In one-of-a-kind production (OKP), a firm usually keeps zero stock for some components of which the demand is unstable. In this work, we model the scheduling problem for the OKP firm who promises due-date-guaranteed delivery to its customers. The firm schedules its jobs to minimize its total cost concerning the constraint of components delivery from its supplier. We develop a pair-wise scheduling algorithm to model. The simulation test shows that the proposed pair-wise scheduling algorithm can find the near optimal solution

Tu, Yiliu

Zhang, Jian

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Association for Information SystemsAIS Electronic Library (AISeL)Eleventh Wuhan International Conference on e-Business Wuhan International Conference on e-Business5-26-2012OKP Supply Chain SchedulingJian ZhangDepartment of Mechanical & Manufacturing, University of Calgary, CanadaYiliu TuDepartment of Mechanical & Manufacturing, University of Calgary, CanadaFollow this and additional works at: http://aisel.aisnet.org/whiceb2011This material is brought to you by the Wuhan International Conference on e-Business at AIS Electronic Library (AISeL). It has been accepted forinclusion in Eleventh Wuhan International Conference on e-Business by an authorized administrator of AIS Electronic Library (AISeL). For moreinformation, please contact elibrary@aisnet.org.Recommended CitationZhang, Jian and Tu, Yiliu, "OKP Supply Chain Scheduling" (2012). Eleventh Wuhan International Conference on e-Business. 23.http://aisel.aisnet.org/whiceb2011/23The Eleventh Wuhan International Conference on E-Business——Operations & Services Management Track 707OKP Supply Chain SchedulingJian Zhang, Yiliu TuDepartment of Mechanical & Manufacturing, University of Calgary, CanadaAbstract: In one-of-a-kind production (OKP), a firm usually keeps zero stock for some components of which the demand isunstable. In this work, we model the scheduling problem for the OKP firm who promises due-date-guaranteed delivery to itscustomers. The firm schedules its jobs to minimize its total cost concerning the constraint of components delivery from itssupplier. We develop a pair-wise scheduling algorithm to model. The simulation test shows that the proposed pair-wisescheduling algorithm can find the near optimal solution.Keywords: one-of-a-kind production, supply chain scheduling, pair-wise algorithm1. INTRODUCTIONNowadays, increasing uncertainties in customer demand drives manufacturing firms to undertakeappropriate management strategies in their production planning. One-of-a-kind production (OKP) is one suchstrategy that allows local firms to provide highly customized products, and at the same time to meet thecustomer’s leadtime requirement. Comparing to mass customization, OKP is featured as a “once” successfulapproach on the product development and production according to specific customer requirements, i.e., noprototype or specimen will be made[1]. Thus, OKP manufacturer usually does not keep inventory of some inputs(e.g., parts) and only makes orders on demand. Concerning the leadtime of the parts delivered by the suppliers,the OKP manufacturer’s production scheduling is further complicated, especially when the manufacturer’scustomers has specific requirements on due date. In this research, we study the supply chain described asfollows.This work relates to the literature of supply chain scheduling, which was initiated by [2]. Due to thecomplexity, most authors study either the supplier-manufacturer (or supplier-assembler) integration, or themanufacturer-deliverer integration.In the research of supplier-manufacturer integration, [3], [4] and [5] has proved that the integrativescheduling between suppliers and manufacturers can significantly increase the supply chain’s responsivenessand lower the total incurred cost. Moreover, they also studied the mechanism which enables the integrationbetween the manufacturer and the suppliers. In the operational level, numbers of works studied the supply chainscheduling under different special conditions. A common method to study the supply chain scheduling is toassume the suppliers and/or the manufacturers as separated machines. [6] studied the case where one supplierserves multiple manufacturers. They reduce their problem to a single machine scheduling problem withdeadlines, where the supplier is the machine and all the products manufactured for the same manufacturerbelongs to the same product family. They developed the algorithm for the supplier to find the optimal scheduleto minimize total setup cost. Both [7] and [8] studied a single-supplier-single-manufacturer supply chain. [7]optimizes the integrative schedule by minimizing the interchange cost and buffer storage cost, where theinterchange cost measures the different between the supplier’s or the manufacturer’s own ideal schedule and theglobal optimal schedule.As for the research in manufacturer-deliverer integration, [9] and [10] can be refereed for detail review. Wepresent the works by [11] and [12], which is most related to our work. They both studied a problem in which the3PL quotes a pricing schedule which maps the shipping time and the price, and the manufacturer sequences the708 The Eleventh Wuhan International Conference on E-Business——Operations & Services Management Trackproduction of orders with different committed due dates, such that the transportation cost are minimized.According to our definition, this type of integration is weak integration. That is, the manufacturer does not knowthe detail scheduling within the deliverer. However, through the price schedule, the manufacturer’s productionschedule is automatically induced to the one which benefits the deliverer. When the manufacturer managed tobalance the total transportation cost and the efficiency of delivery, the scheduling of the manufacturer and thedeliverer is therefore coordinated, indirectly. To solve the computational complexity problem, [11] developedheuristics to find near optimal solution, and [12] improved their algorithm with worst-case concerns. However,they do not consider the constraints incurred by the delivery time from the supplier of the parts.To our best knowledge, there is no research in the literature considering both the supplier’s and themanufacturer’s pricing schedule when studying the manufacturer’s scheduling problem. In our present work, wefirst formulate a basic model, by which the manufacturer minimizes the total procurement cost and the totaldelivery cost. We propose the heuristics so that the near optimal solution can be obtained. Second, we study acase in which a single supplier serves multiple manufacturers, and the supplier and the manufacturers areweakly integrated. In this case, the manufacturers share the demand information from their own customers, i.e.the quantity of the orders and the urgency level of each order. Based on this information, the supplier formulatesthe pricing schedule mapping the leadtime to the price.2. MATHEMATICAL MODELIn the future T periods, the manufacturer has J  jobs to be processed, each of which is indexed by{1,2, J…, }j j: ∈ . In this work, we treat a manufacturer as a one-machine production system. Thus, themanufacturer’s target is to form a job sequence for production. We use {1,2, J}…,i ∈  to index the i ’s job inthe sequence.The starting time for the production of job j  is constrained by the arriving time of its part from thesupplier. For the purpose of simplicity, we assume that the manufacturer’s starting time for processing job j  isconstrained by only one part. This is true in practice when the part is the last one to be delivered and thereforebecomes the bottleneck for scheduling job j . In this work, we study the case where the supplier offers optionaldelivery time option for any single part, and the supplier can only deliver the part at the end of each period. Ifthe manufacturer requires j ja a N: ∈  as delivery time for the part of job j , then the supplier will chargethe manufacturer at expense ( )j ja .When the manufacturer finished the processing of job j , job j  is ready for shipment to the finalcustomer. We study the case that for any job j , the customer has a particular requirement for delivery due date,denoted by jd . In this work, we assume that the customer is indifferent about the delivery time if the thedelivery is within the same period. Thus, jd  can be treated as positive integer, i.e., jd N∈ . The finished jobis shipped to the final customers by the third-party logistics provider (3PL). The shipping cost charged by the3PL is a function of the order’s weight, the distance to the order’s destination and the shipping mode. We denotethe order’s weight and the distance to the order’s destination by j jw w R+: ∈ and j jr r R+: ∈ , respectively.Following to [12], job j ’s shipping mode is represented by its shipping time, which is denoted by j jl l N: ∈ .The Eleventh Wuhan International Conference on E-Business——Operations & Services Management Track 709Given jw , jr  and jl , the manufacturer can compute the cost charged by the 3PL, denoted by ( )j j jl w rΦ , , .We assume that the 3PL only picks up the finished products at the end of each period, and ( )j j jl w rΦ , ,  isdecreasing in jl  but increasing in jw  and jr . If the delivery of the finished product is later than the requireddue date, then a penalty is incurred to the manufacturer. In this work, we assume that the penalty is a function ofthe order’s weight and the amount of tardiness. Saying that the tardiness for delivering job j  is j∆ , then thecost incurred for the penalty is denoted by ( )jwΨ ∆, , which is non-decreasing in ∆ . In this work, themanufacturers we studied are minimizing the total cost which includes the cost charged by the supplier, the costcharged by the 3PL and the penalty for late delivery. The following mathematical model computes the optimalproduction sequence of the orders under the constraints of supplier’s delivery time and the 3PL’s shipping time.Similar as the NPD in [11], in the model, we use three status variable defined as follows:• jf ′ : j Rf +′ ∈ . The finishing time of job j .• if : if R+∈ . The finishing time of the i th job in the scheduled job sequence.• jty : 0 1jty { }∈ , . 1 if job j  finishes in day t ; otherwise 0.The manufacturer’s problem, MP, is formed as:1min [ ( ) ( ) ( ) ( )]Jj W j j j j j j jjja C a l w r t l df=′+ − + Φ , , + Ψ + −∑ (1)1Subject to 1 1, 2,Jijix i J=: = , = …,∑ (2)11 1,2,Jijjx j J== , = …,∑ (3)0 0f = (4)11 1,2,Jj ijji is xf f i Jc=−+ ≤ = …,∑ (5)111,2,JJ j ijjj ij ijs xa x f i Jc==+ ≤ = ,…∑∑ (6)710 The Eleventh Wuhan International Conference on E-Business——Operations & Services Management Track[1 ] 1, 2,i ijj f M x i j Jf ′ − ≤ − , = ,… (7)[1 ] 1, 2,i ijjf M x i j Jf ′− ≤ − , = ,… (8)1, 2,jj t j Jf ′ ≤ = ,… (9){0,1} {0} 1, 2,ij ij j jx y t a N i j J, ∈ , , ∈ ∪ = …, , (10)(1) is the object function which minimizes the total cost, i.e., procurement cost, transportation cost, andtardiness penalty. (2) and (3) describe the job’s position in the scheduled sequence. (5) regulates that the i th jobin the sequence has to start after when 1i − th job finishes. (6) regulates that the i th job in the sequence has tostart after when its part arrives. (7) and (8) regulate that if job j  is the i th job in the sequence, theni jf f ′= . (9) regulates that an order is shipped at the end of the day when it is produced.Since Stecke and Zhao has proved that NPD is a NP-hard problem, then we can conclude that our problemis also NP-hard. Stecke and Zhao developed heuristics for problem NPD to find the near optimal solution, and[12] developed an alternative heuristics to improve the worst case solution. Since they do not consider theimpact of the supplier’s delivery, then we develop a new problem specific heuristics in the next subsection.3. HEURISTICS DEVELOPMENTWe propose a pair-wise algorithm to search the near optimal solution of the model presented in Section 2.In the heuristics, we try to find the optimal solution by iteratively swapping each pair of neighbor jobs. We firstfind the optimal schedule supposing the firm only have one job. Suppose that the firm only have job j , thenthere exits an optimal schedule in which job j  starts at the beginning of jt , such thatarg min ( )j jt Nt C t∈= , (11)where( ) ( 1) (max ( 1 ) ( ) )(max 0 ( 1 ) ( ) ).j j j i j j ji j jC t t {d t s c l r } w r{ t s c l r d }= Γ − + Φ − − + / , , ,  +Ψ , − + / + −  We define ( )j tC′  as ( ) ( 1) ( 1)j j jt C t C tC = + − −′ . Without loss of generality, we set that for any j ,(0)jC M=  where M  is a large positive number. Then, we can have a sequence of jobs, denoted by uQ . InuQ , jobs are sequenced non-decreasingly in jt . Our pair-wise heuristics (PWH) for is developed has follows:The Eleventh Wuhan International Conference on E-Business——Operations & Services Management Track 711The proposed heuristics can be regarded as a modified bubble sorting algorithm. The algorithm’scomplexity is 2( )O n  in worst case, and is ( )O n  in best case.4. PERFORMANCE OF THE PROPOSED HEURISTICSIn order to illustrate the efficiency of our heuristics, we compare the solutions obtained by our heuristicswith the optimal solution obtained by CPLEX.In order to obtain the optimal solution by CPLEX, we constrain the problem to 8J =  and 10T = . Thecost functions are generated based on the following forms, which is consistent with the monotone assumption.1 2 333if( ) ( ) ifj j j j jj jj jj ja aaa   − ≤Γ = ,Γ > (12)where 1 2 3j j j R   +, , ∈  are function parameters which vary between different jobs. In order to make surethat ( ) 0j jaΓ >  for any 0ja > , for each job j , we first randomly generate 1 j  and 3 j  from uniform(50 200)U ,  and (1 )U T, , respectively. Then, we generate 2 j  from 1 3(0 )j jU  , / .1 2 333[ ] if( ) ( ) ifj j j jj j jjr w l ll w ra   − ≤Φ , , = ,Φ >(13)PWH:Input: uQOutput: uQfor ui Q←| |  to 2  step 1−  doSWAPPED False← ;for 1i ←  to 1i −  doif ( )uQ i  starts and ( 1)uQ i +  ends in the same period thenif ( ) ( 1)( ) ( )u uQ i Q it tC C +<′ ′  thenSwap ( )uQ i  and ( 1)uQ i + ;SWAPPED True← ;end ifelseif Swap ( )uQ i  and ( 1)uQ i +  can reduce the total cost thenSwap ( )uQ i  and ( 1)uQ i + ;SWAPPED True← ;end ifend ifend forif SWAPPED= False thenSTOP;end ifend for712 The Eleventh Wuhan International Conference on E-Business——Operations & Services Management Trackwhere 1 2 3 R   +, , ∈  are function parameters. For all the jobs, the shipping cost is in the same form. Same asgenerating ( )j jaΓ , we first randomly generate 1  and 3  from uniform (10 40)U ,  and (1 )U T, ,respectively. Then, we generate 2  from 1 3(0 )U  , / .0 if 0( )if 0j j∆ ≤Ψ ∆ = , ∆ ∆ >(14)where j R +∈  are function parameters. For any job j , j  is randomly generated from (1 500)U ,The other parameters used in the simulation are listed in Table 1. Note that in the simulation, the capacityand each job’s processing time are both in job unit. Thus, c  and js  are both integers.Table 1. Parameter settingsParameter Valuec 5jd random integer from (1 )U T,jr random real from (1 2)U ,jw random real from (1 5)U ,js random integer from (1 15)U ,The simulating result is shown in Table 2. It can be observed that the gap can be constrained to around 1%.Our heuristics appears less efficient in speed and accuracy than the one introduced by [11] because weconsidered the supplier’s delivery and the overdue of the final products. The complexity can be reflected by theCPU times consumed by CPlex, which are much longer than the ones in [11]’s results.Table 2. Simulating resultsOptimal solution Heuristics Average gap betweenCPU time CPU time optimal cost and(seconds) (seconds) heuristic cost(%)1 16.64 0.92 0.972 17.07 0.86 1.033 17.01 0.88 1.084 16.73 0.89 0.925 17.01 0.88 1.026 17.06 0.91 0.967 17.11 0.85 1.038 16.41 0.90 1.049 16.86 0.88 0.9110 16.35 0.88 1.00The Eleventh Wuhan International Conference on E-Business——Operations & Services Management Track 7135. CONCLUSIONSIn this work, we modeled the scheduling problem of an OKP manufacturer whose scheduling is constrainedby its supplier’s delivery. A pair-wised sequencing algorithm is proposed and the simulation test shows thatthe proposed algorithm can find the near optimal solution within a short computation time. The sequencingmethod proposed in this work can be easily extended to fit for various contexts by changing the objectivefunction of the mathematical model.REFERENCES[1] Tu Y, Chu X, Yang W. (2000). Computer-aided process planning in virtual one-of-a-kind production. Computers inIndustry, 41 (1): 99–110.[2] Hall N, Potts C. (2000). Supply chain scheduling: batching and delivery. In: Seventh International Workshop on ProjectManagement and Scheduling (PMS 2000), Osnabruck, Germany.[3] Wei J, Krajewski, L. (2000). A model for comparing supply chain schedule integration approaches. International Journalof Production Research, 38(9): 2099–2123.[4] Hall N G, Potts, C N. (2003). Supply chain scheduling: Batching and delivery. Operations Research, 51(4): 566–584.[5] Chen Z. (2007). Supply chain scheduling: Conflict and cooperation in assembly systems. Operations Research, 55:1072–1089.[6] Cheng, T, Kovalyov, M. (2001). Single supplier scheduling for multiple deliveries. Annals of Operations Research.107(1): 51–63.[7] Agnetis, A, Hall, N G, Pacciarelli, D, (2006). Supply chain scheduling: Sequence coordination. Discrete AppliedMathematics, 154(15): 2044 – 2063.[8] Yeung, W-K, Choi, T-M, Cheng, T C E, (2010). Optimal Scheduling of a Single-Supplier Single-Manufacturer SupplyChain With Common due Windows. IEEE Transactions On Automatic Control, 55(12): 2767–2777.[9] Sarmiento, A M, Nagi, R. (1999). A review of integrated analysis of production distribution systems. IIE Transactions,31 (11): 1061–1074.[10] Chen, Z. (2004). Integrated production and distribution operations: Taxonomy, models, and review. In: Simchi-Levi, D.,Wu, D., Z.J., S. (Eds.), Handbook of Quantitative Supply Chain Analysis: Modeling in the E-Business Era. KluwerAcademic Publishers, Boston, MA.[11] Stecke, K E, Zhao, X. (2007). Production and transportation integration for a make-to-order manufacturing companywith a commit-to-delivery business mode. 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OKP Supply Chain Scheduling

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OKP Supply Chain Scheduling

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