Introduction

The distribution planning decision (DPD) is one of the most comprehensive strategic decision issues that need to be optimised for the long-term efficient operation of a whole supply chain (SC). The DPD involves optimising the transportation plan for allocating goods and/or services from a set of sources to various destinations in a supply chain. An important strategic issue related to the design and operation of a physical distribution network in a supply chain system is the determination of the best sites for intermediate stocking points, or warehouses. The use of warehouses provides a company with flexibility to respond to changes in the marketplace, and can result in significant cost savings due to economies of scale in transportation or shipping costs.

The major task of DPD is the determination of distribution costs, customer service level (safety stock holding), resource (warehouse space) utilisation, and the total delivery time, with reference to multiple warehouse capacities and uncertain forecast demands.

In previous studies, both deterministic and stochastic customers' demands have been considered, but more attention has been paid to the deterministic cases.

Generally, the applied constraints in modelling DPDs are the capacity limitation and single source constraints. In some cases, in addition to the capacity constraints, some other restrictions on the number of covered demands and the service levels of the warehouses are also defined.

Recently some authors have incorporated inventory control decisions into DPD models. For example, Miranda & Garrido, Daskin et al., and Shen et al. present similar versions of the DPD model incorporating the inventory control decisions.

Erlebacher & Meller present a location-inventory model for designing a two-level distribution system serving continuously represented customer locations. They develop a stylised analytical model to provide some intuition and basic results for the problem. The stylised model also motivates bounds for the problem, which they use to develop a heuristic. They show that the heuristic performs very well on test problems that considered variation in customer demand and spatial dispersion.

In these works, the ordering decisions are based on the classic economic order quantity (EOQ) model, and a normal distribution is assumed for the demand pattern.

Additionally, researchers have developed various methods to solve multi-objective DPD problems. Liang develops an interactive fuzzy multi-objective linear programming method for solving the fuzzy multi-objective DPD problem with piecewise linear membership functions. Selim & Ozkarahan suggest an interactive fuzzy goal programming (FGP) for the supply chain distribution network design. The goal of their model is to select the optimum numbers, locations, and capacity levels of the plants and warehouses to deliver the products to the retailers at the least cost while satisfying the preferred customer service level.

In most of the past research studies like Gourdin et al., Jayaraman, Pirkul & Jayaraman, and Tragantalerngsak et al., one major drawback is that they limit the number of capacity levels available to each facility to just one. However, in real case studies, there are usually several capacity levels to choose for each facility. This flexibility in capacity levels makes the problem more realistic and, at the same time, more complex to solve. Another major drawback in some previous studies is that they limit the number of opening facilities to a pre-specified value. Moreover, these studies fail to describe how this value can be determined in advance. Amiri represents a significant improvement over previous research by presenting a unified model of the problem that includes the numbers,- locations, and capacities of both warehouses and plants as variables to be determined in the model. In addition, he develops an efficient heuristic solution procedure based on Lagrangean Relaxation (LR) of the problem, and reports extensive computational tests with up to 500 customers, 30 potential warehouses, and 20 potential plants.

In this paper we develop a new non-linear multi-objective DPD model, consisting of one manufacturer and multiple distribution centres (warehouses), that integrates the location/allocation and distribution plans. In this model, to improve over previous research, we also incorporate tactical/operational decisions - such as inventory control decisions -into the DPD problem. Then we propose an efficient goal programming approach to solve the developed model. We consider three important objective functions:

  • investment in opening distribution centres/warehouses (location costs);
  • total cost of logistics, such as the costs of transporting products from the plant to the opened warehouses, and from the opened warehouses to the retailers, and holding costs (inventories and safety stocks); and
  • total delivery time.

The problem is particularly motivated by consulting work that was done for a large food industry company owning one production site and multiple distribution centres. Since the transportation cost constitutes the main part of the unit cost, and since delivery time and limited storage capacity are also very important, implementing the distribution planning system for the DCs' location and inventory control decisions is of particular interest.

The paper has two important applied and theoretical contributions. First, it presents a new comprehensive and practical, but tractable, optimisation model for distribution network designing. Second, it introduces a novel solution procedure for finding more non-dominated and efficient compromise solutions to a stochastic multi-objective mixed-integer programme. In our literature survey we have found a lack of studies in this field, which is understandable, given that large mixed integer programming is known to be complex even when all data is certain and precise.

The remainder of this paper is organised as follows. In Section 2 we consider a summary of key challenges in agro-food supply chains. In Section 3 we define our notation, state our assumptions, and propose a new multi-objective stochastic non-linear program (MOSNLP) for the proposed DPD problem. After applying appropriate strategies for converting the stochastic model into a multi-objective nonlinear model (MONLP), in Section 4 we propose a novel interactive payoff approach to solve this MONLP and find an efficient compromise solution. The proposed model and the solution method are validated through numerical tests in Section 5. The data for these numerical computations have been inspired by a real life food industrial case study, as well as randomly generated data. Concluding remarks on computational results and further research directions are the subject of Section 6.