Mathematical Problems in Engineering

Volume 2015, Article ID 428925, 22 pages

http://dx.doi.org/10.1155/2015/428925

## A Two-Stage Simulated Annealing Algorithm for the Many-to-Many Milk-Run Routing Problem with Pipeline Inventory Cost

^{1}College of Management & Economics, Tianjin University, 92 Weijin Road, Nankai District, Tianjin 300072, China^{2}Simon Business School, University of Rochester, 304 University Park, Rochester, NY 14620, USA

Received 20 March 2015; Revised 21 July 2015; Accepted 22 July 2015

Academic Editor: Antonios Tsourdos

Copyright © 2015 Yu Lin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In recent years, logistics systems with multiple suppliers and plants in neighboring regions have been flourishing worldwide. However, high logistics costs remain a problem for such systems due to lack of information sharing and cooperation. This paper proposes an extended mathematical model that minimizes transportation and pipeline inventory costs via the many-to-many Milk-run routing mode. Because the problem is NP hard, a two-stage heuristic algorithm is developed by comprehensively considering its characteristics. More specifically, an initial satisfactory solution is generated in the first stage through a greedy heuristic algorithm to minimize the total number of vehicle service nodes and the best insertion heuristic algorithm to determine each vehicle’s route. Then, a simulated annealing algorithm (SA) with limited search scope is used to improve the initial satisfactory solution. Thirty numerical examples are employed to test the proposed algorithms. The experiment results demonstrate the effectiveness of this algorithm. Further, the superiority of the many-to-many transportation mode over other modes is demonstrated via two case studies.

#### 1. Introduction

Concomitant with the accelerated pace of globalization, economic competition has intensified. For better development, companies around the world have committed themselves to reducing costs, of which one of the most important constituents is logistics cost. A sign of this is the flourishing economic and technological development zones in China, where numerous enterprises are collocated in the same area. Other examples include the four major motor cities of the world (Toyota Motor City, Detroit Motor City, Turin Motor City, and Stuttgart Motor City). Accordingly, a multisupplier and multiplant logistics network system based on a factory’s spatial location or functional interaction is formed. Nevertheless, many systems of this kind have not fully developed into a real “cohesive cluster,” especially China’s automotive industry parks. Most automotive parks in China are simply “clustered” based on spatial location instead of the functional interaction among them. Therefore, inbound logistics transportation for this many-to-many () system remains a problem. Without doubt, inbound logistics transportation problems of this kind are more complex. However, proper planning can generate enormous benefits: it can not only strengthen cooperation among enterprises and contribute to their further development, but also reduce the cost of the entire logistics system and mitigate the pressure on logistics and circulation costs for society.

Inbound logistics transportation methods can generally be classified into three types: direct shipping, Milk-run, and cross-docking. Berman and Wang [1] describe these three modes and their respective characteristics. Cross-docking is characterized by long transportation time and large pipeline inventory and is therefore suitable for long-distance transportation. In contrast, direct shipping and Milk-run have the advantage of short transportation time and small pipeline inventory, which make them suitable for short-range transportation. Considering the large number of suppliers and plants and the short distances between them, direct shipping or Milk-run is more applicable for an industrial park. This paper focuses on Milk-run route planning with multiple suppliers and plants within a certain adjacent area.

The name “Milk-run” originated from the traditional system by which milk was sold in the West, in which the milkman simultaneously delivered full bottles of milk and collected the empty ones according to a predefined route. After completing a roundtrip, he returned with the empty bottles to the starting point. This method has subsequently been extensively utilized in miscellaneous industries and the auto manufacturing companies of the world and has developed into a popular way of picking up and delivering goods for multiple suppliers and plants using the same freight car [2]. Nemoto et al. [3] presented two Milk-run methods used by Toyota’s inbound logistics. The first is a many-to-one () mode, in which the car assembly manufacturer dispatches one truck at a specified time to visit various suppliers (i.e., parts suppliers) following a predefined route to collect parts or products and deliver them to the factories. The second is a one-to-many () mode, in which the parts are consolidated at the place of departure in such a way that they are fully loaded on the truck, transported, and unloaded at the various destinations. For both of these logistics modes, managers need to design and plan for specific routes, so as to minimize the overall logistics costs. Taking this rationale a step further, when all suppliers and plants share information with each other, then the many-to-many () Milk-run transportation described in [1] is achieved; that is, trucks pick up products from one or several suppliers and deliver them to one or several plants. Compared with Toyota’s and Milk-run system, this integrated method can significantly reduce the transportation costs to a larger extent for the entire logistics system.

The problem this paper focuses on is based on the classical vehicle routing problem (VRP). The generic VRP and many of its practical occurrences were studied intensively in the literature (Raff [4], Laporte and Nobert [5]). Then some extensive versions of VRP, such as vehicle routing with time windows (VRPTW) [6, 7] and multidepot vehicle routing problems (MDVRP) [8, 9], have been widely studied. These achievements of VRP have been already applied in the Milk-run system. Chuah and Yingling [10] and Jiang et al. [11] studied the common frequency routing problem in the Milk-run system and minimized transportation and inventory costs using column generation combined with the tabu search algorithm. Using genetic algorithm and a robust optimization approach, Sadjadi et al. [2] and Jafari-Eskandari et al. [12, 13] solved general and specific Milk-run issues with time windows and inventory uncertainty. Zhang and Li [14] analyzed the impact of the Milk-run method on supply chains, the conditions for using this method, and the changes in cost before and after adopting the Milk-run plan for a company. They also optimized the Milk-run mode by classifying containers into various carrying capacities based on the integral slice algorithm model. Yun et al. [15] analyzed the relationship between inventory and transportation, established an integrated transportation and inventory optimization model based on Milk-run, proposed genetic algorithm to solve the vehicle scheduling problem, applied a stepwise iterated algorithm to balance inventory and transportation costs, and eventually minimized the total costs for the entire Milk-run logistics system. Du et al. [16] investigated the parameter settings of a real-time vehicle-dispatching system for consolidating Milk-runs. Ma and Sun [17] employed a mutation ant colony algorithm to solve the Milk-run vehicle routing problem with the fastest completion time based on dynamic optimization. However, all of these studies targeted the multiple suppliers to one plant () problem. Effectually, none of them can be directly applied to Milk-run systems with multiple suppliers and plants. Given the status quo, this paper proposes a solution to the Milk-run routing problem involving multiple suppliers and plants ().

A number of researchers have already studied many-to-many transportation problems. The Pickup and Delivery Problem (PDP) is one of the research areas in transportation issues. In general PDP, there is no restriction on the choice of pickup point to provide products for each specified delivery point, as long as the demand of the delivery point is satisfied. However, in practical terms, for inbound logistics systems, a plant’s demand for each supplier is always clearly determined because the plants will place specific orders with each supplier. Thus, the general PDP is not entirely consistent with the actual inbound logistics systems. Furthermore, the set of variables employed to represent the commodities in PDP [18, 19] is not suitable for problems in practical terms. Further, because, in many manufacturing industries, such as auto manufacturing, the number of parts used in an individual assembly plant is more than 3,000 [20], the problem will become more complex if the set is defined based on the types of products. In light of the discussions above, PDP is not applicable for the type of logistics systems focused on in this paper. Therefore, an effective mathematical model with determined demand (hereafter, indicating the total demand of plant for supplier regarding one or more products, which will be described in more detail in Section 2) should be constructed.

The cross-docking system is another transportation mode in inbound logistics involving multiple suppliers and plants. Several scholars have already studied vehicle routing problems in this system. For example, Berman and Wang [1] considered the problem of selecting the appropriate distribution strategy (direct shipping and cross-docking) for delivering a family of products from a set of suppliers to a set of plants so that the total transportation, pipeline inventory, and plant inventory costs were minimized. Lee et al. [21] optimized vehicle routing schedules by planning and scheduling vehicle routes during the pickup and delivery stages of cross-docking. Musa et al. [22] employed the ant colony algorithm to solve the transportation problem of the cross-docking network. Miao et al. [23] employed a hybrid genetic algorithm to solve the multiple crossdocks problem with supplier and customer time windows. Mousavi and Tavakkoli-Moghaddam [24] minimized the total cost by simultaneously taking cross-dock’s location and route planning into consideration. The cross-docking transportation mode has been used extensively in long-distance transportation. However, it is not applicable for short-range transportation [2, 25]. In the cross-docking system, a vehicle must first pick up goods from suppliers and then transport them to the cross-dock, where inbound materials are sorted, consolidated, and stored until the outbound shipment is complete and ready to ship and finally deliver the goods to each plant. This increases the distance and the time needed for each transportation task and, thus, unnecessary pipeline inventory costs. In contrast, the Milk-run system has no transfer system similar to the cross-dock because all goods are directly delivered to the corresponding plant after they are picked up from each supplier. Consequently, all processes, including loading and unloading, are completed in the vehicle within a shorter time, which makes the Milk-run mode more suitable for short-distance transportation. Therefore, it is necessary to develop an effective method to solve the Milk-run routing problem involving multiple suppliers and plants in neighboring regions.

From the discussion above, this paper contributes to solving the Milk-run routing problem involving multiple suppliers and plants in neighboring regions, that is, the many-to-many Milk-run routing problem (MM-MRP). In addition, we also consider the pipeline inventory cost caused by damage to products during the transportation process. It is necessary to consider the pipeline inventory costs because in some specific situations, some products, materials, and commodities, such as parts of some precision instruments, fragile parts, refrigerated foods, and some primary products, are more likely to be damaged with longer transportation time and more frequent loading and unloading times. On the basis of the above discussion, we firstly formulate the many-to-many Milk-run routing problem with pipeline inventory cost (MM-MRP-PIC) and extend the vehicle routing problem (VRP) model to MM-MRP-PIC model.

Because VRP is NP hard, MM-MRP-PIC, an extension of VRP, is surely NP hard. Exact algorithms are difficult to find the optimal solutions within a reasonable time. Many scholars have developed heuristic algorithms to solve the VRP. Alfa et al. [26], Osman [27], and Van Breedam [28] applied simulated annealing to solve VRP and verified its effectiveness. Tabu search was developed by Osman [27] and Gendreau et al. [29] to solve VRP. Furthermore, genetic algorithms [30, 31], ant colony optimization [32–35], and particle swarm optimization [36, 37] are also developed to solve VRP successfully. However, some evolutionary algorithms, such as genetic algorithm, are inefficient to solve MM-MRP-PIC because the problem is so complex that they will spend much time to tackle illegal solutions generated during iterative process. In addition, ant colony can solve the problems whose objective function is only involved with transportation cost while it is difficult to tackle the pipeline inventory cost. Particle swarm optimization is also difficult to be designed for this problem because so various mutations should be considered that the PSO formulas cannot define these mutations together. However, simulated annealing can easily tackle illegal solutions and various mutations simultaneously considering the pipeline inventory cost. Therefore, we employ simulated annealing and make some improvements to solve MM-MRP-PIC. The algorithm is two-stage simulated annealing with limited search scope (TSSALSS). The effectiveness of this algorithm is verified via thirty numerical examples. Compared with traditional SA, TSSALSS not only significantly reduces the computing time, but also improves the quality of the solutions.

The remainder of this paper is organized as follows. In Section 2, we describe the problem and propose the MM-MRP-PIC model. Section 3 is devoted to the solution method (TSSALSS) for this model. Thirty numerical examples are presented to demonstrate the efficacy of our algorithm in Section 4. In Section 5, we present two case studies to demonstrate the validity of the many-to-many transportation mode. Finally, we conclude this paper in Section 6.

#### 2. Formulation

##### 2.1. Problem Description

This paper focuses on logistics systems comprising multiple suppliers and multiple plants in certain neighboring regions. In such a system, every supplier produces one or more parts, and each of the plants places corresponding orders to suppliers. The total demand of plant for the products of supplier () can be indicated by the ratio of the products’ total volume to the truck capacity. The transportation process is organized as follows. A certain number of trucks, each of which has its own flow, are first scheduled for the shipping task. In each flow, the truck sets off from a plant, picks up products at a number of suppliers, delivers those products to corresponding plants, and finally returns to the departure point upon finishing the transportation task. This paper targets route planning for this kind of logistics systems, with the objective of minimizing transportation and pipeline inventory costs for the entire system.

The following three assumptions are made to this problem.

*Assumption 1. *Product quantities cannot be split and each vehicle’s transportation frequency is one.

*Assumption 2. *There are enough vehicles at each plant to complete the transportation task.

*Assumption 3. *All vehicles employed for transportation are of the same type and have cargo capacity .

##### 2.2. Notations

*(1) Constants*. Consider the following. : set of suppliers. : set of plants. : set of nodes, including all suppliers and plants. : set of vehicles employed to complete the transportation task, with each vehicle corresponding to a path; that is, is also the set of flows. : transportation cost for shipping one truckload of products from node to node ,. : the demand of plant for the products of supplier . : pipeline inventory cost per unit time per unit of product . : time spent in transportation from node to node . : capacity of each vehicle. : loading or unloading time per unit of product.

*(2) Decision Variables*. Consider the following:

*(3) Variables That Are Dependent on the Decision Variables*. Consider the following: : pipeline inventory cost for products on vehicle per unit time from node to node ,,. : time spent when vehicle loads products at supplier ,. : time spent when vehicle unloads products at plant ,.

##### 2.3. The Proposed Mathematical Model for MM-MRP-PIC

Consider the following:where subject to

The first part of the objective function is direct transportation cost; the second, third, and fourth parts are pipeline inventory costs. The pipeline inventory costs depend on the road transportation time and residence time for freight loading or unloading at each point. An immediate idea for calculating the pipeline inventory costs for products is multiplying the pipeline inventory costs for products per unit time and the corresponding pipeline transportation time. The pipeline transportation time of is the period starting from the vehicle’s departure time at supplier and ends at the arrival time at plant . It includes transportation time on the way, loading time at various other suppliers, and unloading time at other plants in the same flow (we ignore the loading time of at supplier and the unloading time at plant because the pipeline inventory cost of caused by the loading time at supplier and the unloading time at plant are constant and independent of the decision variables). Although this idea is easy to understand, it is difficult to describe using mathematical models. Therefore, this paper divides the above process into three stages and calculates the respective cost for each part.(1)Pickup stage: the pipeline inventory cost when vehicle picks up products at supplier and goes to the next supplier can be represented by the product of the costs per unit time for the freight () and the sum of the transportation time on road () and the pickup time at supplier (); that is, . is proportional to the loading volume of vehicle at supplier , and the scale factor is .(2)Stage between the final pickup point and the first delivery point: the pipeline inventory cost is determined by the product of the costs per unit time for the freight () and the time spent on transportation (); that is, .(3)Delivery stage: the pipeline inventory cost when vehicle unloads products at plant and goes to the next plant can be represented by the product of the costs per unit time for the freight () and the sum of the transportation time on road () and the unloading time at (); that is, . is proportional to the unloading volume of vehicle at plant , and the scale factor is also assumed to be .

The following is a detailed description of constraints (4)–(17). Equation (4) signifies that the transportation task of vehicle cannot exceed its capacity. Equation (5) means that as long as plant has a demand for products from supplier , a vehicle should be arranged to facilitate the transportation task. Equations (6)–(9) ensure that the path of each vehicle forms a loop. In classical VRP, there can only be one vehicle flow into and out of each node. In contrast, in this paper, more than one truck can go into and out of each node because multiple vehicles can be employed to execute different tasks at the same node. However, every vehicle that flows into a node must also leave that node; that is, the inflow and outflow for each vehicle at each node must be the same. In each flow, a vehicle must first pick up products from suppliers, then deliver goods to corresponding plants, and finally return to the departure point. To ensure that this process is conducted in an orderly manner, (10)-(11) restrict each vehicle to one single route from the plants to the suppliers, and vice versa. Equation (12) ensures that the solution does not contain any illegal isolated subtour. Equation (13) signifies that the pipeline inventory cost from the first plant to the starting supplier is zero. This is obvious because the vehicle sets off from a plant; therefore, it is empty during this period. Equation (14) represents the pipeline inventory cost per unit time for vehicle from node () to node () in the pickup stage, which is the sum of the pipeline inventory cost per unit time for vehicle before node and that of the newly loading freight. Equation (15) is similar to (14); it represents the pipeline inventory cost per unit time for vehicle from node () to node () in the delivery stage. Equations (16)-(17) signify the range of decision variables.

#### 3. Approach

As an extension of VRP, MM-MRP-PIC is NP hard. As a result, a two-stage simulated annealing algorithm with limited search scope (TSSALSS) is developed in this paper. Geng et al. [38] employed an adaptive simulated annealing algorithm with greedy search to solve the traveling salesman problem (TSP) and verified its effectiveness. In this paper, the process of limiting the search scope in TSSALSS is similar to the greedy search in [38], but we use a much simpler method to limit the search scope instead of greedy search which requires calculating the objective function frequently. A detailed solution process for TSSALSS is given below.

Varanelli and Cohoon [39] stated that conventional SA can be initiated at a low temperature in an attempt to improve the heuristic solution. Thus, a faster heuristic (the first stage) is used to replace the SA actions occurring at the highest temperatures in the cooling schedule. In the first stage, an initial satisfactory solution is constructed in the following three steps: (1) Determine an initial feasible task allocation plan. (2) Minimize the total number of vehicle service nodes (calculated using formula (19) or (20)) by adjusting the task allocation for each vehicle. (3) Employ the best insertion heuristic algorithm to determine the route for each vehicle, which is achieved when the transportation cost is relatively optimal. In the second stage, the solution obtained in the first stage is improved by the* Simulated Annealing (SA) algorithm*. In order to improve the search efficiency, in this paper, the scope of the search in the algorithm is limited to a specific range. That is, in the search process, all feasible solutions are subject to the constraint that the total number of vehicle service nodes be not greater than , in which is obtained in the first stage and is a number slightly larger than one; we find to be the most appropriate. Because it is much simpler to calculate (the two neighbor solutions’ difference in the total number of vehicle service nodes, ) than to calculate (the two neighbor solutions’ difference in the objective function value, ), the search process will be much more efficient with the limited search scope. This method can not only reduce the unwanted time caused by searching the overall feasible solution domain, but also prevent the global optimal solution being overlooked by appropriate expansion of the search scope adjusted by coefficient . The structure of the algorithm is shown in Figure 1. The following is a detailed description of the algorithm.