delivery scheduling in Supply Hub in Industrial Park (SHIP)

Int. J. Production Economics 175 (2016) 109–120

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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

Transportation service sharing and replenishment/delivery scheduling in Supply Hub in Industrial Park (SHIP) Xuan Qiu a,n, George Q. Huang b,1 a Department of Industrial Engineering and Logistics Management, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong b Department of Industrial and Manufacturing Systems Engineering, The University of Hong Kong, Hong Kong

art ic l e i nf o

a b s t r a c t

Article history: Received 17 July 2014 Accepted 29 January 2016 Available online 11 February 2016

It becomes increasingly common for an industrial park to provide a supply hub for its member enterprises to share transportation services by dispatching vehicles to circulate multiple manufacturers to cover their delivery requirements. This paper discusses an interactive decision making problem between a Supply Hub in Industrial Park (SHIP) and its member enterprises in transportation service sharing. This problem is modeled as a bilevel program with the SHIP as the leader and manufacturers as followers. The optimal characteristics of the proposed model are analysed, and an algorithm is developed to solve the model. For comparison, a bilevel analytical model is constructed for the problem with direct transportation service. A numerical study is conducted to examine the influence of major parameters and evaluate the value of sharing transportation service. The results show that SHIP's profit will not always increase with the vehicle capacity. Contrary to intuition, manufacturers are found to keep their replenishment and delivery schedules constant when their holding cost rates are high. Compared with direct transportation, SHIP offers a lower delivery price and more frequent delivery service under shared transportation. Sharing transportation service could bring benefits to both SHIP and manufacturers. & 2016 Elsevier B.V. All rights reserved.

Keywords: Supply hub Industrial park Milk-run logistics Bilevel programming

1. Introduction Supply Hub In Industrial Park (SHIP) offers public warehousing and transportation services shared among member enterprises (Qiu and Huang, 2013b). This study focuses on the transportation service sharing, which is one of the typical characteristics of SHIP application. Milk-run service, one type of shared transportation is adopted, where SHIP periodically dispatches a vehicle circulating multiple companies to cover their transportation requirements. Via consolidating small shipments to multiple locations on a single vehicle, the milk-run strategy is capable of acquiring transportation economies of scale. There are numerous successful applications in real life. For instance, Toyota's Kentucky plant uses milk-run vehicles to pickup parts from suppliers for its JIT manufacturing system as often as 16 times daily (Karlin, 2004). Seven–Eleven Japan is another example, where the milk-run strategy significantly reduced its transportation cost and increased the frequency of outbound shipments to its retail stores (Meindl and Chopra, 2004). It can be expected that the sharing of transportation service has considerable n

Corresponding author. E-mail addresses: [email protected] (X. Qiu), [email protected] (G.Q. Huang). 1 Tel.: þ852 28592591.

http://dx.doi.org/10.1016/j.ijpe.2016.02.002 0925-5273/& 2016 Elsevier B.V. All rights reserved.

potentials to improve the performance of not only SHIP but also manufacturing enterprises. This paper considers a special supply chain consisting of one SHIP and multiple manufacturers within the same industrial park. The SHIP leases vehicles from one or more contracted third-party transportation companies to execute the in-park deliveries and pickups. To illustrate the scenario, a manufacturer first orders raw materials from outside suppliers. The raw materials are stored at SHIP, and then delivered to the manufacturer's production line. Milk-run is employed by the SHIP for transporting raw materials. The milk-run cycle starts with loading raw materials on a vehicle according to manufacturers' requirements, followed by unloading the manufacturer's materials at the first stop on the trip. The vehicle then visits other manufacturers, repeating this process at a number of them before returning to the SHIP. To provide the shared transportation service, the SHIP faces two challenges: how to charge the transportation price and how to arrange the milk-run vehicles. It is obvious that the transportation pricing influences individual manufacturer's schedule of raw material deliveries, which in turn affect the vehicle space utilization. Since the deliveries are carried out by milk-run vehicles, high frequent service may result in a waste of vehicle space, while low frequent service may lead to vehicle space insufficiency. Hence, coping with the two challenges above is the key to attain the

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optimal profit of SHIP. In response to SHIP's price and cycle time of shared transportation service, manufacturers enjoy the rights to determine how often, in what quantities the required raw materials are dispatched from external suppliers to SHIP and then to their production lines using milk-run vehicles. Hence, they confront a common problem: how to make optimal replenishment and delivery schedules in order to minimize their individual cost. This paper aims at investigating how the SHIP and manufacturers interact with each other to optimize their individual objective. The SHIP determines the transportation pricing and milk-run service cycle time to maximize its own profit, while each manufacturer determines the raw material replenishment and delivery schedules for the sake of minimizing its overall cost. In this special type of supply chain, individual manufacturers are independent and autonomous enterprises and the SHIP is operated by a third-party logistics provider. Hence, they have their own objectives when making decisions. On the other hand, the decisions of SHIP and manufacturers would impact each other's objectives through sharing common parameters. More specifically, the following research questions are addressed. 1. How do key system parameters affect the SHIP and individual manufacturer's decisions and performance? 2. What is the effect of sharing transportation service on the performance of SHIP and individual manufacturer, and how do various system parameters influence the value of sharing? To answer these research questions, the bilevel programming approach is adopted to model the problem. As SHIP serves as the public service provider in an industrial park, it is treated as the leader, and manufacturers as followers. The SHIP first announces the transportation pricing and the milk-run cycle time. In response to SHIP's decisions, each manufacturer decides the replenishment and delivery schedules of raw materials, while minimizing its overall cost. It is assumed that the SHIP has complete information about manufacturers' cost parameters and raw material demands. Although SHIP and manufacturers are “competitive” in the sense that they are autonomous and independent in making their own decisions, their relationship can be also considered as “cooperate”. Via enjoying the public logistics services provided by SHIP, manufacturers could obtain various benefits e.g. improving land utilization, reducing capital investment, mitigating risks (Qiu and Huang, 2013a, 2013b). Meanwhile, SHIP could also gain profit. In particular, a contract is made between the two parties so as to coordinate their efforts in improving their own performance. Therefore, under the contract, it is reasonable to make the above assumption. Then, taking into account of the response of each manufacturer, the SHIP optimizes its decisions at the beginning. Furthermore, the analytical approach is used to derive the optimal decisions on delivery pricing and individual manufacturer's delivery schedule. To answer the first question, a numerical study is conducted to examine the behavior of SHIP and each manufacturer with respect to various parameters. To answer the second question, a bilevel analytical model is developed for the considered problem with direct transportation service, in which a vehicle is dispatched from SHIP to a firm for raw materials delivery without any stop. Numerical analyses are presented for comparing the performance of individual partner in two supply chains: with and without transportation service sharing under different scenarios constructed by varying key system parameters. The paper is organized as follows. In Section 2, the relevant literature is briefly reviewed. Section 3 formulates the bilevel model with shared transportation, and provides a solution algorithm based on theoretical analyses. Section 4 constructs a bilevel analytical model with direct transportation service. Section 5 presents a numerical study and corresponding sensitivity analyses

to gain important insights. Finally, Section 6 summarizes the contributions and suggests possible directions for future research.

2. Literature review The research literature relevant to the work in this paper falls into two disparate streams: milk-run logistics and bilevel programs in supply chain management. 2.1. Milk-run logistics The concept of milk-run logistics covers a transportation network where all input and output material requirements of several stations are satisfied by one vehicle that visits all these stations and circulates according to a predefined schedule (Baudin, 2005). Among the research on milk-run logistics, the main focus lies in the milk-run vehicle routing problem (Chuah and Yingling, 2005; Du et al., 2007; Jafari-Eskandari et al., 2010; Kilic et al., 2012; Lee et al., 2006; Liu et al., 2003; Peterson et al., 2010; Sadjadi et al., 2009). Chuah and Yingling (2005) consider a JIT supply pickup and delivery system employed with milk-run logistics. They present an inbound inventory routing problem of designing routes, and determining time of routes and frequency of runs for the transportation among suppliers. Peterson et al. (2010) investigate a stochastic milk-run vehicle routing problem with uncertain and splittable customer demands. They design a method to select suppliers into consolidated routes, called flex-runs, aiming at hedging against the transportation volume swings. Instead of investigating a pure milk-run delivery system, Liu et al. (2003) study a mixed truck delivery system where two delivery modes: direct shipment with milk-runs and hub-and-spoke with milkruns, may be adopted depending on the transportation quantity, and geographic locations of suppliers and customers. Via comparing with the two pure delivery systems, they demonstrate that the mixed delivery system can save about 10% of total distance. Taking the same transportation network into account, Du et al. (2007) develop a dynamic vehicle-dispatching system with mixed milk-run service, where suppliers and customers may be served by the same vehicle when orders are placed. Milk-run has also been applied in several studies as an economical approach for inbound and outbound transportations. Chen and Sarker (2009) discuss an integrated procurement–production inventory system with three vendors and one manufacturer, in which the JIT delivery is performed by a shared vehicle going around three vendors to pickup parts and transport them to the manufacturer. Banerjee et al. (2007) adopt the milk-run delivery mode in dispatching finished products from one single manufacturer to multiple retailers in an integrated procurement– production–distribution system. Li and Vairaktarakis (2007) explore the integration of job scheduling with bundle operations and delivery arrangement of finished products under two delivery modes: direct shipment and milk-run deliveries. Milk-run has been considered in a storage assignment problem for transporting materials from a warehouse to multiple manufacturing plants Kovács (2011). Agustina et al. (2014) adopt milk-run for delivering products to various customer locations from a cross-dock center. Zhang et al. (2015) investigate a production network of automobile parts where milk-run delivery is applied for transporting parts from suppliers to assembly factories. Albeit some of the above works consider employing milk-run transportations in the context of a warehouse from where components are delivered to manufacturing departments, both the warehouse and departments are assumed to be owned by a single firm. This paper fills this gap by studying the milk-run deliveries in a typical supply chain in an industrial park with one SHIP and

X. Qiu, G.Q. Huang / Int. J. Production Economics 175 (2016) 109–120

multiple manufacturers, in which all business entities assume the responsibilities of making their own decisions. 2.2. Bilevel programming in supply chain management Bilevel programming is a branch of optimization where a subset of variables is constrained to lie in the optimal set of an auxiliary mathematical program (Marcotte and Savard, 2005). It has been implemented to model the hierarchical decision making processes in supply chains with two non-cooperative decision makers. Huang and Liu (2004) propose a bilevel model to tackle the distribution network design problem, determining the location of distribution center (DC) and the allocation of customers to DCs for reaching a balanced-workload state in the upper and lower level of the model respectively. Wang and Ouyang (2013) build a bilevel model to investigate the facility location planning problem between two competing service providers under the risk of facility disruptions. Ryu et al. (2004), Roghanian et al. (2007) and Calvete et al. (2011) apply the bilevel programming approach for solving productiondistribution planning problems with different decisions and objectives, regarding DC as the leader and manufacturer as the follower. Cao and Chen (2006) present a bilevel programming model to capture the independence relationship of the principal firm and auxiliary plants in a capacitated plant selection problem. The principle firm, as the leader, selects and opens plants while minimizing the cost for opening the plants and the opportunity costs of over-setting the production capacities of the opened plants. The selected plants, as followers, determine the production quantities to minimize their production and operation costs. Marinakis and Marinaki (2008) develop a bilevel programming model for a location routing problem in which the top manager at the upper level determines the optimal location of facilities, and the operational manager at the lower level makes optimal decisions on vehicle routing. In order to cope with the market changes and present competitions, Lukač et al. (2008) model a classical capacitated lot sizing problem as a bilevel mixed 0–1 integer programming problem with the senior manager as the leader who aims at minimizing the sum of total setup time, and the middle manager controlling the machines as the follower whose objective is to minimize total costs. Through switching the leader and follower roles between two decision makers, Gao et al. (2011) formulate a pricing replenishment decision problem as two bilevel problems, and Naimi Sadigh et al. (2012) propose two bilevel programming models to make pricing and advertising decisions in a multiproduct manufacturer-retailer supply chain where demand is a nonlinear function of prices and advertising expenditures. Qiu and Huang (2013a) and Qiu et al. (2014, 2015) study storage pricing problems by using bilevel approach. Wang et al. (2016) construct a bilevel model for coordinating the decision makings of product family architecting and supply chain configuration. Although the bilevel programming approach has been used in studying supply chain management problems, the literature is limited in bilevel models for logistics service sharing decisions. This paper fills this gap by studying a typical supply chain with SHIP where transportation service pricing and cycle time, as well as replenishment and delivery schedules are involved.

111

trip. Manufacturer i demands raw material r (r ¼ 1; …; R) at a constant and deterministic rate of Dir , and makes delivery orders every ki T 0 time interval. The replenishment cycle from external suppliers is N i ki T 0 . This problem is formulated as a bilevel programming model with the SHIP's model and each manufacturer’s model as the upper and lower level programming respectively. The bilevel model is developed on the basis of the following assumptions: (1) The capacity of SHIP is limited, but sufficient enough to satisfy all manufacturers' space requirements. (2) The storage price per unit per time charged by SHIP is fixed and constant over the planning horizon. (3) One unit of goods occupies one unit of space at both SHIP and vehicles. (4) There is an ample stock of raw materials at outside suppliers. The replenishments of raw materials are instantaneous. (5) Raw materials required by one manufacturer share the same delivery frequency. (6) The demand rates of raw materials are deterministic and constant. (7) The initial stock is zero. (8) Shortages are not allowed. (9) A homogeneous fleet of vehicles with limited capacity are leased by SHIP from contracted carriers. (10) Each milk-run tour is carried out by one vehicle, which travels at a steady speed. No traffic problems or vehicle breakdowns occur during the movement. (11) A vehicle visits all manufacturers in one milk-run cycle. (12) All finished products and large raw materials are delivered by direct shipping, which are not discussed in the model. (13) The vehicle capacity exceeds the minimum total delivery quantities of raw materials. (14) The holding cost rate at manufacturers is larger than the storage price charged at SHIP. Assumptions (1)–(10) are widely adopted by researchers in the literature (Chen and Vairaktarakis, 2005; Chowdhury and Sarker, 2001; Cormier and Eldon, 1996; Diponegoro and Sarker, 2006; Du et al., 2007; Lee and Hsu, 2009; Sarker and Diponegoro, 2009; Wu et al., 2016; Zhao et al., 2007). Therefore, they are not explained here. The assumption (11) implies that the milk-run route is predetermined, and hence, the vehicle routing aspect of the problem is disregarded. Similar assumption is used by Kovács (2011) and Banerjee et al. (2007). Since finished products and large size raw materials usually occupy lots of space, it is economical to transport them directly between SHIP and manufacturers. In this paper, shared transportation is applied for delivering small raw materials or components. Hence, the assumption (12) is applied. The minimum total delivery quantities stated in assumption (13) refer to the total raw material requirements in one milk-run cycle. This assumption is made in order to avoid shortage of raw materials. Since manufacturer's holding cost is comprised of capital costs, handling costs, risk costs on inventory and storage space costs, it is generally much higher than SHIP's storage price. Hence, the assumption (14) is made. The notations to be used are listed as follows:

3. The BileveL model with transportation service sharing

Indices: i

This paper considers a supply chain consisting of one SHIP and I manufacturers. Manufacturer i (i ¼ 1; …; I) orders raw materials from outside suppliers, and stores them at SHIP. Every T 0 time interval, the SHIP dispatches a vehicle to deliver raw materials to manufacturers according to their demands by visiting them in one

r Parameters: Di,r demand rate of raw material r at manufacturer i (units/time)

index of finished product and manufacturer (i ¼1,…,I; manufacturer i produces finished product i) index of raw material (r ¼1,…,R)

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fixed cost of initiating a dispatch ($/vehicle trip) holding cost rate of raw material r at manufacturer i ($/unit/time) holding cost rate of raw materials at SHIP ($/unit/time) storage price charged by SHIP ($/unit/ time) market reservation price under full truckload ($/unit) ordering cost per order for manufacturer i to purchase raw material r ($/order) vehicle capacity (units)

FT HRi,r HS SP SR Air

capacity constraint. There are some key points in the objective function which will be explained below. A dynamic transportation pricing strategy is adopted by the SHIP: a freight discount is offered when the delivery quantity is larger. The transportation price function of SHIP is proposed as   VC LP ðqÞ ¼ S ; ð8Þ q where LP ðqÞ is the per unit transportation price charged for a given delivery quantity q ð0 oq rVC Þ, VC is the vehicle capacity, and S denotes the delivery price per unit under full truckload ($/unit). Similar functions are used by Swenseth and Godfrey (2002) and Chen and Sarker (2009) for modeling the freight rate charged by contract carriers. The reasons for implementing the dynamic transportation pricing strategy are discussed as follows. Via charging a lower per unit price for larger shipping quantity, such strategy induces manufacturers to require larger quantity of raw materials each delivery. Hence, the vehicles' space would be effectively utilized. The reduction of the number of vehicles needed due to the high space utilization would not only save the capital expenditure spent on purchasing or leasing vehicles, but also cut the exhaust emissions, which is essential for the sustainable development of the industrial park. All manufacturers demand raw materials at the beginning of the planning horizon, namely, the largest delivery quantity occurs at that time. For the delivery cycle time ki T 0 and the demand rate of raw material Dir, the largest delivery quantity of raw materials I P P R from SHIP is given by r ¼ 1 Dir ki T 0 . Since the delivery quantity

VC Functions: LP(q) transportation price function of SHIP ($/unit) PF profit of the SHIP ($/time) TCMi total cost of manufacturer i ($/time) Decision variables of the SHIP: S transportation price under full truckload ($/unit) T0 milk-run cycle time Decision variables of manufacturer i (i¼1,…,I): ki integer number, where kiT0 represents the delivery time interval from SHIP to manufacturer i Ni number of raw material deliveries from SHIP to manufacturer i per replenishment cycle

i¼1

3.1. SHIP's model The objective of SHIP is to determine the appropriate transportation pricing and milk-run cycle time so that its total profit is maximized. The SHIP's objective function is formulated as follows. Maximize PF ðS; T 0 Þ ¼

 I  X S∙VC i¼1

ki T 0

ð1Þ

þ

I X SP X R D k T ðNi 1Þ r ¼ 1 ir i 0 2 i¼1

ð2Þ



I X HS X R D k T ðN i  1Þ r ¼ 1 ir i 0 2 i¼1

ð3Þ



FT T0

ð4Þ

Subject to: 0 o S r SR ;

ð5Þ

T 0 4 0;

ð6Þ

XI i¼1

XR r¼1

Dir ki T 0 r VC:

ð7Þ

In the objective function, the sum of (1) and (2) is SHIP's total revenue and that of (3) and (4) is SHIP’s total cost. The total revenue consists of the total transportation charge for delivering raw materials (1), and the total storage charge of raw materials (2). The total cost is comprised of the total inventory holding cost of raw materials (3), and the total transportation cost (4). Constraints (5) and (6) show the domain of S and T0. Constraint (7) is the vehicle

is limited by the vehicle capacity, the constraint (7) is obtained. All expressions in the objective function are explained as follows. For the transportation price function (8), the transportation charge for each delivery is S∙VC. Since the delivery time interval is ki T 0 , the transportation charge per unit time for manufacturer i is given by S∙VC , from which the expression (1) is obtained. ki T 0 As discussed above, every ki T 0 time interval, Dir ki T 0 units of raw material r are delivered from SHIP to manufacturer i. Thus, Dir ki T 0 units of raw material r will be stored at SHIP for a time interval of c∙ki T 0 ðc ¼ 1; …; N i  1Þ prior to being delivered to the manufacturer. In one replenishment cycle N i ki T 0 , the total inventory of raw materials stored at SHIP is calculated by XR 2 N ðN  1Þ : ð9Þ D ðk T Þ i i r ¼ 1 ir i 0 2 Then, with the storage price per unit per time SP, the total storage charge of raw materials required by manufacturer i per unit time is given by XR N 1 SP ; ð10Þ D kT i r ¼ 1 ir i 0 2 from which the expression (2) is achieved. Likewise, the expression (3) is obtained. The transportation cost that the SHIP incurs is represented by a fixed cost of initiating a dispatch, which may include a driver's salary, insurance, fixed vehicle leasing/ownership costs, and maintenance costs. Similar transportation cost structure is adopted by Pan et al. (2009) and Geunes et al. (2007). As the milk-run trip is carried out every T0 time interval, the transportation cost per unit time is obtained by expression (4). 3.2. Manufacturer's model The individual manufacturer aims at total cost minimization. Each manufacturer decides on the replenishment schedule of raw materials from outside suppliers, and the delivery schedule of raw materials from SHIP, which are represented by the decision variables Ni and ki respectively. Ni denotes the number of raw material deliveries from SHIP to manufacturer i each replenishment cycle,

X. Qiu, G.Q. Huang / Int. J. Production Economics 175 (2016) 109–120

and ki is an integer where ki T 0 denotes the delivery cycle time of raw materials from SHIP to manufacturer i. Then, the replenishment cycle from suppliers to manufacturer i is N i ki T 0 . With the above notations and definitions, the objective function of manufacturer i (i ¼ 1; …; I) is formulated as: XR Air Minimize TCM i ðki ; N i Þ ¼ ð11Þ r¼1N kT i i 0 þ

S∙VC ki T 0

ð12Þ

þ

SP XR D k T ðN i  1Þ r ¼ 1 ir i 0 2

ð13Þ

þ

XR r¼1

Dir ki T 0 HRir 2

∂2 TCM i ðki ;N i Þ ¼ ∂2 N i

one critical point exists and it

Taking ki as continuous variable, the first derivative of TCM i ðki ; Ni  Þ with respect to ki gives XR ∂TCM i ðki ; N i  Þ S∙VC SP XR Dir T 0 HRir : ¼  Dir T 0 þ 2 r ¼ 1 r¼1 ∂ki 2 ðk i Þ T 0 2 ð19Þ

ð14Þ

ð15Þ

ki ¼

2

ð16Þ

Then, the inventory of raw material r at manufacturer i per unit time is given by Dir k2i T 0 . For the inventory holding cost rate at manufacturer i, HRir , the expression (12) is then obtained. 3.3. The bilevel model



αi ¼

1 T0

sffiffiffiffiffiffiffiffiffiffiffiffiffi 2S∙VC ;

XR r¼1

subject to (5)–(7) where N i and ki ði ¼ 1; …; I Þ for each value of S and T 0 solves: XR Air S∙VC SP XR Min TCM i ðki ; N i Þ ¼ þ þ D k T ðN i  1Þ r¼1N k T r ¼ 1 ir i 0 ki T 0 2 i i 0 XR r¼1

Dir ki T 0 HRir 2

and

ð20Þ

αi

HRir Dir  SP

XR r¼1

Dir :

By substituting Eq. (20) into constraint (7), the domain of S is refined as 0 o S rSmax ; 8 > <

ð21Þ

R

Smax ¼ min S ;  > : 2 PI i¼1

VC PR

r¼1

9 > =

Dffiffiffi ir ffi p

αi

2 > : ;

For the sake of tracking optimal solutions of SHIP's profit function, it is assumed that the SHIP sets an upper bound for T0 (e.g. 24 hours), which is denoted by T U0 . Then,0 o T 0 r T U0 . By making this assumption, this paper focuses on discussing the optimal characteristics of the proposed model in this certain range. According to the definition, the raw material deliveries to manufacturers are restricted by the milk-run trips announced by the SHIP, because the delivery time interval is an integer multiple R E ∂P  dE, then SHIP will suffer from a waste (ki ) of T0. If ΔT ¼  ρ1 E12 CT ∂T  of expenses on empty runs. Hence, at least n one ki ois 1, nameqffiffiffiffiffiffiffi 

S∙VC (see proof ly,min ki ¼ 1. It can be derived that T 0 4min αi i

Taking into account the objective functions of SHIP and manufacturers, the bilevel model (denoted by BM1) for making optimal decisions on transportation pricing, milk-run cycle time, and replenishment/delivery schedules can be formulated as Model BM1:  I  I X X S∙VC SP  HS X FT R Max PF ðS; T 0 Þ ¼ þ D k T ðN i  1Þ  r ¼ 1 ir i 0 ki T 0 2 T0 i¼1 i¼1

þ

Air r ¼ 1 ðN i Þ3 ki T 0 4 0;only

 2 4 0, TCM i as a convex function of ki , Since ∂ TCM∂2i ðkki ;Ni Þ ¼ 2S∙VC i ðki Þ3 T 0  the optimal ki is achieved by setting Eq. (19) equals to zero:

The objective function is comprised of four components: the ordering cost of raw materials from outside suppliers (11), the transportation cost of delivering raw materials from SHIP (12), the rent of storage space for storing raw materials at SHIP (13), and the inventory holding cost of raw materials (14). Constraint (15) shows the domain of Ni and ki, which is obtained based on previous explanations and definitions. Expressions (12) and (13) have been explained in the previous subsection. The expressions (11) and (14) are explained as follows. Since the ordering cost Air is incurred each replenishment cycle N i ki T 0 , the ordering cost per unit time is calculated by NiAkiri T 0 , from which the expression (11) is obtained. According to the explanation of expression (2), the total inventory of raw material r at manufacturer i in one delivery time interval ðki T 0 Þ is calculated as Dir ðki T 0 Þ : 2

PR

is a function of ki . The critical point is the optimal solution of manufacturer i for any given ki , which is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u PR 1 u  t 2 Pr ¼ 1 Air : ð18Þ Ni ¼ ki T 0 SP Rr ¼ 1 Dir

Subject to: N i ; ki 4 0; Ni ; ki A integer:

2

113

i

in Appendix A). Therefore, the range of T 0 is achieved as (sffiffiffiffiffiffiffiffiffiffi) S∙VC min o T 0 r T U0 :

ð22Þ

T 0 can then be expressed by (sffiffiffiffiffiffiffiffiffiffi)! S∙VC T 0 ¼ T U0  γ T U0 min ;

ð23Þ

i

αi

i

αi

0 r γ o 1: The following theorem characterizes the profit function of SHIP.   PR P ðSP  HSÞ D  αi r ffi¼ 1 ir pffiffiffi Theorem 1. Define β ¼ Ii ¼ 1 . If β r 0, then αi ffi 12 0qffiffiffiffiffiffiffiffi pffi 1@ the optimal value of S is Smax . If β 4 0, for a given γ , if VC

FT∙b 2

β

b

a

A

subject to (15). Taking N i as a continuous variable, the first derivative of TCM i ðki ; N i Þ with respect to Ni yields

     qffiffiffiffi   rSmax where a γ ¼ T U0  γ T U0 , b γ ¼ γ min α1i , then the opti-

XR ∂TCM i ðki ; N i Þ Air SP XR ¼ þ D kT : r¼1 r ¼ 1 ir i 0 ∂Ni ðN i Þ2 k i T 0 2

mal value of S is given by 0qffiffiffiffiffiffiffiffiffiffiffi 12 pffiffiffi FT∙b 2 1@ β  aA  S ¼ : VC b

ð17Þ

By setting Eq. (17) equals to zero, the critical point can be obtained with TCM i as a convex function of N i . Because

i

ð24Þ

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Otherwise, 

S ¼ Smax:

ð25Þ

Proof. See Appendix B. This theorem provides guidance for the SHIP operator that the condition of β can be used as a basis for determining the delivery service price to gain the optimal total profit. 3.4. Solution procedure The previous analyses are summarized in the following solution procedure for solving the proposed bilevel model. The algorithm can be briefly described as a three-phase process. In the first phase, SHIP's optimal decision on transportation pricing is calculated, and a candidate milk-run cycle time is generated. In the second phase, each manufacturer's decisions are optimized given the solution of SHIP's model, and then transmitted to SHIP. If a manufacturer has multiple optimal solutions, it is assumed that the optimistic approach is used. That is, the manufacturer is willing to support SHIP by selecting a solution which could maximize the SHIP’s profit generated from it ðPF i Þ: PF i ¼

S∙VC SP  HS XR FT þ D k T ðNi  1Þ  : r ¼ 1 ir i 0 ki T 0 2 T0

ð26Þ

The two phases above proceed iteratively until all candidate solutions of SHIP are enumerated. In the third phase, through selecting the maximum objective of SHIP, the optimal decisions of each supply chain partner are obtained. With regard to the Eq. (23), this paper n assumes that o γ is chosen from a finite and discrete value set γ ¼ γ 1 ; γ 2 ; …; γ J , in which γ 1 ¼ 0, γ J ¼ 1, and γ j ¼ γ j  1 þ J 1 1 ðj ¼ 2; …; J Þ. The number of iterations (value of J) depends on the time basis of dispatching vehicles required by the SHIP. For instance, J is smaller for the time basis of “hour” compared with that of “minute”. The detailed steps of the algorithm are summarized below. Step 1. Initialization: Step 1.1. Set PF max ¼ 0, and j ¼ 1. Step 1.2. Set γ ¼ γ j . Step 2. Calculate the optimal value of S, ki , and N i ði ¼ 1; …; I Þ       (denoted by S γ , ki γ , and  N i γ ) at given γ : Step 2.1. Calculate S γ by Theorem  1; Step 2.2. Calculate   T 0 by taking S γ and γ into Eq. (23); Step 2.3. Take S γ and T 0 into Eq. (20), and denote the result by t ki ;     t t t Step 2.4. Calculate N i ⌈ki ⌉ and N i ⌊ki c by taking ki ¼ ⌈ki ⌉ and T 0 into Eq. (18), and t ki ¼ ⌊ki c and T 0 into Eq. (18) respectively;   t t Step 2.5. Among the four combinations: 〈⌈ki ⌉; ⌈N i ⌈ki ⌉ ⌉i,       t t t t t t 〈⌈ki ⌉; ⌊Ni ⌈ki ⌉ ci, 〈⌊ki c; ⌈N i ⌊ki c ⌉i, and 〈⌊ki c; ⌊N i ⌊ki c ci, choose     the values of  ki γ and Ni γ that minimize TCM i ; If the combination of ki γ and N i γ is not unique, choose the values that could maximize PF i given by Eq. (26).   Step 2.7. Evaluate the capacity constraint (7) by taking ki γ and T 0 . If the constraint is satisfied, go to Step 3. Otherwise, go to Step 5.       Step 3. Calculate PF by taking S γ ,T 0 , ki γ , and Ni γ .   Step 4. If PF Z PFmax  , set PF max ¼ PF, CS ¼ S γ , CT 0 ¼ T 0 ,   Cki ¼ ki γ , CN i ¼ N i γ for i ¼ 1…; I. Step 5. Set j ¼ j þ 1. If j rJ, then go to Step 1.2. Otherwise, go to Step 6. Step 6. Set the optimal PF  ¼ PF max , S ¼ CS , T 0  ¼ CT 0  ,   ki ¼ Cki , and Ni  ¼ CN i  . Step 7. Compute the optimal TCM i  for i ¼ 1…; I by taking the  optimal S , T 0  , ki , and N i  .

Following the above steps, the optimal solutions of SHIP and individual manufacturer are achieved from step 6.

4. The bilevel model with direct transportation service This section considers the scenario where the SHIP does not provide shared transportation service. In this supply chain, raw materials are delivered from SHIP to individual manufacturer by direct shipment, namely, SHIP serves manufacturers separately. Therefore, SHIP only need to determine the transportation pricing, which is represented by SD . In response to SHIP’s transportation service price, manufacturer i ði ¼ 1; …; I Þdecides on the delivery cycle time from SHIP and the replenishment cycle time from D D external suppliers, which are represented by T D i and N i T i respecD D is an integer). Let PF and TCM denote the total profit of tively (ND i i SHIP and manufacturer i per unit time. Assumptions (1)–(9) and (12)–(14), as well as the definitions stated in Section 3 hold in this model. Similar as the model formulation in Section 3, the bilevel model (denoted by BM2) for making decisions on transportation pricing and replenishment/delivery schedules is developed as Model BM2: !     XI SD VC SP  HS XI Max PF D SD ¼ þ D i¼1 i¼1 2 Ti   XR XI FT D T D ND i 1  r ¼ 1 ir i i¼1 D Ti Subject to 0 o SD r SR , PR D r ¼ 1 Dir T i r VC, for i ¼ 1; …; I, D D where T i and N D i for each value of S solves:     SD VC SP XR D D D Min TCM D Dir T D ¼ D þ i T i ; Ni i Ni  1 r ¼ 1 2 Ti XR XR Dir T D Air i þ HR þ ir r¼1 r¼1 D D 2 Ni T i D Subject to N D i 4 0, N i A integer, TD 4 0. i D Since TCM D i has the same structure as TCM i , taking N i as a is obtained similar as continuous variable, the optimal value of ND i Eq. (18): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u PR u2 1 t Pr ¼ 1 Air : ND ð27Þ  ¼ i TD SP Rr ¼ 1 Dir i

From Eq. (20), the optimal value of T D i is obtained by sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2SD VC TD : i ¼

αi

ð28Þ

If the result of Eq. (27) is not integer, ND i  is adjusted by n  j ko   D D D D D D ; ND i  ¼ argmin⌈N D ⌉;⌊ND c TCM i T i ; ⌈N i ⌉ ; TCM i T i ; N i i

i

ð29Þ ⌈ND i ⌉

⌊N D i c

where and denote integers adjacent to the result of Eq. (27). By substituting Eq. (28) into the vehicle capacity constraint, the domain of SD is refined as 0 o SD r SD max ; 99 > => = S max ¼ min S ; mini :   2 >> P > > R : :2 ;; r ¼ 1 Dir D

8 > <

ð30Þ

R

8 > <

αi VC

X. Qiu, G.Q. Huang / Int. J. Production Economics 175 (2016) 109–120

The profit function of SHIP is characterized by the following Theorem.   PR P ðSP  HSÞ D  αi r ffi¼ 1 ir pffiffiffi r0, then the optimal Theorem 2. If β ¼ Ii ¼ 1 αi value of SD is SD max . If β 4 0, then the optimal value of SD is given by PI pffiffiffiffiffi  ¼ 1 FT αi iP qffiffiffiffi P S D ¼  ð31Þ PI pffiffiffiffiffi; R I 1 αi VC ðSP  HSÞ i ¼ 1 r ¼ 1 Dir i¼1 αi  PI pffiffiffiffiffi D i ¼ 1 FT αi  qffiffiffiffi P  when PR P pffiffiffiffiffi r S max ; I 1  D α VC ðSP  HSÞ Ii ¼ 1 ir i r¼1 i¼1 αi and 

SD ¼ SD max ;

ð32Þ

otherwise. Proof. See Appendix C. Theorem 2. demonstrates the existence and uniqueness of the optimal direct transportation service pricing charged by SHIP. This theorem together with Theorem 1 imply that the condition of β plays a pivotal role in determining the optimal delivery price whether SHIP adopts shared transportation service or not.

115

base example are given as follows: Air ¼100, VC¼ 1500, HS¼0.1, HRir ¼2.5, SP ¼1.5, FT¼ 10, SR ¼0.03, TU ¼0.143, D1r ¼ 50,000, D2r ¼ 25,000, D3r ¼ 12,500, D4r ¼9000, D5r ¼5000. The unit time is one week. Assume that the time basis for the milk-run trip is “minute” (e.g. TU ¼ 0.143 week ¼ 1441 min). In order to enumerate all possible milk-run cycle times, the value of J is set to 2000 in this numerical study. 5.1. Sensitivity analysis of key system parameters This section presents a series of numerical examples to study the behavior of SHIP and individual manufacturer with respect to several key parameters: VC, FT, SP, and HR. Results of the sensitivity analysis are summarized in Table 1. Major results are also graphically displayed in Figs. 1–4. Some representative observations are summarized as follows. Fig. 1 depicts the influence of vehicle capacity VC on SHIP's profit PF and individual manufacturer's total cost TCM i (i¼ 1,…,5). As VC varies from 1000 to 2500, PF increases dramatically while TCM i initially increases and then decreases. As VC ranges from

5. Numerical study This section presents a numerical study for four purposes: (1) to demonstrate the results of the proposed bilevel models with and without transportation service sharing, (2) to study how SHIP and individual manufacturer behave when exposed to environmental change, (3) to assess the value of sharing transportation service, and (4) to gain managerial implications for guiding managers of SHIP and member enterprises. For these purposes, a nearlife case is presented, where model parameters are carefully set by following suggestions from our industrial collaborators and other researchers (Sarker and Diponegoro, 2009; Yu et al., 2009; Zhang et al., 2007). As an illustration, the case of five heterogeneous manufacturers and two raw materials are discussed. The input parameters for the

Fig. 1. The influence of the vehicle capacity (VC).

Table 1 Impact of key parameters on the decisions and performance of SHIP and individual manufacturer Parameters

Optimal Solutions T 0 (min)

ki

N i

8.63

55

(1,1,2,2,3)

(10,14,10,11,10)

1000 2000 2500 3000

5.75 1.15 0.06 0.05

38 70 11 11

(1,1,2,2,3) (1,1,2,2,3) (1,1,1,1,1) (1,1,1,1,1)

2 5 20 30

0.06 8.63 8.63 8.63

54 55 55 55

0.9 1.2 1.8 2.1

13.81 11.22 0.01 0.01

1.7 2.1 3 3.5

0.02 5.18 12.95 17.26

S* (10  4) Base case

VC

FT

SP

HR



TCM 1

TCM 2

TCM 3

TCM 4

TCM 5

8139.30

8264.24

5855.47

4132.12

3504.20

2610.46

(14,20,14,16,15) (8,11,8,9,8) (5,7,9,11,15) (5,7,9,11,15)

7430.47 8477.21 8734.51 8734.51

8087.39 8441.69 8340.06 8340.06

5726.01 5987.40 5776.52 5776.52

4043.69 4220.84 4015.90 4015.90

3431.16 3578.01 3389.70 3389.70

2557.88 2610.46 2508.69 2508.69

(1,1,1,1,1) (1,1,2,2,3) (1,1,2,2,3) (1,1,2,2,3)

(10,14,19,23,30) (10,14,10,11,10) (10,14,10,11,10) (10,14,10,11,10)

9784.11 9063.41 6291.08 4442.86

8025.50 8264.24 8264.24 8264.24

5617.27 5855.47 5855.47 5855.47

3942.31 4132.12 4132.12 4132.12

3337.36 3504.20 3504.20 3504.20

2478.31 2610.46 2610.46 2610.46

54 55 70 59

(1,1,2,2,3) (1,1,2,2,3) (1,1,1,1,1) (1,1,1,1,1)

(13,18,13,15,13) (11,15,11,13,11) (7,10,14,16,21) (8,11,15,18,24)

5653.56 6965.75 9495.36 10409.82

6819.41 7595.59 8734.35 9300.46

4845.24 5385.25 6128.02 6543.38

3409.71 3797.80 4306.25 4612.02

2893.82 3223.57 3644.06 3910.33

2155.06 2400.43 2708.25 2910.46

69 55 55 55

(1,1,1,1,1) (1,1,2,2,3) (1,1,2,2,3) (1,1,2,2,3)

(8,11,15,18,24) (10,14,10,11,10) (10,14,10,11,10) (10,14,10,11,10)

8614.89 8128.66 8152.59 8165.89

7832.03 8060.31 8519.17 8774.09

5515.97 5705.63 6042.76 6230.05

3890.63 4030.15 4259.58 4387.04

3299.76 3417.38 3612.72 3721.25

2457.00 2546.09 2690.93 2771.40

PF*

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X. Qiu, G.Q. Huang / Int. J. Production Economics 175 (2016) 109–120

Fig. 2. The influence of the fixed transportation cost (FT).

Fig. 4. The influence of manufacturer’s holding cost rate (HR).

0.9 to 1.5, manufacturers keep their delivery schedule stable (constant ki) while reduces the replenishment cycle time (NikiT0) to reduce the inventory at SHIP. The dominate role played by the increase of SP contributes to the increase of both PF and TCM i . From Fig. 4, with the increase of HR, PF initially decreases and then keeps almost stable, while TCM i keeps increasing. When HR varies from 1.7 to 2.1, T0 decreases from 69 to 55 minutes, leading to the increase of transportation cost. The shorter T0 contributes to the reduction of PF under this condition. Table 1 also shows that when HR increases from 2.1 to 4, T0, ki, and Ni remain constant. The higher TCM i is resulted from the increase of both HR and S. 5.2. The value of sharing transportation service

Fig. 3. The influence of SHIP's storage price (SP).

2500 to 3000, both PF and TCM i keep stable. From Table 1, S goes up when VC varies from 1000 to 2000; while suddenly drops when VC becomes 2500. This is because when VC is large, S is determined by Eq. (25). Table 1 also shows that the milk-run cycle time T0 increases with VC, which yields higher storage charge and lower transportation cost. Due to the dominate role played by the increase of T0, SHIP still gains higher profit even when S becomes significantly low. With regard to manufacturers, the reduction of TCM i is because of the decrease of S. When VC increases beyond 2500, Table 1 shows that T0, ki and Ni are constant. Besides, in this condition, S is determined by Eq. (25) which is reversely proportional to VC, and thus the transportation charge is also constant. This is the reason for the unchanged performance for all partners. As for the influence of fixed transportation cost FT, Fig. 2 illustrates that as FT becomes higher, PF diminishes significantly while TCM i initially increases and then remains invariant with FT. As FT varies from 2 to 5, S increases dramatically from 0.06  10  4 to 8.63  10  4 (from Table 1), leading to the higher TCM i . The raise of S is understandable from Eq. (25) when FT is very low. When FT ranges from 5 to 30, all decision variables remain unchanged. The reduction of PF is solely attributed to the increase of FT. Fig. 3 depicts that both PF and TCM i go up sharply with SP. From Table 1, SHIP reduces S when SP becomes higher. As SP varies from

This section aims at quantifying the value of transportation service sharing to SHIP and individual manufacturer. Since the value of parameters may vary due to uncertainties in any decision making situation, sensitivity analysis is also carried out to examine the impact of selected parameters. The value of sharing transportation service for SHIP is mea D sured by its profit improvement: ΔPF ¼ PF PFDPF  100%, and that for individual manufacturer is measured by the total cost saving: D  TCMi  i  100%. PF  and TCM i  are obtained by solving ΔTCM i ¼ TCMTCM D i

BM1 in Section 3, while PF D and TCM i D are achieved by solving BM2 in Section 4. Table 2 summarizes each supply chain partner's optimal decisions under direct transportation service, and performance differences under the two transportation services. It can be observed from Table 2 that SHIP's profit improvements are positive in most cases. Compared to the direct transportation, the delivery cycle time from SHIP to manufacturers is much shorter under shared transportation. Besides, manufacturers choose a longer replenishment cycle time from suppliers under shared transportation. Hence, SHIP gains higher storage charge which leads to the higher total profit. Table 2 also shows that the total cost savings of manufacturers are all positive and vary little between different manufacturers. The reason is that enterprises save significant amount of holding cost and delivery charge in the shared system due to the shorter delivery cycle time and lower delivery price. It can be seen from Table 2 that SHIP's profit improvement is negative (  9.42%) when the vehicle capacity VC is very small at 1000. In this case, from the comparison of two tables, SHIP offers a cheaper (small S) and more frequent (small T0) shared delivery service. Hence, the lower profit is due to the low delivery charge and high transportation cost. When VC increases from 1500 to

X. Qiu, G.Q. Huang / Int. J. Production Economics 175 (2016) 109–120

117

Table 2 The value of sharing transportation service for SHIP and indivudal manufacturer. Parameters

Base case 1000 VC 2000 2500 3000 2 FT 5 20 30 0.9 SP 1.2 1.8 2.1 1.7 HR 2.1 3 3.5

Optimal decisions in BM2

Profit improvements and cost savings

SD (10-3)

T 1 (min)

T 2 (min)

T 3 (min)

T 4 (min)

T 5 (min)

N D i

ΔPF (%)

ΔTCM 1 (%)

ΔTCM 2 (%)

ΔTCM 3 (%)

ΔTCM 4 (%)

ΔTCM 5 (%)

7.50 5.00 10.00 10.00 8.30 3.30 7.50 7.50 7.50 12.00 9.80 4.70 1.70 1.10 4.50 11.30 15.00

151 101 202 222 222 101 151 151 151 151 151 141 111 131 151 151 151

214 142 285 319 319 142 214 214 214 214 214 202 159 184 214 214 214

302 202 403 451 451 202 302 302 302 302 302 285 226 260 302 302 302

357 238 475 531 531 238 357 357 357 357 357 336 266 306 357 357 357

478 319 638 713 713 319 478 478 478 478 478 451 357 411 478 478 478

3 5 3 2 2 5 3 3 3 3 3 3 4 4 3 3 3

31.31  9.42 2.09 86.59 86.59 18.12 45.44 2.61  26.74 5.49  4.92 34.40 6.02 8.69 31.79 30.73 30.16

11.32 7.57 14.15 17.12 17.12 8.28 11.32 11.32 11.32 19.14 14.5 8.28 3.27 2.15 7.56 15.40 18.90

11.14 7.45 13.89 18.81 18.81 9.21 11.14 11.14 11.14 18.75 14.27 8.99 3.76 2.54 7.46 15.13 18.57

11.32 7.57 14.15 20.18 20.18 9.89 11.32 11.32 11.32 19.14 14.50 9.56 4.07 2.89 7.56 15.40 18.90

11.37 7.57 14.24 20.60 20.60 10.10 11.37 11.37 11.37 19.12 14.47 9.80 4.14 2.83 7.62 15.43 18.93

11.42 7.56 14.32 21.16 21.16 10.43 11.42 11.42 11.42 19.19 14.55 10.06 4.28 2.93 7.66 15.49 19.00

2000, ΔPF drops from 31% to 2%. Because the storage charge under direct transportation increases more dramatically due to the increase of T i D and constant N i D . Table 2 also shows that ΔPF is not sensitive to VC when it increases to a high level of 2500 and 3000. VC also has influence on manufacturers' total cost savings Δ TCM i when it is not very large. As can be observed from Table 2, ΔTCM i increases substantially from 7.57% to 17.12% when VC varies from 1000 to 2500. Compared with shared transportation as shown in Table 1, the delivery cycle time T i D from Table 2 increases more remarkably. For example, T 1 D increases from 0.1 to 0.2 while k1T0 increases from 37 to 70 minutes. This contributes to the more significant increase of holding cost, and thus the total cost under direct transportation. With the increase of FT from 2 to 5, both the profit improvement and cost savings become more significant. It can be seen from Table 2 that the replenishment cycle time of each manufacturer decreases with the increase of FT. For example, manufacturer 1's replenishment cycle decreases from 504 to 454 min. This reduces the storage charge as well as the total profit under direct transportation, which is the reason for the greater profit improvement. For manufacturers, the shorter replenishment cycle time yield higher ordering cost. Hence, the cost saving becomes more remarkable. Differently, when FT increases from 5 to 30, ΔPF diminishes from 31.31% to  26.74% while ΔTCM i is invariant with FT. From Tables 1 and 2, all decision variables are constant in this range of FT. Since the delivery cycle time in direct system is much longer, the increase of transportation cost is less noticeable with the increase of FT. This results in the less remarkable reduction of PFD, and thus the smaller ΔPF. The profit improvement is very sensitive to SHIP’s storage price SP, and could be negative in some cases. For example, ΔPF decreases from 5.49% to  4.92% when SP varies from 0.9 to 1.4. In this range of SP, manufacturers’ decisions are constant under direct service from Table 2, while Ni becomes smaller under shared service from Table 1. Then, the increase of SP yields higher storage charge which is more noticeable in the direct system. Hence, ΔPF becomes less remarkable. Manufacturers' total cost savings drop significantly with the increase of SP. From Table 2, for example, ΔTCM 1 decreases from 19.14% to 14.5% when SP increases from 0.9 to 1.2. The reason is that the ordering cost under shared transportation increases significantly due to the decrease of Ni (from Table 1), while the

ordering cost is unchanged under direct service. Then, the increase of TCM i is more remarkable than TCM D i . With the increase of HR from 1.7 to 3.5, the profit improvement initially increases significantly from 8.69% to 31.79%, and then decreases slowly to 30.16%; and manufacturers' total cost savings increase substantially. When HR varies from 1.7 to 2.1, Table 1 shows that Ni increases while Table 2 shows that Ni D decreases. This leads to the higher storage charge under shared transportation, and thus ΔPF goes up. As HR ranges from 2.1 to 3.5, all decisions except the storage price keep constant under both scenarios (from Tables 1 and 2). The decline of ΔPF and increase of ΔTCM i are all attributed to the more significant increase of storage price under direct transportation. 5.3. Managerial implications Several managerial insights are derived from the numerical studies. First, the SHIP operator should carefully choose the vehicle size and set the delivery cost. It is not always favorable for SHIP to adopt larger vehicles. Using vehicles with significantly large capacity will not increase SHIP's profit. Lowering the delivery cost could improve SHIP's profit while not affect manufacturers' performance; and manufacturers may even save total costs when SHIP’s delivery cost is significantly low. Second, member enterprises should be aware of the importance of the vehicle capacity and SHIP’s storage price, as they have remarkable impacts on their delivery/replenishment decisions and total costs. Using small vehicles could bring benefits to all enterprises. Manufacturers' delivery schedule from SHIP may not be affected when SHIP changes vehicle size or varies its storage price. Third, regarding the influence of holding cost rate at manufacturers, several implications can be drawn for the SHIP operator and member enterprises. When holding cost rate is high, the SHIP operator will be stimulated to charge a high transportation price while enterprises will bear from not only high holding cost but also high delivery charge. Additionally, when the holding cost rate is not significantly low, the variation of it will not affect manufacturers’ delivery and replenishment schedules. Fourth, it is found that SHIP's profit improvement from sharing transportation service is positive in most scenarios when key parameters vary. This finding implies that transportation service sharing could bring benefits to SHIP. The sensitivity of profit improvement with respect to each selected parameter suggests that the benefit of sharing depends on vehicle capacity, delivery

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cost, storage price, and manufacturers' holding cost rate. SHIP could not always gain more substantial profit improvement by using larger vehicles. The increase of delivery cost can lead to more significant profit improvement only when the delivery cost is low. However, SHIP will not benefit from the sharing when using too small vehicles or the delivery cost is too high. Fifth, based on the observations of the impact of transportation service sharing to manufacturers' performance, several managerial guidelines can be drawn for member enterprises. It is always favorable for all enterprises to use the shared transportation service, and the cost savings vary little among enterprises with different raw material demands. The value of sharing to them can be significantly affected by the vehicle capacity, SHIP's storage price, and their holding cost rates. Enterprises could obtain significant cost savings when their holding cost rates are high, SHIP's storage price is low, or large vehicles are used. Enterprises should take these factors into account when evaluating whether to employ the shared transportation service.

6. Conclusion In this paper, transportation service sharing in SHIP is investigated in a supply chain where SHIP and multiple manufacturers interact to determine the optimal transportation pricing, milk-run cycle time, and delivery/replenishment schedules of raw materials. The bilevel programming approach is adopted to model this problem with SHIP as the leader and manufacturers as followers. The optimal characteristics of the bilevel model and each partner's behavior with respect to various parameters are analyzed, and an algorithm is proposed to solve the model. For contrast, the problem with direct transportation service is formulated as a bilevel model, and solved in closed-form. A numerical study is conducted to demonstrate the analytical results, examine the influences of key parameters, and evaluate the value of sharing transportation service. Findings observed from the experiments generate a series of important managerial implications. This paper has made several contributions as follows: 1. Shared transportation is for the first time discussed in the research of SHIP, and the value of sharing transportation service is first quantified in a typical supply chain in an industrial park with SHIP. 2. A bilevel analytical model is first constructed to study the problem with direct transportation service. 3. This paper finds that both SHIP and manufacturers' performance could be unaffected by the vehicle capacity when it is significantly large. A lower delivery cost will always bring more profit for the SHIP while not increase manufacturers' total costs. Contrary to intuition, manufacturers are found to use the same delivery and replenishment schedules no matter how their holding cost rate varies as long as it is not significantly low. 4. Another finding is that transportation service sharing could bring benefits to both SHIP and manufacturers. This is because SHIP offers a low price and high frequent milk-run transportation service which induce manufacturers to adopt a long replenishment cycle time from suppliers. However, the magnitude of benefits is determined by important supply chain parameters. This study has some limitations which may be overcome in future research. First, an assumption has been made upon the direct shipment for transporting finished products. It would be of great interest to consider adopting the shared transportation on the pickup of finished products. Two service modes can be applied: one is the milk-run with backhaul, in which all deliveries

must be completed before any pickup; and the other is simultaneous delivery and pickup, namely, the transportation orders of both raw materials and finished products are covered by one single visit of a vehicle. Second, this paper assumes that all manufacturers are involved in the shared transportation service, and the milk-run route is predetermined. An extension is to develop a new model to determine different transportation service modes for each manufacturer's demand, and perform vehicle routing for the milk-run service. It would be also interesting to incorporate the decisions on sizing and allocating vehicles for each mode. Third, this paper assumes a constant storage price at SHIP. It would be worth investigating the impact of different storage pricing strategies on SHIP and individual manufacturer with transportation service sharing.

Appendix A. nqffiffiffiffiffiffiffio S∙VC The proof of T 0 4 min i αi . nqffiffiffiffiffiffiffiffiffio 

 2S∙VC . It can Let ki ¼ min i ki ¼ 1. From Eq. (20),j ¼ argmini αi qffiffiffiffiffiffiffiffiffi 1 2S∙VC 1 be derived from Eqs. (20) and (21) that 0 o T 0 αj r 1, or 1 o T 0     qffiffiffiffiffiffiffiffiffi   2S∙VC αj o 2 and TCM j N j ; 1 o TCM j N j ; 2 , which is simplified as     qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi , or S∙VC o T 0 o 2S∙VC and TCM j N j ; 1 o TCM j N j ; 2 . T 0 Z 2S∙VC αj  2α j   αj TCM j N j ; 1 o TCM j N j ; 2 gives: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XR XR S∙VC T 0 2SP D A þ þ αj r ¼ 1 jr r ¼ 1 jr T0 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XR XR S∙VC o 2SP D A þ þ T 0 αj; ðA:1Þ r ¼ 1 jr r ¼ 1 jr 2T 0 which is simplified as sffiffiffiffiffiffiffiffiffiffi S∙VC : T0 4

ðA:2Þ

αj

Recall

that

qffiffiffiffiffiffiffi

S∙VC 2αj oT 0 o

qffiffiffiffiffiffiffiffiffi 2S∙VC αj ,

then

qffiffiffiffiffiffiffi

αj o T 0 o

S∙VC

qffiffiffiffiffiffiffiffiffi 2S∙VC αj .

Therefore, when the q smallest deliveryqcycle qffiffiffiffiffiffiffi qffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi time is T 0 , T 0 should S∙VC 2S∙VC 2S∙VC S∙VC oT or T o Z satisfy 0 0 αj αj αj , namely T 0 4 αj , nqffiffiffiffiffiffiffiffiffio 2S∙VC . where, j ¼ argmin αi i

Appendix B. The proof of Theorem 1. Substitute Eqs. (18), (20), and (23) into PF ðS; T 0 Þ yields: PF ðSÞ ¼

0vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi1 pffiffiffiffiffi X R  u u PR S∙VC αi SP  HS XI @t 2 Pr ¼ 1 Air  2S∙VC A pffiffiffiffiffiffiffiffiffiffiffiffiffi þ D ir i¼1 i¼1 r¼1 2 αi 2S∙VC SP Rr ¼ 1 Dir

XI



T U0 

FT  qffiffiffiffiffiffiffi; U γ T 0  min S∙VC αi

ðB:1Þ

i

which is simplified as PF ðSÞ ¼

0vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XR  u u PR S∙VC αi SP  HS XI @t 2 Pr ¼ 1 Air  2S∙VC A þ D i¼1 i¼1 r ¼ 1 ir 2 αi 2 SP Rr ¼ 1 Dir

XI

FT pffiffiffiffiffiffiffiffiffiffi; a þ b S∙VC      qffiffiffiffi   where a γ ¼ T U0  γ T U0 , b γ ¼ γ min α1i . 

i

ðB:2Þ

X. Qiu, G.Q. Huang / Int. J. Production Economics 175 (2016) 109–120

  Substitute Eqs. (27) and (28) into PF D SD yields:

The first derivative of PF ðSÞ with respect to S is given by pffiffiffiffiffiffi ∂PF ðSÞ 1   FT∙b VC ¼ pffiffiffi β þ pffiffiffi ðB:3Þ pffiffiffiffiffiffiffiffiffiffi2 ; ∂S 2 S 2 S a þ b S∙VC

PF

sffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi XR  2VC XI VC αi SP  HS XI β¼ D :  i¼1 i¼1 r ¼ 1 ir 2 2 αi

0

ðB:5Þ

1 pffiffiffiffiffiffi 2 1 B FT∙b VC C FT∙VC∙b pffiffiffi@β þ  2 A þ  3 o 0: p ffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffi 2 S a þ b S∙VC a þ b S∙VC 2

then from Eq. (B.6),

1 ffiffi p 2 S

βþ

pffiffiffiffiffi FT∙b VC pffiffiffiffiffiffiffi 2 ða þ b S∙VC Þ

S

D

¼

> > > > > :

 VC ðSP  HSÞ



SD max

VC ðSP  HSÞ

 o 0.

PI pffiffiffiffi FT αi i ¼ 1 P pffiffiffi PI R 1

PI i ¼ 1

PI

r ¼ 1

αi 

Dir

i ¼ 1

r ¼ 1

αi 

Dir

2

0qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 pffiffiffiffiffi FT∙b VC a 1@ β A : S ¼ b VC

S ¼

> > > > > > > > :

Smax

FT∙b VC a β

b

:

12 A 4Smax

ðB:8Þ

Appendix C. The proof of Theorem 2

i¼1

pffiffiffiffiffi FT αi ffi: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2SD ∙VC

ðC:1Þ



SD ¼

VC



 r SD max

αi

 4 SD max

ðC:6Þ

;

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi   SD U VC , SD is expressed by

ðSP  HSÞ

PI pffiffiffiffiffi ¼ 1 FT αi iP qffiffiffiffi P pffiffiffiffiffi R I 1 αi i¼1 r ¼ 1 Dir i¼1 αi 

PI

ðC:5Þ



Therefore, the optimal SD is obtained as when β 40; and ðC:7Þ

A r Smax

FT∙b VC a β

b

XI

otherwise.

12

0qffiffiffiffiffiffiffiffiffiffi pffiffiffi 1@ β Z 0; or β o0andVC

i ¼ 1

pffiffiffiffi



0qffiffiffiffiffiffiffiffiffiffi pffiffiffi 1@ β o 0andVC

αi

SD ¼ SD max ;

Consequently, the optimal S is obtained as



pffiffiffiffi

Since x ¼

ðB:7Þ

8 0qffiffiffiffiffiffiffiffiffiffi 12 pffiffiffi > FT∙b VC > a > β > 1@ A > > VC b > > <

αi

i ¼ 1

PI pffiffiffiffi FT αi i ¼ 1 P pffiffiffi PI R 1

practical nonsense, and thus will be ignored. Therefore, ∂ ∂PF2 SðSÞ o 0, namely, PF is a concave function of S. By equating Eq. (B.3) to zero, the optimal result is expressed by



sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2SD ∙VC A

0

Hence, from Eq. (B.5), ∂PF∂SðSÞ o0. PF is a decreasing function of S. Since there is no lower bound for S, this condition will lead to a PI 8 pffiffiffiffi FT αi > i ¼ 1  > P pffiffiffi PI pffiffiffiffi > P > I R 1 > D αi < VC ðSP  HSÞ i ¼ 1 αi  r ¼ 1 ir i ¼ 1

¼

0vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi XR  u u PR SD ∙VC αi SP  HS XI @t 2 Pr ¼ 1 Air pffiffiffi þ D ir i¼1 i¼1 r¼1 2 2 SP Rr ¼ 1 Dir

XI

Then, PF D ðxÞ is a concave function of x. By equating PF D ðxÞ to zero, the optimal value of x is obtained as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffi u PI u αi u i ¼ 1 FT 2  u qffiffiffiffi: ðC:4Þ x ¼ t   ffiffiffiffi P q PI PR I αi SP  HS 2 D i¼1 r ¼ 1 ir i¼1 2 αi  2

ðB:6Þ





optimal value of SD is SD max . P qffiffiffiffi pffiffiffiffi P P α R 2 If β 4 0, then Ii ¼ 1 pffiffi2i  SP 2 HS Ii ¼ 1 r ¼ 1 Dir αi o 0, calcu  late the second derivative of PF D SD : rffiffiffiffiffi XI 00 2FT αi PF D ðxÞ ¼  o 0: ðC:3Þ i ¼ 1 x3 2

which is 0

FT∙VC∙b pffiffiffiffiffiffiffi 3 40, ða þ b S∙VC Þ

S

D

pffiffiffiffiffiffiffiffiffiffiffiffiffi SD ∙VC . The first derivative of PF D ðxÞ gives sffiffiffiffiffi pffiffiffiffiffi  2 XI αi SP HS XI XR D0 pffiffiffi  PF ðxÞ ¼ D i¼1 i¼1 r ¼ 1 ir 2 αi 2 rffiffiffiffiffi XI FT αi : ðC:2Þ þ i ¼ 1 x2 2 P qffiffiffiffi pffiffiffiffi PI α SP  HS PI R 2 pffiffii If β r 0, then i¼1 2 i¼1 r ¼ 1 Dir αi Z 0, 2   0 PF D ðxÞ 40. Then, PF D SD is an increasing function of SD . The

then

1 3 pffiffiffiffiffiffi 2 16 1 B FT∙b VC C FT∙VC∙b 7  4 pffiffiffi@β þ  pffiffiffiffiffiffiffiffiffiffi2 A þ  pffiffiffiffiffiffiffiffiffiffi3 5 40; 2S 2 S a þ b S∙VC a þ b S∙VC

As



Let x ¼

ðB:4Þ If 2

D



If β Z 0, then ∂PF∂SðSÞ 4 0. PF ðSÞis an increasing function of S, and the optimal S is obtained by Smax . If β o0, take the second derivative of PF ðSÞ with respect to S provides: 0 1 0 1 pffiffiffiffiffiffi 2 ∂2 PF ðSÞ 1 B FT∙b VC C 1 B FT∙VC∙b C ¼  pffiffiffi@β þ  pffiffiffiffiffiffiffiffiffiffi2 A  2S@ pffiffiffiffiffiffiffiffiffiffi3 A: ∂2 S 4S S a þ b S∙VC a þ b S∙VC ∂2 PF ðSÞ 4 0, ∂2 S

119

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