Research Article A Collaborative Scheduling Model for the ...downloads.hindawi.com/journals/tswj/2014/894573.pdf · Research Article A Collaborative Scheduling Model for the Supply-Hub

Research ArticleA Collaborative Scheduling Model for the Supply-Hub withMultiple Suppliers and Multiple Manufacturers

Guo Li,1 Fei Lv,2 and Xu Guan3

1 School of Management and Economics, Beijing Institute of Technology, Beijing 100081, China2 School of Management, Huazhong University of Science and Technology, Wuhan 430074, China3 Economics and Management School, Wuhan University, Wuhan 430072, China

Correspondence should be addressed to Xu Guan; [email protected]

Received 1 October 2013; Accepted 17 November 2013; Published 15 April 2014

Academic Editors: T. Chen, Q. Cheng, and J. Yang

Copyright © 2014 Guo Li et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper investigates a collaborative scheduling model in the assembly system, wherein multiple suppliers have to deliver theircomponents to the multiple manufacturers under the operation of Supply-Hub.We first develop two different scenarios to examinethe impact of Supply-Hub. One is that suppliers andmanufacturersmake their decisions separately, and the other is that the Supply-Hub makes joint decisions with collaborative scheduling. The results show that our scheduling model with the Supply-Hub is aNP-complete problem, therefore, we propose an auto-adapted differential evolution algorithm to solve this problem. Moreover,we illustrate that the performance of collaborative scheduling by the Supply-Hub is superior to separate decision made by eachmanufacturer and supplier. Furthermore, we also show that the algorithm proposed has good convergence and reliability, whichcan be applicable to more complicated supply chain environment.

1. Introduction

To avoid the supply delay risk caused by any supplier, in prac-tice, manufacturer/assembler normally prefers to outsourceits purchasing business to the third party logistics (TPL) andrequires his suppliers to hold inventory in the warehouseoperated by TPL. Given this trend, a new type of TPL arisesin recent years, which is called Supply-Hub. Investigated bymany scholars [1–3], the Supply-Hub is an integrated logisticsprovider with a series of logistic services (e.g., assemblage,distribution, and warehouse), which is widely applied in theauto and electronic industries to support themanufacturer toimplement the JIT (Just InTime) production, so as to respondto the market changes rapidly. For example, BAX GLOBALand UPS are two typical logistic companies operated underSupply-Hub mode with Chinese auto companies.

For most Supply-Hubs, they are located near the man-ufacturer’s factory, so as to store most of the raw materialsdelivered by the suppliers. According to the agreement, theSupply-Hub will charge the suppliers for the componentsconsumed during a fixed period of time. However, during

this process, it is very hard for the Supply-Hub to coordinatethe production and the delivery among different suppliers,specific to how to precisely determine the supplier’s pro-duction lot, the distribution frequency, and the distributionquantity. In actual business activities, when thematerial flowsare sufficiently large, the coordination and optimization ofproduction and distribution based on the Supply-Hub canbring substantial benefits to themembers in the supply chain.

Our paper is related to the vast literature, which can bedivided into two groups. The first group concerns the orderallocation, vehicle routing, and production planning. Hahmand Yano [4] explore the economic lot and delivery schedul-ing problem. Moreover, Hahm and Yano [5, 6], Khouja [7],and Clausen and Ju [8] consider the problems of determiningthe production scheduling and distribution intervals fordifferent types of components when a supplier providesdifferent kinds of components. Vergara et al. [9] propose thegenetic algorithm to make production scheduling and cycletime arrangements formany kinds of components in a simplemultistage supply chain, where each supplier provides one ora variety of products for the upstream supplier or assembler.

Hindawi Publishing Corporatione Scientific World JournalVolume 2014, Article ID 894573, 12 pageshttp://dx.doi.org/10.1155/2014/894573

2 The Scientific World Journal

Khouja [10] examines production sequencing and distribu-tion scheduling in a single-product andmulti-product supplychain when production intervals equal distribution intervals.However, all of the above literature focus on a simplesupply chain structure, and the problems of productionand distribution in an assembly supply chain with multiplesuppliers andmultiplemanufacturers have not been involved.Pundoor [11] first establishes a cooperative scheduling modelof production and distribution in a multi-suppliers, one-warehouse, and one-customer system.The supplier’s produc-tion and distribution interval and warehouse’s distributioninterval are collaboratively optimized to minimize the unitproduction and logistics cost in the upstream supply chainwherein supplier’s production capability is limited. However,the model does not consider the transportation constraintfrom the warehouse to the manufacturer. Torabi et al. [12]investigate the lot and delivery scheduling problem in asimple supply chainwhere a single supplier producesmultiplecomponents on a flexible flow line (FFL) and delivers themdirectly to an assembly facility (AF). They also develop anew mixed integer nonlinear program (MINLP) and anoptimal enumeration method to solve the problem. Nasoet al. [13] focus on the ready-mixed concrete delivery. Theypropose a novel meta-heuristic approach based on a hybridgenetic algorithm combined with constructive heuristics.Ma and Gong [14] extend the work of Pundoor [11] toa multi-suppliers and one-manufacturer system based onthe Supply-Hub. In their model the production lot anddistribution interval are optimized from either supplier’s ormanufacturer’s perspective.

The second group is about the coordination with theapplication of Supply-Hub. Barnes et al. [1] find that theSupply-Hub is an innovative strategy to reduce cost andimprove responsiveness. They further define the concept ofSupply-Hub and review its development by analyzing thepractical case.They also propose a prerequisite of establishingSupply-Hub and the main way of operating Supply-Hub.Shah and Goh [3] explore the operation strategy of Supply-Hub to achieve the joint operation management betweencustomers and upstream suppliers. Moreover, they analyzehow to manage the supply chain better in a vendor-managedinventory model. Furthermore, they find that the relation-ships between the operation strategies and performanceevaluations of Supply-Hub are complex and nonlinear. Asa result, they propose a hierarchical structure to help theSupply-Hub achieve the balance among different members.Lin and Chen [15] propose a generalized hub-and-spokenetwork in a capacitated and directed network configurationthat integrates the operations of three common hub-and-spoke networks: pure, stopover, and center directs. Theyalso develop an implicit enumeration algorithm with embed-ded integrally constrained multicommodity min-cost flow.Lin [16] studies the integrated hierarchical hub-and-spokenetwork design problem for dual services. They proposea directed network configuration and formulate a link-based integer mathematical model, and also develop a link-based implicit enumeration with an embedded degree andtime constrained spanning tree algorithm. Charles et al.[17] investigate how implement integrated logistics hubs by

considering six independent industrial sectors with refer-ence models and systems. The research results provide afield tested method for deriving integrated logistics hubmodels in different manufacturing economies with notesthat provide sufficient methodological details for repeatingthe construction of logistics hubs in other manufacturingeconomies.

Based on the Supply-Hub, Ma and Gong [14] developcollaborative decision-makingmodels of production and dis-tribution considering the matching of distribution quantitybetween suppliers. The result shows that the total supplychain cost and the production cost of suppliers decreasesignificantly, but the logistics cost of manufacturers andthe operational cost of Supply-Hub increase. In order toexplore the effect of the supply chain design caused by thestructural changes in the assembly system, Li et al. [18]establish several supply chain design models (one withoutsupply center, one with single-stage supply center, and onewith two-stage supply center) according to the character-istics of bill of material (BOM) and the relationships ofmultiple properties among suppliers. With the considerationthat multiple suppliers provide different components to amanufacturer based on the Supply-Hub, Gui and Ma [19]establish an economical order quantity model in such twoways as picking up separately from different suppliers andmilk-run picking up. The result shows that the sensitivity tocarriage quantity of the transportation cost and the demandvariance in different components have an influence on thechoices of two picking up ways. Li et al. [20] give a thoroughreview about collaborative operation and optimization insupply logistics based on the Supply-Hub and point out thathow to coordinate suppliers and share risks is still to beexplored.

Obviously, the above literature does not take the coor-dination issue of Supply-Hub into account. In the actualoperation process of Supply-Hub, the service for multiplesuppliers and multiple manufacturers are often provided bya single Supply-Hub. For example, BAX GLOBAL takes incharge of the logistics services in Southeast Asia for Apple,Dell, and IBM. From the perspective of Supply-Hub, howto integrate resources of multiple suppliers and multiplemanufacturers is the key point of implementing JIT policy inthe supply chain.

2. Problem Definition and Notation

2.1. Problem Description. Let us consider the following oper-ation process: each manufacturer sends its material require-ment plan to the Supply-Hub and corresponding suppliersbased on a rolling plan. After that, the Supply-Hub optimizesand arranges the production and distribution activities foreach supplier based on the information of production costsand inventory status. Finally, the Supply-Hub implements JITdirect-station distribution according to material requirementplan in each week or day provided by eachmanufacturer.Theillustration of the process is shown in Figure 1. It is worthmentioning that the production information is freely sharedamong the suppliers, the Supply-Hub and the manufacturers.

The Scientific World Journal 3

Collaborative scheduling model based on the Supply-Hub

Material requirement plan in each week or day

Supplier 1

Supplier 2

Supplier n

Demandinformation

Delivery

Supply-Hub

Materialrequirement

plan

JITdelivery

Inventory status report

Supplier’s cost information

Manufacturer 1

Manufacturer m

Manufacturer 2

· · ·· · ·

Figure 1: Multi-suppliers and multi-manufacturers operation mode based on the Supply-Hub.

Note that the coordination scheduling is to implementthe JIT distribution of components required by each man-ufacturer with minimal cost. To achieve this goal, themanufacturer’s distribution lot, the supplier’s production lot,and the distribution frequency should be optimized throughintegration of the entire supply chain and logistics operationbased on the Supply-Hub. In Figure 1, the Supply-Hubprovides the service for𝑚manufacturers and 𝑛 suppliers.

For manufacturer 𝑗, where 𝑗 = 1, 2, . . . , 𝑚, the numberof its suppliers is 𝑘𝑗, where 1 ≤ 𝑘𝑗 ≤ 𝑛. It indicates thatthe components required by manufacturer 𝑗 are provided by𝑘𝑗 suppliers. For a certain supplier 𝑖, where 𝑖 = 1, 2, . . . , 𝑛,the number of components required by manufacturer is 𝑙𝑖,where 1 ≤ 𝑙𝑖 ≤ 𝑚. It implies that the components providedby supplier 𝑖 are required by 𝑙𝑖 manufacturer. Therefore, themulti-suppliers and multi-manufacturers system based onthe Supply-Hub considered in our paper is more universaland versatile.

2.2. Assumptions and Notations. The Supply-Hub takescharge in the components purchasing and JIT direct-stationdistribution for 𝑚 manufacturers. Component 𝑖 required bymanufacturer 𝑗 is delivered to manufacturer 𝑗 by the Supply-Hub at suitable interval 𝑅ℎ,𝑗, and component 𝑖 from supplier𝑖 was delivered to the Supply-Hub at regular interval 𝑅𝑖𝑗.According to the distribution lot to the Supply-Hub, thepurchasing lot is determined by supplier 𝑖.

Define

𝑤𝑖𝑗

={

1 if supplier 𝑖 provides component 𝑖 formanufacturer 𝑗0 else.

(1)

2.2.1. Assumptions. The specific assumptions are as follows.

(1) Each supplier provides one kind of the componentfor a manufacturer, and demand for the component isconstant. Note that our results remain unchanged if acertain supplier can provide a variety of components,since it can be actually converted tomultiple suppliersand each provides one component.

(2) The transportation cost of component 𝑖 required bymanufacturer 𝑗 from supplier 𝑖 to the Supply-Hub iscomposed of a fixed cost 𝐹𝑖𝑗 and a variable cost 𝑉𝑖𝑗,and the transportation cost from the Supply-Hub tomanufacturer 𝑗 also contains a fixed cost 𝐹ℎ,𝑖𝑗 and avariable cost 𝑉ℎ,𝑖𝑗.

(3) The lead time for each level of the supply chain isconstant, and it is assumed to be zero without loss ofgenerality.

(4) Shortages are not allowed.(5) Time horizon is infinite.

2.2.2. Notations. The input parameters and decision vari-ables for manufacturers, the Supply-Hub, and suppliers, aredenoted by the subscripts𝑚, ℎ, and 𝑠, respectively.

Manufacturers:

𝑚 is the number of manufacturer; where 𝑗 =

1, 2, . . . , 𝑚

𝑑𝑖𝑗 is the annual demand of manufacturer 𝑗 for thecomponent 𝑖 (units/year);ℎ𝑚,𝑖𝑗 is the manufacturer 𝑗’s holding cost per unit peryear for component 𝑖;


𝐴𝑚,𝑗 is the order cost for manufacturer 𝑗 ($);𝑇𝑗 is the cycle time (year).

The Supply-Hub:

𝐴ℎ is the fixed-order/setup cost per cycle for theSupply-Hub;ℎℎ,𝑖 is the Supply-Hub’s holding cost per unit per yearfor component 𝑖;𝑀ℎ,𝑗 is an integer multiplier to adjust the orderquantity of the Supply-Hub to that of manufacturer𝑗;𝐹ℎ,𝑗 is the fixed transportation cost from the Supply-Hub to manufacturer 𝑗;𝑉ℎ,𝑗 is the variable transportation cost from theSupply-Hub to manufacturer 𝑗.

Suppliers:

𝑛 is the number of suppliers, where 𝑖 = 1, 2, . . . , 𝑛;𝐴 𝑠,𝑖 is the order cost for supplier 𝑖;ℎ𝑠,𝑖 is the supplier’s holding cost per unit per unit peryear for component 𝑖;𝑀𝑠,𝑖𝑗 is an integer multiplier to adjust the order quan-tity of the supplier 𝑖 whose component is required bymanufacturer 𝑗 to that of the Supply-Hub;𝐹𝑠,𝑖𝑗 is the fixed transportation cost for component𝑖 required by manufacturer 𝑗 from supplier 𝑖 to theSupply-Hub;𝑉𝑠,𝑖𝑗 is the variable transportation cost for component𝑖 required by manufacturer 𝑗 from supplier 𝑖 to theSupply-Hub.

3. Model Formulation

3.1. Manufacturer’s Cost Function. Manufacturer 𝑗 orders∑𝑛𝑖=1 𝑑𝑖𝑗𝑇𝑗 units from the Supply-Hub every 𝑇𝑗. The total

annual cost for a manufacturer is the sum of the annual ordercost, 𝐴𝑚,𝑗/𝑇𝑗, and the annual holding cost, ∑𝑛𝑖=1 𝑑𝑖𝑗𝑇𝑗ℎ𝑚,𝑖𝑗/2.The annual cost function for manufacturer 𝑗 is given by

𝐶𝑚,𝑗 =

𝐴𝑚,𝑗

𝑇𝑗

+

∑𝑛𝑖=1 𝑑𝑖𝑗𝑇𝑗ℎ𝑚,𝑖𝑗

2

. (2)

The annual manufacturers’ cost is the sum of 𝐶𝑚,𝑗 for 𝑚manufacturers, and it is given as

𝐶𝑚 =

𝑚

∑

𝑗=1

𝐶𝑚,𝑗 (𝑇𝑗) =

𝑚

∑

𝑗=1

(

𝐴𝑚,𝑗

𝑇𝑗

+


2

) , (3)

where 𝑇𝑗 is a decision variable in (3), and the optimal cycletime for manufacturer 𝑗 is

𝑇∗𝑗 = √

2𝐴𝑚,𝑗

∑𝑛𝑖=1 𝑑𝑖𝑗ℎ𝑚,𝑖𝑗

. (4)

In this paper, an optimal value of decision variable will beindicated by an asterisk (∗).

3.2. Supply-Hub’s Cost Function. The Supply-Hub managesits upstream manufacturers separately; thus, it places anorder for manufacturer 𝑗 every 𝑀ℎ,𝑗𝑇𝑗 and transports thecomponents to manufacturer 𝑗 every 𝑇𝑗. The Supply-Hub’sannual cost to satisfy the demand of manufacturer 𝑗 is

𝐶ℎ,𝑗 (𝑀ℎ,𝑗) =𝐴ℎ

𝑀ℎ,𝑗𝑇𝑗

+

𝑛

∑

𝑖=1

[

ℎℎ,𝑖

2

(𝑀ℎ,𝑗 − 1) 𝑑𝑖𝑗𝑇𝑗]

+

𝐹ℎ,𝑗 + ∑𝑛𝑖=1 𝑑𝑖𝑗𝑇𝑗𝑉ℎ,𝑗

𝑇𝑗

,

(5)

where the terms𝐴ℎ/𝑀ℎ,𝑗𝑇𝑗,∑𝑛𝑖=1[(ℎℎ,𝑖/2)(𝑀ℎ,𝑗−1)𝑑𝑖𝑗𝑇𝑗], and

(𝐹ℎ,𝑗 + ∑𝑛𝑖=1 𝑑𝑖𝑗𝑇𝑗𝑉ℎ,𝑗)/𝑇𝑗 are the Supply-Hub’s annual order

cost, the holding cost, and transportation cost for manu-facturer 𝑚 which requires component 𝑖 from 𝑛 suppliers.Then the Supply-Hub’s total cost is the sum of (5) for 𝑚manufacturers, and it is given as

𝐶ℎ =

𝑚

∑

𝑗=1

𝐶ℎ,𝑗 (𝑀ℎ,𝑗)

=

𝑚

∑

𝑗=1

{

𝐴ℎ


+

𝑛

∑

𝑖=1

[

ℎℎ,𝑖

2


+


𝑇𝑗

} ,

(6)

where𝑀ℎ,𝑗 is a decision variable in (6), and the Supply-Hub’soptimal cycle time for manufacturer 𝑗 is

𝑀ℎ,𝑗 =1

𝑇𝑗

√

2𝐴ℎ

∑𝑛𝑖=1 ℎℎ,𝑖𝑑𝑖𝑗

. (7)

3.3. Supplier’s Cost Function. The Supply-Hub has 𝑛 suppliersto provide all 𝑛 components. When manufacturer 𝑗 placesan order of size ∑𝑛𝑖=1 𝑑𝑖𝑗𝑇𝑗 with the Supply-Hub every 𝑇𝑗and as discussed above, the Supply-Hub determines its orderquantity 𝑀ℎ,𝑗𝑑𝑖𝑗𝑇𝑗 for the supplier 𝑖. In order to fulfill thedemand of manufacturer 𝑗, the order of size 𝑀ℎ,𝑗𝑑𝑖𝑗𝑇𝑗 willbe placed by the Supply-Hub, and shipment will occur every𝑀ℎ,𝑗𝑇𝑗. The annual cost for supplier 𝑖 is written as

𝐶𝑠,𝑖 (𝑀𝑠,𝑖𝑗) =

𝑚

∑

𝑗=1

[

𝐴 𝑠,𝑖

𝑀𝑠,𝑖𝑗𝑀ℎ,𝑗𝑇𝑗

+

ℎ𝑠,𝑖𝑀ℎ,𝑗

2

(𝑀𝑠,𝑖𝑗 − 1) 𝑑𝑖𝑗𝑇𝑗

+

𝐹𝑠,𝑖𝑗 + 𝑉𝑠,𝑖𝑗𝑀ℎ,𝑗𝑇𝑗𝑑𝑖𝑗


] ,

(8)

where the terms𝐴 𝑠,𝑖/𝑀𝑠,𝑖𝑗𝑀ℎ,𝑗𝑇𝑗, (ℎ𝑠,𝑖𝑀ℎ,𝑗/2)(𝑀𝑠,𝑖𝑗−1)𝑑𝑖𝑗𝑇𝑖𝑗,and (𝐹𝑠,𝑖𝑗+𝑉𝑠,𝑖𝑗𝑀ℎ,𝑗𝑇𝑗𝑑𝑖𝑗)/𝑀ℎ,𝑗𝑇𝑗 are, respectively, the annual


order cost, holding cost, and transportation cost for supplier𝑖 to meet the annual demand for components required by theSupply-Hub.Then the collective annual cost for 𝑛 suppliers isgiven as

𝐶𝑠 =

𝑛

∑

𝑖=1

𝐶𝑠,𝑖 (𝑀𝑠,𝑖𝑗)

=

𝑛

∑

𝑖=1

𝑚

∑

𝑗=1

[

𝐴 𝑠,𝑖


+


2


+



] ,

(9)

where 𝑀𝑠,𝑖𝑗 is a decision variable in (9), and the supplier 𝑖’soptimal cycle time for manufacturer 𝑗 is

𝑀𝑠,𝑖𝑗 =1


√

2𝐴 𝑠,𝑖

𝑑𝑖𝑗

. (10)

3.4. Solution Procedures with Decentralized Decision

(1) Each manufacturer 𝑗 determines its optimalcycle time, 𝑇

∗𝑗 = √2𝐴𝑚,𝑗/∑

𝑛𝑖=1 𝑑𝑖𝑗ℎ𝑚,𝑖𝑗,

where 𝑖 = 1, 2, . . . , 𝑛,𝑗 = 1, 2, . . . , 𝑚. Then thecollective annual manufacturers’ cost𝐶𝑚 is computedfrom (3).

(2) The value of 𝑇∗𝑗 is input into (7), 𝑀ℎ,𝑗 = (1/𝑇∗𝑗 )

√2𝐴ℎ/∑𝑛𝑖=1 ℎℎ,𝑖𝑑𝑖𝑗. If 𝐶ℎ,𝑗(⌈𝑀ℎ,𝑗⌉) ≥ 𝐶ℎ,𝑗(⌊𝑀ℎ,𝑗⌋),

then 𝑀∗ℎ,𝑗 = ⌊𝑀ℎ,𝑗⌋. Or else 𝑀∗ℎ,𝑗 = ⌈𝑀ℎ,𝑗⌉.This should be repeated for 𝑚 manufacturers, afterwhich the collective Supply-Hub’s annual cost, 𝐶ℎ =∑𝑚𝑗=1 𝐶ℎ,𝑗, is computed from (6).

(3) The values of 𝑇∗𝑗 and 𝑀∗ℎ,𝑗 are input into (8), and(8) is minimized by searching the optimal value of𝑀𝑠,𝑖𝑗. If 𝐶𝑠,𝑖(⌈𝑀𝑠,𝑖𝑗⌉) ≥ 𝐶𝑠,𝑖(⌊𝑀𝑠,𝑖𝑗⌋), then 𝑀

∗𝑠,𝑖𝑗 =

⌊𝑀𝑠,𝑖𝑗⌋, or else𝑀∗𝑠,𝑖𝑗 = ⌈𝑀𝑠,𝑖𝑗⌉. This may be repeated

for 𝑚 ⋅ 𝑛 times because the component 𝑖 providedby supplier 𝑖 may be required by manufacturer 𝑗,where 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚. Then thecollective supplier’s annual cost 𝐶𝑠 is computed from(9).

(4) The value of optimal 𝑇∗𝑗 ,𝑀∗ℎ,𝑗, and𝑀

∗𝑠,𝑖𝑗 for each side

should be recorded and the total supply chain cost forthe case of no coordination is 𝐶𝑛𝑠𝑐 = 𝐶𝑚 + 𝐶ℎ + 𝐶𝑠,which can be obtained after the above three steps.

4. Supply Chain Coordination

The annual supply chain’s cost is determined by summing (3),(6), and (9) to obtain

𝐶csc = 𝐶𝑚 + 𝐶ℎ + 𝐶𝑠

=

𝑚

∑

𝑗=1

(

𝐴𝑚,𝑗

𝑇𝑗

+


2

)

+

𝑚

∑

𝑗=1

{

𝐴ℎ


+

𝑛

∑

𝑖=1

[

ℎℎ,𝑖

2


+


𝑇𝑗

}

+

𝑛

∑

𝑖=1

𝑚

∑

𝑗=1

[

𝐴 𝑠,𝑖


+


2


+



] .

(11)

This is a centralized decision-making process, in whichthe Supply-Hub tries to schedule and optimize each decisionvariable for the entire supply chain. It is general and practicalthat the Supply-Hub takes charge of distribution frequencyand purchasing frequency for the suppliers and the manufac-turers, respectively.

Note that (11) is convex and differentiable over 𝑇𝑗, where𝜕2𝐶csc/𝜕

2𝑇𝑗 = ∑

𝑚𝑗=1[(2/𝑇

3𝑗 )(𝐴𝑚,𝑗 + (𝐴ℎ/𝑀ℎ,𝑗) + 𝐹ℎ,𝑗 +

∑𝑛𝑖=1(𝐴 𝑠,𝑖/𝑀𝑠,𝑖𝑗𝑀ℎ,𝑗) +∑

𝑛𝑖=1(𝐹𝑠,𝑖𝑗/𝑀ℎ,𝑗))] > 0 for every 𝑇𝑗 > 0,

since 𝐴𝑚,𝑗, 𝐴ℎ, 𝐹ℎ,𝑗, 𝐹𝑠,𝑖𝑗, 𝐴 𝑠,𝑖,𝑀ℎ,𝑗,𝑀𝑠,𝑖𝑗 > 0. Therefore at aparticular set of values for𝑀𝑠,𝑖𝑗 ≥ 1 and𝑀ℎ,𝑗 ≥ 1, where𝑀𝑠,𝑖𝑗and𝑀ℎ,𝑗 are integer, 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚, the firstderivative of (11) should be set to zero and the optimal𝑇∗𝑗 wasobtained. Consider

𝑇∗𝑗 = (2[𝐴𝑚,𝑗 +

𝐴ℎ

𝑀ℎ,𝑗

+ 𝐹ℎ,𝑗 +

𝑛

∑

𝑖=1

𝐴 𝑠,𝑖

(𝑀𝑠,𝑖𝑗𝑀ℎ,𝑗)

+

𝑛

∑

𝑖=1

𝐹𝑠,𝑖𝑗

𝑀ℎ,𝑗

]

× (

𝑛

∑

𝑖=1

𝑑𝑖𝑗ℎ𝑚,𝑖𝑗 + (𝑀ℎ,𝑗 − 1)

𝑛

∑

𝑖=1

ℎℎ,𝑖𝑑𝑖𝑗 +𝑀ℎ,𝑗

×

𝑛

∑

𝑖=1

(𝑀𝑠,𝑖𝑗 − 1) 𝑑𝑖𝑗)

−1

)

1/2

.

(12)

4.1. Complexity Analysis for This Problem. The complexitiesof solving this problem are analyzed as follows. The optimal𝑇𝑗,𝑀ℎ,𝑗, and𝑀𝑠,𝑖𝑗 should be obtained tominimize the supplychain’s cost 𝐶csc, where 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚. If


a certain group of solution to this problem was proved NP-complete, then the whole group of solutions to this problemmust be NP-complete.

Taking supplier 1 as the representative case, whose prob-lem is to minimize 𝐶csc(𝑀𝑠,1𝑗,𝑀ℎ,𝑗, 𝑇𝑗 | 𝑗 = 1, 2, . . . , 𝑚). Wedefine this problem as 𝑃. If the problem 𝑃 can be proved toequal partition problem, then the problem 𝑃 is NP-complete.Partition Problem.Given the positive integer 𝑛,𝐵, and a groupof positive integers 𝐺 = {𝑥1, 𝑥2, . . . , 𝑥𝑛}, then ∑

𝑛𝑖=1 𝑥𝑖 = 2𝐵,

can𝐺 be divided into group𝐺1 and𝐺−𝐺1 tomake∑𝑥𝑖∈𝐺𝑖 𝑥𝑖 =∑𝑥𝑖∈𝐺−𝐺𝑖

𝑥𝑖 = 𝐵.

Lemma 1. Partition is NP-complete; see Garey and Johnson[21].

Proposition 2. The problem 𝑃 is NP-complete.

Proof. We should transform the problem 𝑃 to partition. Letthe sets 𝑇, 𝑀ℎ, 𝑀𝑠,1, with |𝑇| = |𝑀ℎ| = |𝑀𝑠,1| = 𝑚,and 𝑊 ⊆ 𝑇 × 𝑀ℎ × 𝑀𝑠,1 be an arbitrary instance ofproblem 𝑃. Let the elements of these sets be denoted by𝑇 = {𝑇1, 𝑇2, . . . , 𝑇𝑚}, 𝑀ℎ = {𝑀ℎ,1,𝑀ℎ,2, . . . ,𝑀ℎ,𝑚}, 𝑀𝑠,1 ={𝑀𝑠,11,𝑀𝑠,12, . . . ,𝑀𝑠,1𝑚}, and 𝑊 = {𝑊1,𝑊2, . . . ,𝑊𝑞}, where|𝑊| = 𝑞. We should construct a set𝐺 and a size 𝑠 (𝑎) ∈ 𝑍+ foreach 𝑎 ∈ 𝐺, such that 𝐺 contains a subset 𝐺1 satisfying

∑

𝑎∈𝐺1

𝑠 (𝑎) = ∑

𝑎∈𝐺−𝐺1

𝑠 (𝑎) . (13)

The set 𝐺 will contain a total of 𝑞+ 2 elements and will beconstructed in two steps.

The first 𝑞 elements of 𝐺 are {𝑎𝑘 : 1 ≤ 𝑘 ≤ 𝑞},where the element 𝑎𝑘 is associated with the group𝑊𝑘 ∈ 𝑊.The size 𝑠 (𝑎𝑘) of 𝑎𝑘 will be specified by giving its binaryrepresentation, in terms of a string of 0’s and 1’s divided into3𝑚 “zones” of 𝑝 = [log2(𝑞 + 1)] bits each.

Then each 𝑠 (𝑎𝑘) can be expressed in binary with no morethan 3𝑝𝑚 bits; it is clear that 𝑠 (𝑎𝑘) can be constructed fromthe given problem 𝑃 instance in polynomial time; see Gareyand Johnson [21].

If we sum up all elements in any zone, the total can neverexceed 𝑞 = 2𝑝 − 1. Therefore, in adding up∑𝑎∈𝐺1 𝑠 (𝑎) for anysubset 𝐺1 ∈ {𝑎𝑘 : 1 ≤ 𝑘 ≤ 𝑞}, there will never be any “carries”from one zone to the next. If we define 𝐵 = ∑3𝑚−1𝑠=0 2

𝑝𝑠, thenany subset 𝐺1 ∈ {𝑎𝑘 : 1 ≤ 𝑘 ≤ 𝑞} will satisfy

∑

𝑎∈𝐺1

𝑠 (𝑎) = 𝐵. (14)

The last two elements are denoted by 𝑏1 and 𝑏2; that is,

𝑠 (𝑏1) = 2

𝑞

∑

𝑘=1

𝑠 (𝑎𝑘) − 𝐵,

𝑠 (𝑏2) =

𝑞

∑

𝑘=1

𝑠 (𝑎𝑘) + 𝐵.

(15)

Now suppose we have a subset 𝐺1 ∈ 𝐺 such that

∑

𝑎∈𝐺1

𝑠 (𝑎) = ∑

𝑎∈𝐺−𝐺1

𝑠 (𝑎) . (16)

Then both of these sums must be equal to 2∑𝑞𝑘=1𝑠 (𝑎𝑘),

and one of the two sets, 𝐺1 or 𝐺 − 𝐺1, contains 𝑏1 but not𝑏2. It follows that the remaining elements of that set form asubset of {𝑎𝑘 : 1 ≤ 𝑘 ≤ 𝑞} whose sizes sum to 𝐵. Therefore theproblem𝑃 can be transformed to partition, and Proposition 2is proved.

4.2. Solution Procedure. Since the coordination schedulingproblem of multiple suppliers and multiple manufacturersbased on the Supply-Hub isNP-complete, the solutionmay bevery complex. Therefore, the auto-adapted differential evalu-ation algorithmwill be proposed to solve this problem by thispaper. The differential evolution algorithm put forward byRainer Storn and Kenneth Price in 1997 is for meta-heuristicglobal optimization based on population evolutionary andthe real coding, which is originally used to solve the Cheby-shev polynomials. As to more complex global optimizationproblems of continuous space, such as non-linear and non-differentiable problems even without function expression,the differential evaluation algorithm has a better globaloptimization ability and higher convergence performancewith simple operation, less controlling parameters, and betterrobustness, compared to genetic algorithms, particle swarmoptimization, simulated annealing, tabu search, and so forth.

The evolution process of differential evaluation algorithmis similar to genetic algorithms, including population initial-ization, variation, hybridization, and selection. But the maindifferences between these two algorithms are that the processof variation is before hybridization for differential evolutionalgorithm, and evaluation of population depends on compar-isons with testing chromosome and target chromosome. Asa result, the solution procedure of coordination schedulingproblem can be proposed as follows.

4.2.1. Population Initialization. Let 𝑔 stands for the genera-tion of population 𝑃𝑔, and the scale of population is NP; thatis,𝑃𝑔 = {𝑥𝑔

𝑖∗}, where 𝑖∗ = 1, 2, . . . ,NP.𝑥𝑔

𝑖∗is a feasible solution

of the population 𝑃𝑔, which is composed of a vector of 𝐷variables, that is 𝑥𝑔

𝑖∗= (𝑥𝑔

𝑖∗1, 𝑥𝑔

𝑖∗2, . . . , 𝑥

𝑔

𝑖∗𝐷).

As for our scheduling problem, 𝐷 is the number ofdecision variables. Let 𝑥𝑔

𝑖∗= (𝑥𝑔

𝑖∗1, 𝑥𝑔

𝑖∗2, . . . , 𝑥

𝑔

𝑖∗𝐷) = (𝑀ℎ,1,

𝑀ℎ,2, . . . ,𝑀ℎ,𝑚;𝑀𝑠,11,𝑀𝑠,21, . . . ,𝑀𝑠,𝑛1;𝑀𝑠,12,𝑀𝑠,22, . . . ,𝑀𝑠,𝑛2;. . . ;𝑀𝑠,1𝑚,𝑀𝑠,2𝑚, . . . ,𝑀𝑠,𝑛𝑚).

Initialize the population, set 𝑔 = 0, and 𝑥𝑔=0𝑖∗𝑗∗

= 𝑙𝑗∗ +

rand𝑗∗ ⋅ (ℎ𝑗∗ − 𝑙𝑗∗).Where rand𝑗∗ is a real number generated by uniform

random distribution in [0, 1), ℎ𝑗∗ and 𝑙𝑗∗ are the upper andlower boundaries of individual variables, which are randomlydistrusted real numbers.

4.2.2. Variations. The interim of individuals, V𝑔+1𝑖∗

=

(V𝑔+1𝑖∗1, V𝑔+1𝑖∗2, . . . , V𝑔+1

𝑖∗𝐷), should be generated after any individual

𝑥𝑔

𝑖∗is determined in population 𝑃𝑔, where the number of 𝑥𝑔

𝑖∗


is 𝑟 (3 ≤ 𝑟 ≤ NP). Let individual set Ω = {𝜉1, 𝜉2, . . . , 𝜉𝑟} andafter variation the interim individuals V𝑔+1

𝑖∗are

V𝑔+1𝑖∗= 𝜉1 + 𝐹 ⋅ [(𝜉2 − 𝜉3) + (𝜉3 − 𝜉4) + ⋅ ⋅ ⋅ + (𝜉𝑟−1 − 𝜉𝑟)] ,

(17)

where 𝐹 is a differential scale factor. As for our schedulingproblem, the interim individuals V𝑔+1

𝑖∗should be rounded

to the nearest integer since decision variables 𝑀𝑠,𝑖𝑗 and𝑀ℎ,𝑗 must be positive integers, where 𝑖 = 1, 2, . . . , 𝑛, 𝑗 =

1, 2, . . . , 𝑚.

4.2.3. Hybridization. The interim individuals V𝑔+1𝑖∗

shouldbe crossed with current individuals 𝑥𝑔

𝑖∗in probability CR,

where CR ∈ [0, 1]. The proper individuals can be gener-ated after hybridization. Set 𝑈𝑔+1

𝑖∗= (𝑢𝑔+1

𝑖∗1, 𝑢𝑔+1

𝑖∗2, . . . , 𝑢

𝑔+1

𝑖∗𝐷).

𝑢𝑔+1

𝑖∗1, 𝑢𝑔+1

𝑖∗2, . . . , 𝑢

𝑔+1

𝑖∗𝐷is a feasible solution of decision variables.

Consider

𝑢𝑔+1

𝑖∗𝑗∗= {

V𝑔+1𝑖∗𝑗∗

rand𝑖∗𝑗∗ ≤ CRor 𝑗 = rand (𝑖)𝑥𝑔

𝑖∗𝑗∗else

(𝑖∗= 1, 2, . . . ,NP; 𝑗∗ = 1, 2, , 𝐷) ,

(18)

where CR is the cross rate. The larger the CR is, the more the𝑢𝑔+1

𝑖∗𝑗∗can be influenced by V𝑔+1

𝑖∗𝑗∗, which leads the algorithm

to faster convergence with local optimization. In order toincrease the performance of differential evolution algorithm,the auto-adapted cross rate was proposed. Let CR (𝐺𝑡=0) =CRmax. When the differential evolution algorithm in the fixedloop of evaluation does not improve significantly, CR can beautomatically adapted according to

CR (𝐺𝑡+1) = {0.95CR (𝐺𝑡) if 0.95CR (𝐺𝑡) ≥ CRminCRmin else,

(19)

where CRmax and CRmin are the maximum and minimumcrossover probabilities, respectively. 𝐺 is the total evaluationnumber. 𝐺𝑡+1 stands for evaluation value in cycle 𝑡 + 1.The auto-adapted change of CR can improve performanceof the whole algorithm and enhance the ability of globaloptimization algorithms.

4.2.4. Selection. The fitness of candidate individual 𝑈𝑔+1𝑖∗

should be evaluated after hybridization. The candidate indi-vidual𝑈𝑔+1

𝑖∗can be determinedwhether it replaces the current

individuals 𝑥𝑔𝑖∗or not according to

𝑥𝑔+1

𝑖∗= {

𝑈𝑔+1

𝑖∗if 𝐶csc (𝑇, 𝑈

𝑔+1

𝑖∗) ≤ 𝐶csc (𝑇, 𝑥

𝑔

𝑖∗)

𝑥𝑔

𝑖∗else,

(20)

where 𝐶csc(⋅) is the fitness function, which corresponds tothe total cost of (11), and 𝑇 = {𝑇1, 𝑇2, . . . , 𝑇𝑚}, where 𝑇

∗𝑗

(𝑗 = 1, 2, . . . , 𝑚) can be calculated from (12). The processshould be repeated and the best solution should be outputcorresponding to 𝑥𝑔+1

𝑖∗and 𝑇.

Table 1: Stability analysis of the auto-adapted DE algorithm.

𝑚 𝑛 Best value Worst value Mean Standard deviation9 10 38408 38633 38504 427 10 30160 30382 30304 385 10 21618 21783 21706 373 10 12786 12937 12855 313 8 10524 10666 10597 273 6 8267 8390 8311 263 4 5206 5418 5342 31

5. Numerical Analysis

5.1. Parameters Setting. Numerical experiments are con-ducted to examine the computational effectiveness and effi-ciency of the proposed auto-adapted differential evaluationalgorithm by comparing it with the method of decentralizeddecision. The parameters of the auto-adapted DE algorithmare as follows:𝐷 = 𝑚+𝑚∗𝑛, NP = 𝐷, 𝐹min = 0.2, 𝐹max = 0.6,CRmin = 0.2, CRmax = 0.8, and the maximum number ofiterations GenM is set at 500 when 𝑚 = 10 and 𝑛 = 9, 7, 5;GenM is set at 400 when 𝑛 = 3 and𝑚 = 10, 8; GenM is set at300 when 𝑛 = 3 and 𝑚 = 6, 4. The detailed settings for eachtest problem are as follows.

𝐴 𝑠,𝑖 is selected from uniform distribution𝑈 [200, 300].ℎ𝑠,𝑖 is selected from uniform distribution 𝑈 [1, 3].𝐹𝑠,𝑖𝑗 = 30 and 𝑉𝑠,𝑖𝑗 is selected from uniform distribu-tion 𝑈 [10, 20].𝐴ℎ = 50, ℎℎ,𝑖 = 𝛼1 ⋅ ℎ𝑠,𝑖, 𝛼1 = 0.8.𝐹ℎ,𝑗 = 5 and𝑉ℎ,𝑗 is selected from uniform distribution𝑈 [1, 6].If 𝑤𝑖𝑗 = 1, 𝑑𝑖𝑗 is selected from uniform distribution𝑈 [10, 20]; otherwise, 𝑑𝑖𝑗 = 0.ℎ𝑚,𝑗 = 𝛼2 ⋅ (max ℎ𝑠,𝑖), 𝛼2 = 3.𝐴𝑚 is selected from uniform distribution 𝑈 [20, 30].

5.2. Comparative Evaluations. Figures 2, 3, and 4 show theevolution of best solution under 3 different cases, respectively,and we run the proposed auto-adapted DE algorithm underevery case for 100 times and calculate its best solutions, worstsolutions,means, and standard deviations; the result is shownin Table 1. The results of the auto-adapted DE algorithm andthose of the method of decentralized decision are shown inTables 2 and 3. In Table 2, we assume 𝑛 = 10 and𝑚 = 9, 7, 5, 3.In Table 3, we assume 𝑚 = 3 and 𝑛 = 10, 8, 6, 4. In both thetwo tables, C𝑠,𝑖 denotes the cost of supplier 𝑖; Cℎ denotes thecost of Supply-Hub; C𝑚,𝑗 denotes the cost of manufacturer 𝑗;and Csc denotes the cost of supply chain. Table 4 shows thedifference of every cost item in the context of joint decisionand decentralized decision when 𝑛 = 10. Table 5 shows thedifference of every cost item in context of joint decision anddecentralized decision when𝑚 = 3.

From Figures 2, 3, and 4, it can be seen that the auto-adapted DE algorithm is convergent under these 3 cases; in


Table2:Th

eresulto

fjoint

decisio

nanddecentralized

decisio

nwhen𝑛=10.

𝑚Jointd

ecision

Decentralized

decisio

n𝑀ℎ,𝑗

𝑀𝑠,𝑖𝑗

𝑇𝑗

𝐶𝑠,𝑖

𝐶ℎ

𝐶𝑚,𝑗

𝐶sc

𝑀ℎ,𝑗

𝑀𝑠,𝑖𝑗

𝑇𝑗

𝐶𝑠,𝑖

𝐶ℎ

𝐶𝑚,𝑗

𝐶sc

9

6 5 7 6 7 6 6 6 6

4442634074

3453335403

2402533453

2034723442

4053333532

4343634540

3333034533

3333524360

3355033444

0.24

0.28

0.23

0.28

0.23

0.24

0.26

0.26

0.25

3021

2405

2921

3858

2803

2670

3273.5

3397

2800

2465

6177

243

254

266

264

221

226

256

241

206

37968

3 3 3 3 4 3 3 3 3

7675655096

8676756805

6705756985

6067755795

5054656774

6776965890

6566056786

7676756810

07776066910

6

0.23 0.21

0.21

0.22 0.2

0.21

0.23 0.21

0.21

3334

2647

3151

4184

2939

3012

3545

3638

3032

2718

5801

243

245

266

258

219

224

253

234

203

40147

7

8 5 7 6 6 6 8

3332323043

3332423403

3302423553

2033533453

3032334453

4343442350

4333024543

0.24

0.33

0.23

0.27

0.28

0.27 0.21

2392

1696.5

2139

2962

2333

2007

2464

2574

2193

2147

5157

243

269

267

262

230

230

255

29820

3 3 3 3 4 3 3

7675655096

8676756805

6705756985

6067755795

5054656774

6776965890

6566056786

0.23 0.21

0.21

0.22 0.2

0.21

0.23

2638

1895

2355

3280

2517

2303

2719

2756

2365

2365

4779

243

245

266

258

219

224

253

31680

5

7 8 8 8 10

3232222043

3232322302

2302322332

3022322332

2022222332

0.29

0.26

0.25

0.27

0.24

1600

995.5

1360.5

2170

1992

1429

1676

1556

1412

1754.5

3973

250

250

270

262

222

21172

3 3 3 3 4

7675655096

8676756805

6705756985

6067755795

5054656774

0.23 0.21

0.21

0.22 0.2

1836 1151

1544

2457

2202

1704

1937

1721

1566

2017

3483

243

245

266

258

219

22848

38 7 9

3232322052

3232322302

2302222542

0.26

0.29

0.22

955

996

617

1370

1294

968.5

1042 761

713

1016

2330

245

256

266

12786

3 3 3

7675655096

8676756805

6705756985

0.23 0.21

0.21

1100.5

1151

702

1552

1377

1144

1204 834

776.5

1181

2039

243

245

266

13814


Table3:Th

eresulto

fjoint

decisio

nanddecentralized

decisio

nwhen𝑚=3.

Jointd

ecision

Decentralized

decisio

n𝑛

𝑀ℎ,𝑗

𝑀𝑠,𝑖𝑗

𝑇𝑗

𝐶𝑠,𝑖

𝐶ℎ

𝐶𝑚,𝑗

𝐶sc

𝑀𝑠,𝑖𝑗

𝑀𝑠,𝑗

𝑇𝑗

𝐶𝑠,𝑖

𝐶ℎ

𝐶𝑚,𝑗

𝐶sc

108 7 9

3232322052

3232322302

230

222254

2

0.26

0.29

0.22

955

996

617

1370

1294

968.5

1042 761

713

1016

2330

245

256

266

12786

3 3 3

7675655096

8676756805

6705756985

0.23 0.21

0.21

1100.5

1151 702

1552

1377

1144

1204 834

776.5

1181

2039

243

245

266

13814

88 8 8

22222220

32223222

22022223

0.3

0.28

0.28

939.5 979

600

1361

1234 963

1031

742

1980.5

223

234

236

10524

3 3 3

65646450

75766468

56046456

0.25

0.23

0.24

1085

1134.5

693

1536

1362

1127

1187

824

1703.5

220

229

234

11335

67 6 7

322222

323232

220221

0.3

0.3

0.4

948

980

610

1365

1232

947

1570

204.5

205

205.5

8267

3 3 3

656454

656464

550454

0.28

0.26

0.28

1066 1116 682

1512

1344

1105

1380

202

199

200

8806

46 6 8

2222

2222

1201

0.4

0.4

0.5

892

966

594

1304

1057

162

155

154

5284

3 3 3

4453

5344

3403

0.35

0.34

0.37

1030

1079

661.5

1470

895

159.5 152

150

5597


0

3.8

3.7550045040030025020015010050 350

3.85

3.9

3.95

4

4.05

Generation

Best

solu

tion

×104

Figure 2: The evolution of best solution when𝑚 = 9, 𝑛 = 10.

0

1.28

1.2640030025020015010050 350

1.3

1.32

1.34

1.36

1.38

Generation

Best

solu

tion

×104


8300

8200

9000

8400

8700

8800

8900

8500

8600

0 30025020015010050Generation

Best

solu

tion


Table 4: The difference of every cost item when 𝑛 = 10.

𝑚 9 7 5 3Δ𝐶𝑠1 −9.4% −9.3% −12.9% −13.2%Δ𝐶𝑠2 −9.1% −10.4% −13.5% −13.5%Δ𝐶𝑠3 −7.3% −9.2% −11.9% −12.1%Δ𝐶𝑠4 −7.8% −9.7% −11.7% −11.7%Δ𝐶𝑠5 −4.6% −7.3% −9.5% −6%Δ𝐶𝑠6 −11.4% −12.9% −16.1% −15.3%Δ𝐶𝑠7 −7.6% −9.4% −13.5% −13.5%Δ𝐶𝑠8 −6.6% −6.6% −9.6% −8.8%Δ𝐶𝑠9 −7.7% −7.3% −9.8% −8.1%Δ𝐶𝑠10 −9.3% −9.2% −13% −14%Δ𝐶𝑚1 0% 0% 2.9% 0.8%Δ𝐶𝑚2 3.7% 9.8% 2% 4.5%Δ𝐶𝑚3 0% 0.4% 1.5% 0%Δ𝐶𝑚4 2.3% 1.6% 1.6%Δ𝐶𝑚5 0.9% 5% 1.4%Δ𝐶𝑚6 0.9% 2.7%Δ𝐶𝑚7 1.2% 0.8%Δ𝐶𝑚8 3%Δ𝐶𝑚9 1.5%Δ𝐶ℎ 6.5% 7.9% 14.1% 14.3%Δ𝐶𝑠𝑐 −5.4% −5.9% −7.3% −7.4%

Table 5: The change of every cost item when𝑚 = 3.

𝑛 10 8 6 4Δ𝐶𝑚1 0.8% 1.4% 1.2% 1.3%Δ𝐶𝑚2 4.5% 2.2% 3% 2%Δ𝐶𝑚3 0% 0.9% 2.8% 2.7%Δ𝐶𝑠1 −13.2% −13.4% −11.1% −13.4%Δ𝐶𝑠2 −13.5% −13.7% −12.2% −10.5%Δ𝐶𝑠3 −12.1% −13.4% −10.6% −10.3%Δ𝐶𝑠4 −11.7% −11.4% −9.7% −11.3%Δ𝐶𝑠5 −6% −9.4% −8.3%Δ𝐶𝑠6 −15.3% −14.6% −14.3%Δ𝐶𝑠7 −13.5% −13.1%Δ𝐶𝑠8 −8.8% −10%Δ𝐶𝑠9 −8.1%Δ𝐶𝑠10 −14%Δ𝐶ℎ 14.3% 16.3% 13.8% 18.1%Δ𝐶𝑠𝑐 −7.4% −7.2% −6.1% −5.6%

fact, the algorithm is convergent under all these 7 cases inour numerical experiment; we only show the 3 figures dueto the limited space. From Table 1 we can see that even underthe case𝑚 = 9 and 𝑛 = 10, the standard deviation is relativelysmall, sowe can conclude that the auto-adaptedDE algorithmis stable.

FromTables 2, 3, 4, and 5, we can obtain some conclusionsas follows.

(1) When suppliers, the Supply-Hub, and manufacturersmake decisions as a whole, the total cost of supply


chain can be reduced compared to the correspondingcost when they make decisions decentralized. Tables4 and 5 reveal that the total cost of supply chain canbe reduced by 5.4% at least, 7.4% at most.

(2) When suppliers, the Supply-Hub, and manufactur-ers make decisions centralized, every supplier’s costdecreases, but the Supply-Hub’s cost and every man-ufacturer’s cost increases, and the decreased cost ismore than the increased one, so the total cost of sup-ply chain can be reduced. We can see that the Supply-Hub’s cost increases greatly in context of centralizeddecision-making from Tables 4 and 5, so the operatorof the Supply-Hub may be not willing to make deci-sions centralized. In fact, suppliers always sell theirproducts to the manufacturer on consignment underSupply-Hubmode.The inventory holding cost is paidby suppliers when their products are stored in theSupply-Hub, as every supplier’s cost decreases greatlyon the condition of centralized decision-making,so they are willing to pay the increased inventoryholding cost.

(3) The Supply-Hub’s distribution interval and every sup-plier’s distribution interval increase under centralizeddecision-making compared to the results obtained inthe case of decentralized decision-making, but forsupplier’s order interval, some increase and othersdecrease. FromTables 2 and 3, it can be seen that everysupplier’s distribution interval and 𝑀ℎ,𝑗 increase inthe context of centralized decision-making, so theSupply-Hub’s distribution interval for everymanufac-turer also increases under this case.

(4) From Table 2, it can be seen that in case of decentral-ized decision-making, all the suppliers’ and Supply-Hub’s decisions remain the same as the numberof manufacturer increases, but under centralizeddecision-making, their decisions change as the num-ber of manufacturer increases. This is because inthe context of centralized decision-making, everydecisionmaker considers the influence of his decisionon others, and they optimize the whole supply chaincollaboratively.Therefore, as the number of manufac-turer increases, all the suppliers and the Supply-Hubchange their optimal decisions.

6. Conclusions

This paper examines the collaborative scheduling model forthe Supply-Hub consists of multiple suppliers and multiplemanufacturers. We describe the basic operational process ofthe Supply-Hub and formulate the basic decision models.Given two different scenarios of decentralized system andcollaborative system, we first consider the case that theSupply-Hub, the suppliers, and the manufacturers operateseparately in their delivery quantities, production quantities,and order quantities. We next consider the collaborativemechanism, in which the Supply-Hub makes the entiredecisions for all the suppliers and manufacturers. Further-more, we offer the complexity analysis for the collaborative

scheduling model and it turns to be proved NP-complete.Consequently, we propose an auto-adapted differential evo-lution algorithm. The numerical analysis illustrates that theperformance of collaborative decision is superior to thedecentralized decision. All these results demonstrate that theimplementation of Supply-Hub can significantly reduce theoperation cost in the assembly system, and thus improve thesupply chain’s overall performance.

Conflict of Interests

The authors declare no conflict of interests. We declare thatwe have no financial and personal relationships with otherpeople or organizations that can inappropriately influenceour work, there is no professional or other personal interestof any nature or kind in any product, service and/or companythat could be construed as influencing the position presentedin, or the review of, the manuscript entitled, “A CollaborativeScheduling Model for the Supply-Hub with Multiple Suppli-ers and Multiple Manufacturers”.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (nos. 71102174, 71372019, and 71231007),Specialized Research Fund for Doctoral Program of HigherEducation of China (no. 20111101120019), Beijing Philosophyand Social Science Foundation of China (no. 11JGC106),Beijing Higher Education Young Elite Teacher Project (no.YETP1173) and China Postdoctoral Science Foundation (no.2013M542066).

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