Supply chain scheduling to minimize holding costs with outsourcing

Transcription

1 Ann Oper Res (2014) 217: DOI /s Supply chain scheduling to minimize holding costs with outsourcing Esaignani Selvarajah Rui Zhang Published online: 9 January 2014 Springer Science+Business Media New York 2014 Abstract This paper addresses a scheduling problem in a flexible supply chain, in which the jobs can be either processed in house, or outsourced to a third-party supplier. The goal is to minimize the sum of holding and delivery costs. This problem is proved to be strongly NP-hard. Consider two special cases, in which the jobs have identical processing times. For the problem with limited outsourcing budgets, a NP-hardness proof, a pseudo-polynomial algorithm and a fully polynomial time approximation scheme are presented. For the problem with unlimited outsourcing budgets, the problem is shown to be equivalent to the shortest path problem, and therefore it is in class P. This shortest-path-problem solution approach is further shown to be applicable to a similar but more applicable problem, in which the number of deliveries is upper bounded. Keywords Supply chain scheduling Outsourcing Inventory control FPTAS Approximation algorithm Shortest path problem 1 Introduction Supply chain management has been one of the most important topics in manufacturing research. According to the survey paper (Thomas and Griffin 1996), over 11 % of the U.S. Gross National Product is spent on logistics. This underlines the need for research dealing with supply chain problems on the operational level using deterministic models. Hall and Potts (2003) conduct exclusive studies on a series of supply chain scheduling problems, where batch-delivery costs are considered as part of objectives. Similar topics in this research area can be seen in the follow-up papers, (Chen and Vairaktarakis 2005; This research was supported in part by NSERC Discovery Grant E. Selvarajah R. Zhang (B) Odette School of Business, University of Windsor, 401 Sunset Avenue, Windsor N9B 3P4, ON, Canada zhangr6@mcmaster.ca E. Selvarajah selvare@uwindsor.ca

2 480 Ann Oper Res (2014) 217: Agnetis et al. 2006; Chen and Hall 2007; Wang and Cheng 2009). More supply chain scheduling problems with a variety of delivery options and their solution approaches are reviewed by Chen (2010). Considering an in-house production system, a job s completion time is defined as the time when the job is ready to leave the system (or get delivered to customers). Let a job be in the system from time 0. Then, the job s weighted completion time can be interpreted as its holding cost. Due to the fact that delivery costs are usually associated with transferring jobs from production systems to customers, jobs are delivered in batches in order to save delivery costs while making sure that the holding costs due to waiting for deliveries do not cancel the savings from batch-deliveries. Hall and Potts (2003) first introduced supply chain scheduling problems of minimizing the total weighted completion times (i.e., holding costs) and delivery costs on a single machine, which reflect the trade-off between inbound manufacturing and outbound logistics scheduling. Selvarajah and Steiner (2009) and Selvarajah et al. (2011) study similar problems, where jobs are available sequentially, obtaining a1.5-approximation algorithm and a meta-genetic heuristic, respectively. Over the last several decades, an increasing number of suppliers intend to coordinate with outsourcing partners (or third-party suppliers), when they don t have sufficient production capacity. For example, if a supplier cannot get some order (or job) done due to its limited production capacity, the supplier may subcontract part of the job to a third-party supplier rather than expand its own production capacity. In this case, the supplier needs to process only the un-subcontracted part of the job. In scheduling, this can be modeled by controllable processing times. For more details in this topic, the readers are referred to the survey paper (Shabtay and Steiner 2007). As an extreme case, a job s processing time can be either reduced to zero (outsource the whole job to a third-party supplier), or kept as the origin (process the whole job in house). In scheduling, this is called scheduling with rejection. Engels et al. (2003) study single machine scheduling problems of minimizing the total weighted completion times with rejection. In the paper, they present a pseudo-polynomial algorithm and a fully polynomial time approximation scheme (FPTAS) for the problem. Our problem is the batching version of the problem in Engels et al. (2003), in which both limited and unlimited outsourcing (or rejection) budgets are considered. The unlimitedbudget setting simply reflects suppliers efforts to maximize profits (or minimize total costs). However, due to two major disadvantages of outsourcing: concerns of loss of control and issues with confidentiality and security, outsourcing budget might be limited or even set to zero. In private sectors, suppliers may not want to outsource any of their core competencies. In this case, if outsourcing has to be conducted, the budget is upper bounded. In public sectors, it may be more restricted, e.g., by McKenna Long and Aldridge (2004), (The) 2004 Consolidated Appropriations Act contains a provision which limits federal contractors who win outsourced (US) federal contracts from performing the work overseas. As a new emerging area, scheduling with outsourcing attracts more and more attentions from scheduling researchers. Steiner and Zhang (2011) present a pseudo-polynomial algorithm and an FPTAS for the single machine scheduling problem of minimizing the total tardiness, due-date-assignment and rejection costs. However, this paper does not include batching or delivery costs. Qi (2008) study several scheduling measures with outsourcing and logistics coordination. In this paper, the outsourced jobs are required to be sent back to the supplier, and delivered to customers together with the jobs processed in house. However, this paper has unlimited outsourcing budgets. Zhang et al. (2010) propose an NP-hardness proof, a pseudo-polynomial and an approximation algorithm for the scheduling problem of minimizing the makespan with limited outsourcing budgets. More models and results in this research area can be found in the review paper (Shabtay et al. 2013).

3 Ann Oper Res (2014) 217: Our problem can be defined in detail as follows. A supplier is given a job set J = {1, 2,...,n} by a single customer. Each job j J has a processing time p j,aper-timeunit holding cost α j and a fixed outsourcing cost β j. In-house processed jobs are delivered in batches with a flat rate q per batch, no matter how many jobs are in it. Note that the importances of inbound production, outbound deliveries and outsourcing can be reflected by re-setting the parameters α j, q, β j and B accordingly. It is assumed that β j are one-time costs, i.e., customers needs will be fully satisfied by the third-party supplier. In batching scheduling, the completion time of a job j, C j is determined by the completion time of the last processed job of the batch, which includes job j. This paper further adopts the following two assumptions: (1) all the data are non-negative integers and well-defined in advance, and (2) no interruption is allowed once a job starts its in-house processing. Consider a schedule σ, in which the sequencing, batching and outsourcing decisions have been made. Let G(σ ) and H(σ) include the in-house processed and outsourced jobs, respectively, where J = G(σ ) H(σ) and G(σ ) H(σ)=.Letb(σ) be the number of batches. Thus, the scheduling cost can be denoted by j G(σ ) α j C j (σ ) + b(σ)q and the outsourcing cost can be denoted by j H(σ) β j. For simplicity, σ will be omitted unless it is required. With respect to outsourcing budgets, two types of problems, using the three-field notation system (Graham et al. 1979), can be denoted by: P1-type: 1 j H β j B j G α j C j + bq,whereb< is the upper bound of outsourcing budgets; P2-type: 1 j G α j C j + bq + j H β j, where the budgets are unlimited, i.e., B =. A related but more applicable problem, which modifies the P2-type problem to minimize the sum of holding and outsourcing costs, is called the Modified-P2 problem. It is denoted by 1 b R j G α j C j + j H β j,wherer< is the maximum number of deliveries in an feasible schedule. Consider an instance with β j >Bfor the P1-type problem (or with β j q,α j P,where P = n j=1 p j, for the P2-type problem) for all j J. Because it is too costly to outsource any job, there won t be any outsourced job in an optimal schedule, i.e., H = and G = J in all the optimal schedules. Therefore, the P1-type (or P2-type) problem is equivalent to the single-machine, supply chain scheduling problem: minimizing the sum of weighted completion times and delivery costs, denoted by 1 α j C j + bq. Since the 1 α j C j + bq problem is strongly NP-hard (Hall and Potts 2003), the P1-type and P2-type problem are strongly NP-hard as well. Due to the mathematical intractability of the general P1-type and P2-type problem, where the processing times are arbitrary, the rest of paper focuses on the special cases, in which the jobs have identical processing times. The paper is organized as follows. In Section 2, a NP-hardness proof, a pseudopolynomial algorithm and an FPTAS are presented for the P1-type problem (with identical processing times). Section 3 shows that the P2-type problem (with identical processing times) can be formulated and solved as a shortest path problem. By a slight modification, this shortest-path-problem formulation can also be applied to the Modified-P2 problem (with identical processing times). These two problems are in class P. Section 4 includes the final remarks and potentials for future research. 2 The P1-type problem with identical processing times The P1-type problem with identical processing times is denoted by 1 j H β j B, p j = p j G α j C j + bq. Consider a partitioning problem, where there are 2m items

4 482 Ann Oper Res (2014) 217: {a 1,a 2,...,a 2m } such that 2m i=1 a i = 2B, (m + 1)a min >B>(m 1)a max,wherea min = min i=1,...,2m {a i } <a max = max i=1,...,2m {a i }. It is easy to see that the decision version of the problem, if there is a partition Ψ {1, 2,...,2m} such that i Ψ a i = B and Ψ =m (i.e., the number of items in Ψ is m), is NP-complete. Consider an instance of the 1 j H β j B,p j = p j G α j C j + bq problem, where n = 2m, p = 1, q α j,β j,andα j = β j = a j,wherej = 1, 2,...,n.Sinceq is very large, there would be only one batch in any optimal schedules. For any partition H G = {1, 2,...,n} and H G =,alargest j H β j provides a smallest j G α j. Therefore, there is an optimal schedule for the instance with the partition H and G such that j H β j = B and j G α j = B. Since(m + 1)a min >B>(m 1)a max,then H = G =m (i.e., the number of jobs in H (and G) ism). By p = 1, the optimal solution value of the instance is mb + q. Thus, the decision problem if there is a feasible schedule with the objective value of mb + q is equivalent to the decision version of the above partitioning problem. The following theorem was proved. Theorem 1 The 1 j H β j B,p j = p j G α j C j + bq problem is NP-hard. Next, a pseudo-polynomial algorithm is presented to clarify that the 1 j H β j B, p j = p j G α j C j + bq problem is NP-hard only in the ordinary sense. 2.1 Pseudo-polynomial algorithm Proposition 1 There is an optimal schedule σ for the 1 j H β j B, p j = p j G α j C j + bq problem, in which the in-house processed jobs in G are processed in the order of α σ(1) α σ(2) α σ(k) with G =k n. Proof This proposition extends Smith s rule (Smith 1956) to a batching version. The extended part, where two jobs are scheduled in two batches, can be proved using job-exchange arguments. By Proposition 1, let the jobs be indexed such as α 1 α 2 α n. In the following Algorithm A1, let state (j,t,v 1,v 2,v) (0 <j n) beapartial schedule on the jobs {1, 2,...,j},where t is the makespan of in-house processed jobs; v 1 is the sum of per-time-unit holding costs of the jobs in the last batch that is pending for delivering; (Note that the holding costs of this pending batch haven t been realized.) v 2 is the outsourcing cost of the partial schedule; v is the objective value of the partial schedule, excluding the unrealized holding costs of the jobs in the pending batch. (Note that there is at most one pending batch in a partial schedule.) In particular, state (0, 0, 0, 0, 0) is used to represent an empty schedule, in which no job has been scheduled or outsourced yet. Algorithm A1 starts from (0, 0, 0, 0, 0), and schedules or outsources jobs one-by-one in the sequence of 1, 2,...,n.Forstate(j 1,t,v 1,v 2,v), where j = 1, 2,...,n, the next unscheduled job j can be scheduled as the following three alternatives: (1) v 1 > 0andjobj is scheduled in the pending batch and generate a new state (j, t + p, v 1 + α j,v 2,v).

5 Ann Oper Res (2014) 217: (2) Deliver the jobs in the pending batch, schedule job j in a new pending batch, and generate a new state (j, t + p,α j,v 2,v+ q + t v 1 ),whereq is the delivery cost of the new pending batch and t v 1 is the total holding cost of the jobs that are just delivered. Note that this operation also works for the case when (j 1,t,v 1,v 2,v)= (0, 0, 0, 0, 0). (3) v 2 + β j B and job j is outsourced to a third-party supplier and generate a new state (j,t,v 1,v 2 + β j,v). Note that this operation also works for the case when (j 1,t,v 1,v 2,v)= (0, 0, 0, 0, 0). If j = n + 1andv 1 > 0, the partial schedules (n,t,v 1,v 2,v) with n-scheduled jobs are completed into full schedules (n + 1,t,0,v 2,v + t v 1 ) by delivering the pending batch. In order to avoid redundant states, elimination operations are executed during the states generation procedure. Algorithm A1 [Initialization] Set S (0) ={(0, 0, 0, 0, 0)} and S (j) =,forj = 1, 2,...,n+ 1. [Generation] Generate S (j) from the states in S (j 1). For j = 1 to n + 1 Set T = and do the following. For each (j 1,t,v 1,v 2,v) S (j 1) If j n and v 1 > 0, then set T = T (j, t + p,v 1 + α j,v 2,v)/*Alternative (1). If j n,thensett = T (j, t + p,α j,v 2,v+ q + t v 1 ) /*Alternative (2). If j n and v 2 + β j B,thensetT = T (j,t,v 1,v 2 + β j,v)/*alternative (3). If j = n + 1andv 1 > 0, then set T = T (j, t, 0,v 2,v+ t v 1 ) /*A full schedule. If j = n + 1andv 1 = 0, then set T = T (j,t,v 1,v 2,v)/*All jobs are outsourced. Endfor [Elimination] For any two states (j,t,v 1,v 2,v) and (j,t,v 1,v 2,v ),ifv v,then delete (j,t,v 1,v 2,v ) from T and set S (j) = T /*Any later states generated from (j,t,v 1,v 2,v ) won t be better than the ones generated from (j,t,v 1,v 2,v). Endfor [Optimizing] Select the state with the smallest v value in S (n+1) as the optimal solution value and trace back to obtain the corresponding optimal schedule. Theorem 2 Algorithm A1 finds an optimal solution to the 1 j H β j B, p j = p j G α j C j + bq problem in O(n 2 B n j=1 α j ) time. This shows that the problem is NPhard only in the ordinary sense. Proof For each (j 1,t,v 1,v 2,v) in S (j 1) with j n, there are at most three operations. Because v 1 n j=1 α j, v 2 B and at most n + 1 distinct t values (due to p j = p), the [Elimination] procedure limits the number of states in S (j 1) up to O(nB n j=1 α j ). Considering that j takes values from 1 to n + 1, the overall run time is O(n 2 B n j=1 α j ). 2.2 Fully polynomial time approximation scheme Scaling technique developed in Lawler (1982) will be used to convert Algorithm A1 into an FPTAS. Firstly, a p-scaled problem is constructed by setting ˆq = q p and ˆp j = 1, j J, which can be denoted by 1 j H β j B, ˆp j = 1 j G α j Ĉj + b ˆq. Notation Ĉ j is used as ˆp j is the processing time of job j. For any feasible schedules, the difference between the

6 484 Ann Oper Res (2014) 217: objective values of the p-scaled problem and the 1 j H β j B,p j = p j G α j C j + bq problem is exactly a factor of p. Next, a pair of bounds for the optimal solution value of the p-scaled problem is firstly determined. Using these bounds, a K-scaled problem is constructed. Algorithm A1 is shown to be a polynomial algorithm for the K-scaled problem for any given ε>0(fixed).the resulting optimal solution to the K-scaled problem is proved to be a (1 + ε)-approximation solution to the p-scaled problem, and therefore a (1 + ε)-approximation solution to the 1 j H β j B,p j = p j G α j C j + bq problem Bounds determination A pair of bounds of the optimal solution value of the p-scaled problem can be determined by solving an auxiliary problem with zero batch-delivery cost, i.e., q = 0. This auxiliary problem is denoted by 1 j H β j B, ˆp j = 1 max j G {α j Ĉ j },whereα j Ĉ j is the jobholding cost of the in-house processed job j G. Note that Proposition 1 can be applied to this problem. Let the jobs be originally sequenced in the order of α 1 α 2 α n. Let state {j,t,u,w} (j >0) represent a partial (j <n)orafull(j = n) schedule on the jobs {1, 2,...,j}, wheret is the makespan of the in-house processed jobs, u is the total outsourcing cost, and w is the largest job-holding cost so far. Similarly, let state (0, 0, 0, 0) be an empty schedule, in which no job has been scheduled or outsourced yet. The following Algorithm A2 starts from (0, 0, 0, 0), and schedules or outsources jobs one-by-one. The value of w is updated with the largest job-holding cost during the [Generation] procedure. At the end, the optimal solution value, ẑ will be set to be the smallest w value over all the full schedules. Algorithm A2 [Initialization] Set S (0) ={(0, 0, 0, 0)} and S (j) = with j = 1, 2,...,n. [Generation] Generate S (j) from the states in S (j 1). For j = 1 to n Set T = and do the following. For each (j 1,t,u,w) S (j 1) 1. Set T = T (j, t + 1,u,max{w,α j (t + 1)}) /*Schedule job j as an in-house processed job, and update the largest in-house job holding cost (if needed). 2. If u + β j B, thensett = T (j,t,u+ β j,w) /*Send job j to a third-party supplier. Endfor [Elimination] For any two states (j,t,u,w) and (j,t,u,w),ifu u, then delete (j,t,u,w)from T and set S (j) = T /*Any later states generated from (j,t,u,w) won t be better than the ones generated from (j,t,u,w). Endfor [Optimizing] Set ẑ = w as the optimal solution value, where w is the smallest jobholding cost over all the full schedules in S (n). Lemma 1 Algorithm A2 finds an optimal solution to the 1 j H β j B, ˆp j = 1 max j G {α j Ĉ j } problem in O(n 4 ) time. Let ˆv be the optimal solution value of the p- scaled problem. Then, either L =ẑ + q (ẑ > 0) or L =ẑ (ẑ = 0) determines a pair of bounds such that L ˆv nl.

7 Ann Oper Res (2014) 217: Proof Because there are at most n distinct t values and n 2 distinct w values, after the [Elimination] procedure the number of states in S (j 1) is at most n 3. For each (j 1, t,u,w) S (j 1), there are at most two operations. Considering n iterations for j, the overall run time is O(n 4 ). Regarding the correctness, the simplest case is ẑ = 0, which means B n j=1 β j.this gives that the optimal schedules for the p-scaled problem will have all the jobs outsourced to the third-party supplier. Therefore, the objective value ˆv = 0, which satisfies the relation L = 0 ˆv = 0 nl = 0. For the other case, ẑ > 0 means B< n j=1 β j, which implies that there is at least one in-house processed job in any optimal schedule. Let σ be the optimal schedule found by Algorithm A2. Note that σ is also a feasible schedule for the p-scaled problem, whose objective value can be written as ˆv(σ ) =ˆv 1 (σ ) +ˆv 2 (σ ),where ˆv 1 (σ ) is the total holding cost and ˆv 2 (σ ) is the total delivery cost. Knowing that there are at most n jobs and batches gives ˆv 1 (σ ) nẑ and ˆv 2 (σ ) n ˆq. Therefore, ẑ +ˆq ˆv ˆv ( σ ) =ˆv 1 ( σ ) +ˆv 2 ( σ ) nẑ + n ˆq. (1) Thus, L =ẑ +ˆq is determined as the lower bound when ẑ > FPTAS for the p-scaled problem A K-scaled problem, 1 j H β j B, ˆp j = 1 j G α j Ĉj + b ˆq,whereα j = α j K and ˆq = ˆq (K will be determined later), is constructed as a bridge to approximating the optimal K solution value of the p-scaled problem. Considering an optimal schedule φ for the K-scaled problem, let ˆv (φ ) be the calculated objective value. Let ˆv ( ) be the optimal solution value of the 1 j H β j B, ˆp j = 1 j G α j Ĉj + b ˆq problem, where α j = α j K. Thus, ˆv ( ) = ˆv K. (Recall that ˆv is the optimal solution value of the p-scaled problem.) Knowing α j α j gives ˆv (φ ) ˆv ( ). Since the K-scaled and p-scaled problem have the same B and β j, φ is feasible for the p-scaled problem as well. Thus, the objective value of φ with respect to the p-scaled problem can be calculated by ˆv ( φ ) = j G(φ ) ( = K α j Ĉ j ( φ ) + b ( φ ) ˆq j G(φ ) j G(φ ) ( α ) j Ĉj φ + b ( φ ) ) ˆq ( ( K α j + 1 ) ( Ĉ ) j φ + b ( φ ) ) ˆq ( = K j G(φ ) K ˆv ( φ ) + Kn(n+ 1) K ˆv ( ) + Kn(n+ 1) ( α ) j Ĉj φ + b ( φ ) ) ˆq + K j G(φ ) Ĉ j ( φ ) ˆv + Kn(n+ 1). (2)

8 486 Ann Oper Res (2014) 217: Lemma 2 For any given ε>0, by setting K =, where L is determined in Lemma 1, n(n+1) the optimal solution to the K-scaled problem is a (1 + ε)-approximation solution to the p-scaled problem. Proof Since L ˆv gives Kn(n + 1) = εl ε ˆv,the(1+ ε) ratio follows directly from Eq. (2). In order to solve the K-scaled problem, the [Elimination] procedure in Algorithm A1 is modified as follows: (1) Delete the states (j,t,v 1,v 2,v) with v 1 > nl K or v> nl K from T ; (2) For any two states (j,t,v 1,v 2,v) and (j,t,v 1,v 2,v),ifv 2 <v 2, then delete (j,t,v 1,v 2,v) from T ;(3)SetS(j) = T. This modified version is given a name, Algorithm A3. Lemma 3 Algorithm A3 finds an optimal solution to the K-scaled problem in O(n 4 L2 K 2 ) time. Proof Because v 1 nl K, v nl,andatmostn + 1 distinct t values, the modified [Elimination] procedure limits the number of states in each S (j 1) up to O(n 3 L2 ). For each K K 2 (j 1,t,v 1,v 2,v) in S (j 1), as can be seen, there are at most three operations. Therefore, with n + 1 outer iterations, the overall run time of Algorithm A3 is O(n 4 L2 ). K 2 Since the objective value of the p-scaled problem is just a factor of p from the objective value of the 1 j H β j B,p j = p j G α j C j + bq problem, the correctness of following theorem is a consequent result of Lemma 2, Lemma 3 and previous discussions. εl Theorem 3 By setting K =, withagivenε>0, the optimal solution found by Algorithm A3 is a (1 + ε)-approximation solution to the p-scaled problem, which is also a n(n+1) (1 + ε)-approximation solution to the 1 j H β j B,p j = p j G α j C j + bq problem. Note that the run time is O( n8 ). ε 2 Proof By Lemma 3 and K = εl L2,theruntimeisO(n4 ) = O(n 4 L 2 εl /[ n(n+1) K 2 n(n+1) ]2 ) = O( n8 ). ε 2 εl 3 P2-type problem with identical processing times The P2-type problem with identical processing times (p j = p>0) is denoted by 1 p j = p j G α j C j + bq + j H β j. Since Proposition 1 can be applied to this problem, without loss of generality, let the jobs be indexed such that α 1 α 2 α n. Considering time t, let G J (i,k) ={i, i + 1,...,k 1} be the in-house processed jobs from J (i,k),where 1 i k 1 n. The jobs in G (if G ) start their processing at time t. Let H = J (i,k) \G be the outsourced jobs from J (i,k). Thus, the total scheduling, delivery and outsourcing cost of the jobs in J (i,k) can be written by q + ( j G α j )(t + G p) + j H β j, if G > 0, Cost = j J (i,k) β j, if G =0, (3)

9 Ann Oper Res (2014) 217: where G represents the number of jobs in G. Note that it may happen that H = when G > 0, i.e., no job from J (i,k) is outsourced to a third-party supplier. Let Cost (l) be the minimum cost of scheduling the jobs in J (i,k) in such a way that the in-house processed jobs starts at time t and the number of outsourced jobs is l, where 0 l k i. The corresponding partition G (l) H (l) = J (i,k) has H (l) =l and G (l) H (l) =.ByEq.(3), the minimum cost of the two extreme cases with zerooutsourced and (k i)-outsourced jobs can be written as ( k 1 ) Cost (0) = q + [t ] α j + (k i)p j=i (4) and k 1 Cost (k i) = β j, (5) respectively. Let π (l) represent the cost of the partition with l (0 <l<k i) outsourced jobs Φ (l) J (i,k) and Φ (l) π (l) is Δ (l) j=i =l. Thus, the difference between the two values Cost(0) and = Cost(0) π (l) ( ( k 1 ) [t ] ) = q + α j + (k i)p = j=i ( ( q + ( k 1 α j j=i j J (i,k) \Φ (l) j J (i,k) \Φ (l) k 1 ( = lp α j + j=i j Φ (l) α j ) [t + J(i,k) \Φ (l) ] p + j Φ (l) ) [t ] k 1 α j + (k i l)p + lp α j α j ) [t + (k i l)p ] j=i j Φ (l) β j ) j Φ (l) β j β j. (6) In order to determine Cost (l), let the jobs in J (i,k) be indexed as δ [1] δ [2] δ [k i], where δ [j] = α [j] [ t + (k i l)p ] β[j]. (7) It is easy to see that H (l) ={[1], [2],...,[l]} maximizes Δ(l), and therefore minimizes π (l).thus,withg(l) = J (i,k)/h (l) the minimum cost with l outsourced jobs can be written as Cost (l) = q + ( j G (l) α j ) (t + G (l) p) + j H (l) β j. (8)

10 488 Ann Oper Res (2014) 217: Next, a directed acyclic graph G(V, E) is constructed. Let V ={(0, 1)} {(m,m) 2 m n + 1and0 m <m} be the set including all vertices, where (0, 1) is the dummy source vertex and (m,m)denotes a partial schedule on {1, 2,...,m 1} with m jobs from {1, 2,...,m 1} scheduled in house and the rest of jobs outsourced to a third-party supplier. Note that m p is makespan of the in-house processed jobs. Let e (i,i k,k) E represent a directed edge from (i,i) to (k,k),where(i, i), (k,k) V, i<kand i k. Actually, the directed edge e (i,i k,k) can be interpreted as the decision of scheduling G (l(k,i )) (i,k,i p) (i.e., H (l(k,i )) (i,k,i p) =l(k,i )) jobs from J (i,k) into a single batch starting from time i p and outsourcing the rest of jobs from J (i,k) to a third-party supplier, where l(k,i ) = (k k ) (i i ).In particular, l(k,i ) = k k when i = 1andi = 0, i.e., the edges e (0,1 k,k) directed from the dummy source vertex. The weight of e (i,i k,k) is defined by the minimum cost of the above scheduling decision, which is Cost (l(k,i )),i )) (i,k,i p).thevalueofcost(l(k (i,k,i p) can be calculated using Eqs. (4), (5)and(8). Consider a path ω ={(m 0,m 0), (m 1,m 1), (m 2,m 2),...,(m h 1,m h 1), (m h,m h)} on G(V, E), wherem 0 = 0, m 0 = 1andm h = n + 1. This path can be interpreted as a feasible schedule of the 1 p j = p j G α j C j + bq + j H β j problem, in which the k-th batch contains the jobs, G (l(m k,m k 1 )) (m k 1,m k,m J k 1 p) (m k 1,m k ), and the rest of jobs from J (mk 1,m k ), H (l(m k,m k 1 )) (m k 1,m k,m = J k 1 p) (m k 1,m k )\G (l(m k,m k 1 )) (m k 1,m k,m are outsourced to a third-party supplier, where k = 1, 2,...,n+ 1. The objective value of this schedule can be computed by k 1p) summing up the weights of the edges on ω, whichis Cost (l(m 1,m 0 )) (m 0,m 1,m + 0 ) Cost(l(m 2,m 1 )) (m 1,m 2,m 1 p) + +Cost(l(m h,m h 1 )) (m h 1,m h,m h 1p). (9) Note that (m 0,m 0) = (0, 1) denoting the dummy source vertex gives l(m 1,m 0 ) = m 1 m 1. Therefore, the shortest path from vertex (0, 1) to (k, n + 1), wherek = 0, 1,...,n,isan optimal schedule for the 1 p j = p j G α j C j + bq + j H β j problem. The following theorem is straightforward. Theorem 4 The 1 p j = p j G α j C j + bq + j H β j problem can be formulated as a shortest path problem on the directed acyclic graph G(V, E), where V =O(n 2 ) and E =O(n 4 ). This shows that the problem can be solved in polynomial time, i.e., the problem is in class P. Proof Based on the construction of G(V, E), the number of vertices (i,i) V as i = 1, 2,...,n + 1 and i = 0, 1,...,i 1 is exactly (n+2)(n+1), i.e., V =O(n 2 ).Since 2 only the edges representing feasible batch schedules are constructed in G(V, E), there are exactly k i + 1 edges directed from the vertex (i,i) to the vertices (,k),where k = i + 1,i+ 2,...,n+ 1. Thus, the total number of edges is n i=1 i ( n+1 k=i+1 (k i + 1)), i.e., E =O(n 4 ). Regarding the cost of e (i,i k,k), the calculation of δ j for the jobs j J (i,k),byeq.(7), is δ j = α j [ t + (k i li,k )p] β j = α j [ i p + ( (k i) (( k k ) ( i i ))) p ] β j = α j [ k p ] β j. (10)

11 Ann Oper Res (2014) 217: This implies that for each job j,therearen + 1 possible δ j values due to the n + 1 selections of k, i.e., k = 0, 1,...,n. For each given k, a sequence of the jobs in J (i,k) with the nonincreasing order of δ j can be easily determined by Eq. (10). Thus, by pre-determining the n + 1 sequences for the n jobs, which can be done in O(n 3 log n) time, the costs and the set of ordered jobs for the edges in E can be finally established. This demonstrates that G(V, E) can be constructed in polynomial time, and therefore the 1 p j = p j G α j C j + bq + j H β j problem is in class P. Next, the above shortest-path-problem formulation is modified to deal with the special case of the Modified-P2 problem with identical processing times, denoted by 1 p j = p,b R j G α j C j + j H β j. Since the holding costs, outsourcing costs and batching decisions considered in this special case are also considered by the 1 p j = p j G α j C j + bq + j H β j problem, the topology structure of G(V, E) can also used to formulate this special case. Because delivery costs are not included in the special case, q is removed from Eq. (8). With these edge costs, the modified G(V, E) is denoted by Ḡ(V, E), which has the same nodes, edges and scheduling interpretations as G(V, E). Since the number of deliveries is upper bounded, the special case is to find a shortest path on Ḡ(V, E) with R edges at most. This modified shortest path problem is a special case of the delay constrained least cost path (DCLC) problem (Cheng and Ansari 2003) with one-unit time delay on each edge. For the general DCLC problem, Garey and Johnson (1979) proved the NP-hardness and Hassin (1992) presented a dynamic programming algorithm running in O( E T) time, where T is the upper bound of time delays. Since Ḡ(V, E) has E =O(n 4 ) and T = R n in O( E T), the following theorem is proved. Theorem 5 The 1 p j = p,b R j G α j C j + j H β j problem can be formulated as a special case of DCLC problem on Ḡ(V, E) with one-unit time delay on each edge, where V =O(n 2 ) and E =O(n 4 ). Using the dynamic programming algorithm in Hassin (1992), this formulation can be solved in O( E R) = O(n 5 ) time. This shows that the special case of Modified-P2 problem is in class P. 4 Conclusions In this paper, the problem of minimizing the total holding and delivery costs with outsourcing was studied. Since the general problem is strongly NP-hard, two special cases with identical processing times were investigated. The P1-type problem with limited outsourcing budgets was shown to be NP-hard only in the ordinary sense by the presented NP-hardness proof and pseudo-polynomial algorithm. The pseudo-polynomial algorithm was further converted into an FPTAS. By allowing unlimited outsourcing budgets, the P2-type problem was formulated and solved as a shortest path problem. This showed that the P2-type problem is in class P. Regarding the Modified-P2 problem, where the number of deliveries is upper-bounded, its special case with identical processing times was shown to be equivalent to the delay constrained least cost path problem, where the time delay on each edge is one. Thus, this special case can be solved in polynomial time and therefor in class P. For future research, the Modified-P2 problem with arbitrary processing times is open to complexity studies and solution approaches. Knowing that the P1 and P2-type problems with arbitrary processing times are strongly NP-hard, heuristics or approximations are deserve researchers efforts.

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