allocation, denoted by u*(π), as a function of the job sequence π. Lemma 2 For the problem 1pj=wjujk+bt,ρ≤R∑Gjuj, where ρ∈{Cmax,∑Cj,∑Wj, ∑∑∣Ci−Cj∣,∑∑∣Wi−Wj∣,∑(α1Ej+α2Tj+α3dj),∑(α1Ej+α2Tj+α3dj1+α4Dj}, the optimal resource allocation as a function of the job sequence, u*(π), is u[j]*=(Ωj)1k+1(G[j])−1k+1(w[j])kk+1∑j=1n(Ωj)1k+1(w[j]G[j])kk+11k×R−1k,j=1,2,…,n, (24) Ωj are given by Equation (15). Algorithm 2 Calculate Ωj by Equation (15) for j=1,2,…,n. Sequence the jobs according to the HLP rule (see Equation (30)), and the resulting optimal sequence denoted by π*=[J[1],J[2],…,J[n]]. Calculate the optimal resources allocation u[j]*(π*) by Equation (24). Calculate Ωj by Equation (15) for j=1,2,…,n. Sequence the jobs according to the HLP rule (see Equation (30)), and the resulting optimal sequence denoted by π*=[J[1],J[2],…,J[n]]. Calculate the optimal resources allocation u[j]*(π*) by Equation (24). Algorithm 2 Calculate Ωj by Equation (15) for j=1,2,…,n. Sequence the jobs according to the HLP rule (see Equation (30)), and the resulting optimal sequence denoted by π*=[J[1],J[2],…,J[n]]. Calculate the optimal resources allocation u[j]*(π*) by Equation (24). Calculate Ωj by Equation (15) for j=1,2,…,n. Sequence the jobs according to the HLP rule (see Equation (30)), and the resulting optimal sequence denoted by π*=[J[1],J[2],…,J[n]]. Calculate the optimal resources allocation u[j]*(π*) by Equation (24). Proof Obviously, for an optimal solution of the problem 1pj=wjujk+bt,ρ≤R∑Gjuj, ρ=∑j=1nΩjw[j]u[j]k=R is a sufficient and necessary condition. Hence, for any given sequence, the Lagrange function is L(u,λ)=∑j=1nG[j]u[j]+λ(ρ−R)=∑j=1nG[j]u[j]+λ(∑j=1nΩj(w[j]u[j])k−R), (25) where λ is the Lagrangian multiplier. Deriving (25) with respect to the decision variables (u[j] and λ), we have ∂L(u,λ)∂u[j]=G[j]−λkΩj×(w[j])k(u[j])k+1=0,∀j=1,2,…,n. (26) ∂L(u,λ)∂λ=∑j=1nΩj(w[j]u[j])k−R=0. (27) Using (26) and (27), we obtain u[j]=(λkΩj(G[j])−1(w[j])k)1/(k+1) (28) and (λk)k/(k+1)=∑j=1nΩj(w[j])kR(Ωj(G[j])−1(w[j])k)k/(k+1). (29) Finally, inserting (29) into (28), we obtain (24).□ By substituting (24) into (23), we obtain a new unified expression for 1pj=wjujk+bt,ρ≤R∑Gjuj ( ρ∈{Cmax,∑Cj,∑Wj,∑∑∣Ci−Cj∣,∑∑∣Wi−Wj∣,∑(α1Ej+α2Tj+α3dj),∑(α1Ej+α2Tj+α3dj1+α4Dj}) under an optimal resource allocation and as a function of the job sequence: Z(π,u*(π))=R−1k∑j=1n(Ωj)1k+1(w[j]G[j])kk+11k+1. (30) Similar to Section 3, the problem 1pj=wjujk+bt,ρ≤R∑Gjuj can be solved by the following optimization algorithm, where ρ∈{Cmax,∑Cj,∑Wj,∑∑∣Ci−Cj∣,∑∑∣Wi−Wj∣,∑(α1Ej+α2Tj+α3dj),∑(α1Ej+α2Tj+α3dj1+α4Dj}. Theorem 2 The problem 1pj=wjujk+bt,ρ≤R∑Gjujcan be solved by Algorithm2in O(nlogn)time, where ρ∈{Cmax,∑Cj,∑Wj,∑∑∣Ci−Cj∣,∑∑∣Wi−Wj∣,∑(α1Ej+α2Tj+α3dj),∑(α1Ej+α2Tj+α3dj1+α4Dj}. Proof Similar to the proof of Theorem 1.□ Remark 4.1 Obviously, the result can be extended to any linear combination of objective function, for example the problem 1pj=wjujk+bt,α1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣≤R∑Gjuj. Similar to the proof of Sections 3 and 4 , we will show that a large set of scheduling problems which have the common feature that the scheduling objective function can be expressed by using positional penalties (Yedidsion et al. [44]). This unified cost function has the following simple formulation: Z(π,u)=∑j=1nϖjp[j], (31) where ϖj is a position, job-dependent penalty for any job schedule in the jth position. 5. CONCLUSIONS In this paper, we have studied single-machine scheduling problems with deteriorating jobs and convex resource-dependent processing times. Specifically, we considered two versions of the proposed model. The main results are summarized in Table 1. In future research, we plan to explore more realistic settings such as the set of Pareto optimal scheduling (points) (V,Z), where V(π)=∑j=1nGjuj, Z(π)={Cmax,∑Cj,∑Wj,∑∑∣Ci−Cj∣,∑∑∣Wi−Wj∣,∑(α1Ej+α2Tj+α3dj),∑(α1Ej+α2Tj+α3dj1+α4Dj}, and a schedule π with V=V(π) and Z=Z(π) is called Pareto optimal (or efficient) if there does not exist another schedule π′ such that V(π′)≤V(π) and Z(π′)≤Z(π) with at least one of these both inequalities being strict. Table 1. Summary of main results. Problem Complexity Ref. 1∣pj=aj+bt−θuj∣α1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣+α4∑Gjuj O(n3) Wei et al. [30] 1∣pj=aj+bt−θuj∣α1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣+α4∑Gjuj O(n3) Wei et al. [30] 1pj=wjujk+btα1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣+α4∑Gjuj O(nlogn) Wang and Wang [31] 1pj=wjujk+btα1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣+α4∑Gjuj O(nlogn) Wang and Wang [31] 1∣pj=aj+bt−θjuj,CON/SLK/DIF∣∑(α1Ej+α2Tj+α3dj+α4∑Gjuj) O(n3) Wang and Wang [32] 1pj=wjujk+bt,CON/SLK/DIF∑(α1Ej+α2Tj+α3dj+α4∑Gjuj) O(nlogn) Wang and Wang [32] 1pj=wjujk+bt,∑uj≤UZ,Z∈{Cmax,∑Cj,TADC,TADW} O(nlogn) Zhao et al. [33] 1pj=wjujk+bt,∑uj≤U,CON/SLK/DIF/∑(α1Ej+α2Tj+α3dj) O(nlogn) Zhao et al. [33] 1pj=wjujk+bt,∑uj≤U,CONW∑(α1Ej+α2Tj+α3d+α4D) O(nlogn) Zhao et al. [33] 1pj=wjujk+bt,∑Gjuj≤Uα1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣ O(nlogn) Theorem 1 1pj=wjujk+bt,∑Gjuj≤Uα1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣ O(nlogn) Theorem 2 1∣pj=wjujk+bt,∑Gjuj≤U,CON/SLK/DIF/CONW/SLKW/DIFW∣∑(α1Ej+α2Tj+α3dj) O(nlogn) Extensions 1pj=wjujk+bt,α1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣≤C∑Gjuj O(nlogn) Theorem 3 1pj=wjujk+bt,α1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣≤W∑Gjuj O(nlogn) Theorem 4 1∣pj=wjujk+bt,∑(α1Ej+α2Tj+α3dj)≤D,CON/SLK/DIF/CONW/SLKW/DIFW∣∑Gjuj O(nlogn) Extensions Problem Complexity Ref. 1∣pj=aj+bt−θuj∣α1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣+α4∑Gjuj O(n3) Wei et al. [30] 1∣pj=aj+bt−θuj∣α1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣+α4∑Gjuj O(n3) Wei et al. [30] 1pj=wjujk+btα1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣+α4∑Gjuj O(nlogn) Wang and Wang [31] 1pj=wjujk+btα1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣+α4∑Gjuj O(nlogn) Wang and Wang [31] 1∣pj=aj+bt−θjuj,CON/SLK/DIF∣∑(α1Ej+α2Tj+α3dj+α4∑Gjuj) O(n3) Wang and Wang [32] 1pj=wjujk+bt,CON/SLK/DIF∑(α1Ej+α2Tj+α3dj+α4∑Gjuj) O(nlogn) Wang and Wang [32] 1pj=wjujk+bt,∑uj≤UZ,Z∈{Cmax,∑Cj,TADC,TADW} O(nlogn) Zhao et al. [33] 1pj=wjujk+bt,∑uj≤U,CON/SLK/DIF/∑(α1Ej+α2Tj+α3dj) O(nlogn) Zhao et al. [33] 1pj=wjujk+bt,∑uj≤U,CONW∑(α1Ej+α2Tj+α3d+α4D) O(nlogn) Zhao et al. [33] 1pj=wjujk+bt,∑Gjuj≤Uα1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣ O(nlogn) Theorem 1 1pj=wjujk+bt,∑Gjuj≤Uα1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣ O(nlogn) Theorem 2 1∣pj=wjujk+bt,∑Gjuj≤U,CON/SLK/DIF/CONW/SLKW/DIFW∣∑(α1Ej+α2Tj+α3dj) O(nlogn) Extensions 1pj=wjujk+bt,α1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣≤C∑Gjuj O(nlogn) Theorem 3 1pj=wjujk+bt,α1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣≤W∑Gjuj O(nlogn) Theorem 4 1∣pj=wjujk+bt,∑(α1Ej+α2Tj+α3dj)≤D,CON/SLK/DIF/CONW/SLKW/DIFW∣∑Gjuj O(nlogn) Extensions View Large Table 1. Summary of main results. Problem Complexity Ref. 1∣pj=aj+bt−θuj∣α1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣+α4∑Gjuj O(n3) Wei et al. [30] 1∣pj=aj+bt−θuj∣α1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣+α4∑Gjuj O(n3) Wei et al. [30] 1pj=wjujk+btα1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣+α4∑Gjuj O(nlogn) Wang and Wang [31] 1pj=wjujk+btα1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣+α4∑Gjuj O(nlogn) Wang and Wang [31] 1∣pj=aj+bt−θjuj,CON/SLK/DIF∣∑(α1Ej+α2Tj+α3dj+α4∑Gjuj) O(n3) Wang and Wang [32] 1pj=wjujk+bt,CON/SLK/DIF∑(α1Ej+α2Tj+α3dj+α4∑Gjuj) O(nlogn) Wang and Wang [32] 1pj=wjujk+bt,∑uj≤UZ,Z∈{Cmax,∑Cj,TADC,TADW} O(nlogn) Zhao et al. [33] 1pj=wjujk+bt,∑uj≤U,CON/SLK/DIF/∑(α1Ej+α2Tj+α3dj) O(nlogn) Zhao et al. [33] 1pj=wjujk+bt,∑uj≤U,CONW∑(α1Ej+α2Tj+α3d+α4D) O(nlogn) Zhao et al. [33] 1pj=wjujk+bt,∑Gjuj≤Uα1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣ O(nlogn) Theorem 1 1pj=wjujk+bt,∑Gjuj≤Uα1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣ O(nlogn) Theorem 2 1∣pj=wjujk+bt,∑Gjuj≤U,CON/SLK/DIF/CONW/SLKW/DIFW∣∑(α1Ej+α2Tj+α3dj) O(nlogn) Extensions 1pj=wjujk+bt,α1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣≤C∑Gjuj O(nlogn) Theorem 3 1pj=wjujk+bt,α1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣≤W∑Gjuj O(nlogn) Theorem 4 1∣pj=wjujk+bt,∑(α1Ej+α2Tj+α3dj)≤D,CON/SLK/DIF/CONW/SLKW/DIFW∣∑Gjuj O(nlogn) Extensions Problem Complexity Ref. 1∣pj=aj+bt−θuj∣α1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣+α4∑Gjuj O(n3) Wei et al. [30] 1∣pj=aj+bt−θuj∣α1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣+α4∑Gjuj O(n3) Wei et al. [30] 1pj=wjujk+btα1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣+α4∑Gjuj O(nlogn) Wang and Wang [31] 1pj=wjujk+btα1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣+α4∑Gjuj O(nlogn) Wang and Wang [31] 1∣pj=aj+bt−θjuj,CON/SLK/DIF∣∑(α1Ej+α2Tj+α3dj+α4∑Gjuj) O(n3) Wang and Wang [32] 1pj=wjujk+bt,CON/SLK/DIF∑(α1Ej+α2Tj+α3dj+α4∑Gjuj) O(nlogn) Wang and Wang [32] 1pj=wjujk+bt,∑uj≤UZ,Z∈{Cmax,∑Cj,TADC,TADW} O(nlogn) Zhao et al. [33] 1pj=wjujk+bt,∑uj≤U,CON/SLK/DIF/∑(α1Ej+α2Tj+α3dj) O(nlogn) Zhao et al. [33] 1pj=wjujk+bt,∑uj≤U,CONW∑(α1Ej+α2Tj+α3d+α4D) O(nlogn) Zhao et al. [33] 1pj=wjujk+bt,∑Gjuj≤Uα1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣ O(nlogn) Theorem 1 1pj=wjujk+bt,∑Gjuj≤Uα1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣ O(nlogn) Theorem 2 1∣pj=wjujk+bt,∑Gjuj≤U,CON/SLK/DIF/CONW/SLKW/DIFW∣∑(α1Ej+α2Tj+α3dj) O(nlogn) Extensions 1pj=wjujk+bt,α1Cmax+α2∑Cj+α3∑∑∣Ci−Cj∣≤C∑Gjuj O(nlogn) Theorem 3 1pj=wjujk+bt,α1Cmax+α2∑Wj+α3∑∑∣Wi−Wj∣≤W∑Gjuj O(nlogn) Theorem 4 1∣pj=wjujk+bt,∑(α1Ej+α2Tj+α3dj)≤D,CON/SLK/DIF/CONW/SLKW/DIFW∣∑Gjuj O(nlogn) Extensions View Large Also application of job deterioration to the assignment problem in Yedidsion et al. [44], both for linear and convex resource consumption functions can be a subject for future researches. FUNDING This research was supported by the National Natural Science Foundation of China (Grant nos. 71471120 and 71501082), and the Support Program for Innovative Talents in Liaoning University (Grant no. LR2016017). 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Google Scholar CrossRef Search ADS 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)

The Computer Journal – Oxford University Press

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