An Optimal Method for Developing Global Supply Chain Management System

Abstract

Owing to the transparency in supply chains, enhancing competitiveness of industries becomes a vital factor. Therefore, many developing countries look for a possible method to save costs. In this point of view, this study deals with the complicated liberalization policies in the global supply chain management system and proposes a mathematical model via the flow-control constraints, which are utilized to cope with the bonded warehouses for obtaining maximal profits. Numerical experiments illustrate that the proposed model can be effectively solved to obtain the optimal profits in the global supply chain environment.

1. Introduction

Traditionally, supply chain management (SCM) mainly offers different ways to reduce the production and transportation costs such that either the total expenditures in a supply chain can be minimized [1, 2] or the profits can be maximized [3, 4] to enhance the industrial competitiveness. These concepts above have been formed as SCM mathematical models in the last few decades such as supplier’s pricing policy in a just-in-time environment [5], pricing strategy for deteriorating items using quantity discount when customer demand is sensitive [6], and an optimization approach for supply chain management models with quantity discount policy [7].

On the other hand, a part of studies is focused on demand forecasting [8–11] to decrease the impact of the bullwhip effect [12, 13]. Additionally, some studies utilize the neural network method and regression analysis to improve the accuracy for forecasting customer demand [14–16], and furthermore, these connection weights in neural networks are assigned weighting based on fuzzy analytic hierarchy process methods without any tunings [14, 17]. By the previous mentions, we cannot guarantee that the solution are optimal. Based on the reason, this study proposes a deterministic method, an approximate method for linearizing nonlinear time series analysis model, to forecast customer demand in Section 3.2.

Many researchers have suggested that information sharing is a key influence on SCM environments [11, 18] and that it impacts the SCM performance in terms of both total costs and service levels [11, 19, 20]. However, the development of a global SCM model must share information as much as possible. We then suppose that the information in our experimental tests is shared to simplify the SCM situations [21–23].

In order to enhance the competitiveness of industries, a liberalization policy discussed in recent years becomes an important factor. Many developing countries have implemented a liberalization policy with bonded warehouses in the global supply chain environments. The liberalization policy with bonded warehouses in the different countries around the world, makes the goods be stored in the bonded warehouse, the liability of exporter is temporally cancelled, and the importer and warehouse proprietor incur liability. A bonded warehouse is one building or another secured area, where dutiable goods may be stored or processed without payment of duty. It may be managed by the state or by a private enterprise. The liberalization policy with bonded warehouses not only enhances the industries competitiveness but also saves inventory, production and transportation costs. Of course, the objective is to find more bonded warehouses, and more costs can be saved. In this study, the policy has proved the advantage for the exporter obtaining more profit in our numerical examples [24–28].

A global SCM is seen as a network that includes upstream, midstream and downstream sectors linkage, where we consider vendors as upstream, facilities and warehouses as midstream, and retailers (or customers) as downstream. The liberalization policy focuses on bonded warehouses of midstream in different countries around the world is not considered in the current SCM model [7, 29]. In this study, we then propose a global SCM model to be suitable for the international liberalization policy. The objective is to maximize the profits associated with inventory, production, transpiration, and distribution in the period of times. The global SCM model is formed as a linear mixed-integer program and solved by commerce software (i.e., [30]). Then, a flow chart for the proposed procedure is depicted in Figure 1.

Figure 1 

A flow chart of the proposed procedure.

This paper is organized as follows. Section 2 defines the notations in global SCM and forms a basic model of global SCM. Section 3 describes a procedure of global SCM. Section 4 shows the system architecture of a global SCM. Finally, numerical examples demonstrate that the proposed global SCM method can enhance profit ratio in Section 5.

2. A Global SCM Model

There are four stages in the global SCM environment discussed: the vendor, facility, warehouse, and retailer. Each stage may be located in different nations around the world. The general structure of a global supply chain network has been displayed in Figure 2. For simplifying presentation, both income and cost statements occur in the dashed rectangle as follows.(i)Income statement(a)sales of goods in warehouse,(b)transfer price of goods in facility. (ii)Cost statement(a)procurement cost from vendor,(b)transportation cost (i.e., vendor to facility, facility to warehouse, and warehouse, to retailer),(c)inventory cost (i.e., warehouse).

Figure 2 

Schema of a global SCM.

Notations are described in Section 2.1, and a basic model of global SCM is formulated in Section 2.2.

2.1. Notations

There are seven entity sets   𝑁: The set of nations 𝑛 𝑉: The set of vendors 𝑣 𝐹: The set of facilities 𝑓 𝑊: The set of warehouses 𝑤 𝑅: The set of retailers 𝑟 𝑇:   The set of times 𝑡 𝑃: The set of parts (or goods) 𝑝for presenting each entity associated with notations as shown above.

All parameters are composite style, which are composed of uppercase English strings and suffixes with one or more lowercase alphabet characters, and each suffix single lowercase alphabet character presents the element of relative entity sets. These parameters are  𝑃𝑅𝐼𝐶𝐸_𝑤𝑝: The sales price of part 𝑝 in warehouse 𝑤. 𝑇𝑃𝑅𝐼𝐶𝐸_𝑓𝑝: The transfer price of part 𝑝 in facility 𝑓. 𝑃𝑃𝑅𝐼𝐶𝐸_𝑣𝑝: The procurement cost of part 𝑝 from vendor 𝑣. 𝑀𝐶𝑂𝑆𝑇_𝑓𝑝: The production cost of part 𝑝 in facility 𝑓. 𝑇𝐶𝑂𝑆𝑇_𝑣𝑓: The basis of transportation costs from vendor 𝑣to facility 𝑓. 𝑇𝐶𝑂𝑆𝑇_𝑓𝑓′: The basis of transportation costs from facility 𝑓 to facility 𝑓^′. 𝑇𝐶𝑂𝑆𝑇_𝑓𝑤: The basis of transportation costs from facility 𝑓 to warehouse 𝑤. 𝑇𝐶𝑂𝑆𝑇_𝑤𝑟: The basis of transportation costs from warehouse 𝑤to retailer 𝑟. 𝐼𝐶𝑂𝑆𝑇_𝑓: The basis of inventory costs in facility 𝑓. 𝐼𝐶𝑂𝑆𝑇_𝑤: The basis of inventory costs in warehouse 𝑤.  𝑇_𝑣𝑓: The transportation time from vendor 𝑣 to facility 𝑓. 𝑇_𝑓𝑓′: The transportation time from facility 𝑓 to facility 𝑓^′. 𝑇_𝑓𝑤: The transportation time from facility 𝑓 to warehouse 𝑤. 𝑇_𝑤𝑟: The transportation time from warehouse 𝑤 to retailer 𝑟. 𝑇𝑏𝑜𝑚_𝑓𝑝: The production time of part 𝑝 in facility 𝑓.  𝐵𝑂𝑀_𝑝′𝑝: The amount for part 𝑝 reproducing as part 𝑝^′. 𝑃𝑇_𝑝: The weighting of transportation costs for part𝑝.  𝑃𝐼_𝑝: The weighting of inventory costs for part𝑝. 𝐷_𝑡𝑟𝑝: The required quantities of part 𝑝 for retailer 𝑟 at time 𝑡. 𝑇𝐴𝑋_𝑛: The tax in nation 𝑛. 𝐷𝑈𝑇𝑌_𝑛𝑝: The import tax of part 𝑝 in nation 𝑛.  𝐿𝑂𝐶_𝑛𝑣: The vendor 𝑣 in nation 𝑛. 𝐿𝑂𝐶_𝑛𝑓: The facility 𝑓 in nation 𝑛.  𝐿𝑂𝐶_𝑛𝑤: The warehouse 𝑤 in nation 𝑛. 𝐿𝑂𝐶_𝑛𝑟: The retailer 𝑟 in nation 𝑛. 𝑀_𝑈_𝑓𝑝: The supplied upper level of part 𝑝 in the facility 𝑓. 𝑅_𝑈_𝑣𝑓𝑝: The supplied upper level of part 𝑝 from vendor 𝑣 to facility 𝑓. 𝑅_𝑈_{𝑓𝑓′𝑝}: The supplied upper level of part 𝑝 from facility 𝑓 to facility 𝑓^′. 𝑅_𝑈_𝑓𝑤𝑝: The supplied upper level of part 𝑝 from facility 𝑓 to warehouse 𝑤. 𝑅_𝑈_𝑤𝑟𝑝: The supplied upper level of part 𝑝 from warehouse 𝑤 to retailer 𝑟.  𝐼_𝑈_𝑓𝑝: The inventoried upper level of part 𝑝 in facility 𝑓. 𝐼_𝑈_𝑤𝑝: The inventoried upper level of part 𝑝 in warehouse 𝑤.

All decision variables in global SCM are also compositestyle with lowercase suffix alphabet characters. The details of decision variables (integer variables) are described below: 𝑅_{𝑡𝑣𝑓𝑝}: The amount of part 𝑝 transported from vendor 𝑣 to facility 𝑓 at time 𝑡 𝑀_𝑡𝑓𝑝: The amount of part 𝑝 produced in facility 𝑓at time 𝑡 𝑅_{𝑡𝑓𝑓′𝑝}: The amount of part 𝑝 transported from facility 𝑓 to facility 𝑓^′at time 𝑡 𝑅_{𝑡𝑓𝑤𝑝}: The amount of part 𝑝 transported from facility 𝑓 to warehouse 𝑤 at time 𝑡 𝑅_{𝑡𝑤𝑟𝑝}: The amount of part 𝑝 transported from warehouse 𝑤 to retailer 𝑟 at time 𝑡 𝐼_𝑡𝑓𝑝: The inventory of part 𝑝 in facility 𝑓 at time 𝑡 𝐼_𝑡𝑤𝑝: The inventory of part 𝑝 in warehouse 𝑤 at time 𝑡 𝑅𝐵_{𝑡𝑣𝑓𝑝}: The amount of bonded part 𝑝 transported from vendor 𝑣 to facility 𝑓 at time 𝑡 𝑀𝐵_𝑡𝑓𝑝: The amount of bonded part 𝑝 produced in facility 𝑓at time 𝑡 𝑅𝐵_{𝑡𝑓𝑓′𝑝}: The amount of bonded part 𝑝 transported from facility 𝑓 to facility 𝑓^′ at time 𝑡 𝑅𝐵_{𝑡𝑓𝑤𝑝}: The amount of bonded part 𝑝 transported from facility 𝑓 to warehouse 𝑤 at time 𝑡 𝐼𝐵_𝑡𝑓𝑝: The inventory of bonded part 𝑝 in facility 𝑓 at time 𝑡.

2.2. A Proposed Model

We separately use 𝑖𝑏𝑡⁺_𝑡𝑛, 𝑖𝑏𝑡⁻_𝑡𝑛, and 𝑇𝐴𝑋_𝑛 as the symbols of sales profits, losses, and taxes in each nation at different times. The objective of the proposed model is to maximize the sales profits associated with inventory, production, transpiration, and distribution in each nation at a period of times. The linear global SCM model is formulated as follows.

A Global SCM ModelMaximize_{𝑡∈𝑇,𝑛∈𝑁}1−𝑇𝐴𝑋_𝑛𝑖𝑏𝑡⁺_𝑡𝑛−𝑖𝑏𝑡⁻_𝑡𝑛subjectto𝑖𝑏𝑡⁺_𝑡𝑛−𝑖𝑏𝑡⁻_𝑡𝑛=_{𝑝𝑤∈𝑊,𝑟∈𝑅,𝑝∈𝑃}𝑅_{𝑡𝑤𝑟𝑝}∗𝑃𝑅𝐼𝐶𝐸_𝑤𝑝−𝑃𝑇_𝑝∗𝑇𝐶𝑂𝑆𝑇_𝑤𝑟−_{𝑓∈𝐹,𝑤∈𝑊,𝑝∈𝑃}𝑅_{𝑡𝑓𝑤𝑝}∗𝑇𝑃𝑅𝐼𝐶𝐸_𝑓𝑝+𝑃𝑇_𝑝∗𝑇𝐶𝑂𝑆𝑇_𝑓𝑤−_{𝑓∈𝐹,𝑤∈𝑊,𝑝∈𝑃}𝑅_{𝑡𝑓𝑤𝑝}−𝑅𝐵_{𝑡𝑓𝑤𝑝}∗𝑇𝑃𝑅𝐼𝐶𝐸_𝑓𝑝∗𝐷𝑈𝑇𝑌_𝑛𝑝−_{𝑤∈𝑊,𝑝∈𝑃}𝐼_𝑡𝑤𝑝∗𝑃𝐼_𝑝∗𝐼𝐶𝑂𝑆𝑇_𝑤−_𝑤∈𝑊𝐹𝐶𝑂𝑆𝑇_𝑤+_{𝑓∈𝐹,𝑤∈𝑊,𝑝∈𝑃}𝑅_{𝑡𝑓𝑤𝑝}∗𝑇𝑃𝑅𝐼𝐶𝐸_𝑓𝑝+_{𝑓′,𝑓∈𝐹,𝑓′≠𝑓,𝑝∈𝑃}𝑅_{𝑡𝑓𝑓′𝑝}∗𝑇𝑃𝑅𝐼𝐶𝐸_𝑓𝑝−_{𝑓′,𝑓∈𝐹,𝑓′≠𝑓,𝑝∈𝑃}𝑅_{𝑡𝑓′𝑓𝑝}∗𝑇𝑃𝑅𝐼𝐶𝐸_𝑓′𝑝+𝑃𝑇_𝑝∗𝑇𝐶𝑂𝑆𝑇_𝑓′𝑓−_{𝑓′,𝑓∈𝐹,𝑓′≠𝑓,𝑝∈𝑃}𝑅_{𝑡𝑓′𝑓𝑝}−𝑅𝐵_{𝑡𝑓′𝑓𝑝}∗𝑇𝑃𝑅𝐼𝐶𝐸_𝑓′𝑝∗𝐷𝑈𝑇𝑌_𝑛𝑝−_{𝑣∈𝑉,𝑓∈𝐹,𝑝∈𝑃}𝑅_{𝑡𝑣𝑓𝑝}∗𝑃𝑃𝑅𝐼𝐶𝐸_𝑣𝑝+𝑃𝑇_𝑝∗𝑇𝐶𝑂𝑆𝑇_𝑣𝑓−_{𝑣∈𝑉,𝑓∈𝐹,𝑝∈𝑃}𝑅_{𝑡𝑣𝑓𝑝}−𝑅𝐵_{𝑡𝑣𝑓𝑝}∗𝑃𝑃𝑅𝐼𝐶𝐸_𝑣𝑝∗𝐷𝑈𝑇𝑌_𝑛𝑝−_{𝑓∈𝐹,𝑝∈𝑃}𝑀_𝑡𝑓𝑝∗𝑀𝐶𝑂𝑆𝑇_𝑓𝑝−_{𝑓∈𝐹,𝑝∈𝑃}𝐼_𝑡𝑓𝑝∗PI_𝑝∗𝐼𝐶𝑂𝑆𝑇_𝑓−_{𝑓∈𝐹,𝑝∈𝑃}𝐹𝐶𝑂𝑆𝑇_𝑓for𝑡∈𝑇,𝑛∈𝑁, (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) where all decision variables (i.e., 𝑅_{𝑡𝑣𝑓𝑝},𝑀_𝑡𝑓𝑝, etc.) belong to integers.

The objective is to maximize the profits (i.e., expression (1)) in the global SCM model. The revenue of sales (or losses) are obtained from the following statements.

Income Statement (2): sales of part 𝑝 in warehouse 𝑤 minus transportation costs from warehouse 𝑤 to retailer 𝑟, (7): transfer price of part 𝑝 from facility 𝑓 to warehouses 𝑤 (i.e., transfer price), (8): transfer price of part 𝑝 from facility 𝑓 to facility 𝑓^′ (i.e., transfer price),

Cost Statement (3): transportation and ordering costs from facility 𝑓 to warehouse 𝑤,  (4): import duty of part 𝑝 from facility 𝑓 to warehouse 𝑤,  (5): inventory costs of part 𝑝 in warehouse 𝑤,  (6): fixed cost in warehouse 𝑤,  (9): transportation and ordering costs from facility 𝑓 to warehouse 𝑤,  (10): import duty of part 𝑝 from facility 𝑓 to facility𝑓^′,  (11): transportation and ordering costs from vendor 𝑣 to facility 𝑓,  (12): import duty of parts from vendor𝑣 to facility,  (13): production cost in facility𝑓,  (14): inventory cost in warehouse 𝑤,  (15): fixed costs in facilities.

Continuously, there are two kinds of constraints in the proposed model. One is flow conservation and the other is upper-lower bound constraints, which are described below.

Flow Conservation

We have 𝐼_𝑡𝑓𝑝+_𝑣∈𝑉𝑅_{(𝑡−𝑇𝑣𝑓)𝑣𝑓𝑝}+_{𝑓′∈𝐹,𝑓′≠𝑓}𝑅_{(𝑡−𝑇𝑓′𝑓)𝑓′𝑓𝑝}+𝑀_{(𝑡−𝑇𝑏𝑜𝑚𝑓𝑝)𝑓𝑝}−_{𝑝′∈𝑃,𝑝≠𝑝′}𝐵𝑂𝑀_𝑝′𝑝∗𝑀_{𝑡𝑓𝑝′}−_{𝑓′∈𝐹,𝑓′≠𝑓}𝑅_{𝑡𝑓𝑓′𝑝}−_𝑤∈𝑊𝑅_{𝑡𝑓𝑤𝑝}=𝐼_{(𝑡+1)𝑓𝑝}𝐼∀𝑡∈𝑇,𝑓∈𝐹,𝑝∈𝑃,_𝑡𝑤𝑝+_𝑓∈𝐹𝑅_{(𝑡−𝑇𝑓𝑤)𝑓𝑤𝑝}−_𝑟∈𝑅𝑅_{𝑡𝑤𝑟𝑝}=𝐼_{(𝑡+1)𝑤𝑝}∀𝑡∈𝑇,𝑤∈𝑊,𝑝∈𝑃,𝐼𝐵_𝑡𝑓𝑝+_𝑣∈𝑉𝑅𝐵_{(𝑡−𝑇𝑣𝑓)𝑣𝑓𝑝}+_{𝑓′∈𝐹,𝑓′≠𝑓}𝑅𝐵_{(𝑡−𝑇𝑓′𝑓)𝑓′𝑓𝑝}+𝑀𝐵_{(𝑡−𝑇𝑏𝑜𝑚𝑓𝑝)𝑓𝑝}−_{𝑝′∈𝑃,𝑝′≠𝑝}𝐵𝑂𝑀_𝑝′𝑝∗𝑀𝐵_{𝑡𝑓𝑝′}−_𝑓′𝑅𝐵_{𝑡𝑓𝑓′𝑝}−_𝑤𝑅𝐵_{𝑡𝑓𝑤𝑝}=𝐼𝐵_{(𝑡+1)𝑓𝑝}∀𝑡∈𝑇,𝑓∈𝐹,𝑝∈𝑃,𝐼𝐵_𝑡𝑤𝑝+_𝑓∈𝐹𝑅𝐵_{(𝑡−𝑇𝑓𝑤)𝑓𝑤𝑝}−_𝑟∈𝑅𝑅_{𝑡𝑤𝑟𝑝}=𝐼𝐵_{(𝑡+1)𝑤𝑝}∀𝑡∈𝑇,𝑤∈𝑊,𝑝∈𝑃,_𝑤∈𝑊𝑅_{(𝑡−𝑇𝑤𝑟)𝑤𝑟𝑝}=𝐷_𝑡𝑟𝑝∀𝑡∈𝑇,𝑟∈𝑅,𝑝∈𝑃. (16)(17)(18)(19)(20) These flow conservation constraints are described as follows: (16): the limitation of the flow conservation in each facility,  (17): the limitation of the flow conservation in each warehouse,  (18): the limited bonded parts of the flow conservation in each facility,  (19): the limited bonded parts of the flow conservation in each warehouse,   (20): all of the supplied goods satisfy with retailer demand.

Upper-Lower Bound

We have 𝑀_𝑡𝑓𝑝≤𝑀_𝑈_𝑓𝑝,𝐼_𝑡𝑓𝑝≤𝐼_𝑈_𝑓𝑝,𝐼_𝑡𝑤𝑝≤𝐼_𝑈_𝑤𝑝,𝑅_{𝑡𝑣𝑓𝑝}≤𝑅_𝑈_𝑣𝑓𝑝,𝑅_{𝑡𝑓𝑓′𝑝}≤𝑅_𝑈_{𝑓𝑓′𝑝},𝑅_{𝑡𝑓𝑤𝑝}≤𝑅_𝑈_𝑓𝑤𝑝,𝑅_{𝑡𝑤𝑟𝑝}≤𝑅_𝑈_𝑤𝑟𝑝,𝑅𝐵_{𝑡𝑣𝑓𝑝}≤𝑅_{𝑡𝑣𝑓𝑝,}𝑅𝐵_{𝑡𝑓𝑓′𝑝}≤𝑅_{𝑡𝑓𝑓′𝑝},𝑅𝐵_{𝑡𝑓𝑤𝑝}≤𝑅_{𝑡𝑓𝑤𝑝}, (21)(22)(23)(24)(25)(26)(27)(28)(29)(30) for 𝑡∈𝑇,𝑣∈𝑉,𝑓,𝑓^′∈𝐹 and 𝑓≠𝑓^′,𝑝∈𝑃,𝑤∈𝑊, and 𝑟∈𝑅.

The upper-lower bound constraints are noted as follows: (21): capacities in which facilities produce goods, (22): inventory of parts in facilities, (23): inventory of parts in warehouses, (24): quantities of raw parts transported from vendors to facilities, (25): quantities of parts transported from facilities to facilities, (26): quantities of parts transported from facilities to warehouses, (27): quantities of parts transported from the warehouse to retailers, (28): quantities of bonded parts transported from vendors to facilities, (29): quantities of bonded parts transported from facilities to other facilities, (30): quantities of bonded parts transported from facilities to warehouses.

3. A Procedure of Global SCM

There are different procedures in a global supply chain environment such as optimal transportation in logistic management, demand forecasting in preprocessing, bill of materials’ (BOM) table, and considering the liberalization policies with bonded warehouses. These procedures are discussed below.

3.1. Logistic Management

In this study, we assume that a global SCM goes through four stages such as procurement, production, warehousing, and distribution (i.e.,𝑇_𝑣𝑓+𝑇_{𝑓_𝐵𝑜𝑚}+[𝑇_𝑓𝑓′+𝑇_{𝑓_𝐵𝑜𝑚}]+𝑇_𝑓𝑤+𝑇_𝑤𝑟), where the semifinished goods will be transferred to another facility for reproduction (i.e., [𝑇_𝑓𝑓′+𝑇_{𝑓′_𝐵𝑜𝑚}]) to enhance the transfer price of goods. The course of the logistic management needs to go through a phase or a cycle (i.e., 𝑡=1,2,…,𝑛). The processing flow of global SCM is presented in Figure 3.

Figure 3 

Processing flow of the global SCM.

In order to avoid delaying the demand of retailers in the future, in the beginning, we let the quantities of inventories equal zero in each stage in the first phase. Then, we will add 𝑚times for utilizing demand forecast method. The period of 𝑛+𝑚 times will be calculated in this global. Occurring in the period from (𝑛+1) to (𝑛+𝑚) times, all of the goods in transportation and processing will be restored to the inventories of the adjacent warehouses.

3.2. Demand Forecast Method—Time Series Analysis

As mentioned above, we need to forecast the quantity of retailers’ (i.e., customer) demand. Time series forecasting is a well-known method [31, 32] to forecast future demands based on known past events.

By referring to Li’s method [33], we then have the following nonlinear time series analysis’ model.

A Nonlinear Demand Forecasting Model (NDF Model)Min^𝑇_𝑡=1𝑆_𝑡−𝑎+𝑏𝑄_𝑡𝑦_𝑡s.t.𝑦_𝑡=𝑦_𝑡′∀𝑡,𝑡^′and𝑡≠𝑡^′,whichmean𝑡and𝑡^′belongtothesamequarter_𝑘𝑡=1,𝑦_𝑡𝑦=𝑘,_𝑡≥0,𝑆_𝑡,𝑎,and𝑏arefreesignvariables, (31)(32)(33)(34) where 𝑘 is sum of seasons (usually 𝑘=4) and 𝑆_𝑡, 𝑄_𝑡, and 𝑦_𝑡 denotes the sales, quarter numbers, and seasonal indexes, respectively. For example, if 𝑇=8, then 𝑡=(1,5) represent the first quarter, 𝑡=(2,6) represent the second quarter, 𝑡=(3,7) represent the third quarter, and 𝑡=(4,8) represent the fourth quarter.

A NDF model is nonlinear because of existing absolute functions and nonlinear terms. Following linear systems can linearize them. The linear approximation of the function 𝐹_1,𝑡(𝑎,𝑦_𝑡)=𝑎𝑦_𝑡 is a continuous function 𝑓_1,𝑡(𝑎,𝑦_𝑡), which is equal to 𝐹_1,𝑡(𝑎,𝑦_𝑡) (i.e., 𝑓_1,𝑡(𝑎_𝑙1,𝑎_𝑡,𝑙2)=𝐹_1,𝑡(𝑎,𝑦_𝑡) at each grid point (𝑙₁,𝑙₂)) and a linear function of its variables (𝑎,𝑦_𝑡) on the interior or edges of each of the indicated triangles. To describe the linear approximation of the function 𝐹_1,𝑡(𝑎,𝑦_𝑡), we then have the following linear system (referring to the models in Babayev [34]).

Let 𝑓_1,𝑡𝑎,𝑦_𝑡=𝑎𝑦_𝑡≅𝛼_𝑡=^𝑚+1_𝑙1=1^𝑚+1_𝑙2=1𝐹_1,𝑖𝑑_1,𝑙1,𝑑_2,𝑡,𝑙2𝜔_𝑙1,𝑙2,𝑎=^𝑚+1_𝑙1=1^𝑚+1_𝑙2=1𝑑_1,𝑙1𝜔_𝑙1,𝑙2,𝑦_𝑡=^𝑚+1_𝑙1=1^𝑚+1_𝑙2=1𝑑_2,𝑡,𝑙2𝜔_𝑙1,𝑙2,_{𝑚+1𝑙1=1𝑚+1𝑙2=1}𝜔_𝑙1,𝑙2=1,_{𝑚𝑙1𝑚=1𝑙2=1}𝑢_𝑙1,𝑙2+𝑣_𝑙1,𝑙2𝜔=1,_𝑙1,𝑙2+𝜔_{𝑙1,𝑙2+1}+𝜔_{𝑙1+1,𝑙2+1}≥𝑢_𝑙12,𝑙for𝑙₁𝑙=1,…,𝑚,₂𝜔=1,…,𝑚,_𝑙1,𝑙2+𝜔_{𝑙1+1,𝑙2}+𝜔_{𝑙1+1,𝑙2+1}≥𝑣_𝑙1,𝑙2for𝑙₁𝑙=1,…,𝑚,₂=1,…,𝑚, (35)(36)(37)(38)(39)(40)(41) where𝜔_𝑙1,𝑙2≥0 and 𝑢_𝑙1,𝑙2,𝑣_𝑙1,𝑙2∈{0,1}.

Expressions (36)~(38) define the point (𝑎,𝑦_𝑡) as the convex linear combination of the vertices of the chosen triangle. Expressions (39)~(41) imply that if 𝑢_𝑙1,𝑙2=1, then 𝜔_𝑙1,𝑙2+𝜔_{𝑙1,𝑙2+1}+𝜔_{𝑙1+1,𝑙2+1}=1 and 𝑓_1,𝑡(𝑎,𝑦_𝑡)=𝐹_1,𝑡(𝑑_1,𝑙1,𝑑_2,𝑡,𝑙2)𝜔_𝑙1,𝑙2+𝐹_1,𝑡(𝑑_1,𝑙1,𝑑_{2,𝑡,𝑙2+1})𝜔_{𝑙1,𝑙2+1}+𝐹_1,𝑡(𝑑_1,𝑙1+1,𝑑_{2,𝑡,𝑙2+1})𝜔_{𝑙1+1,𝑙2+1}; or if 𝑣_𝑙1,𝑙2=1, then 𝜔_𝑙1,𝑙2+𝜔_{𝑙1+1,𝑙2}+𝜔_{𝑙1+1,𝑙2+1}=1 and 𝑓_1,𝑡(𝑎,𝑦_𝑡)=𝐹_1,𝑡(𝑑_1,𝑙1,𝑑_2,𝑡,𝑙2)𝜔_𝑙1,𝑙2+𝐹_1,𝑡(𝑑_1,𝑙1+1,𝑑_2,𝑡,𝑙2)𝜔_{𝑙1+1,𝑙2}+𝐹_1,𝑡(𝑑_1,𝑙1+1,𝑑_{2,𝑡,𝑙2+1})𝜔_{𝑙1+1,𝑙2+1}.

The mentions above give a basic formulation satisfying the piecewise linear approximation of 𝐹_1,𝑡(𝑎,𝑦_𝑡)=𝑎𝑦_𝑡≅𝛼_𝑡. Similarly, 𝐹_2,𝑡(𝑏,𝑦_𝑡)=𝑏𝑦_𝑡≅𝛽_𝑡 can be also linearized by the same way (for the proof, refer to Babayev [34]).

On the other hand, we use additional binary variables to treat the absolute function. A piecewisely linear demand forecasting model can be reformulated as follows.

A Piecewisely Linear Demand Forecasting Model (PLDF Model)Min^𝑇_𝑡=1Φ_𝑡()(),𝑢s.t.32and33_𝑡−1𝑀+𝑆_𝑡−𝛼_𝑡+𝛽_𝑡𝑄_𝑡≤Φ_𝑡≤𝑆_𝑡−𝛼_𝑡+𝛽_𝑡𝑄_𝑡+1−𝑢_𝑡𝑀,−𝑢_𝑡𝑀+𝛼_𝑡+𝛽_𝑡𝑄_𝑡−𝑆_𝑡≤Φ_𝑡≤𝛼_𝑡+𝛽_𝑡𝑄_𝑡−𝑆_𝑡+𝑢_𝑡𝑀, (42)(43)(44)(45) where 𝑀 is a big enough constant, 𝑢_𝑡∈{0,1}, Φ_𝑡≥0, 𝛼_𝑡, and 𝛽_𝑡 are piecewisely linear variables of 𝑎𝑦_𝑡 and 𝑏𝑦_𝑡, respectively, referred to(35)–(41).

Considering the following situations we know that the nonlinear objective function can be linearized by linear system, that is, (44) and (45).(i)if 𝑢_𝑡=1, then Φ_𝑡=𝑆_𝑡−(𝛼_𝑡+𝛽_𝑡𝑄_𝑡), which is forced by (44) because of Φ_𝑡≥0,(ii)if 𝑢_𝑡=0, then Φ_𝑡=(𝛼_𝑡+𝛽_𝑡𝑄_𝑡)−𝑆_𝑡, which is forced by (45) because of Φ_𝑡≥0.

3.3. Bill of Materials (BOM)

A bill of materials’ (BOM) table is expressed as the “parts list” of components. In order to understand the BOM for this global SCM model, the symbols of𝑝and𝑝^′could be expressed as a material or goods (i.e., parameter 𝐵𝑂𝑀_𝑝′𝑝). Hence, we can utilize a two-dimensional matrix to express the relationship between partial and goods. For example, a 𝑝₄ with a 𝑝₅ can produce a 𝑝₃, and a 𝑝₃ with two 𝑝₂ can produce a 𝑝₁. The BOM table of the sparse matrix is expressed in Figure 4. The aim is to design a friendly interface for our system design.

Figure 4 

BOM figure.

3.4. Liberalization Policy with Bonded Warehouses

This study proposes a global SCM model which is capable of treating liberalization policy for bonded warehouses in the different countries around the world [24–28]. A bonded warehouse is one building or any other secured area, where dutiable goods may be stored or processed without payment of duty. It may be managed by the state or by private enterprise.

While the goods are stored in the bonded warehouse, the liability of exporter is temporally cancelled, and the importer and warehouse proprietor incur liability. These goods are in the bonded warehouse, and they may be processed by assembling, sorting, repacking, or others. After processing and within the warehousing period, the goods may be exported without the payment of duty, or they may be withdrawn for consumption upon payment of duty at the rate applicable to the goods in their manipulated condition at the time of withdrawal. Bonded warehouses provide specialized storage services such as deep freeze, bulk liquid storage, commodity processing, and coordination with transportation and are an integral part of the global supply chain.

The liberalization policy with bonded warehouses not only enhances the industries competitiveness but also saves inventory, production, and transportation costs. Of course, the objective aims to find more bonded warehouses, and more costs could be saved.

4. System Development

Currently, global supply chain management (global SCM) system always needs experts offering the qualities of goods in bonded warehouses to calculate the production capacity since there is a lack of kernel technique (i.e., deterministic mathematical model) for developing global SCM system. At the same time, considering the policy of trade liberalization with bonded warehouses by manual is a difficult job. Therefore, this study develops a global SCM system with optimization technique to cope with the bonded warehouses for obtaining the maximal profits, automatically.

4.1. System Design

This study designs a system to apply for a global SCM system based on a proposed mathematical model. The architecture of the global SCM system includes an application server in the middle tier to distribute all of the processing jobs. Establishing a Broker Server can not only distribute the jobs into different application servers, but also the jobs work much more efficient. The basic architecture of the global SCM system is shown in Figure 5.

Figure 5 

Architecture of the multitier web server.

Furthermore, the system architecture includes three tiers which are composed of various modules. In the first tier, GUI layer, an intuitive and friendly interface for users is built. In the second tier, the application (AP) server layer, is a kernel in the global SCM system whose base is based on the Java programming language and an optimal software component (i.e., LINGO’s Solver). In the last tier, database layer stores all of the data such as the username, password, environment setting, and logistics information into database system. Moreover, the data must be backed up to the other database to protect it from hacking or crashing. Then, the system architecture is shown in Figure 6.

Figure 6 

Architecture of the global SCM system.

5. Numerical Examples

In our numerical examples, we assume that the situation occurred in three different overseas locations. There are three vendors (𝑣₁,𝑣₂,𝑣₃), four facilities (𝑓₁,𝑓₂,𝑓₃,𝑓₄), four warehouses (𝑤₁,𝑤₂,𝑤₃,𝑤₄), three retailers (𝑟₁,𝑟₂,𝑟₃). Under the liberalization policies, these limited quantities of bonded warehouses are listed in the Appendix. There are three parts (𝑝₂,𝑝₄,𝑝₅) supplied from the vendor to the facility, and the parts be produced another two parts (𝑝₁,𝑝₃) in different facilities. The period of time includes two demands: one is actual customer demand (i.e., 𝑡_𝑛=15), and the other is forecasting demand (i.e., 𝑡_𝑚=2), so that the total period of time is 17 (i.e., 𝑡=1 to 17). The other parameters are expressed in Tables 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, and 26 of the Appendix. The numerical example was solved via a global SCM system; these logistic results at different times 𝑡 are depicted in Figures 7, 8, 9, 10 and 11.

Figure 7 

The logistic flow at time = 1.

Figure 8 

The logistic flow at time = 2.

Figure 9 

The logistics flow at time = 6.

Figure 10 

The logistic flow at time = 7.

Figure 11 

The logistic flow at time = 8.

Solving the example, the objective value is $603,010 with consideration of the bonded warehouses. We then calculate them without bonded warehouses again, and the objective value is 514,680. Comparing both of them, the profit ratio increases to 14.6% as shown in Table 1.

Table 1 

Profit ratio.

Table 2 

Vendor location.

Table 3 

Facility location.

Table 4 

Warehouse location.

Table 5 

Retailer location.

Table 6 

The weighting of transportation costs of parts.

Table 7 

The weighting of inventory costs of parts.

Table 8 

The duty of parts in different nations.

Table 9 

The tax in different nations.

Table 10 

Sales price of each part for each vendor.

Table 11 

Maximum supplied limitation in each vendor.

Table 12 

Production costs and limitations for each facility.

Table 13 

Transportation costs and lead time from each vendor to each facility.

Table 14 

Amount of part 𝑝 needed for producing part 𝑝^′.

Table 15 

The lead time of the facility for producing part 𝑝^′ and sales price in the facility.

Table 16 

The base of inventory costs.

Table 17 

Maximum stock level in each facility.

Table 18 

Transportation costs and lead time from one facility to another facility.

Table 19 

Maximum transferable quantity from one facility to another facility.

Table 20 

The base of inventory costs and fixed costs.

Table 21 

Transportation costs and lead time from the facility to the warehouse.

Table 22 

Maximum transferable quantity from the facility to the warehouse.

Table 23 

Transportation costs and lead time from the warehouse to the retailer.

Table 24 

Maximum transferable quantity from the warehouse to the retailer.

Table 25 

Sales price of each part and limitation in each warehouse.

Table 26 

Retailer demand and forecasting at each time.

In order to observe the advantages of the proposed method, this study experiments different numerical examples by generating parameters below.(i)The total locations are randomly generated from a uniform distribution according to 2 ≤ vendors ≤ 20, 3 ≤ facilities ≤ 20, 4 ≤ warehouses ≤ 20, and 2 ≤ retailers ≤ 40.(ii)The period of time is randomly generated from the uniform distribution over the range of [2, 16].(iii)The capacities of bonded warehouses are according to the capacities of original warehouses based on uniform distribution from 50% to 90%.(iv)In order to balance between the supply and demand, all of the coefficients follows feasible sets for the global SCM models.According to the computational results followed rules (i) to (iv), we observe that the liberalization policy with bonded warehouses directly affects the activities of the global SCMs. When we fixed the number of bonded warehouses to 5, the profit ratio approaches 15%. When we take the number of bonded warehouses being 40 as an example, the profit ratio enhances to 20%. Finally, the profit ratio is close to 25% at 50 bonded warehouses. That is, considering that more nodes have bounded constraints, there is a greater cost saving. The tendency of the other 10 tests is depicted in Figure 12.

Figure 12 

The tendency of the profit ratios for different numbers of warehouses.

The proposed model considers the liberalization policy with bonded warehouses as flow-control constraints to obtain optimal profits. Additionally, numerical experiments also illustrate that the proposed model can enhance the profit ratio when the number of bonded warehouses is large.

6. Conclusions

This study proposes a global SCM model which is capable of treating liberalization policy with bonded warehouses. All of the constraints about bonded warehouses can be added to the proposed global SCM model. Additionally, the proposed model is embedded into a global SCM system we developed. The system can automatically calculate the profit ratio to further enhance the competitiveness. Numerical experiments’ reports illustrate that the proposed model can be effectively solved to obtain the optimal profit.

Future research can consider different quantity discount policies, artificial neural network method for solving the uncertainty demand forecast, heuristic methods for accelerating solving speed, and linearization technique for resolving the proposed demand forecasting model.

Appendix

For more details, see Tables 2–26.

Acknowledgments

The authors would like to thank the editor and anonymous referees for providing most valuable comments for us to improve the quality of this paper. This research was supported by the project granted by ROC NSC 101-2811-E-009-011.