Mathematical Problems in Engineering

Volume 2015, Article ID 304981, 11 pages

http://dx.doi.org/10.1155/2015/304981

## Three-Phase Methodology Incorporating Scatter Search for Integrated Production, Inventory, and Distribution Routing Problem

Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia

Received 21 January 2015; Revised 2 June 2015; Accepted 8 June 2015

Academic Editor: Matteo Gaeta

Copyright © 2015 Noor Hasnah Moin and Titi Yuliana. 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

This paper proposes the use of scatter search metaheuristic to solve an integrated production, inventory, and distribution routing problem. The problem is based on a single production plant that produces a single product that is delivered to *N* geographically dispersed customers by a set of homogenous fleet of vehicles. The objective is to construct a production plan and delivery schedule to minimize the total costs and ensuring each customer’s demand is met over the planning horizon. We assumed that excess production can be stored at the plant or at customer’s sites within some limits, but stockouts due to backordering or backlogging are not allowed. Further testing on a set of benchmark problems to assess the effectiveness of our method is also carried out. We compare our results to the existing metaheuristic algorithms proposed in the literature.

#### 1. Introduction

Two substantial components to improve the timeliness and consistency of delivery are integrated logistics management and product availability. Integrating production and distribution decisions can have a significant impact on setup, holding, and delivery costs. In general, the problem of coordinating production and transportation is called the production-inventory-distribution routing problem (PIDRP) [1]. The research on PIDRP involves some different areas such as vehicle routing, production, and inventory [2]. PIDRP is most similar to the inventory routing problem and periodic routing problem, because it requires multiple visits to each customer’s sites over the planning horizon.

Chandra and Fisher [3] studied about a single plant, multicustomers in a multiple period by comparing two different methods to investigate the value of coordinating production and distribution. The problem is to minimize the total cost of production, transportation, and inventory. Two different alternative solution approaches are presented to manage this operation, one in which the production scheduling and vehicle routing problems are solved separately and another in which they are coordinated within a single model. The computational results reported a reduction in total operating cost from coordination ranged from 3% to 20%.

Lei et al. [1] presented multifacility, heterogeneous fleet version of the PIDRP that was motivated by a chemical manufacturer with international customers. The problem is to coordinate the production, inventory, and transportation schedules to minimize the total cost over the planning horizon. They proposed a two-phase methodology, by solving a restricted version of the problem by eliminating the routing constraints in phase one and proposed a routing heuristic based on an extended optimal partitioning procedure in phase two to transform the less-than transporter load assignments obtained in phase one into more efficient delivery schedules.

Boudia et al. [4] proposed integer linear model to solve the PIDRP but it failed to solve the large instances, and then a Greedy Randomized Adaptive Search Procedure (GRASP) is developed to tackle the production and distribution decisions simultaneously. Another two improved versions using either a reactive mechanism or a path-relinking process embedded in GRASP are also developed and the results are better than the GRASP alone.

Boudia and Prins [5] presented an alternative method to solve the PIDRP. They solved the problem with memetic algorithm population management (MAPM) and compared with a two-phase heuristics and GRASP. Computational testing showed that MAPM can tackle large instances and gave better results compared to two-phase approach and GRASP.

Bard and Nananukul [2, 6] developed a mixed integer programming (MIP) model aimed at minimizing production, inventory, and delivery costs. The objective of the problem is to minimize the total cost over the planning horizon without incurring any stockouts at the customer sites. The problem includes a production plant, multiple customers with time varying demand, a finite planning horizon, and a fleet of homogenous vehicles. They developed a three-phase methodology centered on tabu search and using allocation model to find a good initial solution. They also combined the features of reactive tabu search algorithm and branch-and-price algorithm by taking efficiency of the tabu search heuristic and the precision of the branch-and-price algorithm. Nananukul [7] extended the idea by improving the clustering of the customers by creating adaptive core clusters in the reactive tabu search algorithms which are used in the clustering process instead of the original data points thus enabling the algorithm to be efficient. Unfortunately the detailed computational results were not given.

Armentano et al. [8] presented two tabu search variants for PIDRP. The first variant involves construction of a short-term memory and integrating a path relinking procedure, while another one incorporates a longer term memory and integrate the first variant. The algorithms are tested on generated instances and on instances taken from Boudia et al. [4] which involved a single product. Computational results showed that the two variants of tabu search yield good tradeoffs between solution quality and computational time and successfully outperformed Boudia and Prins [5] and Bard and Nananukul [2, 9] in all instances.

Adulyasak et al. [10] improves upon the results of Armentano et al. [8] by proposing adaptive large neighborhood search heuristic to take care of binary variables representing the setup and routing variables whilst the continuous variables associated with inventory, production, and quantities delivered are handled by solving a network flow problems. The results outperformed all other know heuristics for PIDRP.

Two inventory replenishment policies, order-up-to level and maximum level were considered in Adulyasak et al. [11] for inventory routing problem and PIDRP. By using adaptive large neighborhood search to obtain the initial solutions and the branch-and-cut algorithm were proposed to solve the different formulations. The authors managed to solve to optimality relatively small instances, 50 customers, three periods, and three vehicle on parallel computers.

Torabi et al. [12] proposed a two-step solution approach to solve an integrated multisites production planning, procurement, and distribution plans. The first step restricts the vehicle routings into direct shipment and solved the full model as mixed integer problem. Scatter search algorithm is employed in the last step by solving the associated consolidation problem in order to improve the solutions. Computational testings showed that this method gives the comparable or improved solution compared to the best solution for original model for 8 out of 10 cases and failed to get a better solution for 2 other problems. However the algorithms were not performed on benchmark instances.

We refer the readers to a review of formulations and solution algorithms in PIDRP by Adulyasak et al. [13] for the state of the art.

In particular, the PIDRP considered in this paper is very similar to that of Bard and Nananukul [2, 6] which involves a production plant, multiple customers with time varying demand, a finite planning horizon, and a fleet of homogenous vehicles that delivers the product from the production plant to the customers’ sites. Excess production can be stored either at the plant or at the customer sites within some limits, but inventory cannot be transferred between sites and stockouts are not permitted. The objective is to minimize a combination of setup, holding, and routing costs both at the production facility and at customers, without incurring any stockouts at the customer sites. We also propose a three-phase methodology and our algorithm differs from Bard and Nananukul [2, 6] in Phase 2 and Phase 3. In Phase 2, we employ the Giant Tour procedure [14], sweep, and savings algorithms which have been shown to be efficient routing procedures and Phase 3 introduces the scatter search algorithm, embedding the new inventory updating mechanism as the improvement methodology.

The remainder of this paper is organized as follows. In Section 2, we present the description of PIDRP and its mathematical formulation. In Section 3, the three-phase scatter search method we employed to solve the PIDRP is discussed in detail and followed by the presentation of computational experiments and results in Section 4. Finally, conclusions are drawn in Section 5.

#### 2. Problem Description and Mathematical Formulation

We consider a production, inventory, and distribution routing problem similar to the one proposed by Bard and Nananukul [2, 9]. It consists of a single production plant that produces a single product and distributes it to a set of customers with nonnegative demand in period where and and limited number of items can be produced in period and a limited number of inventories can be stored by incurring unit holding cost at production plant. A fleet of homogeneous vehicles with capacity delivers the items to the customer’s sites, and each vehicle can make at most one trip per period. A limited amount of inventory can be stored at customers’ sites with unit holding cost of , and each customer can only be visited at most once per period.

Furthermore, it is assumed that at the end of planning horizon all inventories (both at the production facility and at customer’s site) are required to be zero. The objective is to construct a production plan and delivery schedule which minimizes production, inventory, and distribution costs while fulfilling each customer’s demand requirement. Let denote the production plant (depot) and be a set of customers. is a decision variable denoting the amount to be delivered to customer in period . The traveling distance (cost) from customer to customer is denoted by . equals 1 if there is a route from customer to customer and 0 otherwise. represents the total quantity on a vehicle before delivering to customer in period .

The maximum inventory level for production plant is denoted by and for customers’ sites is denoted by . In this study, the initial inventories at customers’ sites are assumed to be zero.

The integrated production, inventory, and distribution routing problem (PIDRP) can be formulated as follows [2]:subject towhere and .

The objective function comprises the transportation costs, production setup costs, holding costs at the warehouse and holding costs at the customer sites. Equations (2) and (3) represent the inventory flow balance equations for production facility and customers, respectively. Equation (5) limits production on period to the capacity of the factory, and (6) allows production in period 0. The total amount available for delivery on period is limited by the amount of inventories at the factory on period as formulated in (4). Equation (11) limits the amount delivered to each customer and (7) ensures that if customer is serviced on period , then it must have a successor on its route, while route continuity is enforced by (8). Equation (9) limits the number of vehicles that leaves the factory at period , and (10) keeps track of the load on the vehicles. We note that Adulyasak et al. [10] use slightly different approach in the formulation.

In this study, extending the idea from Bard and Nananukul [6] we propose a three-phase methodology to solve the PIDRP. Our algorithm differs from Bard and Nananukul [6] in Phase 2 and Phase 3. Starting from Phase 1 that solves the allocation model which is the simplified version (relaxed) of the model to determine the amount to be delivered to each customer in each period, Phase 2 routes the customers using the Giant Tour procedure [14], sweep, and savings algorithms to determine the delivery routes for each period. In phase 3, we develop scatter search method by creating composite decision rules and surrogate constraints to improve the initial solutions. We also incorporate the inventory updating based on the forward and backward transfer.

We identify the initial solution in Phase 1 by solving the allocation model as a mixed integer programming to get a set of feasible allocations. The routing variables and routing constraints (7)–(10) are removed and aggregated vehicle capacity constraints are introduced to the allocation model. Since we already deleted the routing constraints in the allocation model, we need an alternative representation for the cost term to determine the approximated cost which is needed to make a delivery to customer in period . represents the fixed cost of making delivery to customer on period , denotes the variable cost of delivering one item to customer in period , and is valued to 1 if delivery is made to customer in period and 0 otherwise.

As in Bard and Nananukul [2], we divide the problem into two cases, for problem instances with and for instances with . Since all the instances we considered are , we introduce the variable cost term , where is approximated by the cost of making a delivery to customer directly from the depot divided by the total demand of customer in period .

The allocation model of the PIDRP is formulated as follows:

Additional new constraints are as follows:

Allocation model modifies the objective function in the full model and replaces the routing variables with additional parameter to represent the costs (the second term in ). The term is the approximated transportation cost replacing the term , the actual distance (transportation) cost. Allocation model also uses the same constraints but eliminating the routing constraints (constraints (7)–(9)) and adding additional two constraints. Equation (14) limits the total amount that can be delivered on period to a fixed percentage of the total transportation capacity. The parameter testing by Bard and Nananukul [2] showed that the percentage value of 80% always yielded feasible solutions. Additional variable is included in constraint (15) to keep track of whether customer receives a delivery in period . Constraint (15) was a modification of constraint (11) by replacing the routing variables with .

#### 3. Scatter Search

The scatter search metaheuristic has successfully been applied to a widespread variety of vehicle routing problems. Corberán et al. [15] proposed a scatter search to solve a real-life problem with multiple objectives. Two different heuristics were used to construct the initial trial solutions in the scatter search. Two simple exchange procedures, insertion and swap, are used to improve the solutions. The combination method is based on a voting scheme. The algorithms tested on real data show that scatter search can solve the practical problem efficiently. A more recent application of scatter search is by Mota et al. [16], who presented a scatter search for vehicle routing problem with split demands. Local search is adopted as the improvement method. Four kinds of critical clients are defined to produce new solutions. The algorithm was tested on a set of benchmark instances and it was found that scatter search algorithm always produces the least number of vehicles compared the tabu search developed by the authors.

Scatter search is an evolutionary metaheuristic that operates on a set of solutions, which the scatter search literature refers to as the reference set (*Refset*). The evolution of the* Refset* is achieved by way of combining reference solutions to yield trial solutions with combination of attributes not present in the previous set of solutions. The Refset is a collection of “good” solutions found during the search, where the meaning of “good” is not limited to quality as measured by the objective function value. For instance, a solution may be good because it provides diversity with respect to other solutions in the reference set. In fact, some implementations of scatter search divide the* Refset* into two subsets, consisting, respectively, of solution quality and diversity. Scatter search was first introduced by Glover [17]. Glover made a template in a version customized for nonlinear optimization problems with continuous variables. Laguna and Marti [18] published the first book on scatter search, containing introductory tutorials and advanced techniques such as the use of memory and path relinking.

The scatter search terminology that is used in this paper is similar to Laguna and Marti [18]. The algorithm is made up of several distinct steps: A diversification generator, an improvement method, a reference set update, a subset generation method which operates on the reference set in order to produce a subset of its solutions as a basis for creating combined solutions, and a solution combination method which transforms a given subset of solutions produced by the subset generation method into one or more combined solutions vectors.

The scatter search procedure stops when a termination criterion—either the maximum number of iterations,* MaxIter*, is reached, or the reference set does not change, or improvement does not warrant further iterations.

The scatter search algorithm can be formally stated as shown in Algorithm 1.