هزینه کلی لجستیک در انتخاب تامین کننده تحت شرایط منابع متعدد، معیارهای متعدد و محدودیت ظرفیت

Little attention is given in the literature to decisions on the appropriate selection of suppliers, and on assigning order quantities to these suppliers, in the case of multiple sourcing, with multiple criteria and with suppliers’ capacity constraints. Only a few mathematical programming models to analyse such decisions have been published to date, and these have tended to consider only net price as the cost of purchasing, although the costs of transportation, ordering and storage may be significantly important to the decision. In this paper a mixed integer non-linear programming model is presented to solve the multiple sourcing problem, which takes into account the total cost of logistics, including net price, storage, transportation and ordering costs. Buyer limitations on budget, quality, service, etc. can also be considered in the model. An algorithm is proposed to solve the model, and the model is illustrated using a numerical example.

In most industries the cost of raw materials and component parts constitutes the main cost of a product, such that in some cases it can account for up to 70% [1]. In high technology firms, purchased materials and services represent up to 80% of total product cost [2]. Thus the purchasing department can play a key role in an organization's efficiency and effectiveness because it has a direct effect on cost reduction, profitability and flexibility of a company. Selecting the right suppliers significantly reduces the purchasing cost and improves corporate competitiveness, which is why many experts believe that the supplier selection is the most important activity of a purchasing department [3] and [4]. In spite of the importance of supplier selection problems only a few articles have addressed the decision making. Weber and Current [5] stated that only 10 articles analysed the problem up to the time of their review. A comprehensive review of the articles which have addressed the problem can be found in Ghodsypour [6] and Ghodsypour and O’Brien [7]. The most important articles are described below. Moore and Fearon [8] stated that price, quality and delivery are important criteria for supplier selection and they explained that linear programming can be applied to this decision making. They also discussed other applications of computer technology in the purchasing area. Gaballa [9] is the first author who applied mathematical programming to vendor selection in a real case. He used a mixed integer programming model to formulate this decision making problem for the Australian Post Office. The objective of this programming is to minimize the total discounted price of allocated items to the vendors, under constraints of vendors’ capacity and demand satisfaction. Anthony and Buffa [10] developed a single objective linear programming model to support strategic purchasing scheduling (SPS). The linear model minimizes total cost by considering limitations of purchasing budget, vendor capacities and buyer's demand. Price and storage cost are included in the objective function. The costs of ordering, transportation and inspection are not included in the model. Buffa and Jackson [11] presented a multi-criteria linear goal programming model for supplier selection. In this model two sets of factors are considered: (1) supplier attributes, which include quality, price, service experience, early, late and on-time deliveries and (2) the buying firm's specification, including material requirement and safety stock. Bender et al. [12] applied single objective programming to develop a commercial computerized model for vendor selection at IBM. They used mixed integer programming, to minimize the sum of purchasing, transportation and inventory costs by considering multiple items, multiple time periods, vendors’ quality, delivery and capacity. In this model quantity discount also is included. No mathematical formulations were presented and they did not indicate the kind of discount. Narasimhan and Stoynoff [13] applied a single objective, mixed integer programming model to a large manufacturing firm in the Midwest, to optimize the allocation procurement for a group of vendors. The objective of this model is to minimize the sum of the shipping and the penalty costs. The model constraints are related to vendors’ production capabilities and demand. Kingsman [14] stated that one of the most important problems which has received little attention from OR practitioners is the purchasing of materials whose prices are continually fluctuating in a stochastic manner over time. He discussed conceptually linear programming and dynamic programming as tools for purchasing raw materials with fluctuating prices. Turner [15] presented a single objective linear programming model for British Coal. This model minimized the total discounted price by considering the vendor capacity, maximum and minimum order quantities, demand, and regional allocated bounds as constraints. Pan [16] proposed multiple sourcing for improving the reliability of supply for critical materials, in which more than one supplier is used and the demand is split between them. Most purchasing managers agree that buying from more than one vendor will protect the buying firm in the case of shortages. Pan [16] used a single objective linear programming model to choose the best suppliers, in which three criteria are consiered – price, quality and service. The total cost is taken into account as an objective function and quality and service are considered as constraints. Sharma et al. [17] proposed a non-linear, mixed integer, goal programming model for supplier selection. They considered price, quality, delivery and service in their model, in which all criteria are considered as goals. The cost goal is decreased in relation to the increase in purchased quantity and is raised in relation to the increase in quality level. Seshadri et al. [18] developed a probabilistic model to represent the connection between multiple sourcing and its consequences, such as number of bids, the seller's profit and the buyer's price. Only one criterion, cost, is considered in this model and the authors stated that the user should transfer the other criteria such as quality, delivery, etc., into an equivalent price. Benton [19] developed a nonlinear program and a heuristic procedure using Lagrangian relaxation for supplier selection under conditions of multiple items, multiple suppliers, resource limitations and quantity discount. The model objective is to minimize the sum of purchasing costs, inventory carrying costs and ordering cost. Storage and investment limitations are considered as constraints. Hong and Hayya [20] analysed the JIT purchasing environment. As the need for small lots is an important issue in this system, they discussed splitting a large order quantity into multiple deliveries or multiple suppliers to reduce the lot size. Their main objective was reducing the cost, hence they solved the problem by considering two important assumptions, the first is that the ordering cost of N suppliers is equal to, or less than, N times one supplier's ordering cost. The second assumption is that the purchasing price must be less than a fixed value. As they solved the problem for the special case, it cannot be used for general situations Chaudhry et al. [21] developed linear and mixed integer programming for supplier selection. In their model price, delivery, quality and quantity discount are included. The objective of the model is to minimize aggregate price by considering both cumulative and incremental discounts. Quality and delivery are included as constraints. Weber and Current [5] used multiobjective linear programming for supplier selection to systematically analyse the trade-off between conflicting factors. In this model aggregate price, quality and late delivery are considered as goals, and two sets of constraints are taken into account: (1) systems’ constraints, which are defined as the constraints which are not directly under the control of the purchasing managers such as vendor capacities, demand satisfaction, minimum order quantities established by the vendors and the total purchasing budget; and (2) policy constraints, including maximum and/or minimum order quantities purchased from a particular supplier, and the maximum and/or minimum number of vendors to be employed. Current and Weber [22] proposed that mathematical constructs of facility location modeling can be applied to supplier selection. They did not solve any special vendor problem but they showed the similarities between the vendor selection problem and facility layout models. The complexity of both location models and supplier selection problems indicates that fitting these two methods together cannot be easy. Rosenthal et al. [23] developed a mixed integer programming model to solve the vendor selection with bundling, in which a buyer needs to buy various items from several vendors whose capacity, quality and deliveries are limited and who offer bundled products at discounted prices. They used single objective programming and considered price, quality, delivery and suppliers’ capacity as criteria in their model. Ghodsypour and O’Brien [7] developed a decision support system (DSS) for reducing the number of suppliers and managing the supplier's partnership. They used integrated analytical hierarchy process (AHP) with mixed integer programming and considered suppliers’ capacity constraint and the buyers’ limitations on budget and quality etc. in their DSS. Ghodsypour and O’Brien [24] proposed a model to deal with supplier selection, multiple sourcing, multiple criteria and discounted price. They considered the effects of limitations on budget, quality and suppliers’ capacity. Ghodsypour and O’Brien [25] developed an integrated AHP and linear programming model to help managers consider both qualitative and quantitative factors in their purchasing activity in a systematic approach. They proposed an algorithm for sensitivity analysis to consider different scenarios in this decision making. Most of these articles considered net price as the cost of logistics in their models, although the storage, transportation and ordering costs are also important in this decision making [26] and [27]. Only two articles [19] and [20] involved ordering and storage costs in their models. Benton [19] did not consider the supplier's capacity and quality constraints. Hong and Hayya [20] discussed reducing lot size in the JIT environment, which is different from a general model for the supplier selection problem. These two articles considered single objective model in their work, which consider one criteria as the objective and the other criteria as the constraint in the programming. In this situations the criteria which are considered as constraints are weighted equally which rarely happens in practice. In this present article, first a single objective model is developed to minimize the total cost of logistics, including aggregate price, ordering, and inventory costs, subject to suppliers’ capacity constraints and the buyers’ limitations on budget, quality, delivery, etc. Second, a multiple objective programming approach is discussed to take into account different weights for various criteria.

Supplier selection is one of the most important activities of purchasing managers in which cost, quality, delivery, etc., should be considered in selecting the best suppliers. Shortage of suppliers’ capacity makes the problem difficult, and considering the total cost of purchasing makes it more complicated. This paper has described a non-linear integer programming model which has been developed to help managers in this decision making. In order to solve the non-linear integer programming, it is necessary to solve 2n pure non-linear programs. Although the model should be run 2n times for n suppliers, the model solution should not take too long because in most practical cases there are usually a maximum of 12 vendors [21] and also because some cases are omitted, as they cannot satisfy the demand constraint.