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AACIMP 2009 Summer School lecture by Boris Goldengorin. "Logistics" course. 3rd - 4th hour. Part 4.

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- 1. All Minimal and Maximal Open Single Machine Scheduling Problems are Polynomially Solvable Harmen W. Bouma 1345982 April 8, 2009
- 2. Master’s Thesis Operations Research Supervisor: Prof. Sc.D. B. Goldengorin Co-assessor: Prof. dr. G. Sierksma
- 3. All Minimal and Maximal Open Single Machine Scheduling Problems are Polynomially Solvable H.W. Bouma Abstract In this thesis, Boolean Linear Programming (BLP) models are presented for three single machine scheduling problems with equal-length jobs and diﬀerent release dates, and it is proven that they are polynomially solvable. The objec- tive of the ﬁrst problem, in which preemption is allowed, is to minimize the total weighted completion time. The objective of the other two problems is to minimize the total weighted tardiness. The second problem is preemptive, the third is not. To this date, the complexity status of these problems has remained open. The open complexity status was mentioned by Labetoulle et al. [21] in 1984. The BLP models are based on the Assignment Problem formulation, which is a well-known polynomially solvable problem in combinatorial optimization. The Assignment Problem can be used to solve many problems, including for example the single machine scheduling problems in which the total weighted tardiness is minimized and in which all jobs have unit processing times and diﬀerent release dates. In order to solve the three problems mentioned above only preoptimized permutations are incorporated into the Assignment Problem formulation. The proof that the problems are polynomially solvable is based on the notion of Total Dual Integrality. Keywords: (Non-)preemptive scheduling; Equal-length jobs; Release dates; Weighted completion time; Weighted tardiness; Total dual integrality.
- 4. Preface Since the ﬁfties of the twentieth century Scheduling Theory has been a popular research area. In the early years pioneers came up with algorithms to solve simple job scheduling problems. In the years that followed, many algorithms and heuristics were found to solve both basic and more complicated problems. Nowadays, the complexity status of many basic scheduling problems is known, although there are still some problems with an open complexity status. In this thesis I will show that three of these open problems are polynomially solvable. Two of the problems are about ﬁnding a preemptive schedule in a single machine setting where the jobs have equal processing times and diﬀerent release dates. The objectives are to minimize the total weighted completion time and the total weighted tardiness, respectively. The third problem is the non-preemptive version of the total weighted tardiness problem. All problems will be modeled as Assignment Problems with side constraints. The thesis is organized as follows. In Chapter 1 an overview of single ma- chine scheduling problems is given and some basic concepts of computational complexity and Integer Linear Programming are discussed. In Chapter 2 the problems are deﬁned and an informal description of 1|pmtn; pj = 2; rj | wj Cj is presented which motivates the use of the Assignment Problem formulation. A similar approach is used for the other open problems. Chapter 3 contains a discussion of problems that are modeled as a variation on the Assignment Problem as well. In Chapter 4, the Boolean Linear Programming (BLP) model for the problem 1|pmtn; pj = 2; rj | wj Cj is given and it is proven that the corresponding system of equations is Totally Dual Integral. This implies that the problem is polynomially solvable. In Chapter 5 these results are generalized to the case pj = p. In Chapter 6 the BLP models are presented for the total weighted tardiness scheduling problems and it is proven that they are also poly- nomially solvable. In Chapter 7 some computational results are presented and Chapter 8 contains recommendations for future research and a short summary. I wish to thank my supervisor Prof. Sc.D. B. Goldengorin for giving me the opportunity to do this research. He advanced the idea of solving the problem by a model based on the Assignment Problem. I would like to thank him for his support and advice he provided me with during my research. Furthermore, I would like to thank Prof. dr. G. Sierksma, who was willing to be co-assessor of this research project, and Guido Diepen of Paragon Decision Technology for his technical support with AIMMS. Groningen, Harmen Bouma
- 5. Contents 1 Introduction 13 1.1 Single Machine Scheduling Problems . . . . . . . . . . . . . . . . 14 1.2 Computational Complexity . . . . . . . . . . . . . . . . . . . . . 16 1.3 Overview of Complexity Status of Single Machine Scheduling Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3.1 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3.2 Related Problems . . . . . . . . . . . . . . . . . . . . . . . 21 1.4 Linear and Integer Programming . . . . . . . . . . . . . . . . . . 23 1.4.1 Total Unimodularity and Total Dual Integrality . . . . . . 24 2 Problem Formulation 27 2.1 Minimizing the Total Weighted Completion Time . . . . . . . . . 27 2.1.1 The Assignment Model . . . . . . . . . . . . . . . . . . . 28 2.2 Minimizing the Total Weighted Tardiness . . . . . . . . . . . . . 31 2.2.1 The Preemptive Problem . . . . . . . . . . . . . . . . . . 31 2.2.2 The Non-Preemptive Problem . . . . . . . . . . . . . . . . 32 2.3 Outline of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . 33 3 Variations on the Assignment Problem 35 3.1 Finding Subdigraphs with Prescribed Degrees . . . . . . . . . . . 35 3.2 A Polynomial Solvable Case of the Axial Three-Dimensional As- signment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3 The Imbalanced Time Minimizing Assignment Problem . . . . . 38 3.4 Resource-Constrained Assignment Scheduling . . . . . . . . . . . 38 4 Minimizing the Total Weighted Completion Time when p = 2 41 4.1 The Number of Time Intervals between the First and Second Part of a Job . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 The Primal Model . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.1 BLP Model . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2.3 Integer and LP Relaxations . . . . . . . . . . . . . . . . . 44 4.2.4 Total Unimodularity . . . . . . . . . . . . . . . . . . . . . 45 4.3 The Dual Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . 46 4.3.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.4 Total Dual Integrality . . . . . . . . . . . . . . . . . . . . . . . . 49 4.4.1 Integrality of Solutions to the Dual Problem . . . . . . . . 50 7
- 6. 4.4.2 A Polynomial Time Algorithm . . . . . . . . . . . . . . . 53 4.5 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . 57 4.5.1 Complementary Slackness . . . . . . . . . . . . . . . . . . 57 4.5.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.5.3 Optimality Conditions . . . . . . . . . . . . . . . . . . . . 59 5 Minimizing the Total Weighted Completion Time when pj = p 63 5.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 Computational Complexity . . . . . . . . . . . . . . . . . . . . . 65 5.2.1 A Polynomial Time Algorithm . . . . . . . . . . . . . . . 65 5.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6 Minimizing the Total Weighted Tardiness 69 6.1 The Preemptive Case . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.1.1 Computational Complexity . . . . . . . . . . . . . . . . . 70 6.1.2 A Polynomial Time Algorithm . . . . . . . . . . . . . . . 70 6.1.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.2 The Non-Preemptive Case . . . . . . . . . . . . . . . . . . . . . . 71 6.2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.2.2 Computational Complexity . . . . . . . . . . . . . . . . . 73 6.2.3 A Polynomial Time Algorithm . . . . . . . . . . . . . . . 73 6.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 7 Computational Results 75 8 Summary and Concluding Remarks 81 A Complexity Results for Single Machine Problems 85 B Proofs from Chapters 5 and 6 87 B.1 The Dual Model of Chapter 5 . . . . . . . . . . . . . . . . . . . . 87 B.1.1 Integrality of the Dual Optimal Solution . . . . . . . . . . 89 B.2 The Dual Model of Section 6.2 . . . . . . . . . . . . . . . . . . . 92 B.2.1 Total Dual Integrality . . . . . . . . . . . . . . . . . . . . 93 C Instances used for Computational Results 97 D List of Symbols and Abbreviations 103 8
- 7. List of Figures 1.1 The EDD and optimal schedule for an instance of 1|rj |Lmax . . . 19 1.2 The preemptive WSPT and the optimal schedule for an instance of 1|pmtn; rj | wj Cj . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.1 Graphical representation of a special case of the three-dimensional assignment model. . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Graphical representation of the multidimensional assignment prob- lem for 1|pmtn; pj = 2; rj | wj Cj with n = 3. . . . . . . . . . . 37 4.1 Schedule S1 , where d ∈ {2, 4, 6, . . .}, 0 < λ < 1 . . . . . . . . . . . 55 4.2 Schedule S2 , where d1 , d2 ∈ {1, 3, 5, . . .}, 0 < λ < 1. . . . . . . . . 55 4.3 Schedule S3 , where d1 , d2 ∈ {1, 3, 5, . . .}, 0 < λ < 1. . . . . . . . . 56 4.4 Schedule S4 , where d1 , d2 ∈ {1, 3, 5, . . .}, 0 < λ < 1. . . . . . . . . 56 4.5 Schedule S5 , where d1 ∈ {2, 4, 6, . . .}, d2 , d3 ∈ {3, 5, 7, . . .}, d2 > d1 , d3 > d2 , 0 < λ < 1. . . . . . . . . . . . . . . . . . . . . . . . . 56 4.6 Schedule S6 , d1 ∈ {2, 4, 6, . . .}, d2 , d3 ∈ {3, 5, 7, . . .}, d2 > d1 , d3 > d2 , 0 < λ < 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.1 The diﬀerences in terms of percentages between the optimal val- ues of the problems 1|pmtn; pj = p; rj | wj Cj and 1|pj = p; rj | wj Cj . 78 7.2 The diﬀerences in terms of percentages between the optimal val- ues of the problems 1|pmtn; pj = p; rj | wj Tj and 1|pj = p; rj | wj Tj . 79 9
- 8. List of Tables 1.1 Overview of complexity results of problems that are related to the open problems that are solved in this thesis. . . . . . . . . . 26 2.1 The cost matrix W , representing the costs of assigning job parts to time intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2 The cost matrix W , representing the costs of assigning job parts to time intervals when wj Tj is the objective. . . . . . . . . . . 32 4.1 The way in which dual variables p, q and s correspond to the rows and columns of the cost matrix W . . . . . . . . . . . . . . . 48 4.2 Cost matrix W adjusted by the dual variables. . . . . . . . . . . 49 7.1 CPU time in seconds and optimal solutions for the problem 1|pmtn; pj = p; rj | wj Cj , for p = 2. . . . . . . . . . . . . . . . . . . . . . . . 75 7.2 CPU time in seconds and optimal solutions for the problem 1|pmtn; pj = p; rj | wj Cj , with p = 3 and p = 4. . . . . . . . . . . . . . . . . 76 7.3 CPU time in seconds and optimal solutions 1|pmtn; pj = p; rj | wj Tj , for p = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.4 CPU time in seconds and optimal solutions for the problem 1|pmtn; pj = p; rj | wj Tj , with p = 3 and p = 4. . . . . . . . . . . . . . . . . 77 7.5 CPU time in seconds and optimal solutions 1|pj = p; rj | wj Tj , for p = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 7.6 CPU time in seconds and optimal solutions for the problem 1|pj = p; rj | wj Tj , with p = 3 and p = 4. . . . . . . . . . . . . . . . . 78 7.7 Optimal solutions for the problem 1|pj = p; rj | wj Cj . . . . . . 78 8.1 The problems that are solved in this thesis and problems that are closely related to them. . . . . . . . . . . . . . . . . . . . . . . . 83 C.1 Instances with 20 jobs and p = 2. . . . . . . . . . . . . . . . . . . 97 C.2 Instances with 50 jobs and p = 2. . . . . . . . . . . . . . . . . . . 98 C.3 Instances with 100 jobs and p = 2. . . . . . . . . . . . . . . . . . 99 C.4 Instances with 50 jobs and p = 3. . . . . . . . . . . . . . . . . . . 100 C.5 Instances with 30 jobs and p = 4. . . . . . . . . . . . . . . . . . . 101 11
- 9. Chapter 1 Introduction In many manufacturing and service industries planning and scheduling belong to the day-to-day activities. Whether it is about a copy shop holder who needs to determine which order to print ﬁrst, or the type of aircraft that should be assigned to a particular ﬂight, the decisions made can substantially aﬀect the eﬃciency or proﬁtability of a company. Consider for example a copy shop holder, who serves several business clients. Some orders are urgent, while others are not. A report that is just ﬁnished and should be presented the following morning has higher priority for a ﬁrm than the quarterly staﬀ magazine. The urgency of an order is given by the diﬀerence between the deadline of an order and the point in time at which the order actu- ally comes in at the copy shop. This is the so-called time window. Furthermore, each order has a diﬀerent processing time, depending on its size, its quality etc. Last but not least, each order has a certain weight. When the copy shop holder does not meet a deadline, the client receives a discount based on the amount of delay and the weight of the order. Naturally, urgent and large orders have higher weights than low priority and small jobs. Given all orders, the copy shop holder would like to determine the printing schedule such that the total amount of discount is as small as possible. Now consider the airline company. It needs to assign airplanes to ﬂights such that none of the ﬂights is canceled. The cost of assigning a certain type of air- craft to a particular ﬂight is known in advance. Each ﬂight is characterized by a departure and arrival time and by the expected number of passengers. Each airplane has a certain capacity. Obviously, it is desirable that an airplane has suﬃcient capacity to carry all passengers, since otherwise a certain amount of revenue is missed. On the other hand, when there is overcapacity the revenues may not be suﬃcient to cover the ﬂight’s costs. Given the departure and ar- rival time, the cost of a ﬂight, the expected number of passengers per ﬂight and the revenue per passenger, the airline is interested in assigning its airplanes to ﬂights such that total proﬁts are maximized. These examples show that scheduling problems arise frequently in prac- tice and often involve the core business of a company. Therefore, considerable amounts of attention have been paid to scheduling techniques in literature for 13
- 10. over ﬁfty years. Pinedo [28] deﬁnes scheduling as the allocation of limited re- sources to activities that have to be done over time. This allocation needs to be such that a company optimizes its objectives and achieves its goals. In the examples stated above, the photocopiers in the copy shop and the ﬂeet of the airline company are the resources. The activities are the orders and the ﬂights. The objectives are to minimize the total amount of discount and to maximize proﬁts, respectively. Scheduling techniques rely heavily on mathematical models, exact algo- rithms and heuristics. For many problems eﬃcient algorithms have been found to solve the problems to optimality. However, there are still many problems that have not been solved yet. In this thesis suitable models for three of these unsolved scheduling problems will be presented. In Chapter 2 the problem is formulated and in Chapter 3 some background information is given about the modeling techniques that are used in Chapters 4 to 6. In the remainder of the Introduction an overview of scheduling problems is given in which only one type of resource is involved, the so-called single machine models. Furthermore, the concept of computational complexity and the mathematical techniques that are used in this thesis are shortly discussed. 1.1 Single Machine Scheduling Problems Most of the scheduling problems that were solved during the ﬁfties of the twen- tieth century involved only one machine. Although single machine problems do not arise frequently in practice, a study of them is valuable for two reasons. Firstly, the solution methods for single machine models provide insights into solution techniques that could be used in more complex situations with sev- eral machines. Secondly, in many production settings there is one stage that is crucial for the overall performance of a schedule. This stage is called the bottleneck. Optimizing the schedule for the bottleneck, which could well be a single machine, then optimizes the schedule for the entire problem (Chen, Potts and Woeginger [11]). In a single machine model there is a certain resource which is generally referred to as a ‘machine’. This resource is needed to fulﬁll a number of tasks, usually called ‘jobs’. The machine can only handle one job at a time. Each job may have diﬀerent characteristics, such as the time that is needed to complete a job or the point in time at which a job becomes available for processing. Finally, there is an objective that has to be optimized. For example, one might be interested in minimizing the total number of tardy jobs, or the point in time at which all jobs are completed. Most types of scheduling problems can be described by a triple α|β|γ which is widely used in the literature and was introduced by Graham et al. [17]. Here, α describes the machine environment, β the job characteristics and γ the objective criterion to be minimized. Hence, for single machine models it follows that α = 1. An overview of several job characteristics and objective criteria is given below. It should be noted that this overview is not exhaustive. However, it is suﬃcient to deﬁne most basic scheduling problems. 14
- 11. Job Characteristics. Assume that there are n jobs to be scheduled and let the subscript j refer to a particular job. Examples of job characteristics are: Processing time (pj ). The time that is needed to fully process job j. Release date (rj ). The time at which job j becomes available for processing. Due date (dj ). The time at which job j is committed to be completed. When this due date is not met, a penalty is incurred. When the due date must be met at all costs, it can replaced by a (strict) deadline Dj . Weight (wj ). The weight of a job, which is a priority factor. It reﬂects the importance of job j relative to other jobs. Preemption (pmtn). When preemption is allowed, a job can be removed from a machine before it is fully processed. At a later stage it can then be resumed. In case of preemption, processing that has already been done is not lost. Precedence constraints (prec). When a job cannot be processed before an- other job is completed, the problem is subject to so-called precedence constraints. Precedence constraints can be depicted by acyclic graphs, where each node is a job and each arc a precedence constraint. When the graph consists of chains, prec is replaced by chains in the β-ﬁeld. Similarly, when the precedence constraints can be represented by a series- parallel graph, prec is replaced by sp-graph. It should be noted that pj only appears in the β-ﬁeld when pj = p, i.e., when all processing times are equal. In general, wj and dj do not appear at all in the β-ﬁeld since it is clear from the objective whether there are weights and due dates. This is not the case for strict deadlines Dj . Note that it could well be that the β-ﬁeld is empty. Objective Criteria. Let the time at which job j is ﬁnished be denoted by Cj , the completion time of job j. Deﬁne the lateness Lj and the tardiness Tj of job j as follows: Lj = Cj − dj , Tj = max{Cj − dj , 0}. The lateness of job j indicates the amount of time that a job is delayed. Note that a negative value implies that a job is early. The tardiness indicates the amount of delay of a job, and is zero when a job is early or exactly in time. Furthermore, let 1 if Cj > dj Uj = 0 otherwise. When job j is delayed Uj is one, otherwise it is zero. This criterion can be used to count the number of tardy jobs. However, the amount of delay is not expressed by it. The objective criteria can now be given by: 15
- 12. Makespan (Cmax = maxj Cj ). The makespan is the maximum completion time, or the time at which all jobs are processed. When the makespan is minimized, the schedule tends to ﬁnish processing as early as possible. Total completion time ( j Cj ). This objective sums the completion time of all jobs. Minimizing the total completion time can be used as a surrogate for minimizing the total Work-In-Process, since it tends to ﬁnish each job as quickly as possible. Total weighted completion time ( j wj Cj ). This objective sums the completion time of all jobs, after they have been adjusted for their weights. When total weighted completion time is minimized it becomes advantageous to ﬁnish jobs with high weights early relative to jobs with low weights. The total weighted completion time can be used to minimize the total value of the Work-In-Process. Maximum lateness (Lmax = maxj Lj ). The maximum lateness records the largest delay in a schedule. The maximum lateness objective minimizes in a way the worst performance of a schedule, since all delays are kept as small as possible. Total number of tardy jobs ( j Uj ). This objective counts the number of tardy jobs. Minimizing the number of tardy jobs can be used to maximize the percentage of on-time deliveries. Total weighted number of tardy jobs ( j wj Uj ). This objective counts the cu- mulative weight of all tardy jobs. This objective can be used to maximize the percentage of on-time deliveries adjusted for their weights. Total tardiness ( j Tj ). The total tardiness criterion sums all delays. Minimiz- ing the total tardiness may be preferable to minimizing the total number of tardy jobs. When the total number of tardy jobs is minimized, some jobs may get seriously delayed in order to keep the number of tardy jobs low. This in turn may lead to unacceptable situations in practice and does not occur when the total tardiness objective is used. Total weighted tardiness ( j wj Tj ). The total weighted tardiness sums the total amount of tardiness adjusted for the weights of the jobs. It can be used when some delays are more expensive than other delays. 1.2 Computational Complexity One of the most important questions in scheduling theory concerns the complex- ity status of a problem. The complexity status roughly describes the relation- ship between the size of a problem and the computation time that is needed to solve that problem. The problem size is determined by the input data and it is generally assumed that the input data is binary encoded. The following can be found in any textbook on Complexity Theory (see e.g. Garey and Johnson [14]). For problem size n, let TA (n) denote the worst-case time complexity when a problem is solved by algorithm A. That is, TA (n) is the maximum number of computational steps, over all input sizes n, that are required for the execution 16
- 13. of algorithm A. When the computational complexity is determined, it is crucial to look at the order of the function TA (n). Consider for example algorithm A1 with TA1 (n) = 0.5n2 and algorithm A2 with TA2 (n) = 5n. Although algorithm A1 is faster than algorithm A2 for 1 ≤ n ≤ 9 and equally fast for n = 10, algorithm A2 is faster for all n > 10. This has to do with asymptotic growth and is caused by the fact that TA2 (n) is of lower order than TA1 (n). Given two functions F (n) and G(n), with n ∈ N, the order of F (n) is said to be lower than or equal to the order of G(n) if F (n) ≤ K · G(n) for all n > n0 , where K and n0 are two positive constants. When the order of F is lower than or equal to the order of G, it is denoted F = O(G). Hence TA1 (n) is O(n2 ) and TA2 (n) is O(n). The worst-case time complexity of an algorithm is important in order to determine whether it is eﬃcient or not. Complexity theory is used to classify algorithms either as eﬃcient or ineﬃcient. Any O(P )-algorithm, where P is a polynomial in the problem size, is an eﬃcient algorithm. All problems that can be solved by an O(P )-algorithm are said to be polynomially solvable and belong to the class P. A problem is said to be pseudo-polynomially solvable, when the running time is polynomial in the numerical value of the input (using unary encoding), but non-polynomial with respect to binary encoding. Problems which can only be solved by non-polynomial algorithms belong to the class N P. It has been shown that the majority of the problems belonging to this class are equivalent to each other. This implies the following: if an eﬃ- cient algorithm could be developed that solves one problem in this class, it can be used to solve all problems that are in this class. Conversely, when it can be proven that no such algorithm possibly exists for one of these problems, it immediately follows that no eﬃcient algorithm exists for any problem in this class. These problems are called NP-complete. A more formal deﬁnition will be given below. However, it is necessary to introduce the diﬀerence between optimization and recognition problems ﬁrst. An optimization problem is concerned with the solution that actually min- imizes (maximizes) a certain objective function. A recognition problem deals with the question whether or not there exists a certain instance such that the objective function is at most (at least) k, where k is an arbitrary constant. Hence, a recognition problem can be answered by YES or NO. An instance of a recognition problem is a YES instance if the answer to this problem is YES, and a NO instance if it is NO. Note that the optimization problem can be solved by repetitive application of the recognition problem. To deﬁne NP-complete and NP-hard problems the following is needed as well: A problem P1 polynomially reduces to a problem P2 if some polynomial-time algorithm that solves P1 uses the algorithm for solving P2 at unit cost. Here, at unit cost means that the algorithm for P2 requires unit time to execute. From this deﬁnition the following property arises: If P1 polynomially reduces to P2 and some polynomial-time algorithm solves P2 , then some polynomial time-algorithm solves P1 . 17
- 14. A special form of problem reduction is called problem transformation. A problem P1 polynomially transforms to problem P2 if for every instance I1 of problem P1 an instance I2 of problem P2 can be constructed in polynomial time such that I1 is a YES instance of P1 if and only if I2 is a YES instance of P2 . Now, a recognition problem P1 is said to be NP-complete if 1. P1 ∈ N P, and 2. all other problems in the class N P polynomially transform to P1 . A recognition problem P1 is said to be NP-hard if all other problems in the class N P polynomially reduce to P1 . Note that this class of problems is broader than the class NP-complete, because it includes the class N P as well as problems that are not in this class. Note that these deﬁnitions of NP-completeness and NP-hardness are based on the assumption of binary encoded problem sizes. A problem that is NP- complete or NP-hard could well be pseudo-polynomially solvable. When a problem is NP-complete with respect to unary encoding as well, it is said to be strongly NP-complete. A similar result holds for strongly NP-hard problems (see e.g. Leung [24]). For a more detailed description of computational com- plexity, the reader is referred to Garey and Johnson [14] or Leung [24]. 1.3 Overview of Complexity Status of Single Ma- chine Scheduling Models Many single machine models have already been shown to be (pseudo-)polyno- mially solvable or NP-hard. On a website, Brucker and Knust [10] keep track of the complexity results of scheduling problems. Problems that were solved in the ﬁfties of the previous century are for example 1||Lmax and 1|| wj Cj . In the ﬁrst problem (Jackson [18]) the objective is to ﬁnd a schedule that min- imizes the maximum lateness. The β-ﬁeld is empty, which implies that all jobs have diﬀerent processing times and due dates, but that preemption is not al- lowed and there are no release dates, precedence constraints, etcetera. This problem is solved by the Earliest Due Date (EDD) ﬁrst rule, which orders all jobs by their due date (from low to high) and schedules them accordingly. This algorithm runs in O(n2 ). In 1|| wj Cj (Smith [33]) the schedule that minimizes the total weighted completion time needs to be determined. All jobs have diﬀerent processing times and weights. This problem is solved by the Weighted Shortest Processing Time (WSPT) ﬁrst rule, where job j is scheduled before job k if wj wk > . pj pk The WSPT algorithm has running time O(n log(n)). 18
- 15. Another example of a scheduling problem that is polynomially solvable is 1|pj = p; rj | wj Cj . In this problem the schedule that minimizes the total weighted completion time needs to be determined. All jobs have the same processing time and diﬀerent release dates. The problem can be solved by a dynamic programming algorithm that was introduced by Baptiste [4]. The time complexity of the algorithm is O(n7 ). Examples of problems that are NP-hard are: 1|rj |Lmax and 1|pmtn; rj | wj Cj . The problem 1|rj |Lmax is a generalization of the problem 1||Lmax . In this problem, the jobs have diﬀerent release dates, processing times and due dates. In Lenstra, Rinnooy Kan and Brucker [23], it is proven that this problem is strongly NP-hard. To see why the EDD rule does not necessarily solve this problem to optimality, consider the following example: job j 1 2 3 4 pj 4 1 6 4 rj 0 0 1 3 dj 13 14 7 11 The EDD schedule is depicted in Figure 1.1a. It is not diﬃcult to see that Lmax = L3 = L4 = 3. In Figure 1.1b, the optimal schedule is depicted, with Lmax = L1 = L2 = 2. Figure 1.1: (a) The EDD schedule with Lmax = 3. (b) The optimal schedule with Lmax = 2. Now consider the problem 1|pmtn; rj | wj Cj , which is a generalization of 1|| wj Cj . A possible approach for this problem would be to check whether the preemptive version of the WSPT rule leads to an optimal solution. This rule compares the ratios of the weight of a job and the remaining processing time for all available jobs and schedules the job with highest ratio ﬁrst. When a new job is released, or when a job is ﬁnished, this procedure is repeated until all jobs are processed. However, the preemptive WSPT rule does not always lead to an optimal solution. Consider the following example: 19
- 16. job j 1 2 3 pj 2 3 4 rj 0 1 2 wj 1 4 10 The preemptive WSPT schedule is depicted in Figure 1.2a and has objective value wj Cj = 101. However, the optimal schedule is depicted in Figure 1.2b and has objective value wj Cj = 98. The proof that 1|pmtn; rj | wj Cj is strongly NP-hard is given in Labetoulle et al. [21]. Figure 1.2: (a) The preemptive WSPT schedule with wj Cj = 101. (b) The optimal schedule with wj Cj = 98. In Appendix A a complete overview of the complexity status of single ma- chine problems is given. 1.3.1 Open Problems Although the complexity status of most single machine scheduling problems is known, there still exist problems with an open complexity status. This means that it is not known whether they are polynomially solvable or not. One of these problems is the preemptive equal-length job scheduling problem with release dates 1|pmtn; pj = p; rj | wj Cj , in which the total weighted completion time is minimized. This problem is similar to two of the problems that were discussed in the preceding paragraph, namely 1|pj = p; rj | wj Cj and 1|pmtn; rj | wj Cj . The non-preemptive problem 1|pj = p; rj | wj Cj , which is an easier case of the open problem, is polynomially solvable. It can be shown that this is the case for all easier cases of the open problem. Therefore, 1|pmtn; pj = p; rj | wj Cj is said to be minimal open. The problem 1|pmtn; rj | wj Cj is slightly harder than the open problem, since the processing times of the jobs may diﬀer. As was already indicated, this problem is strongly NP-hard. However, 1|pmtn; pj = p; rj | wj Cj is not maximal open, which implies that there are harder cases that are not known to be NP-hard. Consider for example the open problem with strict deadlines, 1|pmtn; pj = p; rj ; Dj | wj Cj . This problem is harder than the open problem under consideration and its complexity status is open as well. The problem can be solved by the model presented in Chapter 5. Two problems that are very similar to 1|pmtn; pj = p; rj | wj Cj are 1|pmtn; pj = p; rj | wj Tj and 1|pj = p; rj | wj Tj . 20
- 17. In both problems the total weighted tardiness is minimized instead of the to- tal weighted completion time. The ﬁrst problem allows for preemption, the second does not. According to the website of Brucker and Knust [10], the non- preemptive problem is both minimal and maximal open. This implies that all easier problems are polynomially solvable and all harder cases are NP-hard. Apparently, Brucker and Knust do not regard the preemptive problem as an easier or harder case. This is in agreement with a observation in Pinedo [29], which says that allowing preemptions usually simpliﬁes the analysis of a prob- lem, but not always. On the website of Brucker and Knust [10] it is also stated that the preemptive problem is maximal open, but not minimal. However, it will be shown in the following section that all easier problems (i.e., with equal job weights or without release dates) are known to be polynomially solvable. Since 1|pj = p; rj | wj Tj is not regarded as an easier problem, this implies that 1|pmtn; pj = p; rj | wj Tj must be minimal open as well. 1.3.2 Related Problems In this paragraph an overview of problems that are related to the open problems introduced in Section 1.3.1 is given. It should be noted that some complexity results have been obtained for the general class of problems with objective func- tion fmax = maxj {fj (Cj )}. Here fj is a monotone, non-decreasing cost func- tion. Both the (weighted) completion time and the (weighted) tardiness can be described by such functions fj . In Table 1.1, the overview is given schematically. As was already shown at the beginning of Section 1.3, the problem 1|| wj Cj (Smith [33]) is polynomially solvable. The same result holds true for 1||fmax , which can be solved by the EDD rule. When the objective is replaced by the to- tal (weighted) tardiness, the problem becomes (strongly) NP-hard (Lawler [20], Du and Leung [12]). However, Lawler [20] shows that there exists a pseudo- polynomial algorithm to solve 1|| wj Tj if the weighting of jobs is agreeable. This is the case when pi < pj implies that wi ≥ wj . Note that it immediately follows that the equal weight problem 1|| Tj is also pseudo-polynomially solv- able. The simpliﬁed total weighted tardiness problem in which the jobs have equal lengths, 1|pj = p| wj Tj , is polynomially solvable. The schedule for this prob- lem can be divided into parts of length p. At each of these time intervals, a job is scheduled. For each job and time interval combination the corresponding costs are known. By means of the Assignment Problem, which is polynomially solvable and discussed in more detail in Chapter 2, the optimal assignment of jobs to time intervals can be easily found. When release dates are added to the polynomially solvable problems 1||Lmax and 1|| Cj , they both become strongly NP-hard. This is shown by Lenstra et al. [23]. From the fact that 1|| Tj is NP-hard it immediately follows that 1|rj | Tj is NP-hard as well. An example of 1|rj |Lmax is given at the beginning of Section 1.3. Note that the problem 1|rj |Cmax is polynomially solvable, since every non-delay schedule is optimal. A schedule is called non-delay if a machine is not kept idle while one or more jobs are available for processing. When the scheduling problems with release dates are relaxed by the assump- tion of equal processing times pj = p, they become polynomially solvable when 21
- 18. Lmax and wj Cj are minimized. The latter problem was solved by Baptiste [4]. Simons [32] showed that the problem P |pj = p; rj |Lmax is polynomially solv- able. Here, P stands for m parallel machines, which is a generalization of the single machine problem. It immediately follows that 1|pj = p; rj |Lmax is poly- nomially solvable as well. The problem in which the total weighted tardiness is minimized for equal-length jobs with diﬀerent release dates is one of the open problems that is solved in this thesis. The version with equal job weights as well as the version with unit processing times pj = 1 are polynomially solvable. The former case is proven by Baptiste [4], the latter can be solved by the Assignment Problem (see Chapter 2 for a discussion of the Assignment Problem). Hence, the model of Baptiste [4] solves the equal-length job scheduling problem with release dates when the total weighted completion time or the total tardiness is minimized, but not when the total weighted tardiness is minimized. This is caused by the fact that the model only holds true when the function (fi − fj ) is monotone. This implies that either (fi − fj )(t1 ) ≥ (fi − fj )(t2 ) ∀t1 ∀t2 > t1 or (fi − fj )(t1 ) ≤ (fi − fj )(t2 ) ∀t1 ∀t2 > t1 . When fj (t) = wj max{0, t − dj } it can be shown that (fi − fj ) is not monotone. Consider, for example, the case with d1 = 2, d2 = 3, w1 = 1 and w2 = 10. It follows that (f1 − f2 )(2) = 0, (f1 − f2 )(3) = 1 and (f1 − f2 )(4) = −8. Hence (f1 − f2 )(2) < (f1 − f2 )(3) but (f1 − f2 )(2) > (f1 − f2 )(4). Now consider the scheduling problem in which jobs have diﬀerent release dates, are subject to precedence constraints and where preemption is allowed. Baker et al. [2] present an algorithm that solves the problem 1|prec; pmpn; rj |fmax in polynomial time. However, the counterparts in which the total completion time and the total tardiness are minimized are NP-hard, which follows from Lenstra and Rinnooy Kan [22]. Baptiste et al. [5] show that the relaxed problem 1|prec; pmpn; pj = p; rj | Cj has a non-preemptive optimal schedule, and can therefore be solved by the algorithm developed by Simons [32] that solves 1|prec; pj = p; rj | Cj in polynomial time. Let us now return to the problems without precedence constraints. Baker et al. [2] give a polynomial time algorithm that solves 1|pmtn; rj |fmax . However, both 1|pmtn; rj | wj Cj (Labetoulle et al. [21]) and 1|pmtn; rj | wj Tj are strongly NP-hard. Because 1|| wj Tj is strongly NP- hard, it follows directly that 1|pmtn; rj | wj Tj is strongly NP-hard as well. Now look at the problem in which the total completion time and not the total weighted completion time is minimized: 1|pmtn; rj | Cj . This problem is poly- nomially solvable (Baker [1]). Likewise, the problem 1|pmtn; pj = p| wj Cj is polynomially solvable. Since all jobs are released at the same time the optimal schedule is non-preemptive. Therefore, the result- ing problem is just a special case of 1|| wj Cj , which is polynomially solvable. However, when release dates are added it is not known whether the resulting problem is polynomially solvable or not. In conclusion we have that there ex- ist algorithms that solve 1|pmtn; pj = p| wj Cj , 1|pj = p; rj | wj Cj and 1|pmtn; pj = p; rj | Cj in polynomial time, but that none of these algorithms solves the problem when respectively release dates, preemption or unequal job 22
- 19. weights are added to the problem. Now consider the scheduling problems in which the total weighted tardi- ness has to be minimized. The problem 1|pmtn; rj | Tj is NP-hard. This follows directly from the NP-hardness of 1|| Tj ; the addition of preemp- tion and/or release dates does not simplify the problem. However, when the jobs have equal processing times the problem becomes easier. The problem 1|pmtn; pj = p| wj Tj is equivalent to 1|pj = p| wj Tj since preemption is not advantageous when the release dates are equal. It has already been shown that this problem is polynomially solvable. Furthermore, Tian, Ng and Cheng [34] show that 1|pmtn; pj = p; rj | Tj is polynomially solvable. The fact that their model does not solve the problems in which the total weighted tardiness or the total weighted completion time is minimized, is a direct conse- quence of the addition of job weights. In a note, Goldengorin [16] shows why the introduction of job weights gives rise to diﬃculties. He states that although it might be locally optimal to preempt job j in favor of job k, it does not mean that this leads to a globally optimal solution. To put it diﬀerently, it often is optimal to interrupt job j by job k only if job j can be completed before a critical point in time t0 . Otherwise, job j should not be interrupted by job k. This restriction does not hold true when the total tardiness is minimized. In that case, when it is locally optimal to preempt job j in favor of job k, it imme- diately follows that it is globally optimal as well. Summarizing, it follows that 1|pmtn; pj = p| wj Tj and 1|pmtn; pj = p; rj | Tj are polynomially solvable, but that 1|pj = p; rj | wj Tj and 1|pmtn; pj = p; rj | wj Tj are open problems. A problem that is related to the preemptive open problems that are solved in this thesis is 1|pmtn; pj = p; rj | wj Uj . When Uj does not denote the total weighted number of tardy jobs, but the total weighted throughput that has to be maximized, i.e., 0 if Cj > dj Uj = 1 otherwise, the problem is polynomially solvable (Baptiste et al. [6]). However, when wj Uj denotes the weighted number of tardy jobs that needs to be mini- mized, the problem can be seen as a special case of 1|pmtn; pj = p; rj | wj Tj (see Chapter 6) and can be solved accordingly. 1.4 Linear and Integer Programming Many decision problems - like the scheduling problems of the preceding section - can be modeled by Linear Programming (LP) models. An LP model consists of an objective function and a number of constraints which are all linear functions of the decision variables. When the decision variables can only take integer val- ues, the resulting model is called an Integer Linear Programming (ILP) model. Similarly, when the decision variables can only take boolean values (i.e., 0 or 1), the model is called a Boolean Linear Programming (BLP) model. The LP model that will be used in this thesis has the form min{wx | Ax ≥ b, x ≥ 0}, 23
- 20. where A is a m × n-matrix, w and x are vectors in Rn and b is a vector in Rm . Every LP model has a dual, which is a LP model itself. The original LP model is called the primal model. When the primal model is a minimization problem, the dual is a maximization problem and vice versa. Each decision variable (constraint) of the primal model corresponds to a constraint (decision variable) of the dual model. The dual of the primal LP model presented above is given by max{yb | yA ≤ w, y ≥ 0}, where y is a vector in Rm . By the strong duality theorem of Linear Programming (see e.g. Schrijver [30]), the optimal solution of the primal model is equal to the optimal solution of the corresponding dual model, provided that both solutions are feasible and bounded. A nice relation between the optimal solutions of the primal and dual model is called Complementary Slackness (CS), which states that x and y are both optimal solutions (provided that they are feasible) if and only if yi (ai x − bi ) = 0 for all i, (wj − yAj )xj = 0 for all j. Here, ai denotes the ith row of A and Aj the jth column of A. It is generally known that LP models are polynomially solvable. However, ILP models are NP-complete. Therefore, no eﬃcient algorithms are known to solve ILP problems. A possible way to solve an ILP model is by solving the corresponding LP relaxation. That is, the model is solved without the restriction of integrality. When the optimal solution of the LP relaxation appears to be integer, it is the optimal solution to the ILP model as well. However, this occurs only in certain special cases and sometimes just by coincidence. 1.4.1 Total Unimodularity and Total Dual Integrality The set of all possible solutions to a LP model is a polyhedron. The polyhedron P corresponding to the model presented above is P = {x ∈ Rn | Ax ≥ b, x ≥ 0}. Polyhedra are closed, convex sets. To deﬁne what a face of a polyhedron is, we need the following: A hyperplane is a set of the form Hw,d = {x ∈ Rn | wx = d}, where w ∈ Rn is a nonzero row vector and d ∈ R. The hyperplane Hw,d is a supporting hy- perplane of P if max{wx | x ∈ P } = d. Now, F is called a face of P if F = P or F = P ∩ H for some supporting hyperplane H of P . Furthermore, a point p ∈ P is a vertex of P if {p} is a face of P (see e.g. Schrijver [30]). A polyhedron P is said to be integral, if each face of P contains integral vectors (see e.g. Schrijver [30]). In that case, there always exists an optimal solution to the corresponding LP relaxation that is integral. This is very useful 24
- 21. when one is trying to solve an ILP model. When the polyhedron is integral, it suﬃces to solve the LP relaxation corresponding to the ILP model. Sometimes it is possible to tell from the properties of the constraint matrix A that the polyhedron P = {x ∈ Rn | Ax ≥ b, x ≥ 0} is integral for each integral vector b. This is the case if and only if A is Totally Unimodular (TU). A matrix A is called TU if each subdeterminant of A is 0, 1, or −1. In particular, each entry of A is 0, 1, or −1. There exist polynomial time algorithms to check whether A is TU (see e.g. Schrijver [30]). However, when the constraint matrix A is not TU, the integrality of the polyhedron P = {x ∈ Rn | Ax ≥ b, x ≥ 0} for a particular integral vector b can still be guaranteed by the notion of Total Dual Integrality (TDI). The system Ax ≥ b, x ≥ 0 is TDI if the dual has an integral optimal solution y for each integral vector w for which the maximum is ﬁnite. Furthermore, it can be shown that when Ax ≥ b, x ≥ 0 is a TDI system, and when b is integral, the polyhedron P = {x ∈ Rn | Ax ≥ b, x ≥ 0} is integral (see e.g. Schrijver [30]). The diﬀerence between a TDI system in which A is not TU, and a TDI system in which A is TU, is that in the former case an integral optimal solution is only guaranteed when w is an integer vector, while in the latter case this restriction is not needed. The concepts of Total Unimodularity and Total Dual Integrality are very useful to determine the complexity status of ILP problems. As was already indicated, ILP problems are in general NP-complete. However, when the con- straint matrix is TU, or when the system Ax ≥ b, x ≥ 0 is TDI, there exists an optimal solution to the LP relaxation that is an optimal solution to the ILP model as well. In case of a Total Unimodularity, the optimal solution of the LP relaxation that is found by a regular LP solver is integral. In case of Total Dual Integrality, this does not need to be the case. However, in that case the frac- tional optimal solution can be used to construct an integral optimal solution. Furthermore, there exists sophisticated solvers (e.g. CPLEX) that make use of presolvers such that an ILP is solved very eﬃciently in case of TDI. 25
- 22. Table 1.1: Overview of complexity results of problems that are related to the open problems that are solved in this thesis. 1||fmax in P 1|| wj Cj in P 1|| wj Tj (*) NP-hard 1|| Tj (**) NP-hard 1|pj = p| wj Tj in P 1|rj |Lmax NP-hard 1|rj | Cj NP-hard 1|rj | Tj NP-hard 1|rj |Cmax in P 1|pj = p; rj | wj Cj in P 1|pj = p; rj | wj Tj open 1|pj = p; rj |Lmax in P 1|pj = 1; rj | wj Tj in P 1|pj = p; rj | Tj in P 1|pmtn; prec; rj |fmax in P 1|pmtn; prec; rj | Cj NP-hard 1|pmtn; prec; rj | Tj NP-hard 1|pmtn; prec; pj = p; rj | Cj in P 1|pmtn; prec; pj = p; rj | Tj NP-hard 1|prec; pj = p; rj | Cj in P 1|prec; pj = p; rj | Tj NP-hard 1|prec; pj = p| Tj NP-hard 1|pmtn; rj | wj Cj NP-hard 1|pmtn; rj | wj Tj NP-hard 1|pmtn; rj | Cj in P 1|pmtn; rj | Tj NP-hard 1|pmtn; pj = p; rj | wj Cj open 1|pmtn; pj = p; rj | wj Tj open 1|pmtn; pj = p| wj Cj in P 1|pmtn; pj = p| wj Tj in P 1|pmtn; pj = p; rj | Tj in P (*) Pseudo-polynomially solvable for special cases (**) Pseudo-polynomially solvable
- 23. Chapter 2 Problem Formulation In this chapter the three scheduling problems that are solved in this thesis will be deﬁned. By means of an example, a motivation is given for the use of the AP formulation. In Section 2.3 the structure of the proof of polynomial solvability will be explained. 2.1 Minimizing the Total Weighted Completion Time Consider the problem 1|pmtn; pj = p; rj | wj Cj , in which n jobs with process- ing time p have to be scheduled. Each job j has a certain release date rj and cannot be ﬁnished before rj + p. Without loss of generality, assume that the release dates are such that r1 ≤ r2 ≤ . . . ≤ rn . Furthermore, assume that all entries are nonnegative integers and that there exists a schedule without idle time, i.e., the machine remains busy until all jobs are completed. It immedi- ately follows from the assumption of integral entries that in an optimal schedule preemption occurs only at unit points in time. Hence, the time horizon can be divided into pn time intervals of unit length and each job j needs to be assigned to p of these time intervals. This problem presented above is an open problem. At the website of D¨rr [13] u two attempts are presented to solve the (more general) problem 1|pmtn; pj = p; rj | wj Cj . The ﬁrst is a LP model (Baptiste et al. [8]), the second is a dynamic program- ming model (Baptiste et al. [7]) for the special case when p > max rj . The LP model will be shortly discussed below. The BLP Model. Let the indices be given by j ∈ {1, 2, . . . , n}, a ∈ {0, 1, 2, . . . , n − 1}, t ∈ {0, 1, 2, . . . , T }. 27
- 24. Let the boolean decision variable xjt denote how much of job j is completed at time t. The BLP model reads min wj txjt j,t T subject to t=0 xjt = 1 j = 1, . . . , n j≥i t<ri +(a+1)p xjt ≤ a a = 0, . . . , n − 1 i = 1, . . . , n − a + 1 xjt ∈ {0, 1} for all j, t. The ﬁrst set of constraints states that every job should eventually be fully processed. The second set states that in an interval strictly less than (a + 1)p time units at most a jobs that are released at or after the starting time of that interval can be entirely executed and completed. Baptiste et al. [8] show that the polyhedron determined by the LP relaxation contains vertices which are not integral. Therefore, it cannot be assured that the optimal solution of the LP relaxation is integral. Although this is not a proof that the LP relaxation does not solve the problem in polynomial time, another observation could be made that explain the incorrectness of the model. The LP relaxation of the model presented above allows for solutions that do not correspond to a feasible schedule. Consider for example the case with n = p = 3 and rj = j − 1. A feasible solution is given by 1 x13 = x19 = x26 = x29 = x35 = x37 = , 2 and all other variables are zero. Hence, after 2 time units nothing of job 1 is 1 produced yet, but after 3 time units it is already processed for 1 2 time units. Al- though Baptiste et al. [8] mention this example and show that it is not optimal, they do not say anything about the infeasibility of the corresponding schedule. In the models presented in Chapters 4 to 6, all feasible solutions - integral and fractional - correspond to feasible schedules. 2.1.1 The Assignment Model In order to show that the problem 1|pmtn; pj = p; rj | wj Cj is polynomially solvable the problem is modeled as an Assignment Problem (AP) with additional constraints. The AP is widely discussed in the literature and can be found in most textbooks on combinatorial optimization and integer lin- ear programming (see e.g. Sierksma [31]). The ﬁrst polynomial time algorithm for the AP was suggested by Kuhn [19]. The AP can be formulated as follows: Consider two sets U and V , both containing T elements and a cost matrix W where wjt is the cost of assigning element j ∈ U to t ∈ V . An assignment is a permutation π = (π(1), π(2), . . . , π(T )) of V in which the jth element of U is assigned to element tj = π(j) from V . An optimal assignment minimizes 28
- 25. T j=1 wjπ(j) . The Assignment Problem can be stated as a BLP model as follows: T T min wjt xjt j=1 t=1 T subject to j=1 xjt = 1 t = 1, . . . , T T t=1 xjt = 1 j = 1, . . . , T xjt ∈ {0, 1} for all j, t. When the boolean constraint is replaced by 0 ≤ xjt ≤ 1, the LP relaxation is obtained. Since the constraint matrix of the AP is TU (see e.g. Sierksma [31]), there always exists an integral optimal solution to the LP relaxation. This implies that the AP is polynomially solvable. In order to illustrate how to use the AP formulation for the problem 1|pmtn; pj = p; rj | wj Cj , consider the following example for the special case with p = 2. Example. Let there be 5 jobs with processing time 2 that have to be sched- uled. The release dates and weights of the jobs are given in the following table: job j 1 2 3 4 5 rj 0 1 2 3 4 wj 1 3 7 16 33 Processing of the jobs starts at time 0 and is ﬁnished at time T = 10. Hence, there are 10 unit time intervals to which the jobs have to be assigned. Since the processing time of each job is 2, they can be divided into two parts such that each of the ten evolving job parts is assigned to exactly one time interval. It should be noted that when job j is released at time rj , the ﬁrst part can be assigned to time interval rj + 1 or any later time interval. The second part can only be processed at time interval rj + 2 or at any later time interval. Now consider the AP in which ten job parts have to be assigned to ten time intervals against minimum costs, where the costs of assigning each job part j to a certain time interval t are given in Table 2.1. Here, 1.1 denotes the ﬁrst part of job 1, 2.1 the ﬁrst part of job 2, etcetera. Since the objective value of the scheduling problem, wj Cj , depends on the completion time of the jobs, it follows that the assignment of the ﬁrst part of a job to a certain time interval does not come with any costs. However, since a job can not be processed before it is released and since processing has to start before the ﬁnal time interval T , the costs for t < rj + 1 and t = T are set to inﬁnity. Consequently, the ﬁrst part of a job will never be assigned to one of these intervals. If the second part of a job is assigned to time interval t the costs are wj t. As the second part of a job cannot be processed before time interval rj + 2, it 29
- 26. Table 2.1: The cost matrix W , representing the costs of assigning job parts to time intervals. Time Intervals t 1 2 3 4 5 6 7 8 9 10 1.1 0∗ 0 0 0 0 0 0 0 0 ∞ J 2.1 ∞ 0∗ 0 0 0 0 0 0 0 ∞ o 3.1 ∞ ∞ 0 0 0 0 0∗ 0 0 ∞ b 4.1 ∞ ∞ ∞ 0 0 0 0 0∗ 0 ∞ p 5.1 ∞ ∞ ∞ ∞ 0 0 0 0 0∗ ∞ a 1.2 ∞ 2 3 4 5 6 7 8 9 10∗ ∗ r 2.2 ∞ ∞ 9 12 15 18 21 24 27 30 t 3.2 ∞ ∞ ∞ 28∗ 35 42 49 56 63 70 s 4.2 ∞ ∞ ∞ ∞ 80∗ 96 112 128 144 160 5.2 ∞ ∞ ∞ ∞ ∞ 198∗ 231 264 297 330 follows that the corresponding costs can be set to inﬁnity as well. An optimal solution for the Assignment Problem is given by the asterisks in Table 2.1. Note that, although the resulting schedule S = (1, 2, 2, 3, 4, 5, 3, 4, 5, 1) is feasible, the objective value corresponding to this schedule ( wj Cj = 493) is not equal to the objective value of the AP, which is 325. The reason for this diﬀerence is that in the AP the ﬁrst parts of jobs 3, 4 and 5 are scheduled after their second parts, which has a positive eﬀect on the costs, but which should not be possible in a correct model. It is not diﬃcult to see that the schedule S cannot be optimal; it is possible to assign job 3 to the 4th and 6th time interval and to assign job 5 to the 7th and 9th interval. The completion time of job 5 does not change by this operation while the completion time of job 3 decreases. Hence, the resulting schedule is better than S. A natural question that comes to mind is whether the solution of the AP can be used to construct an optimal solution of the scheduling problem under consideration. A possible way to do this would be the following: 1. Let the sequence in which the second job parts are scheduled be as in the optimal solution of the AP. 2. Starting at time interval 1, assign each ﬁrst job part that was originally scheduled after its second part to the time interval right before this second part. Note that the second job part and all subsequent job parts move to the next time interval. 3. Repeat step 2 until all ﬁrst job parts are scheduled before the correspond- ing second job parts. Applying these steps to the solution of the example leads to the schedule S = (1, 2, 2, 3, 3, 4, 4, 5, 5, 1) 30
- 27. with objective value 460. Unfortunately, it can be shown that S is not optimal either. It is not diﬃcult to see that interchanging jobs 4 and 5 leads to a better schedule. Although the AP formulation clearly does not lead to an optimal solution of the problem 1|pmtn; pj = 2; rj | wj Cj , the example shows that this formu- lation could be used when additional constraints are added which ensure that the second part of a job can not be processed before its ﬁrst part. However, it is desirable to maintain the nice properties of the AP that assure an inte- gral optimal solution of the LP relaxation. In general, these nice properties are destroyed when side constraints are added to the AP. In the literature, there can be found many examples that have to contend with these diﬃculties. A selection of them is treated in Chapter 3. However, ﬁrst the other two open problems are formulated. 2.2 Minimizing the Total Weighted Tardiness The open problems in which the total weighted tardiness is minimized consist of a preemptive and a non-preemptive version. They are very similar to the problem deﬁned in Section 2.1 as will be shown below. 2.2.1 The Preemptive Problem Consider the problem 1|pmtn; pj = p; rj | wj Tj , in which n jobs with process- ing time p have to be scheduled. Each job has a certain release date and due date and can not be ﬁnished before rj + p. Without loss of generality, assume that the release dates are such that r1 ≤ r2 ≤ . . . ≤ rn . Furthermore, assume that all entries are nonnegative integers and that there exists a schedule without idle time intervals, i.e., all pn time intervals will be assigned without any time interval remaining idle. This implies that at least one job is released at time zero and all jobs are completed at time T = pn. Just like the problem 1|pmtn; pj = p; rj | wj Cj , this is an open problem. It should be noted that the problems are very similar. To see this, consider the example of Section 2.1.1 with due dates. Recall that in this example p = 2. Example job j 1 2 3 4 5 rj 0 1 2 3 4 wj 1 3 7 16 33 dj 5 8 7 7 6 A similar cost matrix as depicted in Table 2.1 can be constructed. Note that in this case, assigning the second part of job j to time interval t does only come with any costs if dj > t. An optimal solution to the AP is given by the asterisks in Table 2.2. The corresponding optimal value is 5. This solution leads to the following schedule 31
- 28. Table 2.2: The cost matrix W , representing the costs of assigning job parts to time intervals when wj Tj is the objective. Time Intervals t 1 2 3 4 5 6 7 8 9 10 1.1 0∗ 0 0 0 0 0 0 0 0 ∞ J 2.1 ∞ 0∗ 0 0 0 0 0 0 0 ∞ o 3.1 ∞ ∞ 0∗ 0 0 0 0 0 0 ∞ b 4.1 ∞ ∞ ∞ 0 0 0 0 0 0∗ ∞ p 5.1 ∞ ∞ ∞ ∞ 0∗ 0 0 0 0 ∞ a 1.2 ∞ 0 0 0 0 1 2 3 4 5∗ r 2.2 ∞ ∞ 0 0 0 0 0 0∗ 3 6 t 3.2 ∞ ∞ ∞ 0∗ 0 0 0 7 14 21 s 4.2 ∞ ∞ ∞ ∞ 0 0 0∗ 16 32 48 5.2 ∞ ∞ ∞ ∞ ∞ 0∗ 33 66 99 132 (1, 2, 3, 3, 5, 5, 4, 2, 4, 1). Note that the AP schedules the second part of job 4 be- fore its ﬁrst part. Therefore, the corresponding objective value is not equal to the total weighted tardiness, which is in fact 37. This schedule is not optimal. By in- spection, it is possible to ﬁnd the better schedule (1, 1, 3, 3, 5, 5, 4, 4, 2, 2) with ob- jective value 22. Although the AP does not lead to an optimal solution of the problem 1|pmpn; pj = p; rj | wj Tj , the example shows that the AP formulation can be used when extra constraints are added that ensure that the job parts are scheduled in the right order. 2.2.2 The Non-Preemptive Problem Consider the problem 1|pj = p; rj | wj Tj , in which n jobs with processing time p have to be scheduled. Each job has a certain release date and due date. Without loss of generality, assume that jobs are ordered by their release date, r1 ≤ r2 ≤ . . . ≤ rn and that there is always a job available for processing. When this assumption does not hold, the problem can be split up into two separate problems. Furthermore, assume that all entries are nonnegative integers. Just as in the preemptive case, the time horizon can be divided into time intervals of unit length. However, since preemption is not allowed, it should be noted that an optimal schedule might contain idle time intervals. Consider for example the following instance with p = 2: job j 1 2 3 rj 0 1 4 dj 8 3 6 wj 1 1 1 It is not diﬃcult to see that the schedule 2 2 3 3 1 1 with objective value 0 is optimal, and that no optimal schedule with less idle time exists. 32
- 29. Note that the maximum number of idle time intervals is (n−1)(p−1). To see this, consider the case in which the machine remains idle during the ﬁrst (p − 1) time intervals before it starts processing job j = 2. When this job is completed, the machine remains idle for another p − 1 time units until it starts processing job j = 3. This can be repeated until job j = n is completed and processing of job j = 1 is started. It follows that all job parts have to be assigned to time intervals t = 1, . . . , T , where T = (n − 1)(p − 1) + np. This is an open problem. In Chapter 6 it is shown that it can be solved by the AP with side constraints. The cost matrix is similar to the one in Table 2.2. Since there are more time intervals than job parts, the constraints of the AP require that each time interval is assigned at most (but not exactly) one job part. The side constraints require that as soon as the ﬁrst job part is scheduled, all other job parts are scheduled successively and in the right order. 2.3 Outline of the Proof In order to prove that the preemptive problems discussed in this thesis are polynomially solvable the following line of reasoning is used: First, a crucial property for any optimal preemptive schedule is introduced. This property is used to formulate a BLP model. Then, it is shown that the constraint matrix corresponding to this BLP model is not Totally Unimodu- lar. Therefore, it can not be concluded directly that the polyhedron determined by the LP relaxation is integral. Nevertheless, it can be proven that the sys- tem of (in)equalities determined by the LP relaxation is Totally Dual Integral. Since the LP relaxation is polynomially solvable, this directly implies that the corresponding scheduling problem is polynomially solvable. The proof of Total Dual Integrality is based on the dual of the LP relax- ation. After the dual is formulated, it is proven that it always has an integral optimal solution. Since all job weights are assumed to be integral, it then im- mediately follows from the deﬁnition of Total Dual Integrality that the system of (in)equalities determined by the LP relaxation is Totally Dual Integral. The model for the non-preemptive problem in which total weighted tardiness is minimized is based on its preemptive counterpart. Since preemption is not allowed, many constraints can be simpliﬁed. Following the same line of reasoning as in the preemptive case, it can be easily proven that the resulting system of (in)equalities is Totally Dual Integral. 33
- 30. Chapter 3 Variations on the Assignment Problem In this chapter four problems will be discussed that can be modeled using the framework of the Assignment Problem. From the examples it follows that the nice properties of the AP are not always preserved when extra constraints are added; the ﬁrst two problems are polynomially solvable, the others are not. 3.1 Finding Subdigraphs with Prescribed De- grees The following is a special case of the problem discussed in Bang-Jensen and Gutin [3, p. 142]. Let there be a directed weighted graph D. Suppose one would like to identify the minimum cost subgraph of D, if one exists, with prescribed degrees on the vertices. This could be of interest for example when the cheapest cycle factor of D is required. This problem can be stated as follows: Given the weighted graph D = (V, A) with V = {v1 , v2 , . . . , vn } and inte- gers a1 , a2 , . . . , an , b1 , b2 , . . . , bn , ﬁnd the cheapest subgraph D = (V, A∗ ) of D which satisﬁes d+ (vi ) = ai and d− (vi ) = bi for each i = 1, 2, . . . , n, or show D D that no such subdigraph exists. Here, d+ (vi ) and d− (vi ) denote the out- and, D D respectively, in-degree of vertex i in subgraph D . This problem can be formulated as a BLP model as follows. Let wij denote the weight of arc (i, j). Assume that ai ≤ d+ (vi ) and bi ≤ d− (vi ) for all D D n n i ∈ {1, 2, . . . , n} and i=1 ai = i=1 bi . Furthermore, deﬁne the following decision variable: 1 if (i, j) ∈ A∗ xij = 0 otherwise. The model reads: n n min wij xij i=1 j=1 35
- 31. subject to j:(i,j)∈A xij = ai i = 1, . . . , n i:(i,j)∈A xij = bj j = 1, . . . , n xij ∈ {0, 1} for all i, j. This is a special case of the AP since many possible assignments are excluded beforehand (only existing arcs can be included in the subdigraph). One could also say that extra constraints which ensure that xij = 0 when (i, j) ∈ A need / to be added to the AP. Note also that ai and bj are not necessarily equal to one. Bang-Jensen and Gutin [3] show that this problem can be solved in polynomial time. 3.2 A Polynomial Solvable Case of the Axial Three-Dimensional Assignment Model The following is taken from Gilbert and Hofstra [15]. Consider the problem in which there are p jobs, q workers and r machines, p ≤ q ≤ r. Each job has to be assigned to a worker and a machine, but each worker and each machine can be assigned to at most one job. Suppose the cost of assigning job i to worker j and machine k can be expressed as the cost of assigning job i to worker j plus the cost of assigning job j to machine k, i.e., wijk = wij + wjk . This problem can be formulated by means of the (two-dimensional) Assignment Problem. Deﬁne the following decision variables: 1 if job i is assigned to worker j xij = 0 otherwise, for i = 1, . . . , p and j = 1, . . . , q, 1 if worker j is assigned to machine k yjk = 0 otherwise, for j = 1, . . . , q and k = 1, . . . , r, 0 if worker j is assigned a job zj = 1 otherwise. The BLP model can now be formulated as: p q q r min wij xij + wjk xjk i=1 j=1 j=1 k=1 q subject to j=1 xij = 1 i = 1, . . . , k p i=1 xij + zj = 1 j = 1, . . . , q r k=1 yjk + zj = 1 j = 1, . . . , q q j=1 yjk ≤ 1 k = 1, . . . , r xij , yjk , zj ∈ {0, 1} for all i, j, k. In Figure 3.1 the two-dimensional representation of this problem is given. Note that the cells in the upper left quadrant correspond to the variables xij , which assign jobs to workers. The cells in the lower right quadrant correspond to 36
- 32. the variables yjk , which assign workers to machines. The cells on the diagonal of the lower left quadrant correspond to the zj variables. All black cells are infeasible. Since this problem is now formulated as a classical AP, it immediately follows that it is polynomially solvable. Figure 3.1: Graphical representation of a special case of the three-dimensional assignment model. A suggestion would be to use this formulation for the problem 1|pmtn; pj = 2; rj | wj Cj , by assigning each ﬁrst job part to a certain time interval and each time interval to another (idle) time interval. This second time interval can then be assigned to a second job part. A graphical representation with a possible assignment is given in Figure 3.2 for n = 3. Figure 3.2: Graphical representation of the multidimensional assignment prob- lem for 1|pmtn; pj = 2; rj | wj Cj with n = 3. However, it appears that this model does not prevent the second part of a job to be scheduled before the ﬁrst part. Therefore, it has no advantages compared to the classical AP formulation. In fact, it is worse because the number of variables is increased. 37
- 33. 3.3 The Imbalanced Time Minimizing Assign- ment Problem This problem is taken from M¨ller, Camozzato and Bassi de Ara´jo [27]. Con- u u sider the problem of assigning n jobs to m machines, n > m, each job with a distinct processing time for each machine. One job is assigned to exactly one machine and each machine is assigned to at least one job. Hence, this is the problem R||Cmax , where R stands for unrelated parallel machines, with the constraint that each machine performs at least one job. Let pij denote the processing time of job j on machine i. Deﬁne the following decision variable: 1 if job j is assigned to machine i xij = 0 otherwise. Furthermore, let t be a positive integer. The BLP model is given by: min Z = t m subject to i=1 xij = 1 j = 1, . . . , n n j=1 xij ≥ 1 i = 1, . . . , m n j=1 pij xij ≤ t i = 1, . . . , m xij ∈ {0, 1} for all i, j. Constraint 1 assures that every job is assigned to exactly one machine. The second constraint assures that every machine is assigned to at least one job. The third constraint imposes that every machine ﬁnishes processing not later than time t, which is the time to be minimized. Note that this problem is equivalent to the recognition problem where the question is posed whether there exists a feasible schedule for R||Cmax such that the maximum completion time is at most t. This question can be answered by YES or NO. If there is a polynomial time algorithm that solves the latter problem, it can be applied iteratively for smaller and smaller values of t until the answer is NO. The solution to the former problem is then equal to the solution to the latter problem where t is chosen such that the answer is just YES. However, this problem is proven to be NP-hard. It is already diﬃcult to solve the model for relatively short instances. 3.4 Resource-Constrained Assignment Schedul- ing The following is taken from Mazzola and Neebe [26]. Consider the problem in which personnel has to be scheduled to tasks. There are often departmental resource constraints involving characteristics such as budgetary considerations, degree of technical training of department personnel, or rank of personnel. Be- sides minimizing the total time required to handle all tasks by the personnel, these constraints have to be taken into account. Suppose that n workers i have to be assigned to n jobs j. The time needed for worker i to complete job j is denoted by wij . The cost incurred by department k, k ∈ {1, . . . , s}, when 38
- 34. worker i is assigned to job j is denoted by rijk . The budget of department k is denoted by bk . Similar to the AP, let xij be the decision variable indicating whether worker i is assigned to job j or not. The Assignment Problem with Side Constraints (APSC) is given by: n n min wij xij i=1 j=1 n subject to i=1 xij = 1 j = 1, . . . , n n j=1 xij ≥ 1 i = 1, . . . , n n n i=1 j=1 rijk xij ≤ bk k = 1, . . . , s xij ∈ {0, 1} for all i, j. The APSC is shown to be NP-complete. Lieshout and Volgenant [25] consider the case with only one side constraint (k = 1). For some special cases, this problem is polynomially solvable. However, in general, the addition of one side constraint already results in a NP-complete problem. 39
- 35. Chapter 4 Minimizing the Total Weighted Completion Time when p = 2 In this chapter, the BLP model for the problem 1|pmtn; pj = 2; rj | wj Cj will be presented, as well as its Integer and LP relaxations. By means of the dual problem, it will be shown that the system of (in)equalities determined by the LP relaxation is TDI. The immediate consequence is that the problem 1|pmtn; pj = 2; rj | wj Cj is polynomially solvable. The chapter will be con- cluded with the optimality conditions that evolve from Complementary Slack- ness. 4.1 The Number of Time Intervals between the First and Second Part of a Job Consider the following simple example with 2 jobs, r1 = 0, r2 = 1 and p = 2. Depending on the job weights either the schedule (1, 2, 2, 1) or (1, 1, 2, 2) is optimal. The feasible schedule (1, 2, 1, 2) can never be optimal since (1, 1, 2, 2) has a strictly better objective value irrespective of the weights. It appears that, when it is proﬁtable to interrupt job 1, job 2 is fully processed before job 1 is resumed. A generalization of this result is given in Theorem 1. Theorem 4.1 In an optimal schedule of the problem 1|pmtn; pj = 2; rj | wj Cj , there are either zero or an even number of time intervals between the intervals to which the ﬁrst and second part of a particular job are assigned. Proof: By contradiction. Suppose a schedule S - in which there is an odd number of time intervals between the ﬁrst and second part of job j - is optimal. Hence, we have that job j is interrupted by some job k, but job j is completed before job k is fully processed. Note that job k does not need to interrupt job j directly. It might be that some other jobs are fully processed after the ﬁrst part of job j and before the ﬁrst part of job k. 41
- 36. Suppose part 1 of job j is scheduled at time interval t, its second part is scheduled at time interval t + d, d ∈ {2, 4, 6, . . .}, and part 1 of job k is sched- uled somewhere in between at time interval t + c, c ∈ {1, 3, 5, . . .}, c < d. Now interchange the positions of part 2 of job j and part 1 of job k. All other jobs remain in the same position. Note that the number of time intervals between the ﬁrst and second part of job j is now zero or even. Call the new schedule S . The total weighted completion time of all jobs completed before job j is not aﬀected by the interchange, nor is the total weighted completion time of the jobs ﬁnished after job j. Since the weighted completion time of job j is strictly less in schedule S than in schedule S, it follows that S cannot be optimal. This com- pletes the proof of the theorem. Theorem 4.1 appears to be crucial for the implementation of the side con- straints in the AP formulation. It makes it possible to formulate the extra constraints such that the system of (in)equalities remains TDI. It should be noted that Theorem 4.1 is a direct consequence of Lemma 2.1 in- troduced by Tian et al. [34]. This lemma states that in a p-active schedule, job i is either fully processed before job j is started, or job i is interrupted by job j and not resumed before job j is completed. This implies that if job i is interrupted, it is interrupted for p = 2 or a multiple of p time units. Note that a preemptive schedule is p-active if no job’s completion time can be advanced without postponing the completion time of some other job. It is not diﬃcult to show that an optimal preemptive schedule is p-active when the total weighted completion time is minimized. 4.2 The Primal Model The following indices, parameters and decision variables are used in the BLP model. Indices j ∈ {1, . . . , n} job index; t ∈ {1, . . . , T } time index; k ∈ {1, 3, 5, . . . , T − 1} the set of all odd time indices; l ∈ {2, 4, 6, . . . , T − 2} the set of all even time indices up to and including T − 2. Parameters wjt cost of part 1 of job j when it is processed at time interval t; wjt cost of part 2 of job j when it is processed at time interval t; rj release date of job j. In the BLP model the regular weight wj will not be used. Instead, wjt and wjt are introduced. They are deﬁned as: 0 for all j, for T − 1 ≥ t ≥ rj + 1 wjt = ∞ otherwise. 42
- 37. wj t for all j, for t ≥ rj + 2 wjt = ∞ otherwise. Decision Variables 1 if job j’s ﬁrst part is assigned to time interval t xjt = 0 otherwise. 1 if job j’s second part is assigned to time interval t xjt = 0 otherwise. Furthermore, the following subsets of time intervals, Tk and Ul , are used. Tk = {1, 3, 5, . . . , k} The subset of odd time intervals up to and including interval k; Ul = {2, 4, 6, . . . , l} The subset of even time intervals up to and including interval l. 4.2.1 BLP Model The BLP model can now be formulated as: n T n T min wjt xjt + wjt xjt j=1 t=1 j=1 t=1 T subject to xjt ≥ 1 j = 1, . . . , n (4.1) t=1 T xjt ≥ 1 j = 1, . . . , n (4.2) t=1 n xjt + xjt = 1 t = 1, . . . , T (4.3) j=1 xjt − xj,t+1 ≥ 0 j = 1, . . . , n (4.4) t∈Tk t∈Tk k = 1, 3, 5, . . . , T − 1 xjt − xj,t+1 ≥ 0 j = 1, . . . , n (4.5) t∈Ul t∈Ul l = 2, 4, 6, . . . , T − 2 xjt , xjt ∈ {0, 1} j = 1, . . . , n t = 1, . . . , T. (4.6) Constraints (4.1) and (4.2) ensure that each ﬁrst respectively second part of a job is processed for the duration of at least one time interval. Constraints (4.3) take care of the fact that each time interval is assigned to exactly one job part. Since the number of job parts is equal to the number of time intervals, it immediately follows from (4.1) and (4.2) that each job part is assigned to exactly one time interval. This can be seen as follows: Suppose xjt = 1 and xj t = 1 are part of a feasible solution. Constraint (4.1) ˆ is satisﬁed for job j. However, by constraints (4.1), (4.2) and (4.3) there are 43
- 38. T − 1 job parts that have to be assigned to the remaining T − 2 time intervals. Since at most one job part can be assigned to each time interval by constraints (4.3), this contradicts the feasibility of xjt = 1 and xj t = 1. ˆ Constraints (4.4) and (4.5) together assure that one cannot start processing the second part of a job before the ﬁrst part is processed. Furthermore, they assure that the number of time intervals that lies between the processing of the ﬁrst and second part of a job is either zero or even; when part 1 is assigned to time interval t, part 2 can only be assigned to time interval t+d, d ∈ {1, 3, 5, . . .}. Constraints (4.6) are boolean constraints. 4.2.2 Example Consider the example of Chapter 2. job j 1 2 3 4 5 rj 0 1 2 3 4 wj 1 3 7 16 33 The optimal solution is given by x11 = 1, x29 = 1, x33 = 1, x47 = 1, x55 = 1, x12 = 1, x2,10 = 1, x34 = 1, x48 = 1, x56 = 1, and all other variables are zero. Hence, it follows that (1, 1, 3, 3, 5, 5, 4, 4, 2, 2) is the optimal schedule and the objective value wj Cj is 386. 4.2.3 Integer and LP Relaxations Consider the BLP model presented in Section 4.2.1. Note that constraints (4.6) may be replaced by xjt , xjt ∈ Z+ for all j, t. (4.7) Constraints (4.7) require that the decision variables are non-negative integers. Note that it is suﬃcient to require that the decision variables are non-negative; it is not necessary to add xjt , xjt ≤ 1. By constraints (4.3), we already have that the decision variables xjt and xjt can not take any value larger than one. When (4.6) is replaced by (4.7), an ILP model is obtained. From the ILP, it immediately follows that the LP relaxation is obtained when constraints (4.6) are replaced by xjt , xjt ≥ 0 for all j, t. From now on, the LP relaxation will be denoted by (P). 44

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