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The Computer Journal
, Volume Advance Article – Dec 29, 2017

9 pages

/lp/ou_press/single-machine-con-slk-due-date-assignment-scheduling-with-zIknapuJvl

- Publisher
- Oxford University Press
- Copyright
- © The British Computer Society 2017. All rights reserved. For permissions, please email: journals.permissions@oup.com
- ISSN
- 0010-4620
- eISSN
- 1460-2067
- D.O.I.
- 10.1093/comjnl/bxx121
- Publisher site
- See Article on Publisher Site

Abstract This paper addresses resource allocation scheduling problems with CON/SLK due date assignment and job-dependent learning effects on a single machine. Two different resource consumption functions under common (denoted by CON) and slack (denoted by SLK) due date assignment are considered. The objective is to determine the optimal due dates of jobs, a sequence for jobs and resource allocation of jobs to minimize a total cost function. The optimality properties for all problems and polynomial time algorithms are proposed to solve these problems. 1. INTRODUCTION Scheduling with resource allocation (Shabtay and Steiner [1]) and/or learning effects (Biskup [2]) has been extensively studied in the literature. Recently, Wang and Wang [3] considered single-machine scheduling problem with convex resource allocation model. Yin et al. [4] considered single-machine resource allocation problem with common due-date assignment and batch delivery costs. Yin et al. [5] considered single-machine resource allocation bi-criterion problem with slack due-window assignment method. Yang et al. [6] and Oron [7] considered scheduling problems with controllable processing times under a deteriorating environment. Oron [8] considered scheduling problems with controllable processing times and position-dependent workloads. Pei et al. [9, 10] considered single-machine serial-batching scheduling with learning effects. Wang et al. [11] considered single-machine resource allocation problems with truncated job-dependent learning and deterioration effects. Zhao et al. [12] and Wang et al. [13] considered single-machine scheduling problems with resource-dependent maintenance activity. For most scheduling problems, due date assignment methods have drawn increasing attention (Gordon et al. [14–16], Yin et al. [17] and Shabtay [18]), where job scheduling on machines in a ‘Just-In-Time (JIT)’ production is adopted. ‘In a JIT system, jobs are to be completed neither too early nor too late, which leads to the scheduling problem involving both earliness and tardiness costs and the cost of assigning due dates. Completing a job early means having to bear the costs of holding unnecessary inventories, while finishing a job late results in a contractual penalty and a loss of customer goodwill’ (Lu et al. [19], and Liu et al. [20]). Brucker [21] considered the single-machine common (CON) due date scheduling problem, the objective function is to minimize the total cost that comprises the total weighted absolute value in lateness and CON due date cost, where the weight is a position-dependent weight. Using the extended three-field notation scheme (Graham et al. [22]), the problem can be denoted by 1∣CON,dopt∣∑i=1nωi∣Lπ(i)∣+ω0dopt, where dopt is the common due date, π(i) is a job scheduled in position i on the machine, ωi ( i=0,1,2,…,n) is the non-negative weight of the i th position in a sequence. Brucker [21] proved that the problem 1∣CON,dopt∣∑i=1nωi∣Lπ(i)∣+ω0dopt is solved in O(nlogn) time. Liu et al. [20] considered the same scheduling model as Brucker [21], for the slack (SLK) due date scheduling problem, Liu et al. [20] proved that the problem 1∣SLK,qopt∣∑i=1nωi∣Lπ(i)∣+ω0qopt can also be solved in O(nlogn) time, where di=piA+qopt ( i=1,2,…,n) is the due date di for job Ji, and the common flow allowance qopt is a decision variable to be determined. Liu et al. [20] proved that the problem 1∣SLK,qopt∣∑i=1nωi∣Lπ(i)∣+ω0qopt is solved in O(nlogn) time. ‘However, the phenomena of resource allocation and learning effects occurring simultaneously can be found in many real-life situations. For example, in the chemical industry, the processing time of a chemical compound can be changed by increasing the amount of catalysts, which entails some extra costs’ (Wang and Cheng [23]). ‘Clearly, compressing jobs would be rational and possible only if the additional cost is compensated by the gains from job completion at an earlier time. The scheduling problem with controllable processing times is concerned with determining not only the job sequence, but also the amount of compression for each job so as to minimise the total cost. On the other hand, the learning effects reflect that the workers become more skilled to operate the machines through experience accumulation. For this situation, considering scheduling problems with learning effects and resource allocation is both necessary and reasonable’ (Wang et al. [24]). Wang and Wang [25, 26] and Li et al. [27] considered single-machine scheduling problems with resource allocation and learning effects simultaneously. Wang and Wang [25] considered common due window (CONW) assignment single-machine scheduling problems. For linear and and convex resource consumption functions, they proved that some problems can be solved in polynomial time. Wang and Wang [26] considered due date assignment single machine scheduling problems. For convex resource consumption function and three due date assignment methods (CON/SLK/DIF, where DIF denotes unrestricted (different) due dates assignment), they proved that some problems can be solved in polynomial time. Li et al. [27] considered single-machine slack due window (SLKW) assignment scheduling problems. It is natural and interesting to continue the work of Brucker [21] and Liu et al. [20], but consider the CON/SLK due date assignment scheduling problems with resource allocation and job-dependent learning effects. It is showed that the single-machine problems for the linear and convex resource consumption functions can be solved in polynomial time. Section 2 formulates the model. Sections 3 and 4 give some basic results for the CON due date assignment method and the SLK due date assignment method. In Section 5, conclusions are presented. 2. PROBLEM DESCRIPTION There is a set N={J1,J2,…,Jn} of n independent and non-preemptive jobs to be processed on a single machine, and all the jobs are available at time 0. Biskup [28] considered the following job-independent learning index model where the actual processing time of job Ji is piA=pirα where piA, pi and α≤0, respectively, denote the actual processing time, the normal processing time (i.e. the processing time without any resource allocation and learning effects), the position-dependent learning index and r is the position of job Ji which is scheduled in a sequence. Mosheiov and Sidney [29] considered the job-dependent learning index model where the actual processing time of job Ji is piA=pirαi, where αi≤0 ( i=1,2,…,n) denote the job-dependent learning index. Shabtay and Steiner [1] considered the linear resource consumption function model where the actual processing time of job Ji is piA=pi−βiui, where βi ( i=1,2,…,n) denote the positive compression rate of job Ji and ui is the amount of resource that can be allocated to job Jj, with 0≤ui≤u¯i<piβi, u¯j is the upper bound on the amount of resource that can be allocated to job Jj. Shabtay and Steiner [1] considered the convex resource consumption function model where the actual processing time of job Ji is piA=(pi/ui)η, where η is a positive constant. Following Wang et al. [24] and Lu et al. [19], for linear resource consumption function, the actual processing time of job Ji is piA=pirαi−βiui, (1) where βi ( i=1,2,…,n) denote the positive compression rate of job Ji, r is the position of job Ji which is scheduled in a sequence and ui is the amount of resource that can be allocated to job Jj, with 0≤ui≤u¯i<pinαiβi, u¯j is the upper bound on the amount of resource that can be allocated to job Jj. Following Wang et al. [24] and Lu et al. [19], for convex resource consumption function, the actual processing time of job Jj is piA=pirαiuiη, (2) where η is a positive constant. 3. CON (COMMON) DUE DATE ASSIGNMENT For the CON (common) due date assignment method, the due dates are equal to each other, i.e. di=dopt, i=1,2,…,n, where di is the due date of job Ji, the common due date dopt is a decision variable to be determined. The problem is to determine the sequence of jobs and dopt so that the following objective function: ∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i)=∑i=1nωi∣Cπ(i)−dπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i) (3) being minimized, where π(i) is the job scheduled in position i on the machine, ωi ( i=0,1,2,…,n) is the non-negative weight of the i th position in a sequence, vi ( i=1,2,…,n) is the per time unit cost associated with the resource allocation. Let Ci be the completion time of job Ji, Li=Ci−di be the lateness of job Ji, using the three-field notation of Graham et al. [22], the problems can be denoted as 1∣CON,dopt,piA=pirαi−βiui∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i) and 1∣CON,dopt,piA=(pirαi/ui)η∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i). First, some lemmas for the 1∣CON,dopt,piA∈{pirαi−βiui,(pirαi/ui)η}∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i) problem are given. Lemma 1 (Brucker [21] and Liu et al. [20]). In an optimal schedule, there exists no-idle time between the processing of jobs and the first job starts at time zero. It is convenient to introduce a dummy job J0 with processing time p0=0 and weight ω0 which is always scheduled at time 0, i.e. π(0)=0, then ∑i=1nωi∣Cπ(i)−dπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i)=∑i=0nωi∣Cπ(i)−dπ(i)∣+∑i=1nvπ(i)uπ(i) We conclude that an optimal schedule is given by a sequence [π(0),π(1),…,π(n)], where π(0)=0. Lemma 2 (Brucker [21] and Liu et al. [20]). For a given sequence π=[π(0),π(1),…,π(n)], the optimal value of doptis equal to the completion time of a job, i.e. dopt=Cπ(k)=∑i=0kpπ(i)A, where kis a median for the sequence ω0,ω1,…,ωn, ∑j=0k−1ωj≤∑j=knωjand∑j=0kωj≥∑j=k+1nωj. (4) Lemma 3 The objective function of the 1∣CON,dopt,piA∈{pirαi−βiui,(pirαi/ui)η}∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i)problem can be written as ∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i)=∑i=1nWipπ(i)A+∑i=1nvπ(i)uπ(i), (5) where the positional weight of position j in the schedule is given by Wi=∑h=0i−1ωh,fori=1,…,k;∑h=inωh,fori=k+1,…,n. (6) Proof For a given sequence π=[π(0),π(1),…,π(n)] and qopt=Cπ(k), we have ∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i)=ω0Cπ(k)+∑i=1kωi(Cπ(k)−Cπ(i))+∑i=k+1nωi(Cπ(i)−Cπ(k))+∑i=1nvπ(i)uπ(i)=∑i=0kωi∑h=i+1kpπ(h)A+∑i=k+1nωi∑h=k+1ipπ(h)A+∑i=1nvπ(i)uπ(i)=∑h=1kpπ(h)A∑i=0h−1ωi+∑h=k+1npπ(h)A∑i=hnωi+∑i=1nvπ(i)uπ(i)=∑i=1nWipπ(i)A+∑i=1nvπ(i)uπ(i), where Wi=∑h=0i−1ωh,fori=1,…,k;∑h=inωh,fori=k+1,…,n. □ 3.1. The problem 1∣CON,dopt,piA=pirαi−βiui∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i) From Lemma 3 and pπ(i)A=pπ(i)iαπ(i)−βπ(i)uπ(i), we have ∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i)=∑i=1nWipπ(i)A+∑i=1nvπ(i)uπ(i)=∑i=1nWi(pπ(i)iαπ(i)−βπ(i)uπ(i))+∑i=1nvπ(i)uπ(i)=∑i=1nWipπ(i)iαπ(i)+∑i=1n(vπ(i)−Wiβπ(i))uπ(i). (7) From (7), for a given sequence, the optimal resource allocation of a job in a position with a non-positive (positive) vπ(i)−βπ(i) should be its upper (lower) bound on the amount of resource u¯π(i) (0), i.e. the optimal resource allocation of job Jπ(i) is uπ(i)*=u¯π(i),ifvπ(i)−Wiβπ(i)≤0,0,ifvπ(i)−Wiβπ(i)>0. (8) For a given sequence, from (8), we can obtain the optimal resource allocation. In order to determine the optimal sequence, we formulate the 1∣CON,dopt,piA=pirαi−βiui∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i) problem as a linear assignment problem. Let Wji=Wipjiαj,ifvj−Wiβj>0,Wipjiαj+(vj−Wiβj)u¯j,ifvj−Wiβj≤0. (9) From (7) and (8), we can formulate the problem 1∣CON,dopt,piA=pirαi−βiui∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i) as the following linear assignment problem: Minimize∑j=1n∑i=1nWjixji (10) subjectto∑j=1nxji=1,i=1,2,…,n, (11) ∑i=1nxji=1,j=1,2,…,n, (12) xji∈{0,1},j=1,2,…,n;i=1,2,…,n, (13) where xji∈{0,1} such that xji=1 if job Jj is scheduled in position i, and xji=0, otherwise. Based on the results of our analysis, the problem 1∣CON,dopt,piA=pirαi−βiui∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i) can be optimally solved by the following algorithm. Theorem 1 Algorithm1solves the 1∣CON,dopt,piA=pirαi−βiui∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i)problem in O(n3)time. Algorithm 1 Step 1. By Lemma 2, calculate k. Step 2. Calculate the Wji by using Equation (9). Step 3. Solve the linear assignment problem Equations (10)–(13) to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (8), and the optimal processing times by using Equation (1). Step 5. Set dopt=∑i=0kpπ(i)A. Step 1. By Lemma 2, calculate k. Step 2. Calculate the Wji by using Equation (9). Step 3. Solve the linear assignment problem Equations (10)–(13) to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (8), and the optimal processing times by using Equation (1). Step 5. Set dopt=∑i=0kpπ(i)A. Algorithm 1 Step 1. By Lemma 2, calculate k. Step 2. Calculate the Wji by using Equation (9). Step 3. Solve the linear assignment problem Equations (10)–(13) to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (8), and the optimal processing times by using Equation (1). Step 5. Set dopt=∑i=0kpπ(i)A. Step 1. By Lemma 2, calculate k. Step 2. Calculate the Wji by using Equation (9). Step 3. Solve the linear assignment problem Equations (10)–(13) to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (8), and the optimal processing times by using Equation (1). Step 5. Set dopt=∑i=0kpπ(i)A. Proof The correctness of the algorithm follows from the above analysis. Steps 1, 4 and 5 require linear time, Steps 2 and 3 require O(n3) time. Thus, the overall computational complexity of the algorithm is O(n3). □ 3.2. The problem 1∣CON,dopt,piA=(pirαi/ui)η∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i) From Lemma 3 and pπ(i)A=(pπ(i)iαπ(i)/uπ(i))η, we have ∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i)=∑i=1nWipπ(i)A+∑i=1nvπ(i)uπ(i)=∑i=1nWipπ(i)iαπ(i)uπ(i)η+∑i=1nvπ(i)uπ(i). (14) By taking the first derivative of the objective given by (14) with respect to uπ(i),i=1,2,…,n, equating it to zero and solving it for uπ(i), we have Lemma 4 For the 1∣CON,dopt,piA=(pirαi/ui)η∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i)problem, the optimal resource allocation as a function of the job sequence, that is uπ(i)*=ηWivπ(i)1η+1×(pπ(i)iαπ(i))ηη+1, (15)where Wi(i=1,2,…,n)are given by Equation (6). By substituting (15) into (14), we have ∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i)=η−ηη+1+η1η+1×∑i=1n(vπ(i)pπ(i)iαπ(i))ηη+1(Wi)1η+1, (16) where Wi(i=1,2,…,n) are given by Equation (6). Let ϒji=η−ηη+1+η1η+1×(vjpjiαj)ηη+1(Wi)1η+1. (17) As in Section 3.1, the 1∣CON,dopt,piA=(pirαi/ui)η∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i) problem can be solved by the following linear assignment problem: Minimize∑j=1n∑i=1nϒjixjisubjectto(11),(12),(13). (18) Similar to Section 3.1, for the 1∣CON,dopt,piA=(pirαi/ui)η∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i) problem, we can propose the following algorithm: Theorem 2 Algorithm2solves the 1∣CON,dopt,piA=(pirαi/ui)η∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i)problem in O(n3)time. Algorithm 2 Step 1. By Lemma 2, calculate k. Step 2. Calculate the ϒji by using Equation (17). Step 3. Solve the linear assignment problem Equations (18), (11)–(13) to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (15), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. Step 1. By Lemma 2, calculate k. Step 2. Calculate the ϒji by using Equation (17). Step 3. Solve the linear assignment problem Equations (18), (11)–(13) to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (15), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. Algorithm 2 Step 1. By Lemma 2, calculate k. Step 2. Calculate the ϒji by using Equation (17). Step 3. Solve the linear assignment problem Equations (18), (11)–(13) to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (15), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. Step 1. By Lemma 2, calculate k. Step 2. Calculate the ϒji by using Equation (17). Step 3. Solve the linear assignment problem Equations (18), (11)–(13) to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (15), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. Lemma 5 (Hardy et al. [30]). The sum of products ∑j=1nxjyjis minimized if sequence x1,x2,…,xnis ordered non-decreasingly and sequence y1,y2,…,ynis ordered non-increasingly or vice versa, and it is maximized if the sequences are ordered in the same way. From (16), for a special case αi=α (i.e. job-independent learning index), i=1,2,…,n, we have ∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i)=η−ηη+1+η1η+1×∑i=1nνiμπ(i), (19) where νi=(Wi)1η+1(iα)ηη+1 (20) and μπ(i)=(vπ(i)pπ(i))ηη+1. (21) The term (19) can be minimized by Lemma 5, hence 1∣CON,dopt,piA=(pirαi/ui)η,αi=α∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i) can be solved by the following algorithm: Theorem 3 Algorithm3solves the 1∣CON,dopt,piA=(pirαi/ui)η,αi=α∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i)problem in O(nlogn)time. Algorithm 3 Step 1. By Lemma 2, calculate k. Step 2. Calculate νi and μi for i=1,2,…,n by Equations (20) and (21). Step 3. By using Lemma 5 to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (15), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. Step 1. By Lemma 2, calculate k. Step 2. Calculate νi and μi for i=1,2,…,n by Equations (20) and (21). Step 3. By using Lemma 5 to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (15), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. Algorithm 3 Step 1. By Lemma 2, calculate k. Step 2. Calculate νi and μi for i=1,2,…,n by Equations (20) and (21). Step 3. By using Lemma 5 to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (15), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. Step 1. By Lemma 2, calculate k. Step 2. Calculate νi and μi for i=1,2,…,n by Equations (20) and (21). Step 3. By using Lemma 5 to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (15), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. Proof The correctness of the algorithm follows from above analysis. Steps 1, 2, 4 and 5 can be performed in linear time and Step 3 requires O(nlogn) time. Thus, the overall computational complexity of the algorithm is O(nlogn).□ 3.3. The problem 1∣CON,dopt,piA=(pirαi/ui)η,∑i=1nui≤U∣∑i=1nωi∣Lπ(i)∣+ω0dopt In this subsection, we study a constrained version, where there exists an upper bound on the amount of resource available, i.e. ∑i=1nui≤U, where U is an upper bound on the amount of resource available. Obviously, ∑i=1nui=U is a necessary and sufficient condition for an optimal solution to the problem 1∣CON,dopt,piA=(pirαi/ui)η,∑i=1nui≤U∣∑i=1nωi∣Lπ(i)∣+ω0dopt. Lemma 6 For the 1∣CON,dopt,piA=(pirαi/ui)η,∑i=1nui≤U∣∑i=1nωi∣Lπ(i)∣+ω0doptproblem, the optimal resource allocation as a function of the job sequence, that is uπ(i)*=(Wi)1η+1(pπ(i)iαπ(i))ηη+1∑i=1n(Wi)1η+1(pπ(i)iαπ(i))ηη+1×U,i=1,2,…,n, (22) where Wi(i=1,2,…,n) are given by Equation (6). Proof For any given sequence, from (14), the Lagrange function is L(uπ(i),λ)=∑i=1nωi∣Lπ(i)∣+ω0dopt+λ∑i=1nuπ(i)−U=∑i=1nWipπ(i)iαπ(i)uπ(i)η+λ∑i=1nuπ(i)−U, (23) where λ is the Lagrangian multiplier. Deriving (23) with respect to uπ(i) and λ, we have ∂L(uπ(i),λ)∂λ=∑i=1nuπ(i)−U=0 (24) ∂L(uπ(i),λ)∂uπ(i)=λ−ηWi×(pπ(i)iαπ(i))η(uπ(i))η+1=0,∀i=1,2,…,n. (25) Using (24) and (25), we have uπ(i)=(ηWi(pπ(i)iαπ(i))η)1/(η+1)(λ)1/(η+1) (26) and (λ)1/(η+1)=∑i=1n(ηWi)1/(η+1)(pπ(i)iαπ(i))η/(η+1)U. (27) From (26) and (27), we have (22).□ By substituting (22) into (14), we have ∑i=1nωi∣Lπ(i)∣+ω0dopt=U−η∑i=1n(Wi)1/(η+1)(pπ(i)iαπ(i))η/(η+1)η+1, (28) where Wi(i=1,2,…,n) are given by Equation (6). Let Ψji=(Wi)1/(η+1)(pjiαj)η/(η+1). (29) As in Section 3.1, the 1∣CON,dopt,piA=(pirαi/ui)η,∑i=1nui≤U∣∑i=1nωi∣Lπ(i)∣+ω0dopt problem can be solved by the following linear assignment problem: Minimize∑j=1n∑i=1nΨjixjisubjectto(11),(12),(13). (30) Similar to Section 3.1, for the 1∣CON,dopt,piA=(pirαi/ui)η,∑i=1nui≤U∣∑i=1nωi∣Lπ(i)∣+ω0dopt problem, we can propose the following algorithm: Theorem 4 Algorithm4solves the 1∣CON,dopt,piA=(pirαi/ui)η,∑i=1nui≤U∣∑i=1nωi∣Lπ(i)∣+ω0doptproblem in O(n3)time. Algorithm 4 Step 1. By Lemma 2, calculate k. Step 2. Calculate the Ψji by using Equation (29). Step 3. Solve the linear assignment problem Equations (30), (11)–(13) to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (22), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. Step 1. By Lemma 2, calculate k. Step 2. Calculate the Ψji by using Equation (29). Step 3. Solve the linear assignment problem Equations (30), (11)–(13) to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (22), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. Algorithm 4 Step 1. By Lemma 2, calculate k. Step 2. Calculate the Ψji by using Equation (29). Step 3. Solve the linear assignment problem Equations (30), (11)–(13) to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (22), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. Step 1. By Lemma 2, calculate k. Step 2. Calculate the Ψji by using Equation (29). Step 3. Solve the linear assignment problem Equations (30), (11)–(13) to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (22), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. From (28), for a special case αi=α, i=1,2,…,n, we have ∑i=1nωi∣Lπ(i)∣+ω0dopt=U−η∑i=1n(Wi)1/(η+1)(pπ(i)iα)η/(η+1)η+1=U−η∑i=1nϕiφπ(i)η+1, (31) where ϕi=(Wi)1η+1(iα)ηη+1 (32) and φπ(i)=(pπ(i))ηη+1. (33) Similar to Section 3.2, the problem 1∣CON,dopt,piA=(pirαi/ui)η,αi=α,∑i=1nui≤U∣∑i=1nωi∣Lπ(i)∣+ω0dopt can be solved by the following algorithm: Theorem 5 Algorithm5solves the 1∣CON,dopt,piA=(pirαi/ui)η,αi=α,∑i=1nui≤U∣∑i=1nωi∣Lπ(i)∣+ω0doptproblem in O(nlogn)time. Algorithm 5 Step 1. By Lemma 2, calculate k. Step 2. Calculate νi and μi for i=1,2,…,n by Equations (32) and (33). Step 3. By using Lemma 5 to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (22), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. Step 1. By Lemma 2, calculate k. Step 2. Calculate νi and μi for i=1,2,…,n by Equations (32) and (33). Step 3. By using Lemma 5 to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (22), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. Algorithm 5 Step 1. By Lemma 2, calculate k. Step 2. Calculate νi and μi for i=1,2,…,n by Equations (32) and (33). Step 3. By using Lemma 5 to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (22), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. Step 1. By Lemma 2, calculate k. Step 2. Calculate νi and μi for i=1,2,…,n by Equations (32) and (33). Step 3. By using Lemma 5 to determine the optimal job sequence. Step 4. Calculate the optimal resources by using Equation (22), and the optimal processing times by using Equation (2). Step 5. Set dopt=∑i=0kpπ(i)A. 4. SLK DUE DATE ASSIGNMENT For the slack due date assignment method, the due date di for job Ji is equal to his actual processing time plus the common flow allowance, i.e. di=piA+qopt, i=1,2,…,n, where the common flow allowance qopt is a decision variable to be determined. The problem is to determine the sequence of jobs and qopt so that the objective function ∑i=1nωi∣Lπ(i)∣+ω0qopt+∑i=1nvπ(i)uπ(i)=∑i=1nωi∣Cπ(i)−dπ(i)∣+ω0qopt+∑i=1nvπ(i)uπ(i) (34) being minimized. Using the three-field notation of Graham et al. [22], the problems can be denoted as 1∣SLK,qopt,piA=pirαi−βiui∣∑i=1nωi∣Lπ(i)∣+ω0qopt+∑i=1nvπ(i)uπ(i) and 1∣SLK,qopt,piA=(pirαi/ui)η∣∑i=1nωi∣Lπ(i)∣+ω0qopt+∑i=1nvπ(i)uπ(i). Lemma 7 (Liu et al. [20]). For a given sequence π=[π(0),π(1),…,π(n)], the optimal value of qoptis equal to the completion time of a job, i.e. qopt=Cπ(k)=∑i=0kpπ(i)A, where kis a median for the sequence ω0,ω1,…,ωn, ∑j=0kωj≤∑j=k+1nωjand∑j=0k+1ωj≥∑j=k+2nωj. (35) Lemma 8 The objective function of the problem 1∣SLK,qopt,piA∈{pirαi−βiui,(pirαi/ui)η}∣∑i=1nωi∣Lπ(i)∣+ω0qopt+∑i=1nvπ(i)uπ(i)can be written as ∑i=1nωi∣Lπ(i)∣+ω0qopt+∑i=1nvπ(i)uπ(i)=∑i=1nVipπ(i)A+∑i=1nvπ(i)uπ(i), (36)where the positional weight of position jin the schedule is given by Vi=∑h=0iωh,fori=1,…,k;∑h=i+1nωh,fori=k+1,…,n−1;0,fori=n. (37) Proof Similar to Lemma 3.□ 4.1. The problem 1∣SLK,qopt,piA=pjrαj−βjuj∣∑i=1nωi∣Lπ(i)∣+ω0qopt+∑i=1nvπ(i)uπ(i) From Lemma 7 and pπ(i)A=pπ(i)iαπ(i)−βπ(i)uπ(i), we have ∑i=1nωi∣Lπ(i)∣+ω0qopt+∑i=1nvπ(i)uπ(i)=∑i=1nVipπ(i)A+∑i=1nvπ(i)uπ(i)=∑i=1nVipπ(i)iαπ(i)+∑i=1n(vπ(i)−Viβπ(i))uπ(i). (38) Similar to Section 3.1, from (38), let Vji=Vipjiαj,ifvj−Viβj>0,Vipjiαj+(vj−Viβj)u¯j,ifvj−Viβj≤0, (39) we can formulate the the problem 1∣SLK,qopt,piA=pirαi−βiui∣∑i=1nωi∣Lπ(i)∣+ω0qopt+∑i=1nvπ(i)uπ(i) as the following linear assignment problem: Minimize∑j=1n∑i=1nWjixjisubjectto(11),(12),(13). (40) Theorem 6 The 1∣SLK,qopt,piA=pirαi−βiui∣∑i=1nωi∣Lπ(i)∣+ω0qopt+∑i=1nvπ(i)uπ(i)problem can be solved in O(n3)time. 4.2. The problem 1∣SLK,qopt,piA=(pirαi/ui)η∣∑i=1nωi∣Lπ(i)∣+ω0qopt+∑i=1nvπ(i)uπ(i) From Lemma 8 and pπ(i)A=(pπ(i)iαπ(i)/uπ(i))η, we have ∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i)=∑i=1nVipπ(i)A+∑i=1nvπ(i)uπ(i)=∑i=1nVipπ(i)iαπ(i)uπ(i)η+∑i=1nvπ(i)uπ(i). (41) Lemma 9 For the 1∣SLK,qopt,piA=(pirαi/ui)η∣∑i=1nωi∣Lπ(i)∣+ω0qopt+∑i=1nvπ(i)uπ(i)problem, the optimal resource allocation as a function of the job sequence, that is uπ(i)*=ηVivπ(i)1η+1×(pπ(i)iαπ(i))ηη+1, (42)where Vi(i=1,2,…,n)are given by Equation (37). By substituting (42) into (41), we have ∑i=1nωi∣Lπ(i)∣+ω0qopt+∑i=1nvπ(i)uπ(i)=η−ηη+1+η1η+1×∑i=1n(vπ(i)pπ(i)iαπ(i))ηη+1(Vi)1η+1, (43) where Vi(i=1,2,…,n) are given by Equation (37). Let Φji=η−ηη+1+η1η+1×(vjpjiαj)ηη+1(Vi)1η+1. (44) As in Section 3.1, the 1∣CON,dopt,piA=(pirαi/ui)η∣∑i=1nωi∣Lπ(i)∣+ω0dopt+∑i=1nvπ(i)uπ(i) problem can be solved by the following linear assignment problem: Minimize∑j=1n∑i=1nΦjixjisubjectto(11),(12),(13). (45) Theorem 7 The 1∣SLK,qopt,piA=(pirαi/ui)η∣∑i=1nωi∣Lπ(i)∣+ω0qopt+∑i=1nvπ(i)uπ(i)problem can be solved in O(n3)time. Theorem 8 The 1∣SLK,qopt,piA=(pirαi/ui)η,αi=α∣∑i=1nωi∣Lπ(i)∣+ω0qopt+∑i=1nvπ(i)uπ(i)problem can be solved in O(nlogn)time. 4.3. The problem 1∣SLK,qopt,piA=(pirαi/ui)η,∑i=1nui≤U∣∑i=1nωi∣Lπ(i)∣+ω0qopt Similar to Section 3.3, we have Lemma 10 For the 1∣SLK,qopt,piA=(pirαi/ui)η,∑i=1nui≤U∣∑i=1nωi∣Lπ(i)∣+ω0qoptproblem, the optimal resource allocation as a function of the job sequence, that is uπ(i)*=(Vi)1η+1(pπ(i)iαπ(i))ηη+1∑i=1n(Wi)1η+1(pπ(i)iαπ(i))ηη+1×U,i=1,2,…,n, (46)where Vi(i=1,2,…,n)are given by Equation (37). Theorem 9 The 1∣SLK,qopt,piA=(pirαi/ui)η,∑i=1nui≤U∣∑i=1nωi∣Lπ(i)∣+ω0qoptproblem can be solved in O(n3)time. 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The Computer Journal – Oxford University Press

**Published: ** Dec 29, 2017

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