Online Appendix for Coordination of Outsourced Operations at a Third-Party Facility Subject to Booking, Overtime, and Tardiness Costs

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1 Submitted to Operations Research manuscript OPRE Online Appendix for Coordination of Outsourced Operations at a Third-Party Facility Subject to Booking, Overtime, and Tardiness Costs Xiaoqiang Cai Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, xqcai@se.cuhk.edu.hk George L. Vairaktarakis Weatherhead School of Management, Department of Operations, Case Western Reserve University, Euclid Avenue, Cleveland, OH , gxv5@case.edu Appendix 1. Minimizing the Number of Tardy Jobs We present, in this appendix, additional structural properties and optimal algorithms for the special case of (OP) with β j = β for every j N. This special case is of interest in its own right because it is often that tardiness penalties are not dependent on the jobs or the manufacturers but on the mode of transportation used to express ship late orders. Alternatively, β may represent the cost of a lost sale which is expected to be nearly identical for players who produce the same product. Moreover, the constant penalty case allows for structural insights that are not available for the general case. These insights allow the development of a strongly polynomial time algorithm. Towards this end, observe that the optimal cost for 1 βu i differs by a factor β from the optimal cost for 1 U i. The later problem is solved in O( N log N ) time due to the EDD ordering in the MH algorithm below. Therefore, the pseudopolynomial algorithm DP can be replaced by MH so as to solve 1 β U i in polynomial time. Algorithm MH (Moore, 1968) 1. Order jobs in N in EDD order; let L be the resulting ordered set. 2. Schedule the first job in L, say j, next to the last non-tardy job in N L. Let C j be the resulting completion time and set L := L {j}. 1

2 2 Article submitted to Operations Research; manuscript no. OPRE If C j > d j then select a non-tardy job in N L with longest processing time, say j 0, declare it tardy and we schedule it last. 4. If L then goto step 2 else Stop. Note that, whenever the currently scheduled job becomes tardy, the longest preceding job is made tardy in step 3 of algorithm MH, so as to allow as many of the jobs left in L to complete on-time. When the jobs to be scheduled are specified, it is convenient to use the following notation: MH(W)= the schedule produced when MH is applied to the single machine problem associated with transformations (1) from the main article, applied to the jobs scheduled for processing in windows W. Z(W)= the sum of booking, overtime, plus tardiness costs corresponding to schedule MH(W). In what follows we assume that an optimal collection WO has been determined and one needs to construct an optimal schedule σ for (OP ), including the decision of booking overtime. The following 2 schedules facilitate this construction. σ R : Optimal schedule when no overtime is used and schedule MH(WO) is obtained assuming that the last window in WO has unlimited capacity. σ F : Optimal schedule when the overtime of every window in WO is fully utilized before allocating work to the following window. Note that when no overtime is used in the windows of WO, we may need additional capacity to process all the workload. This is the reason for allowing unlimited capacity in the last window of WO in σ R. Let T R and E R be the set of tardy and non-tardy jobs in σ R respectively. Similarly, let T F and E F be the corresponding sets in σ F and T, E the corresponding sets in σ. The next two results are important for the identification of T which in turn is key in finding WO. Lemma 1. T F T T R Proof: Suppose that jobs are indexed according to the EDD order and J i T F. We will prove that J i T. If not, let J i be the l-th tardy job when MH obtained σ F. From the mechanics of MH there must exist r > i such that the completion time of J r is C r > d r when J r is considered for scheduling. Such J r must exist otherwise J i wouldn t become tardy by MH. Let A l be the set of jobs scheduled in σ F prior to C r when J r is first introduced and A l the set of jobs scheduled in σ when J r is first introduced. The total workload prior to C r in σ F is no less than the corresponding workload in σ when J r is first considered for scheduling. This is because the total overtime used prior to C r in σ F is no less than in σ. Therefore, A l A l. Without loss of generality suppose that in constructing σ F and σ ties are broken by MH so

3 Article submitted to Operations Research; manuscript no. OPRE that the set A l has as many jobs in common with A l as possible. This maximality property implies that, whenever possible, the same longest job is made tardy in σ F and σ. Since A l A l, we have max{p j : J j A l } max{p j : J j A l } = p i. If the latter inequality holds strictly, then MH must have already rendered J i tardy in σ prior to scheduling J r, i.e., J i T. On the other hand, if J i A l, the maximality of A l other jobs in A l should also be in A l implies that all because their processing time is no longer than p i and the fact that MH always removes the longest among the jobs scheduled so far. Equivalently, A l A l and hence A l = A l. But then, J r must be tardy when introduced to σ since it is already tardy when introduced to σ F which uses at least as much overtime as σ prior to introducing J r. Then, MH would render J i tardy in σ because it is the longest job in A l. In either case, J i T. This completes the left hand side of the lemma. The proof for the right hand side is analogous to our argument above except that MH is applied to windows of σ R. In light of Lemma 1, finding T is equivalent to identifying jobs in T R T F that must be processed early in σ. Suppose that the SPT order of T R T F is T R T F = {J i1, J i2,..., J ir } and E = E R + E where E T R T F. The following lemma indicates that T {J i1, J i2,..., J ir }. consists of the first few jobs in the ordered set Lemma 2. Suppose that T R T F = {J i1, J i2,..., J ir }. If J ik+1 E then J ik E for 1 k < r. Proof: Suppose that J ik+1 is early in σ but J ik is tardy. Because of the SPT ordering we have p ik p ik+1. If d ik d ik+1 then swap J ik+1 and J ik. Then, J ik becomes non-tardy because so is J ik+1 even though its processing time is not less than p ik. Hence, there is an optimal schedule such that if J ik+1 E then J ik E. On the other hand, if d ik < d ik+1, then consider the EDD order of jobs in E R +E +{J ik } {J ik+1 }. The amount x of overtime needed to process jobs in E R + E + {J ik } {J ik+1 } is no more than the amount x needed to process jobs in E R + E because p ik p ik+1. Also, none of the jobs in E R + E + {J ik } {J ik+1 } is tardy because this is the case when x units of overtime are used in σ F (because J ik / T F ). Therefore, there is an optimal schedule where J ik E and J ik+1 may or may not be tardy. This completes the proof of the lemma. We can now present an optimal algorithm for (OP) with constant penalties for the jobs. The algorithm examines all possible schedules W f,f that satisfy properties i, ii, iv, and v where F denotes the number of peak windows and f the total number of windows booked. Number f cannot exceed w = i p i. These 2 numbers together with properties i and ii completely specify the L

4 4 Article submitted to Operations Research; manuscript no. OPRE collection W f,f. Then, algorithm CP Opt below addresses problem (OP) with constant penalties by identifying the optimal non-tardy set E = E R + E for E {J i1, J i2,..., J ir }. Algorithm CP Opt Input : Collections W R and W P, associated booking costs and values O, α, β and d j : j N Output : Optimal collection W O and schedule σ Begin Let E = and Z O = [1] For F = 0 to min{w, W P } do For f = F to w do begin [2] W f,f := {F earliest windows in W P } {f F earliest windows in W R } [3] Apply MH(W f,f ) when every window has length L + O; let T F be the tardy set Apply MH(W f,f ) when every window has length L except the last that has infinite capacity; let E R, T R be the resulting early and tardy sets Order T := T R T F = {{J i1, J i2,..., J ir }} in SPT order For k = 1 to r do begin Find smallest integer k 0 such that P W k = j E R p j + k l=1 p i l k 0 (L + O) If k 0 > f then Goto [9] [4] Schedule jobs in {J ik+1,..., J ir } as late as possible in windows in W f,f [5] Schedule jobs in E R + {J i1,..., J ik } in EDD order using the next available window in W f,f and overtime whenever a job becomes tardy. Let σ be the resulting schedule [6] If Z O (σ) Z O then W O = W f,f, Z O = Z(σ), σ = σ and E = E R + {J i1,..., J ik } end [7] end End The complexity of CP Opt is dominated by the O(w 2 ) loops in [1] and the MH applications in [3] that take O(n log n) time. The resulting complexity is O(w 2 N log N ). When no overtime is available, we know that F = w and hence the above algorithm can be modified to solve (3P) with constant penalties in O(w N log N ) time. These observations verify Theorem 2 in Section 3 of the main article.

5 Article submitted to Operations Research; manuscript no. OPRE Appendix 2. Minimizing the Total Weighted Tardiness In this appendix we explore the relationship of our model to the case where the lateness penalty β j U j (C j ) is replaced by β j (C j d j ) +, where C j is the completion time of job j, for every j N. This is commonly referred to as the problem of minimizing the total weighted tardiness (TWT). For arbitrary due dates, minimizing the TWT on a single processor is known to be strongly N P - hard (Lawler, 1977). McNaughton (1959) has also shown that allowing preemption of jobs does not affect the optimum tardiness cost. Because of the strong N P -hardness of TWT, it is unlikely to get a polynomial-time algorithm to compute its exact solution. There is a research result, however, which shows that a pseudopolynomial algorithm exists for the TWT problem as long as the number of distint due dates is finite; see Kolliopoulos and Steiner (2006). It is interesting to note that our model falls naturally into this pseudo-polynomially solvable setting, because of the structure of our model that the due date of every job coincides with the end of a manufacturing window, whereas the number of windows is finite. Consequently, following from Kolliopoulos and Steiner (2006), we have Theorem 1. There is an algorithm with time complexity O(( N K! ( ) N +K K ( N T ) K )O(K2 ) 2 ) which optimally solves the TWT problem in our model, where N is the number of jobs, K is the number of windows, and T is an upper bound on the maximum tardiness of jobs in N. In our problem, a possible upper bound is T = b K + Ō. Hence, given a window collection W(r), the algorithm of Kolliopoulos and Steiner (2006) can be adapted to yield the booking plus total weighted tardiness cost, referred to as the problem (B-TWT). Having such an optimal algorithm, players can determine their optimal booking and sequencing strategies, and 3P can compute a globally optimal schedule. Moreover, the coordinating mechanism developed in Section 4 carries over to the problem (B-TWT). Although the algorithm of Kolliopoulos and Steiner (2006) is psedudo-polynomial, its complexity is still too high. This is less of an issue for the problem with agreeable weights where p i < p j implies β i > β j. On a single processor, the TWT problem with agreeable weights was considered by Lawler (1977) who presented a dynamic programming algorithm that takes O( N 4 P ) time, where P is the total processing time. Lawler s algorithm can also be adapted to yield the booking plus total weighted tardiness cost for our problem (B-TWT) when the weights meet the agreeable condition. If one does not seek to find the exact optimal solution, the dynamic programming scheme we propose in Section 3 can be applied to yield an approximate solution for the problem (B-TWT), based on a property established in Kolliopoulos and Steiner (2006) regarding a relationship between

6 6 Article submitted to Operations Research; manuscript no. OPRE the TWT problem and the problem of minimizing total weighted late work. First, consider the case without overtime. Let σ, σ T be optimal solutions for the problems (RP) and (B-TWT), respectively, and let Z(.) be the total cost. Following Kolliopoulos and Steiner (2006) we have Theorem 2. If there exists polynomial f( N ) such that P = O(f( N )), then Z(σ T ) f( N )Z(σ ). In other words, algorithm Opt provides an approximation for (B-TWT). Interestingly, the above theorem does not depend on regular or overtime production and hence the result extends to the case with overtime. Note that the dynamic programming scheme we proposed in Section 3 exhibits quite acceptable computational complexity, and thus provides a powerful approximation approach to the problem (B-TWT). The savings allocation scheme (4) presented in Section 4 of the main article is applicable because the weighted tardy penalty β j U j (C j (σ)) and the total weighted tardiness β j (C j (σ) d j ) + of a job j are both nondecreasing functions of the completion time C j (σ) of j in σ. For problem (B-TWT), the savings v T (S) for a coalition S M are v T (S) = Z T (σ 0 (S)) Z T (σ (S)), where σ 0 (S) is the initial schedule σ 0 on players in S, and σ (S) the corresponding optimal schedule for players in S subject to the admissibility constraint, and Z T (σ(s)) = i S k W i (σ) ρ(h k + α k O k (σ)) + j N(S) β j (C j (σ) d j ) +. The last expression accounts for regular and overtime booking costs and total weighted tardiness costs. Appendix 3. Algorithm Coalition In what follows we describe, based on algorithm Opt presented in Section 3, an algorithm Coalition to compute savings v(s) associated with an arbitrary coalition S M. Note that after the booking process is complete, the jobs in N form a sequence σ 0 = (σ 0 (1), σ 0 (2),, σ 0 ( N )) where σ 0 (i) denotes the i-th job in the sequence. Let G{a, b} = {σ 0 (a), σ 0 (a + 1),..., σ 0 (b)}, for 1 a b N be a coalition of jobs. A window is booked by coalition G{a, b} if it is occupied by any job in the coalition. Upon completion of the initial bookings, let W e and W l be the earliest and the latest booked windows of coalition G{a, b}, respectively. By definition, note that windows W e and W l may process jobs outside G{a, b}. This cannot be the case for windows between W e and W l. As a result, any window between W e and W l can generate booking savings by consolidating the workload of jobs in

7 Article submitted to Operations Research; manuscript no. OPRE G{a, b} and releasing unused windows. Windows W e, W l however, may not yield booking savings to G{a, b} if they are also occupied by jobs outside the coalition. Coalition G{a, b} faces the problem of rescheduling jobs over the windows booked originally, so as to achieve the minimum combined booking, overtime, and tardiness cost. Denote this problem as P Coalition. The main difference between P Coalition and (OP) is that windows W e and W l may not be released and hence the booking charges h e, h l have to be paid. We introduce the following notation: Q ab = j G{a,b} p j; the total processing time (workload) of jobs in G{a, b}, Q e : the amount of window time in W e not belonging to G{a, b}, 0 Q e L, Q l : the amount of window time in W l not belonging to G{a, b}, 0 Q l L, Evidently, Opt can be modified to produce optimal schedules that satisfy the constraints imposed on collection W(r, Ō) of Opt. Permanently including W e or W l forces us to include one or more of these windows in W(r, 0). Permanently excluding a window simply prevents us from introducing that window in the next candidate collection. Moreover, Opt is not effected if W e and/or W l allow for different maximum regular or overtime production time. Hence, P Coalition can be solved by making the following adjustments on the regular and overtime capacity availability of windows W e, W l, and revised booking costs h e, h l: W e : Regular: max{l Q e, 0}, Overtime: L + O max{q e, L}. W l : Regular: max{l Q l, 0}, Overtime: L + O max{q l, L}. These capacity limits simply reflect the remaining regular and overtime capacity in windows W e, W l after accounting for the workload of jobs not in G{a, b} that are processed in W e, W l respectively. Given these availabilities, let W (G) denote an optimal collection of windows for coalition G{a, b}. The following constraints indicate when W e, W l are candidate or required windows of W (G). In each case the revised cost of the window is included. Q e (0, L), h e = h e, W e W (G) Q l (0, L), h l = h l, W l W (G) Q e = 0, h e = h e, W e candidate Q l = 0, h l = h l, W l candidate Q e L, h e = 0, W e candidate Q l L, h l = 0, W l candidate. To verify these constraints, observe that, when Q e (0, L) window W e must be in W (G) to process jobs not in coalition G{a, b}, and hence the booking cost remains unchanged. When Q e = 0, then W e is used exclusively by jobs in G{a, b} and hence can be replaced by any other window if this is profitable. The resulting booking saving in this case is h e = h e. If on the other hand Q e L, window W e must be included in W (G) to process jobs not in coalition G{a, b}, and hence the cost to jobs in G{a, b} is 0. Similarly for window W l. The exhaustive enumeration nature of this modified version of Opt (referred to as algorithm Coalition) produces an optimal solution for

8 8 Article submitted to Operations Research; manuscript no. OPRE problem P Coalition. The time complexity of Coalition is the same as that for Opt. Appendix 4. Counterexample Consider an instance where manufacturers 1, 2, 3 arrive to the system in the order 1, 2, 3, each of which owns a single job with p j = 1, d j = 9 for j = 1, 2, 3, L = 3, h 1 = h 3 = 0, h 2 = 100, Ō = 0, ρ = 1 and arbitrary positive β j s. Recall that all parameters are rescaled so that the 3 manufacturing windows have no idle time between them (i.e., window t + 1 starts at time a t+1 = b t ). Before coordination, manufacturers 1 and 2 (occupying windows W 1 and W 3 respectively) incur no cost while manufacturer 3 (occupying W 2 ) incurs cost of 100 and all jobs finish early. After coordination all 3 jobs are scheduled in window W 1. Consider coalitions S = {1, 3} and T = {2, 3}. Then, v(t ) = v(s)=100 because both release W 2 thus earning refund savings of 100. The same is true for coalition S T = {1, 2, 3}. However, v(s T ) = v({3}) = 0 because manufacturer 3 cannot create savings all by himself. Thus, v(s T ) + v(s T ) = 100 < 200 = v(s) + v(t ). This example shows that the cooperative game is not convex.