Coordination of Outsourced Operations to Minimize Weighted Flow

MANUFACTURING & SERVICE OPERATIONS MANAGEMENT

informs

Vol. 12, No. 2, Spring 2010, pp. 236–255 issn 1523-4614 eissn 1526-5498 10 1202 0236

®

doi 10.1287/msom.1090.0265 © 2010 INFORMS

Coordination of Outsourced Operations to Minimize Weighted Flow Time and Capacity Booking Costs Tolga Aydinliyim

Decision Sciences Department, Charles H. Lundquist College of Business, University of Oregon, Eugene, Oregon 97403, [email protected]

George L. Vairaktarakis

Department of Operations, Weatherhead School of Management, Case Western Reserve University, Cleveland, Ohio 44106, [email protected]

A

set of manufacturers outsources certain operations to a single third party following the announcement of a booking price for each available day of production. Knowing these costs, manufacturers book available production days in a first-come-first-serve order to optimize their individual cost. The cost for each manufacturer consists of booking and work-in-progress costs, as expressed by the weighted flow time. When window booking is completed, the third party identifies a schedule that minimizes the total cost incurred by all manufacturers. This coordination reduces the total cost but may result in higher costs for a subset of manufacturers. For this reason, the third party devises a savings sharing scheme with which the monetary benefit for each manufacturer is greater. In this article we present algorithms for the problem considered, as well as savings-sharing schemes that make coordination a better alternative for all parties. The highlight of our experiments is that the costs of the production chain can be reduced by an average of 32% if one-third of the members let the third party cover their increased work-in-progress cost in exchange for 38%–53% of the total savings. Key words: coordination; cooperative games; incentives; outsourcing; supply chain management; production planning and scheduling History: Received: January 31, 2006; accepted: April 13, 2009. Published online in Articles in Advance September 14, 2009.

1.

Introduction and Literature Review

their operations, which require little or no setup between them but have different processing time requirements (e.g., finishing operations, testing operations). The setup times between the jobs that belong to different manufacturers are also insignificant. The third party announces his capacity availability as well as the booking cost for each day of production, referred to as a manufacturing window. The booking price of each window might reflect peak demand periods or timeliness with respect to the start of the planning horizon. Knowing the availability and the costs of the manufacturing windows, manufacturers book third-party capacity independently in a firstcome-first-serve (FCFS) basis; their objective is to minimize their individual costs expressed as the total weighted flow time plus booking costs. Therefore, for each manufacturer there is a trade-off between booking expensive early windows and cheap late windows. When all bookings are finished, the third party offers a coordinated schedule by rescheduling all jobs

Today’s business-to-business world is made up of continuously growing supply chains that create a complex business environment that is hard to control. Huge networks of competing suppliers, manufacturers, and distributors create situations where individual decisions are affected by the decisions of other parties in the network. The quality of the decisions depends on the amount of information the decision makers have about other members of the supply chain. Therefore, many industry sectors have recently created portals where key information is shared among the members of a supply chain, which allows those members to cooperate to reduce their individual costs as well as the cost of the entire supply chain. In this article, we model such cooperation and provide incentive schemes. Specifically, we consider models whereby a group M of manufacturers book third-party capacity for 236

Aydinliyim and Vairaktarakis: Coordination of Outsourced Operations

237

Manufacturing & Service Operations Management 12(2), pp. 236–255, © 2010 INFORMS

Figure 1

Order Fulfillment Process of the Cisco’s Networked Supply Chain and Our Model (Manufacturers M1 M2 MM and the Third Party 3P )

Customer order

Demand planning

Build

M1

M2

………

Testing

Testing

3P

………

………

Manufacturing partner

………

M|M|

Payment


Supplier partner

Supplier partner

Ship and deliver

Manufacturing partner Manufacturing partner

Supplier partner

Test

Testing


as if the jobs from different manufacturers belonged to a single party. Overall cost reductions are obvious and significant, but some manufacturers might end up with higher individual costs. Therefore, the third party provides incentives to each manufacturer so that everyone benefits from coordination and proposes an allocation of savings that exceeds potential savings the manufacturers can achieve by forming smaller groups. The ability to implement coordinated production schedules depends on the information-sharing capabilities of companies. Implementing information technologies to increase such capabilities has become the priority of many vertically-integrated companies. One such portal, known as eHub, is in use at Cisco Systems, which is a private trading e-marketplace that provides a central point for planning and execution of tasks across the company’s extended manufacturing supply chain (Grosvenor and Austin 2001). Orders from Cisco’s customers are stored in this enterprize resource planning database and sent to the related manufacturing partners over a virtual private network. These manufacturing partners correspond to

the group of manufacturers M in our model (see Figure 1). Within Cisco’s supply chain there are contractors who can see information on the network because their own production systems are also connected to eHub. For example, consider a contractor in the network who performs the testing operations for the assemblies or components produced by the manufacturing partners. This situation is equivalent to a model where manufacturing partners M outsource their testing operations to a single contractor in Cisco’s network, who plays the role of the single third-party in our model (see Figure 1). Recall that the production schedules of all parties involved are transparent to everyone because they are all connected to Cisco’s information-sharing portal. Therefore, coordinated capacity and production planning opportunities exist within this framework. In our model, we consider such coordination possibilities and provide incentive schemes to make everyone benefit from coordination. The operational protocol of the Semiconductor Product Analysis and Design Enhancement (SPADE)

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Center1 illustrates our model more precisely. SPADE provides to semiconductor companies services such as the analysis and optimization of designs and products when their silicon prototypes are available in the form of silicon wafers or silicon dies. SPADE has a group of specialized facilities, including Focused Ion Beam, Emission Microscope, Electrostatic Discharge (ESD) Tester, Backside Preparation System (Chip UnZip), Laser Cutting System, etc., which can be booked at varying charges. SPADE announces the available booking periods for the upcoming three weeks (similar to our finite number of manufacturing windows), and each manufacturer books capacity in a FCFS manner.2 Rules of the charging scheme are available to customers3 and include the following: (a) jobs performed on Sundays and public holidays will be charged at double the basic rate, and (b) jobs performed during nonoffice hours will be charged at 1.5 times the basic rate. These rules show that there are different types of charging rates corresponding to the arbitrary booking prices in our model. At SPADE Center, jobs can be preempted by others and resume at a later processing window. SPADE also provides pricing mechanisms for reoptimization by individuals or groups of partners. Theses include the following: (c) depending on the timing of a cancelation, full or partial refunds are provided according to the job cancelation policy.4 The success of SPADE’s scheduling rules in facilitating coordination among manufacturers is unknown; however, in this article we propose a game theoretic coordination mechanism that leads to the voluntary cooperation of all parties involved. As a result of the FCFS capacity booking practice at the third party, the following two sources of inefficiencies arise: (a) idle time may be left at the end of the last window booked by each manufacturer. The combined idle time of all players may exceed the length of one complete window; and (b) a manufacturer that books late in the FCFS order may have to schedule a high-priority job in one of the latter windows because all earlier windows are already 1

http://www.ust.hk/spade.

2

See item 4 in Terms and Conditions at http://www.ust.hk/spade/ request/index.html. 3

http://www.ust.hk/spade/pricelist.html.

4

http://www.ust.hk/spade/request/cancellation.html.



booked. This leads to increased weighted flow time costs. Therefore, centralization can eliminate inefficiencies by making improvements along these two dimensions. The third party can also benefit from coordination among the manufacturers. Specifically, some of the manufacturing windows are released and some others that were not reserved in the FCFS schedule may be booked in the coordinated schedule. The third party is entitled to keep a fraction of the booking costs associated with the released windows (see item (c) above in SPADE’s charging scheme) and return the rest to the manufacturers in the form of booking refunds. Furthermore, he can generate additional benefits for himself by rebooking these windows. In this paper we incorporate the booking refunds into our formulation and recognize potential rebooking as an additional benefit to the third party. In our model, we assume complete information sharing among all manufacturers and the third party. The extent to which information can be fully shared depends on the type of information and the infrastructure that facilitates information sharing. For example, the booking costs of the manufacturing windows may be announced openly (e.g., SPADE), and some job characteristics such as the processing time requirements of the outsourced jobs can easily be verified once the initial bookings are completed. However, some parameters, such as the priority that manufacturers place on jobs, may not be as easily observed in all applications. Our model captures the case where the manufacturers are members of the same network (as the manufacturers in Cisco’s supply network, all of which use eHub) or the players are divisions of the same parent company (as in the problem of scheduling the use of a single repair and maintenance facility for the divisions of a single firm). Therefore, the players are encouraged to share vital information, such as unit weighted flow time costs, because cooperation would lead to reduced costs and improved delivery performance across the supply chain. It is still possible for the manufacturers to report inaccurate values if they are seeking advantage in the coordinated schedule. Studying incomplete or imperfect information structures, or both, as well as truthrevealing mechanisms is a fruitful research direction.



In our model, we assume that players book windows optimally when they enter the system without anticipating the decisions of future players. Clearly, more strategic firms would take the subsequent rescheduling game into consideration when choosing windows. Specifically, if a player expects that a particular window is critically important for future players, he might prefer to reserve it at a higher individual cost, hoping for a portion of the savings that will be created by rescheduling. Such games are beyond the scope of the paper. In this paper, we address the issue of coordination among manufacturers by modeling their interactions as a cooperative sequencing game. Sequencing games are at the interface of sequencing and scheduling of operations and cooperative game theory; they involve solving two types of problems: (a) a series of combinatorial optimization problems, where a schedule that minimizes a certain cost objective is found for any given subset of players; and (b) a game theoretic problem that models the interactions of the players as a cooperative game. Each player optimizes his schedule according to his individual objective function and has the option to act solely on his own behalf or join a group of other players who can coordinate their activities to maximize the group savings. To ensure the cooperation of all players, the coordinated solution needs to be accompanied by an incentive payment scheme. The subset of allocations where savings are distributed to players in such a way that no subset of players can be better off by seceding from the rest of the players is called the core of a game. Sequencing games were introduced by Curiel et al. (1989). They considered the simplest case of a single machine with no restrictions on the jobs. The cost criterion is weighted flow time, and it is proved that the corresponding sequencing games are convex. For such games Shapley (1971) showed that a core allocation is guaranteed to exist. For arbitrary regular cost criteria and for a special class of games (referred to as 0 -component additive games), Curiel et al. (1994) proposed a core allocation known as the -rule. Curiel et al. (2002) presented a survey of all seminal papers since 1989 on sequencing games, core allocations, and convexity issues. Literature on sequencing games in which each player has more than one job to be processed is quite limited. Calleja et al. (2004) studied

239

a single-machine setting in which each player might have more than one job to process and each job might be of interest to more than one player. They showed that the core of these games is nonempty only if the underlying cost functions are additive with respect to the initial order of jobs. Calleja et al. (2002) considered a two-machine problem, where each player has precisely two operations, each processed on a dedicated machine. The cost criterion is the maximal weighted completion time of the two operations. It is shown that these games are not convex but a core allocation exists. Hamers et al. (1995) considered singlemachine games for jobs with release times and the weighted flow time cost criterion and showed that they are not convex except for instances with unit processing times or unit weights. Nevertheless, they proved that the core of these games is nonempty. Borm et al. (2002) considered problems with job due dates. For three different due date-related cost criteria they showed that the core is nonempty. Convexity was proven for only a special subclass. Hamers et al. (2002) imposed precedence constraints on the jobs and proved that the corresponding games are convex when the precedence network consists of parallel chains. Hamers et al. (1999) considered games with m parallel identical machines and no restrictions on the jobs. The cost criterion was weighted flow time. They proved the existence of core allocations for m = 2 and for some special cases when m > 2. Recently, Cai and Vairaktarakis (2007) studied a production planning setting similar to ours, which involves a single third party and multiple manufacturers each outsourcing a subset of jobs to a third party. The cost criterion for each manufacturer is job-specific tardiness penalties plus booking costs that can take on one of two values that reflect peak and off-peak production. The authors also allow for overtime capacity provided by the third party on an hourly basis. Similar to this paper, they consider a cooperative game and provide a core allocation of savings. Despite the similarities in the production planning setting studied and the sequence of events, our paper significantly differs from Cai and Vairaktarakis (2007) in the following dimensions. First, we consider arbitrary booking costs at the third party. This allows the use of a more flexible pricing scheme, so that the third party can charge earlier windows relatively more

240



expensively than later windows, or he can increase the booking prices for nonregular hours. In addition, the second part of our objective function is the total weighted flow time cost, which enables us to study the timeliness of the delivery of the outsourced workload back to each manufacturer. This is a major difference from Cai and Vairaktarakis (2007), where for their tardiness-related objective a delayed job matters only if it is finished after its due date. Furthermore, our objective captures inefficiencies associated with delays that affect the next steps in the production process. This is extremely important in order fulfillment processes similar to Cisco’s, where a manufacturing partner has to wait for the testing process to be completed before the final product (all outsourced and inhouse produced components of it) is shipped to the customer (see Figure 1). Not surprisingly, considering this objective function requires completely different algorithms and solution procedures than those that are developed in Cai and Vairaktarakis (2007). Finally, the savings-allocation scheme that we propose differs in two dimensions: (a) our rule allocates savings based on the ownership of the windows in the initial sequence, as opposed to Cai and Vairaktarakis (2007) who allocated savings based on the ownership of jobs; and (b) our allocation rule distinguishes between idle time savings and other savings from resequencing of jobs and released windows in the coordinated schedule. Cai and Vairaktarakis (2007) allocate the overall savings according to an incremental savings-allocation scheme. Literature at the interface of scheduling and cooperative game theory is quite limited. However, the issue of coordination and the inefficiencies that arise from decentralization—i.e., decentralization cost, when multiple parties compete for common resources—have been studied in different contexts. See, for example, congestion games introduced by Rosenthal (1973) and its applications to managing highways (Sandholm 2002) and communication networks. Researchers in various fields have studied different mechanisms to eliminate the decentralization cost, such as contract design in supply chains (Cachon 2003), equilibrium pricing and auction design (Wellman et al. 2001), and scheduling-based coordinating mechanisms (Bukchin and Hanany 2007) in production capacity allocation.

In this study, we take an approach similar to the cooperative sequencing games literature and devise a savings allocation scheme to encourage coordination and hence to eliminate the decentralization cost. In our model, we allow each manufacturer to schedule multiple jobs, as in Calleja et al. (2004), and manufacturing costs include weighted flow time costs, as in Curiel et al. (1989). Our main modeling contribution is the addition of a more realistic production planning setting to the cooperative sequencing games literature, where the third-party capacity is represented by noncontiguous manufacturing windows with limited capacity and a composite objective function, which is the sum of the work-in-process (WIP) and the booking costs; plus the booking costs of the manufacturing windows is considered. The rest of the article is organized as follows. We start the next section with a formal definition of our model and assumptions. In §3, we discuss coordination results and present a savings-sharing scheme for a “fair” allocation of savings. In §4, we present three heuristics that provide near-optimal solutions to the centralized problem and two different lower bounding schemes that help assess the quality of the heuristics. We present the results of our computational experiments and report our insights in §5 and make concluding remarks in §6.

2.

Model Formulation and Preliminaries

Let M be a set of manufacturers who outsource their jobs, say N = J1 Jn , to a single third party, 3P . The set of jobs that belong to manufacturer m ∈ M is denoted by Nm . Each job has known processing time pj and weight wj , i.e., the unit WIP cost of Jj . Party 3P has T available days of production (each referred to as a manufacturing window). The set of manufacturing windows is = W1 WT . Window Wk costs hk to book, as announced by 3P at time 0, for 1 ≤ k ≤ T . The manufacturing windows are noncontiguous; i.e., there is a down time G between two consecutive windows. Following the announcement of the available manufacturing windows and the associated booking costs, the manufacturers make reservations on a FCFS basis, say in the order 1 2 3 M.


241


Let 0m be an initial optimal schedule for manufacturer m and 0m be the corresponding window collection. Manufacturer m jointly optimizes the schedule 0m of his jobs and the subset 0m ⊆ of windows selected. The joint optimization is made so as to minimize the total cost TC0m , which is the total flow m time cost Jj ∈Nm cj 0 plus the total booking cost Wk ∈ m hk . Every job is shipped as soon as processing 0 is completed; this protocol is referred to as immediate shipments. Hence, cj = wj · Cj , where Cj is the completion time of Jj in schedule and wj is the unit WIP cost associated with job Jj . Following the determination of 0m for each m ∈ M, 3P reschedules all the jobs in N as if they belong to one manufacturer to obtain the best schedule ∗ and associated collection ∗ of windows. Schedule ∗ minimizes the total cost over all manufacturers, so it is no worse than the schedule 0 obtained by concatenating M 01 02 03 0 . Every manufacturer m ∈ M would agree to go along with ∗ only if his savings by coordinating with all manufacturers are no less than the savings by coordinating with a subset of manufacturers. Therefore, 3P must find an allocation of the savings produced by ∗ such that every manufacturer is better off through this coordination. Normally, ∗ is expected to use fewer windows than 0 because of better use of idle times across all manufacturers. Party 3P refunds a fraction (0 ≤ ≤ 1) of the booking savings and keeps the rest for himself. We now state the assumptions of our model. All jobs are assumed to be of similar nature (e.g., testing operations, maintenance operations, generic assembly, fabrication operations, etc.), and therefore the changeover time between jobs that belong to different manufacturers is assumed to be 0. All announcements by 3P about the booking costs and the reservations by manufacturers are made before the start of the first manufacturing window in the planning horizon (time 0). Also, all jobs are assumed to be available for processing at time 0. The FCFS ordering of the manufacturers implies that 0m ⊆ W1 WT \ 01 ∪ 02 ∪ 03 ∪ · · · ∪ 0m−1

(1)

Expression (1) means that manufacturer m optimizes his initial schedule over the windows that have not

been booked by manufacturers 1 m − 1. Hence, in schedule 0 , each booked window includes jobs from only one manufacturer. We assume that the total processing requirements of jobs in Nm do not exceed the available processing hours of windows in W1 WT \01 ∪ 02 ∪ 03 ∪ · · · ∪ 0m−1 . Otherwise, manufacturer m could not be served by 3P and would seek another third party. Also, if a job is preempted at the end of a manufacturing window, it resumes normally in the next booked window. Furthermore, we assume that every manufacturing window Wk ∈ has precisely L hours of production for k = 1 T . The number of manufacturing windows required to process jobs in N is , where = Jj ∈N pj /L . As described earlier, each manufacturer, as well as 3P , faces the same optimization problem on a different set of available windows. For ease of presentation, we solve the centralized problem faced by 3P and assume that the manufacturers and subgroups of manufacturers adopt the same algorithms. Specifically, 3P schedules the jobs in N to minimize the total cost by utilizing windows in . Every window booked by a player processes a batch of jobs—those completed within the window. Therefore, the centralized problem involves batching of jobs in N into windows in . Suppose is a single machine sequence of jobs in N and Fj is the completion time of the jth job in . We want to partition in parts and find the total weight of the completed jobs in each part. This is done as follows. Let ti = i · L, and define si =

Jj ∈N ti−1 w, then there can be no x = xA xB xC that satisfies (8) and (9) at the same time. Therefore, for a very general case, where the booking costs are larger then unit WIP costs, it is impossible to find a core allocation. At this point we define a set of admissible rearrangements for jobs. The following definition and one component of the subsequent allocation rule are similar to those proposed by Curiel et al. (1994). Definition 1. Let S be a schedule of jobs in NS . A schedule S is said to be admissible with respect to 0 S if every Jj ∈ N \NS has the same predecessors in both S and 0 S. We make the following assumptions about the cooperative game among the manufacturers: Assumption 1. The optimal schedule for any coalition S ⊆ M, namely ∗ S, must be admissible with respect to 0 S. Subject to Assumption 1, we define the cooperative savings game (M -) among the manufacturers where -& 2M → R, - = 0, ∗ S is admissible with respect to 0 S and ∗

-S = TC0 S − TC S = *cj 0 S − cj ∗ S+ + m∈S Jj ∈Nm

·

Wk ∈0 S

hk −

Wk ∈ ∗ S∩0

hk

−

Wk ∈ ∗ S\0

hk

(10)

Evidently, -S& S ⊆ M represents the total savings that can be attained when all manufacturers in S coordinate to process jobs in NS optimally subject to Assumption 1. Now observe that, to reach ∗ S, each coalition solves a maximization problem with objective function -S subject to ∗ S being admissible with respect to 0 S. This objective function has the following sunk cost cj 0 S + · hk m∈S Jj ∈Nm

Wk ∈0

and hence, the optimization problem reduces to the minimization of cj ∗ S + · hk + hk

m∈S Jj ∈Nm

Wk ∈ ∗ S∩0

Wk ∈ ∗ S\0

subject to ∗ S being admissible with respect to 0 S. Note that this is a special case of the centralized problem with the objective function (3) and differs from that problem in that the job set is NS instead of N , the booking costs for the windows are hk

if Wk ∈ \

· hk

if Wk ∈ ∗ S ∩

+

if Wk ∈ 0 M\S

(11)

and ∗ S is admissible with respect to 0 S. Therefore, the algorithms that we propose in §4 can be applied to the savings maximization problem of any coalition S ⊆ M. 3.2. Structural Results of the Cooperative Game In this section we develop properties of game (M -) that lead to the development of an allocation rule that satisfies xm ≥ -S ∀ S ⊆ M xm = -M (12) m∈S

m∈M

so that every manufacturer m ∈ M prefers the grand coalition M to a smaller coalition S ⊆ M. In Shapley (1971) it is shown that convex games have a nonempty core. Definition 2. A game is said to be convex when ∀ S T ⊆ M

-S ∪ T + -S ∩ T ≥ -S + -T


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The following example shows that (M -) is not necessarily convex. Example 2. Consider a four-player game in which each player has one job to process, M = 1 2 3 4, N1 = J1 , N2 = J2 , N3 = J3 , N4 = J4 , p1 = 2, p2 = 1, p3 = 3, p4 = 1, w1 = w2 = w3 = w4 = 1, = 4, L = 3, = 1, h1 = h2 = h3 = h4 = 0 and each window starts three time units after the completion of the previous one (G = 3). The initial schedule costs 1 · 2 + 1 · 7 + 1 · 15 + 1 · 19 = 43. The optimal schedule for M costs 1 · 1 + 1 · 2 + 1 · 7 + 1 · 13 = 23. Therefore, -M = -1 2 3 4 = 20. Similar calculations show that -1 2 3 = 11, -2 3 4 = 12, and -2 3 = 2. Now consider S = 1 2 3, T = 2 3 4, S ∪ T = 1 2 3 4, and S ∩ T = 2 3. Then -1 2 3 + -2 3 4 = 23 > 22 = -1 2 3 4 + -2 3; hence, the game is not convex. Although M - is not convex, we may still find a core if the game is superadditive. Definition 3. Game M - is said to be superadditive when ∀ S T ⊆ M with S ∩ T =

-S ∪ T ≥ -S + -T

This means that two disjoint coalitions can do no worse by forming a larger coalition that is just the union of the two, but still a careful allocation of such savings is required. In what follows we investigate the three different means of savings that the grand coalition can generate, present superadditivity results for each of these games, and propose a core allocation rule. 3.3. The Savings from Cooperation Recall that in §2, we expressed the objective function for the centralized problem as

TC =

wj · *Dk − L + Fj − ti−1 + · yji · zki

Wk ∈ i=1 Jj ∈N

+

Wk ∈

hk

i=1

zki

(13)

where is the single machine sequence of the jobs. Rewrite the above expression as j

k TC = p i w j + si k−1G+k−iLzi Jj ∈N

+

i=1

Wk ∈

hk

i=1

i=1

zki

Wk ∈

(14)

where Jj denotes the jth job in (hence, the batch indicator yji is no longer needed) and si is the sum of weights of jobs in batch Bi as defined in §2. The first term in (14) captures WIP costs of processing jobs in N on a single machine according to sequence , and the rest capture booking and WIP costs for the particular collection of windows booked in . On the other hand, the total cost of the initial schedule 0 includes an extra term reflecting the idle time It in the last booked window Wt for each manufacturer, as follows: TC0 =

j i−1 pi wj + si 0 It

Jj ∈N

+

i=1

i=1

+

si 0

Wk ∈

hk

i=1

Wk ∈

i=1

t=1

k − 1G + k − iLzki

zki

(15)

where si 0 terms represent the sum of weights processed in each batch Bi according to the initial sequence followed by each manufacturer in his FCFS booking of 3P capacity. To illustrate, consider the following example. Example 3. Suppose manufacturer 1 has one job with weight w1 and processing time p1 < L, and in the initial schedule 01 this job is processed in Wa . Similarly, suppose manufacturer 2 has one job with weight w2 and processing time p2 < L and in the initial schedule 02 this job is processed in Wb , where a is strictly smaller than b. The total cost of this schedule is w1 · Da − L + p1 + w2 · Db − L + p2 + ha + hb and can be rewritten as w1 ·p1 +w2 p1 +p2 +w2 ·Ia +w1 ·*La−1+Ga−1+ +w2 ·*Lb −2+Gb −1++ha +hb or, equivalently, *w1 · p1 + w1 · *La − 1 + Ga − 1+ + ha + + *w2 p1 + p2 + w2 · Ia + w2 · *Lb − 2 + Gb − 1+ + hb + where the two brackets correspond to the players 1 and 2, respectively. A coordinated schedule for manufacturers 1 and 2 in Example 3 may lead to savings from (a) idle times and (b) better job sequencing and choice of windows


245


of the entire workload. Next, we analyze the cooperation game in terms of two games corresponding to the sources (a) and (b) of savings for any coalition S ⊆ M. We define 1. -id S to be the savings from idle times, and 2. -net S to be the savings from job interchanges and reallocation of windows, such that -S = -id S + -net S

M -id ,

M -net ,

xmnet =

where 1 ≤ a1 < b1 < a2 < · · · < ar < br ≤ T . We refer to *a1 b1 + *a2 b2 + etc. as maximally connected components of the coalition 0 S of windows. Observe that, from Assumption 1, the savings that coalition S can attain is the sum of the savings that can be attained by the maximally connected components of 0 S. Therefore, we define *a b+ as the maximum savings that can be attained by rescheduling the jobs originally scheduled in *a b+ and rewrite the savings functions -S, -id S, -net S in terms of the maximally connected components of 0 S; i.e.,

k=1

*ak bk + -net S =

r k=1

-id S =

Wk ∈0 m

1 2

·Ik

Wt ∈*k+1T +∩0

1 2

st + ·sk

Wt ∈*1k−1+∩0

It (19)

and

0 S = *a1 b1 + ∪ *a2 b2 + ∪ · · · ∪ *ar br +

r

=

and

For ease of presentation, rather than working with coalitions of manufacturers, we work with the corresponding window coalition 0 S. Let *a b+ denote the set of windows {Wa Wa+1 Wa+2 Wb . Then, one can express 0 S as

-S =

xmid

(16)

We have the following result: Proposition 1. Games M - are superadditive.

Next, we develop allocation rules that make every manufacturer better off as part of the grand coalition than it would be if manufacturers formed smaller coalitions. For 0 ≤ 4 ≤ 1, define

r k=1

id *ak bk + (17)

net *ak bk +

Savings produced by the windows in *a b+ consist of savings id *a b+ from better use of idle times and savings net *a b+ from job rearrangements and reallocation over windows. Therefore, *a b+ = id *a b+ + net *a b+

(18)

Wk ∈0 m

4· net *1k+−net *1k−1+ +1−4 · net *kT +−net *k+1T +

(20)

The intuition behind these allocations is as follows. For idle time savings, window Wk contributes Ik units of idle time to jobs completing in subsequent windows. The total weight of these jobs is Wt ∈*k+1 T +∩ st , and 0 therefore the resulting contribution of Ik to the WIP savings is Ik · Wt ∈*k+1 T +∩ st . Half of those savings are 0 allocated to the manufacturer who owns Wk , leaving the rest for the owners of the booked windows that follow (and that can hence use it). By symmetry, all jobs in Wk (whose total weight is sk ) benefit by start ing Wt ∈*1 k−1+∩ It time units earlier. The owner of 0 Wk receives part of those savings as well. Note that the availability of Ik units of idle time may result in completing a job at an earlier window, in which case the completion time decreases by at least Ik + G. The contribution of G to the savings is accounted for in xmnet . The rationale behind xmnet is that the owner of window Wk receives portion 4 of the marginal savings (net of idle time savings) achieved by Wk when it joins the coalition of all preceding windows plus portion 1 − 4 of the marginal savings (net of idle time savings) achieved when it joins the coalition of all following windows. Figure 2 highlights the different components of the allocation rule proposed in Expressions (19) and (20). The idle time savings are split equally between the owner of the windows and the other manufacturers.


246

Figure 2


Components of the Savings Allocated to the Owner of Window Wk

Idle time savings 1/2 of savings for allowing later jobs to use the idle time at Wk , i.e., 1/2 · Ik st Wt ∈k+1 T ∩

0

1/2 of savings for processing the workload of W k during the idle times at earlier windows, i.e., 1/2 · sk It

Wk

Idle

Wt ∈1 k−1 ∩

0

Idle

Idle

Wk

Savings from job rearrangements and booking costs fraction of the incremental savings (net of idle time savings) allocated to Wk for joining coalition of earlier windows

net 1 k − net 1 k − 1

Wk 1 − fraction of the incremental savings (net of idle time savings) allocated to Wk for joining coalition of later windows

The rationale for this split is that value creation by accommodating future jobs by using the idle time of Wk is equally important to value creation using the idle time of earlier windows to process the workload in Wk . Note that the ownership of a window in the initial sequence qualifies a particular manufacturer for idle time savings in two ways (see Figure 2). In contrast, the savings from job rearrangements and booking savings (i.e., savings net of idle time savings) are split in proportion to 4 and 1 − 4. Here, we allow differentiating between the marginal benefits that result by adding Wk to the tail of the window coalition *1 k − 1+ rather than to the head of the window coalition *k + 1 T +. In other words, adding Wk to *k + 1 T + provides different savings opportunities than adding Wk to *1 k − 1+. The former action may provide flexibility for the jobs in *k T +, whereas the latter provides flexibility to the jobs in *1 k+. Recall that the third party keeps a fraction of the booking costs of the released windows in the

1 − net k T − net k + 1 T

coordinated schedule to himself5 —i.e., 1 − · * Wk ∈ hk − Wk ∈ ∗ ∩ hk +—which could be thought 0 0 of as rents collected by facilitating (or allowing) rescheduling. Furthermore, if the grand coalition chooses to book new windows in the coordinated schedule, the third party is paid Wk ∈ ∗ \ hk for 0 these windows. Finally, let xm be defined as xm = xmid + xmnet

(21)

We can now state the following result: Theorem 2. Expressions in (19), (20), and (21) provide core allocations for the games M -id , M -net , and M -, respectively. The savings allocation vector x = x1 xM sat isfies the core inequalities, m∈S xm ≥ -S, ∀ S ⊆ M. 5 The third party can resell these windows, which we recognize as a potential benefit that is not taken into account in our formulation.


247


Intuitively, by joining the grand coalition, each manufacturer would save at least as much as it could save if it joined a smaller coalition. Therefore, x creates a Pareto improvement (because x also satisfies m∈M xm = -M) for every manufacturer not only with respect to his initial state as a result of the FCFS booking (a manufacturer cannot achieve any savings by himself; i.e., -m = 0, ∀ m ∈ M), but also with respect to his other choices of forming smaller coalitions. Note that our proposed rule defines one set of transfers that is a core allocation subject to Assumption 1. Also recall that, with no restrictions on the rescheduling of the manufacturers’ workloads, one cannot find a core allocation (see Example 1). Therefore, it may be possible to impose a different restriction and find another form of transfers that would result in a core allocation.

4.

Heuristics

In §2, we proved that the associated production planning problem is NP-hard in Theorem 1. In light of this result, we will develop algorithms to attain approximate solutions and lower bounds. In what follows we propose three heuristics for the problem of minimizing weighted flow time plus booking costs. All three heuristics start by arranging the jobs in weighted-shortest-processing-time (WSPT) order (simply putting jobs in decreasing wj /pj order) and proceed in two steps: (a) producing a sequence (which in turn yields batching B1 B2 B3 B ) and then (b) finding an optimal collection of windows to allocate the batches. Our heuristics differ from one another with respect to step (a). For step (b), they all use the same optimal dynamic program. We start by describing algorithms for producing batches B1 B2 B3 B . The first batching procedure takes the WSPT order of jobs and batches them by applying (2) without further processing. The second batching procedure is called -Partition and solves a series of − 1 knapsack problems described as follows. -Partition Input: Processing times pj and weights wj for Jj ∈ N = J1 Jn Output: Batches B1 B2 B3 B [0] Let k = − 1, B1 = B2 = B3 = · · · = B = and NU = N

[1] Solve knapsack problem: Mk = max Jj ∈N U wj · xj s.t. Jj ∈N U pj · xj ≤ k · L, xj ∈ 0 1, where xj = 1 if Jj is in the knapsack and 0 otherwise. [2] Set Bk+1 = Jj & xj = 0, N U = N U − Bk+1 If k = 0, then STOP, else set k = k − 1 and go to [1]. Our third batching procedure is called -RatioPartition. It is identical to -Partition except for the objective function of the knapsack problem in line [1], which now becomes max Jj ∈N U wj /pj · xj . The number of batches is bounded by T ; hence, at most T knapsack problems will be solved for heuristic -Partition. Solving a knapsack problem requires On Jj ∈N pj time and therefore heuristic -Partition is pseudo-polynomial with complexity OnT Jj ∈N pj , as is -Ratio-Partition. Given B1 B2 B3 B , the following dynamic programming formulation, referred to as BA, optimally assigns batches to manufacturing windows. Let fi k be the minimum total booking plus WIP costs for B1 Bi when Bi is processed in window Wk , and B1 Bi−1 are processed prior to Wk . Recursive relation: fi k = min

i−1≤r

percentage booking savings from coordination, and +

A =

m ∈ M&

Jj ∈Nm

cj H < M

Jj ∈Nm

cj 0m

100%>


251


Table 2

Percentage Total Cost and Booking Savings from Coordination (C and B) and the Percentage of Players Whose WIP Costs Improve After Coordination A+

Parameters

P /T = 15%

P /T = 30%

Nm

pj

C

B

A+

C

B

A+

4

20

8

10

1 4 5 8 1 4 5 8

3073 2454 3982 3084

1040 468 2119 1059

7250 6750 6875 6375

2816 2327 3939 3689

1065 439 2070 1010

6500 6750 6875 6375

3273

1172

6813

3193

1146

6625

M

Averages

percentage of manufacturers whose WIP costs decrease after coordination, where H is the best among the three heuristic schedules produced. In our experiments, we allow for full refund; i.e., = 1. In Table 2, we aggregate savings statistics over the combinations 4 20, 8 10 for M Nm . We make the following observations: • On average, coordination results in 32.33% savings in total costs =C, which increase with M (at a decreasing rate) and with L/p¯j . From the averages for =C in Table 2 (32.73% and 31.93%), we can conclude the average percentage savings in total costs after coordination is 32.33%. As the number M of manufacturers increases, the savings from booking and job rearrangements increase because more idle time is freed up and more rearrangement opportunities are created. The overall average savings aggregated over M = 4 and 8 are 27.93% and 36.74%, respectively. Evidently, =C increases at a decreasing rate, because when M doubles =C does not. Even though it is not apparent from Table 2 because of aggregation, for L = 16 and pj ∈ *1 4+ or pj ∈ *5 8+, we have L/p¯j taking on the values 6.40 and 2.46, respectively, with corresponding overall average =C values 34.53% and 30.14%. Hence, =C increases with L/p¯j . • Booking savings =B increase with M and decrease with pj but do not depend on P /T . As the number M of manufacturers increases, booking savings from coordination increase because of better use of idle times and the release of unused windows. For M = 4 8 the overall average booking savings are 7.53% and 15.65%, respectively. In contrast, for pj ∈ *5 8+ the average =B value is 7.44%. This is about half the corresponding average for pj ∈ *1 4+,

which is 15.74. The above holds true for both P /T values. • On average, after coordination, 67.19% of the manufacturers incur lower WIP costs. As indicated by A+ , under all scenarios, approximately two-thirds of the manufacturers improve over their initial schedules in terms of their WIP costs before the savings are allocated. These manufacturers are primarily latecomers in the FCFS order because after coordination they use windows booked by early comers that were not available when they arrived to the system. Generally, the manufacturers that initially booked the later windows benefit from coordination as they gain access to earlier windows. Similarly, manufacturers that had the chance to book earlier windows might be delayed in the coordinated schedule, but the deterioration in their service level is compensated by the allocation of the savings, as demonstrated in the following subsection. 5.2.2. Allocation Results. The second part of our analysis is meant to provide insights on the value of the allocation scheme to individual manufacturers. Individual manufacturers’ benefits from coordination are summarized in Table 3 (where columns correspond to players in the FCFS order) using the Table 3

Percentage of Savings Allocated to the Players x id x net x

and the Changes in the WIP Costs After Coordination as a Percentage of the Initial Total Costs WIP/TC

pj = 1 4

M = 4 id

x x net x WIP/TC

pj = 5 8

1

2

3

3006 3938 3863 −5383

1650 2862 2764 586

2549 2152 2184 3341

4

1

2795 3368 1048 4179 1189 4133 5348 −12433

2

3

4

2163 2811 2775 549

2116 2031 2036 3367

2353 979 1055 5325

6

7

8

1352 1115 1139 4949

1275 836 881 4998

1253 509 585 5596

6

7

8

1295 989 1012 4872

1154 798 826 5195

1285 463 527 5663

pj = 1 4 M = 8 id

x x net x WIP/TC

1

2

1395 982 1852 1491 1805 1447 −5791 −1547

3

4

1121 1484 1445 −512

1315 1460 1439 2826

5 1307 1253 1259 3187

pj = 5 8 M = 8 id

1

2

x 1333 1489 x net 1905 1855 x 1860 1826 WIP/TC −16451 −4711

3

4

1320 1599 1577 −602

1041 1220 1206 2044

5 1083 1173 1166 2368


252


following statistics: m =WIP Jj ∈Nm *cj 0 − cj H + = 100% TC Jj ∈Nm cj 0 + Wk ∈ hk 0

(WIP savings as a percentage of total savings for manufacturer m after coordination). xmid , xmnet , and xm are defined in §3. We report each of these statistics as a percentage of the total savings in each category in the rows of Table 3 indicated by xid , xnet , and x, respectively (e.g., xid row reports xmid / m∈M xmid · 100 values). We make the following observations: • xnet and x decrease with the manufacturer index m. About 32.81% of the manufacturers have worse WIP performance in the coordinated schedule. Hence it is fair that the portion of WIP savings allocated to them is higher than that for latecomers in the FCFS order. For these manufacturers, observe that the WIP savings as a percentage of the initial cost =WIP/TC increase with m (see Table 3). Because latecomers improve dramatically with respect to WIP performance (e.g., more than 50% for m = M), the portion of savings allocated to them is small (e.g., about 5%–12% for M = 4 8, respectively) and most of the savings are allocated to early comers. Yet xnet and x decrease as M increases because the total savings are distributed among more manufacturers. • The idle time savings xid decrease linearly with M. For small M, the xid values follow a convex pattern for m = 1 2 M, that disappears when M increases. The overall average xid values for M = 4 8 are 25.00% and 12.50%, respectively. For M = 4 the idle time savings seem to follow a convex pattern and for M = 8 no pattern is evident (see Table 3). The latter is because every manufacturer has several preceding and/or following windows that have idle time in the original schedule, so each manufacturer is allocated about the same amount of savings irrespective of its original position. In this case the savings from idle times range between 9.82% and 14.89% for M = 8, whereas for M = 4 they range between 16.50% and 33.68%. • The percentage of the total savings allotted to the losers (those whose WIP costs increase after coordination) ranges between 38% and 53%.

The total savings allotted to losers may be computed by adding the percentage allocations in Table 3 associated with negative =WIP/TC values. These percentages range from 38.63% for player 1 when M = 4 and pj = *1 4+ (see row x in Table 3) to as high as 52.63% (from row x in Table 3, we have 18 60% + 18 26% + 15 77% = 52 63%) for players 1, 2, 3 when M = 8 and pj = *5 8+.

6.

Conclusion

In this paper, we studied a production chain coordination model in which manufacturers outsource similar operations to a single third party. We presented three heuristics and two lower bounds for the associated optimization problem. Given the initial schedule, savings can be generated by coordinating production over the windows booked by all manufacturers in M. We devised a savings sharing scheme to allocate savings from coordination so that every manufacturer agrees to follow the centralized schedule. The highlight of our findings is as follows: The costs of the production chain can be reduced by an average of 32% if one-third of its members let a third party cover their increased WIP cost in exchange for 38%–53% stake in the total savings. This observation summarizes the first and the last bullet points in §5.2.1 and the last bullet point in §5.2.2. Acknowledgments

The authors thank the editor-in-chief, the associate editor, and four anonymous referees for their insightful comments and suggestions. Previous versions of this paper were titled “Coordination of Outsourcing Operations.”

Appendix Proof of Theorem 1. The proof is based on a reduction from 3-Partition, which can be stated as follows. Given 3n integers in A = a1 a3n such that B/4 < aj < B/2 and 3n j=1 aj = nB, is there a partition of A to n disjoint sets each having three elements that sum up to B? Given an instance of 3-Partition, construct the following instance of the immediate shipment problem. wj = pj = aj L = B

for j = 1 3n h1 = · · · = hn = 0

Let G be the time lag between two consecutive manufacturing windows. We will show that there exists a solution


253


for the immediate shipment problem with cost zIS if and only if there exists a solution to 3-Partition. First, suppose a solution for the immediate shipment problem exists. In this instance the booking costs are zero. Let sk be the sum of the weights of the jobs in Wk . Then the problem of minimizing WIP costs with immediate shipment can be defined as follows: j 3n n wi · wj + sk · k − 1 · G min zIS = j=1

i=1

k=1

s.t. s1 ≤ L s1 + s2 ≤ 2L s1 + s2 + s3 ≤ 3L s1 + s2 + s3 + · · · + sn = nL The first double summation in the objective function is constant irrespective of the processing order of the jobs, because wj /pj = 1 by construction. The following is true for the second summation: n n n n sk ·k−1·G = sk ·k·G− sk ·G = G· sk ·k −n·L·G k=1

k=1

k=1

k=1

n Therefore, z = min zIS is attained when k=1 sk · k is minimized subject to the above constraints. Clearly, z∗ = 3n j ∗ ∗ j=1 i=1 wi · wj + n − 1nLG/2 if and only if s1 = s2 = ∗ ∗ s3 = · · · = sn = L. This means that the total weight and the total processing time requirements of the jobs that complete in each window are L. Because the constructed instance imposes the restriction L/4 < pj = wj < L/2 on the jobs, the optimal solution can be attained only if there are exactly three jobs finished in each window. Thus, a solution for the immediate shipment problem induces a 3-Partition. In contrast, given a solution for 3-Partition, each part will have three elements with total processing time equal to L. Then the three jobs associated with each part have total processing L; i.e., s1 = s2 = s3 = · · · = sn = L. The total delivery j cost of the resulting schedule is z∗ = 3n j=1 i=1 wi · wj + n − 1nLG/2 Thus, a solution for 3-Partition implies a solution for the immediate shipment problem. Proof of Proposition 1. Consider coalitions S T ⊆ M with S ∩ T = . Let 0 S , 0 T be the window coalitions booked by players in S, T , respectively. Let -id S be the cost savings incurred when S eliminates all the idle times from the unused portions of the windows in 0 S . Evidently, -id S is the difference between the total cost expressions (14) and (15). Let -id T be defined similarly. Consider the union of jobs that belong to S ∪T . Clearly, S ∪ T can further benefit by the remaining idle time (if any) after the last job in S, and hence -id S ∪ T ≥ -id S + -id T . For the superadditivity of M -net , note that for any S ⊆ M, we have -net S = -S − -id S. Consider the schedule of jobs in NS after all savings from -id S are realized. ∗

The remaining savings, -net S, will be realized as the jobs in S are put into a better order and are processed in the best combination of windows (where sequence optimization and reallocation to the optimal set of windows are achieved jointly) to attain a total cost TC ∗ S, as in (14). Similarly, let TC ∗ T be the optimal cost of processing jobs in NT by utilizing windows ∗ T . Coalitions S and T attain savings of -net S and -net T , respectively. Now consider the coalition S ∪ T and the concatenation of schedules ∗ S, ∗ T . Of course, the concatenation achieves the same savings -net S+-net T when evaluated over ∗ T ∪ ∗ T . By shifting jobs in NS ∪ NT earlier within already occupied windows, S ∪ T can generate further savings beyond -id S ∪ T . Hence, S ∪ T can at least make up for the losses starting from the concatenation of ∗ S and ∗ T . Furthermore, there may be pairs of jobs Ji ∈ NS and Jj ∈ NT that can generate savings over the same windows, ∗ T ∪ ∗ T . Additional cost reductions may be attained by reallocating the jobs to a new collection of windows. One can iteratively improve on the previous schedule until no further gains are possible, such that the jobs NS ∪ NT are processed over the set of windows ∗ S∪T with cost TC ∗ S ∪ T . Therefore, -net S ∪ T ≥ -net S + -net T . By definition, -S = -id S + -net S for S ⊂ M, and hence the superadditivity of M -id and M -net guarantees that M - is superadditive. This completes the proof. Proof of Lemma 1. Define BWi = Mi − Mi − 1 from step [1] of batching procedure -Partition for i = 1 2 . Suppose the dynamic program in (24) uses windows ending at D1 D2 D3 D to process batch weights BW 1 BW 2 BW 3 BW . Likewise, suppose that an optimal schedule for the batch shipment problem uses windows with completion times D1∗ D2∗ D3∗ D∗ to process batch weights W1∗ W2∗ W3∗ W∗ , respectively. We want to show that the total cost ZMaxKnap attained by MaxKnap is lower than the total cost associated with the optimal schedule Z ∗ ; i.e., ZMaxKnap = BW1 · D1 + · · · + BW · D + h1 + · · · + h ≤ W1 ∗ · D1∗ + · · · + W∗ · D∗ + h∗1 + · · · + h∗ where hi and h∗i denote the booking cost of the ith window booked by the heuristic and the optimal schedules, respectively. Batch weights BW1 BW2 BW3 BW are the maximum weight that can be processed within L 2L 3L L hours, and hence the following inequalities hold: wj − BW 1 ≤ wj − W1∗ BW 1 ≥ W1∗ ⇒ Jj ∈N

BW 1 + BW 2 ≥ W1∗

+ W2∗

⇒

Jj ∈N

≤

Jj ∈N

Jj ∈N

wj − BW 1 − BW 2 wj − BW 1 − BW 2 (32)


254


BW 1 +···+BW ≥ W1∗ +···+W∗ ⇒ wj −W1∗ −···−W∗ Jj ∈N

≤

Jj ∈N

wj −W1 ∗ −···−W∗

Furthermore, the dynamic program in (24) finds the best windows to process batch weights BW 1 BW . Therefore, MaxKnap finds the minimum total cost of processing BW 1 BW , and hence the following inequality holds: BW1 · D1 + · · · + BW · D + h1 + · · · + h

≤ BW1 · D1∗ + · · · + BW · D∗ + h∗1 + · · · + h∗ Otherwise, the solution of (24) could be improved by selecting the same windows as those used in the optimal schedule. The right-hand side of the above inequality can be rewritten as follows: BW1∗

· D1∗

+ · · · + BW∗

· D∗

+ h∗1

+ · · · + h∗

+

∗ − D−1 + h∗1

· D∗ ≤

Jj ∈N

+

+

+ · · · + h∗

∗ w − W · D2∗ − D1∗ j Jj ∈N 1

Jj ∈N

∗ ∗ wj − W1∗ − · · · − W−1 · D∗ − D−1

+ h∗1 + · · · + h∗

due to 32

= W1∗ + W2∗ + W3∗ + · · · + W∗ · D1∗ + W2∗ + W3∗ + · · · + W∗ · D2∗ − D1∗ ∗ + W∗ · D∗ − D−1 ∗ = W1∗ · D1∗ + · · · + W∗ · D∗ + h∗1 + · · · + h∗ = Zbatch

The proof is completed.

Ik

0

1 ·I 2 k W ∈*k+1T +∩

t

Wt ∈*k+1T +∩ S 0

0 S

t

0

1 st + ·sk 2 W ∈*1k−1+∩ t

It

0 S

st

m∈S

net xm =

=

m∈S Wk ∈ m 0

4 · net *1 k+ − net *1 k − 1+

+ 1 − 4 · net *k T + − net *k + 1 T + 4 · net *1 k+ − net *1 k − 1+

Wk ∈*ab+

wj · D1∗

t

The second line of the above expression becomes equality when S = M, and hence all (in)equalities are satisfied for M -id . To prove (ii) we have

1 1 ·I s + ·s I 2 k W ∈*k+1T +∩ t 2 k W ∈*1k−1+∩ t

= id *ab+ = -id S

Wk ∈ S 0

wj − BW1 − · · · − BW−1

Jj ∈N

m∈S Wk ∈ m 0

=

∗ + BW · D∗ − D−1 + h∗1 + · · · + h∗ = wj · D1∗ + wj − BW1 · D2∗ − D1∗

=

id xm

m∈S Wk ∈ m 0

+ BW 2 + BW 3 + · · · + BW · D2∗ − D1∗

Jj ∈N

m∈S

≥

= BW 1 + BW 2 + BW 3 + · · · + BW · D1∗

Jj ∈N

Proof of Theorem 2. The result will be proved in three steps: id (i) We will show that xm & m ∈ M satisfies the core (in)equalities (12) when 0 S = *a b+, and -S is replaced by -id S = id *a b+. net (ii) We will show that xm & m ∈ M satisfies the core (in)equalities (12) when 0 S = *a b+ and -net S = id net + xm = net *a b+. Adding by parts would imply that xm xm & m ∈ M satisfies (12) for 0 S = *a b+. (iii) We will extend parts (i) and (ii) to arbitrary window coalitions 0 S = *a1 b1 + ∪ · · · ∪ *ar br +. To prove (i) note that

+ 1 − 4 · net *k T + − net *k + 1 T + = 4 · net *1 a+ − net *1 a − 1+ + 1 − 4 · net *a T + − net *a + 1 T + + 4 · net *1 a + 1+ − net *1 a+ + 1 − 4 · net *a + 1 T + − net *a + 2 T +

+ 4 · net *1 b − 1+ − net *1 b − 2+ + 1 − 4 · net *b − 1 T + − net *b T + + 4 · net *1 b+ − net *1 b − 1+ + 1 − 4 · net *b T + − net *b + 1 T + = 4 · net *1 b+ − net *1 a − 1+ + 1 − 4 · net *a T + − net *b + 1 T +



≥ 4 · net *a b+ + 1 − 4 · net *a b+ = net *a b+ = -net S The inequality holds because of superadditivity of M vnet due to Proposition 1. Note that *1 a − 1+ ∪ *a b+ = *1 b+ and *a b+ ∪ *b + 1 T + = *a T +. Therefore, net *1 a − 1+ +net *a b+ ≤ net *1 b+ and net *a b+ + net *b + 1 T + ≤ net *a T +. For a = 1 and b = T and assuming that net *1 0+ = net *T + 1 T + = 0, the above expression implies the following: net xm = 4 · net *1 T + + 1 − 4 · net *1 T + m∈M

= net *1 T + = -net M net } is in the core of M -net when Therefore, {x1net xM 0 S = *a b+. Steps (i) and (ii) yield that {x1 xM } is in the core of M - when 0 S = *a b+. For (iii), consider coalition S such that 0 S = *a1 b1 + ∪ · · · ∪ *ar br +. Admissible rearrangements should preserve the set of predecessors for jobs that belong to nonmembers of the coalition (see Assumption 1). For example, no interchange is allowed between jobs that complete in *a1 b1 + with jobs from *a2 b2 + because this would change the set of predecessors that complete in Wb1 +1 . Therefore, WIP savings are additive over maximally connected window coalitions. net id ≥ rk=1 net *ak bk + and m∈S xm ≥ Equivalently, m∈S xm r k=1 id *ak bk + with equalities holding when S = M. This net & m ∈ M is in the core of M - and the proves that xm theorem is complete.

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