Bicriteria models to minimize the total weighted number of tardy jobs with convex controllable processing times and common due date assignment

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1 MISTA 2009 Bicriteria models to minimize the total weighted number of tardy jobs with convex controllable processing times and common due date assignment Dvir Shabtay George Steiner Abstract We study a single-machine scheduling problem in a flexible framework where both job processing times and due dates are decision variables to be determined by the scheduler. The model can also be applied for quoting delivery times when some parts of the jobs may be outsourced. We analyze the problem with common due date assignment and a convex resource consumption function. We provide a bicriteria analysis where the first criterion is to minimize the total weighted number of tardy jobs plus due date assignment cost, and the second criterion is to minimize total weighted resource consumption. We consider four different models for treating the two criteria. Although the problem of minimizing a single integrated objective function can be solved in polynomial time, we prove that the three bicriteria models are NP-hard. We also present special cases which are polynomially solvable and frequently occur in practice. 1 Introduction Manufacturing or service organizations must frequently quote delivery dates (due dates) to clients. Customers demand that suppliers meet contracted delivery dates or face substantial penalties. For example, Slotnick and Sobel [14] cite contracts from the aerospace industry, which may impose tardiness penalties of millions of dollars on subcontractors for aircraft components. In order to avoid tardiness penalties, including the possibility of losing customers, suppliers are under increasing pressure to quote attainable delivery dates for customer orders. Naturally, longer due dates are easier to meet, but promising delivery dates too far into the future may not be acceptable to the customer. At the same time, shorter due dates increase the probability that the order will be delivered late. Thus there is an important tradeoff between assigning relatively shortduedatestocustomerordersandavoidingtardinesspenalties,whichcreatesthe Dvir Shabtay Department of Industrial engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel dvirs@bgu.ac.il George Steiner DeGroote School of Business, McMaster University, Hamilton, Ontario, Canada L8S 4M4 steiner@mcmaster.ca 301

2 need for methodology that allows firms to quote attainable delivery dates and obtain efficient schedules at the same time. While traditional scheduling models considered due dates as given by exogenous decisions (see Baker and Scudder [1]), in a more agile system where due date assignment and scheduling are integrated, they are determined by taking into account the system s ability to meet the quoted delivery dates. For this reason, an increasing number of recent studies viewed due date assignment as part of the scheduling process and showed how the ability to control due dates can be a major factor in improving system performance. The most frequently studied model is the common due-date assignment method (denoted by CON ), for which all jobs are assigned the same due date, that is d j = d for j =1,..., n, whered j is the due date of job j and d 0 is a decision variable [5]. Similarly to due dates, job processing times in scheduling were traditionally considered to be fixed parameters. However, a supplier may have several alternative ways to supply and deliver products to the customer. For example, the supplier may outsource part of the production to subcontractors if this allows them to quote an earlier delivery date. This type of outsourcing can be modeled by assuming that the jobs have controllable processing times and the job processing times can be shortened by employing additional resources via outsourcing. There are also many other situations where the scheduler can shorten the processing time of a job by allocating more resources to it, see e.g., Janiak [6] and [7], which describes an interesting application of a scheduling model with controllable processing times in steel mills. Due to the large variety of applications, there is extensive literature on the subject of scheduling with controllable processing times. A survey of results up to 1990 is provided by Nowicki and Zdrzalka [11], and more recent ones are given by Chudzik et al. [2], Shabtay and Steiner [13] and Janiak et al. [8]. In this paper, we study scheduling problems in a flexible environment where both the job processing times and due date assignment are controllable by the scheduler. The decisions to be made incur two types of costs: the scheduling costs, which include the costs of due date assignment and penalties for tardy jobs, and the cost of controlling the job processing times. In [12] we studied a model minimizing the sum of all these costs. However, there are many situations where these two types of costs are not comparable or additive. For example, we may have a preset budget for the resource consumption costs or the maximum amount of resource that is available, or the maximum amount that can be subcontracted, and thus these costs have to be controlled separately from the scheduling costs. This results in having to schedule with two separate criteria. 1.1 Problem Definition Our bicriteria scheduling problem is formally stated as follows: a set of n independent, non-preemptive jobs, J = {1,..., n}, are available for processing at time zero and are to be processed by a system modeled as a single machine. The processing time of job j, p j, is a continuous convex function of the amount of resource that is allocated to the processing of the job given by the following resource consumption function: p j (u j )= θ j /u j k, j =1,...,n, (1) where n is the number of non-preemptive jobs, u j is a decision variable that represents the amount of resource allocated to the processing of job j, θ j is a positive parameter, 302

3 which represents the workload of the processing operation for job j and k is a positive constant. We note that many scheduling models use linear resource consumption functions, however, these models fail to reflect the law of diminishing marginal returns. This law states that productivity increases at a decreasing rate with the amount of resource employed. In order to model this, many studies on scheduling with resource allocation assume that the job processing time is a convex decreasing function of the amount of resource allocated to the job, described by eq. (1). This model has been used extensively in continuous resource allocation theory [13]. The due dates are assignable by the CON method. A schedule S is defined by the sequence in which the jobs are processed, a resource allocation vector u =(u 1,u 2,..., u n ), andacommonduedate d. The quality of a schedule S is measured by two criteria: The scheduling criterion, which includes (weighted) penalties for tardy jobs and the cost of due date assignment, defined by Z(S) = n n α j U j + β d j ; (2) and the second criterion is the total resource consumption cost (or amount) given by V (S) = n v j u j, (3) where for schedule S, C j (S) is the completion time of job j and U j is the tardiness indicator variable for job j, i.e., U j =1if C j (S) >d j and U j =0if C j (S) d j. In addition, α j is the cost of job j being tardy, β isthecostofoneunitofdeliverytime quotation per job and v j is the cost of one unit of resource allocated to job j. Note that for the CON method, eq. (2) can be converted to Z(S) = n α j U j + βnd. (4) Since our problem has two criteria, four different optimization problems can arise: The first one, which we denote by P1, is to minimize F l (Z, V )= n α j U j + β n d j + n v j u j. Using the scheduling notation introduced in [15], this problem can also be referred to as 1 d j = d unknown,conv F l (Z, V ); The second one, denoted by P2, is to minimize n α j U j + β n d j subject to n v j u j V ; where V is a limitation on the total resource consumption cost. Following the notation in [15], we refer to this problem as 1 d j = d unknown,conv (Z/V ); The third one, which we denote by P3, is to minimize n v j u j subject to n α j U j + β n d j K, where K is a given upper bound on the scheduling cost. We refer to this problem by 1 d j = d unknown,conv (V/Z) ( based on [15]); The last one, denoted by P4 (and referred to by 1 d j = d unknown,conv #(V/Z)), is to identify the set of Pareto-optimal schedules (points) (V,Z), where a schedule S with V = V (S) and Z = Z(S) is called Pareto-optimal (or efficient) if there does not exist another schedule S 0 such that V (S 0 ) V (S) and Z(S 0 ) Z(S) with at least one of these inequalities being strict. 303

4 It should be noted that solving P4 also solves P1-P3 as a by-product. Note also that all problem types involve both continuous decisions (the due date and processing time values) and combinatorial (sequencing) decisions. Similar bicriteria models were analyzed in [10] with linear resource consumption functions. A linear time optimization algorithm was presented in [12] for the P1-type scheduling problem 1 d j = d unknown,conv F l (Z, V ). The complexity and practical solvability of the P2-P4 bicriteria problems, however, remained an open question until now. The main goal of this paper is to provide answers for these open questions. 1.2 Organization of the Paper The rest of the paper is organized as follows. In Section 2 we determine the optimal continuous resource allocations as functions of the job partition into early and tardy jobs and thus reduce the P2 problem to a pure combinatorial problem. In contrast with the polynomial solvability of the P1 problems, we prove in Section 3 that all P2-P4 problems are NP-hard. In Section 4 we give a polynomial time algorithm for special cases of these problems that are important in practice. 2 Reducing the P2 Problem to a Partition Problem With fixed processing times, our P2 problem becomes a 1 d j = d unknown n α j U j + βnd due date assignment problem. This problem was analyzed by De et al. [3] and Kahlbacher and Cheng [9], who reduced it to a partition problem where set J is to be partitioned into two disjoint subsets: (a) Subset E, that will be sequenced at the beginning of the schedule, containing the early jobs whose total processing time is used to set the common due date value. (b) Subset T, consisting of the tardy jobs, which will be sequenced after the completion of subset E. They presented the following O(n) time optimization algorithm for the solution of this partition problem. Algorithm 1 : Optimization algorithm for the CON due date assignment problem with fixed job processing times. Step 1. Define E = {j J p j α j /(βn)} as the set of early jobs and T = JÂE as the set of tardy jobs. Step 2. SchedulesetE before set T, where the sequence within each set is immaterial. Step 3. The optimal due date coincides with the completion time of set E, i.e., d = p j. We can see from Algorithm 1 that the optimal due date assignment and job partition into E T are determined by the processing times. For our problem with controllable processing times, this means that they depend on the resource allocation strategy. Furthermore, the value of the scheduling objective in eq. (4), under an optimal due date assignment strategy, can be represented as follows: Z(τ,u) = α j + βn p j (u j )= α j + βn k θj /u j, (5) j T j T 304

5 where τ is the partition of the jobs into the two sets E and T (also dependent on u). Thus, our P2 problem is reduced to finding τ and u which minimize the objective in eq. (5) subject to n v j u j V. In the following Lemma, we present the optimal resource allocation strategy for the P2 problem as a function of an assumed job partition into early and tardy subsets. Lemma 1 For an arbitrary partition τ = E T, the optimal resource allocation u (τ) =(u 1(τ),u 2(τ),..., u n(τ)) for the 1 d j = d unknown,conv (Z/V ) problem is u j (τ) = v 1/(k+1) j θ k/(k+1) j V if j E i E (v iθ i ) k/(k+1) 0 if j T Proof For an arbitrary partition τ, our problem reduces to a continuous resource allocation problem of minimizing βn p j(u j ) = βn k θj /u j subject to n v j u j V. Since the objective value is independent of u j for j T,itimplies that u j =0for j T. In addition, since each p j(u j ) is a decreasing continuous function of u j, the n v j u j V constraint will be satisfied as an equality. Thus, the optimal resource allocation for each job in set E can be obtained by applying the Lagrangian method (described in detail in the Appendix). Since both the objective and the constraint are convex functions, the resulting resource allocation in eq. (6) is optimal. By substituting (6) into the objective in (5), we obtain that the objective value under an optimal resource allocation strategy and as a function of τ can be written as Z(τ,u (τ)) = α j + βn k+1 k/k+1 θj V k v j, (7) j T reducing our problem to the purely combinatorial problem of finding the partition τ which minimizes (7). (6) 3TheNP-hardness of Problems P2-P4 Let us first define the following decision problem associated with our optimization problems (P2-P4): DVP1: Given a single-machine scheduling problem where job due dates are assignable according to the CON method, job processing times are controlled by the convex resource consumption function (1), and parameters K and V.Isthereaschedulewith n α ju j + βnd K and n v ju j V? It is easy to see that DVP1 is in NP, since if we have a job partition τ and a resource allocation vector u =(u 1,u 2,..., u n), we can easily check in polynomial time whether both n α j U j + βnd K and n v j u j V. We next show that DVP1 is NP-complete by reducing the NP-complete Partition problem to it. Partition: Given a finite set A = {w 1,w 2,...,w h } of positive integers, where h w j = W. Can the set A be partitioned into two disjoint subsets, A 1 and A 2, so that j A i w j = W/2 for i =1, 2? 305

6 Theorem 1 DVP1 is NP-complete even if v j =1for j =1,..., n. Proof We construct the following instance of DVP1 from an instance of Partition: There are n = h jobs, where the job processing times follow eq. (1) with θ j = wj 2 for j =1,..., n. The penalty for completing job j after its due date is α j = w j and the resource consumption cost is identical for all jobs with v j =1for j =1,..., n. In addition, k =1, β is an arbitrary positive integer and the limitations are K =3W/4 and V = βnw. We will first show that if we have a YES instance of Partition, then there exists a schedule for the corresponding instance of DVP1 with n α j U j +βnd K =3W/4 and n u j V = βnw. Let E be the set of jobs corresponding to set A 1 of the solution of Partition, and let T be the set of all other jobs. Substituting eq. (6), the optimal resource allocation strategy with the given V = βnw is u 2βnwj if j E j (τ) = (8) 0 if j T. It is clear that j J u j (τ) = u j (τ) = 2βnw j = βnw. In addition, byeq.(7),wehave Z(τ,u (τ)) = w j w j = W W 2 + W 4 = 3W 4. (9) j T Conversely,weshowthatifwehaveaNOinstanceof Partition, then there is no schedule for the corresponding instance of DVP1 with n α j U j + βnd K = 3W/4 and n u j V = βnw. Let us assume that, contrary to the statement we want to prove, there is a schedule for the corresponding instance of DVP1 with n α j U j + βnd 3W/4 and n u j V = βnw, and let E and T denote the set of early and tardy jobs, respectively, in this schedule. Since we have a NO instance of Partition, theneither w j >W/2 or w j <W/2. If w j >W/2, then let = w j W/2 > 0. Forthiscase,wehave Z(τ,u (τ)) = W 2 + W 1 (W/2+ )2 = W 2 + W 1 W 2 /4+W + 2 = 3W W. This value is greater than 3W/4 since >0, which contradicts our assumption. If w j <W/2, then let = W/2 w j > 0. For this case, we have Z(τ,u (τ)) = W W 1 (W/2 )2 = W W 1 W 2 /4 W + 2 = 3W W, which is again greater than 3W/4, and thus contradicts again our assumption and completes the proof. Note that DVP1 is the decision version of both the P2-type 1 d j = d unknown, conv (Z/V ) and the P3-type 1 d j = d unknown,conv (Z/V ) problem. In addition, since the P4-type 1 d j = d unknown,conv #(V/Z) problem is at least as hard as the P2-type and P3-type problems, we have the following corollary. Corollary 1 The P2-type 1 d j = d unknown,conv (Z/V ), P3-type 1 d j = d unknown,conv (V/Z) and the P4-type 1 d j = d unknown,conv #(V/Z) problems are all NP-hard even if all v j =1. 306

7 4 Polynomially solvable cases Lemma 2 Consider jobs l and m satisfying the following conditions: v l θ l v m θ m and α l α m.thenforany V value, there is an efficient schedule in which job l is scheduled before m. Proof Consider an efficient schedule S, let jobs l and m satisfy the conditions of the lemma and assume that m is before l in S. Interchange l and m to get schedule S. If both l and m belong to E or to T in S, then changing their order does not affect the optimality of the schedule since the scheduling objective value in (5) and the optimal resource allocation given by (6) are independent of the internal sequence within each set. If l and m belong to different sets, then by Step 2 of Algorithm 1, we can assume that m E and l T in schedule S, and by Lemma 1 that u l (S) = 0. Definetheresourceallocationsin S according to eq. (6). Since the optimal resource allocations defined by (6) satisfy (13) of the Appendix, we also have n v j u j ( S)= ( S) v ju j ( S)=V, where E( S)=E\{m} {l}. Let E 0 = E( S)\m. Then based on (7), the change in the scheduling objective function is Z(S) Z( S)=α l α m + βn V k [(B +(v mθ m ) k/k+1 ) k+1 (B +(v l θ l ) k/k+1 ) k+1 ], where B = 0(v jθ j ) k/k+1. Since v l θ l v m θ m and α l α m, we have that Z(S) Z( S) 0. This means that S is also an efficient schedule. Lemma 2 implies that we can restrict our search for efficient schedules in which everypairofjobssatisfyingthelemma sconditionsalsosatisfies the precedence constraint l ¹ m, i.e., l is scheduled before m. It also follows that in any algorithm that tries to find the set E of an efficient schedule, we can consider job l before job m for inclusioninthesete. In the following we present a polynomial time optimization algorithm to determine the set of all efficient schedules for the special case where the partial order ¹ becomes a linear (complete) order, i.e., when the two ordersappearinginlemma2areagreeable and order all the jobs in the same sequence. These conditions will be satisfied in many applications, since they simply say that for shorter jobs (jobs with lower workload and resource consumption cost), the supplier has to pay larger tardiness penalties if they are late, and this coincides with the typical customer expectation that it should be easier to meet a due date for shorter jobs than for longer ones. For the agreeable case, Lemma 2 implies that the early jobs must form an initial subsequence of jobs ordered into a chain by ¹. The following algorithm finds the efficient frontier for the agreeable case, starting with the largest possible early set E = J. Algorithm 2 : Optimization algorithm for the 1 d j = d unknown,conv,agreeable #(V/Z) problem. Step 0. Preprocessing: Sort and renumber the jobs according to ¹ generated by the agreeable condition. Step 1. Initialization: Set q = n where q denotes the number of early jobs in the current segment of efficient schedules, E = {1,..., n}, T = φ. Let V0 lim =0and Vn+1 lim =, where Vj lim,j =1,...,n+1, denotes the minimum total weighted resource consumption needed for an efficient schedule that has exactly the first j jobs scheduled early in the ¹ order. 307

8 Step 2. Calculate the minimum amount of total weighted resource consumption for job q (but not job q +1)tobeinsetE by k+1 k+1 Vq lim = βn q 1/k q 1 k/(k+1) vj θ j k/(k+1) vj θ j. α q (10) Step 3. Theq-th segment of the Pareto curve is given by n V, α j + βn q k+1 k/k+1 θj V k v j (11) j=q+1 for any V satisfying Vq lim <V Vq+1. lim The optimal resource allocation for any point in this segment is given by eq. (6), and the optimal common due date is computed by d = k+1 q q p j (u j )= p j (u j )= k/k+1 θj v j V k, j=q where the last equality can be derived by substituting (6) into (1). Step 4. Set q = q 1. Ifq =0, then stop. Otherwise go to step 2. Algorithm 2 starts by renumbering the jobs according to the optimal job sequence that corresponds to the agreeable condition. In step 2 the algorithm determines the minimum amount of total weighted resource consumption for job q so that in the corresponding efficient schedule E = {1, 2,..., q}. The algorithm starts with the largest initial segment in ¹, i.e., with E = J. Using eq. (7), Vq lim is computed as the value of the total weighted resource consumption at which we are indifferent between assigning job q to be the next tardy job or leaving it in the early set, i.e., Vq lim is the value at which these two scenarios have equal scheduling costs. Thus, Vq lim satisfies k+1 k+1 n α j + βn q 1 k/k+1 n θj V lim k v j = α j + βn q k/k+1 θj q V lim k v j, q j=q+1 whose solution yields the Vq lim in eq. (10). Thus, for any Vq lim includes jobs 1,...,q while the tardy set consists of jobs q+1,..., n. Therefore, according toeq.(7),theefficient curve segment for V lim q <V V lim q+1 the early set <V V lim q+1 is given by eq. (11). Theorem 2 Algorithm 2 solves the 1 d j = d unknown,conv,agreeable #(V/Z) problem in O(n log n) time. Proof The correctness of Algorithm 2 follows from Lemmas 1, 2 and the preceding discussion. Step 0 requires sorting and thus takes O(n log n) time;steps2and3can be performed in linear time in the first iteration. In all other iterations, however, these steps require only constant time. Since Step 4 also takes constant time, and there are n iterations of Steps 2-4, the overall complexity of Algorithm 2 is O(n log n) indeed. Corollary 2 The efficient frontier (Pareto curve) for the 1 d j = d agreeable #(V/Z) problem contains at most n curve segments. unknown,conv, Proof Each iteration through Step 2 and Step 3 constructs one curve segment of the tradeoff curve, and there are at most n such iterations before the algorithm ends. 308

9 5 Conclusions and Future Research We considered four different versions of the problem of minimizing the weighted number of tardy jobs plus due date assignment cost and minimizing the total weighted resource consumption in scheduling a single machine. We assume that each job processing time is a bounded, decreasing convex function of the number of resource units that are allocated to the job and all jobs share the same common due date, which is also a decision variable. In an earlier study, Shabtay and Steiner [12] showed that the problem of minimizing an integrated objective (formed as the sum of the two criteria) is solvable in polynomial time. In contrast to this result, we prove that all of the other three versions of the problem are NP-hard. It is very interesting to notice that the polynomial solvability of P1 and the NP-hardnessofP4leadtotheconclusionthat in fact the enumeration of the supported strictparetooptimaispolynomialandthe enumeration of non-supported strict Pareto optima is hard (see [15]) for more details). This is quite an unusual result in multicriteria scheduling. In future research, we may seek approximation and exact algorithms to solve problems P2-P4. Acknowledgements This research was partially supported by the Paul Ivanier Center for Robotics and Production Management, Ben-Gurion University of the Negev and by the Natural Sciences and Engineering Research Council of Canada under grant No. OG References 1. Baker, K.R. and Scudder, G.D., Sequencing with earliness and tardiness penalties: a review, Operations Research, 38, (1990). 2. Chudzik, K., Janiak, A., and Lichtenstein, M., Scheduling problems with resource allocation. In A. Janiak, ed., Scheduling in computer and manufacturing systems, WKL, Warszawa (2006). 3. De, P., Ghosh, J.B., and Wells, C.E., Optimal delivery time quotation and order sequencing, Decision Sciences, 22, (1991). 4. Graham, R.L., Lawler, E.L., Lenstra, J.K. and Rinnooy Kan, A.H.G., Optimization and approximation in deterministic sequencing and scheduling: a survey, Annals of Discrete Mathematics, 5, (1979). 5. Gordon,V., Proth, J.M., and Chu, C.B., A survey of the state-of-the-art of common due date assignment and scheduling research, European Journal of Operational Research, 139, 1-25 (2002). 6. Janiak, A., One-machine scheduling with allocation of continuously-divisible resource and with no precedence constraints, Kybernetika, 23(4), (1987). 7. Janiak. A., Minimization of the makespan in a two-machine problem under given resource constraints, European Journal of Operational Research, 107, (1998). 8. Janiak, A., Janiak, W., and Lichtenstein, M., Resource management in machine scheduling problems: a survey, Decision Making in Manufacturing and Services, 1, (2007). 9. Kahlbacher, H.G., and Cheng, T.C.E., Parallel machine scheduling to minimize costs for earliness and number of tardy jobs, Discrete Applied Mathematics, 47, (1993). 10. Leyvand,Y.,Shabtay,D.,andSteiner,G.,Optimaldeliverytimequotationtominimize total tardiness penalties with controllable processing times, submitted (2008). 11. Nowicki, E., and Zdrzalka, S., A survey of results for sequencing problems with controllable processing times, Discrete Applied Mathematics, 26, (1990). 12. Shabtay, D., and Steiner, G., Optimal due date assignment and resource allocation to minimize the weighted number of tardy jobs on a single machine, Manufacturing & Service Operations Management, 9(3), (2007). 13. Shabtay, D., and Steiner, G., A survey of scheduling with controllable processing times, Discrete Applied Mathematics, 155(13), (2007). 309

10 14. Slotnick, S.A. and Sobel, M.J., Manufacturing lead-time rules: customer retention versus tardiness costs, European Journal of Operational Research, 169, (2005). 15. T kindt, V., and Billaut, J.-C., Multicriteria scheduling: theory, models and algorithms. 2nd edition, Springer, Berlin (2006). Appendix - Optimal Resource Allocation Strategy as a Function of τ Since under an optimal resource allocation strategy, the total weighted resource consumption constraint is satisfied as equality, we can apply the Lagrangian method to solve the resource allocation problem. The Lagrangian function is L(u 1,u 2,..., u n,µ)=βn k θj /u j + µ v j u j V, (12) where µ is the Lagrangian multiplier. Differentiating eq. (12) with respect to the decision variables (u j for j =1,...,n and µ), the necessary and sufficient conditions for an optimal solution are: L(u 1,u 2,..., u n,µ) µ = v j u j V =0; (13) L(u 1,u 2,...,u n,µ) u j Using eq. (14) we obtain that = kβnθk j u k+1 j + µv j =0 for j E. (14) Thus, θ k i v i u k+1 i u j = = θk j v j u k+1 = µ for i, j E. (15) kβn j k θj θ i k+1 v i v j Inserting eq. (16) into eq. (13) we obtain that or equivalently that u i θ k θj v j θ i 1 k+1 ui for i, j E. (16) k+1 v i v j k k+1 i v 1 k+1 i Rearranging eq. (18) we obtain the solution in eq. (6). 1 k+1 ui = V, (17) θj v j k k+1 = V. (18) 310