امکان قیمت گذاری چند هدفه و مدل تعیین اندازه دسته تولید با طبقات تقاضای چندگانه

We address an inventory-marketing system to determine the production lot size, marketing expenditure and selling prices where a firm faces demand from two or more market segments in which the firm can set different prices. Considering pricing, marketing and lot-sizing decisions simultaneously, the model maximizes the profit and return on inventory investment under multiple time varying demand classes. The model is formulated as a fuzzy non-linear multi-objective one where some parameters are ill-known and modeled by fuzzy numbers. A hybrid possibilistic-flexible programming approach is proposed to handle imprecise data and soft constraints concurrently. After transforming the original model into an equivalent multi-objective crisp model, it is then converted to a classical mono-objective one by a fuzzy goal programming method. An efficient solution procedure using particle swarm optimization (PSO) is also provided to solve the resulting non-linear problem.

The integration/coordination of production and marketing decisions has been known to be crucial in practice for diminishing their conflicts and increasing a firm’s profit by reducing opportunity losses incurred from separate or independent decision-making [1–6]. To do so, one important area is developing joint pricing and lot-sizing models (JPLM), where demands are assumed to be constant but price-dependent. As a result, item’s price and economic lot size or economic order quantity (EOQ) are determined simultaneously to maximize a firm’s total profit over a planning horizon. In this regard, it is noteworthy to mention that marketing expenditures, which include the advertisement and promo- tion, directly affect the demand of an item. Marketing effort motivates sales and influences potential consumers with an immediate reason to buy [7]. Understanding and differentiating customers by their needs and responses to marketing mixes play a vital role in managing customer relationships. This can be achieved by market segmentation that has been applied in almost every marketing research area including both the consumer and the firm behaviors. One of the underlying principles of revenue management (RM) is to divide a single market into multiple sub-markets/segments and then set different prices in each sub-market. For instance, many firms differentiate customers by leading them to different channels, such as online versus retail store, where firms set one price in the retail channel but offer discounts to online purchasers. Market segmentation generally increases revenue and hence profit; however, different prices in various market segments stir some customers to switch between segments. For example, a customer might visit a retail store to “touch and feel” a product but goes home and buys it online at a lower price. As Phillips [8] mentioned, price differentiation is a powerful way for sellers to improve their profitability. Traditionally, profit maximization or cost minimization has been considered in numerous papers as the objective function for designing and analyzing inventory-marketing models. In another relevant research stream, there are a number of studies on optimization of inventory systems under return on investment (ROI) maximization. Return on inventory investment (ROII) maximization has also been examined by Schroeder and Krishnan [9], Otake et al. [10], Otake and Min [11], Li et al. [12] andWee et al. [13]. Despite these studies, there are few works focusing on the profit and ROII maximization simultaneously when modeling inventory systems [13]. More surprisingly, the application of fuzzy set theory in JPLM and channel pricing areas is more limited. It is often a difficult task to determine the exact values for the parameters of an inventory system, such as cost and demand parameters, which is the primary concern in the previous reports. Therefore, they are assumed to be imprecise, i.e., fuzzy in nature [14]. Most of the research developing inventory-marketing strategies formulate the market uncertainty (e.g., uncertain demands) by using probability distributions that need sufficient historical data. However, the stochastic models may not be the best choice whenever statistical data are insufficient and unreliable or even non-obtainable. In these situations, fuzzy set theory provides an alternative approach to deal with the epistemic uncertainty (i.e., lack of knowledge) in the inventory and marketing related parameters. Furthermore, a decision maker (DM) often has vague goals such as “This profit function should be larger than or equal to a certain value.” For such cases, fuzzy set theory and flexible programming methods should be used [13]. Negligence of some issues such as inherent uncertainty in critical input data (i.e., market demands and unit costs) and imposing the decisions made at a managerial level as a hard constraint to an operational level without allowing any deviation; often result in poor efficiency of JPLMs in practice. In this regard, we formulate a novel JPLM and channel pricing model in a fuzzy environment. Therefore, the main purpose of this paper is to develop a fuzzy multi-objective integrated pricing and lot-sizing model to efficiently handle different demand classes in an inventory system under uncertainty. The two important objective functions, i.e., profit and return on inventory investment (ROII), are considered to find item’s prices and economic production quantity. The proposed model integrates the marketing-inventory and price discrimination decisions into a single model while maximizing the total profit and ROII concurrently. As mentioned earlier, in an integrated inventory and marketing planning framework, objective functions’ goals, unit costs, marketing parameters, etc., are often assumed to be crisp and defined with certainty. However, this rarely happens in practice where the goals and parameters are normally vague and imprecise. Therefore, this paper presents a fuzzy multi-objective programming method to capture this inherent fuzziness in the critical data and goals. The rest of this paper is organized as follows. In the next section, the relevant literature is reviewed. Problem descrip- tion, assumptions and formulation are presented in Section 3. Then, by applying efficient defuzzification strategies, the resultant crisp multi-objective model is dealt with fuzzy goal programming in Sections 4 and 5. In Section 6, a tailored particle swarm optimization is employed to solve the resultant non-linear problem followed by an illustrative example in Section 7. Finally, concluding remarks and some future research directions are given in Section 8.

This paper attempts to formulate a fuzzy multi-objective integrated pricing and lot-sizing model for an inventory- marketing problem with multiple market segments under uncertainty. It is assumed that the demand function is not only sensitive to price and marketing expenditure, but it also varies with respect to time. Furthermore, the proposed fuzzy modeling is more effective for handling the real situations where exact data is not available for marketing and lot-sizing problems. In this formulation, fuzzy parameters and goals are represented by appropriate linear membership functions and after the defuzzification process, the equivalent crisp multi-objective model is converted to a single-objective non- linear one by the well-known additive fuzzy goal programming method. A tailored particle swarm optimization (PSO) is also proposed to solve the resultant crisp non-linear program. This paper contributes to the literature by revealing this fact that fuzzy approaches can effectively be used for formulation of integrated pricing and lot-sizing problems with fuzzy customer demands and inventory costs. For future researches, the following areas are recommended: (a) In some pricing and inventory decision making situations, life cycle of products is short and finite. Hence, one of the main directions is to develop a finite planning horizon case of our proposed model. (b) The study can be extended to adapt other fuzzy mathematical programming-based approaches such as methods given in [42,43] based on the possibility and necessity representation of constraints. (c) Other meta-heuristic search algorithms such as simulated annealing combined with gradient-based approaches may also be employed accompanied by a numerical comparative analysis among different solution methods. (d) Extending the current model to the multi-product case is another direction for further work. (d) Additionally, accounting for linguistic terms quoted by the decision maker along with if–then rules when model formulation makes the model more convenient in market analysis. (e) Finally, the discussed problem could be generalized to allow for promotion, temporary discount or even cancel- lation opportunity for customers.