تجزیه و تحلیل مسئله فروش چند محصوله سر روزنامه با قید بودجه
|کد مقاله||سال انتشار||تعداد صفحات مقاله انگلیسی||ترجمه فارسی|
|22648||2005||12 صفحه PDF||سفارش دهید|
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Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : International Journal of Production Economics, Volume 97, Issue 3, 18 September 2005, Pages 296–307
This is a sequel of an earlier paper entitled “Exact, approximate, and generic iterative models for the multi-product newsboy problem with budget constraint’’ (Abdel-Malek et al., 2004) that appeared in this journal. Motivated by Lau and Lau's (Eur. J. Oper. Res., 94 (1996) 29) observation where infeasible ordering quantities (negative) were obtained when applying existing methods, the extension here examines the solution space of the problem in order to provide the necessary insight into this phenomenon. The resulting insight shows that the solution space can be divided into three regions that are marked by two distinct thresholds. The first region is where the budget is large and the solution is the same as the unconstrained problem. The second region is where the budget is medium and the constraint is binding, however the newsboy can order all the products on the list. The third region is where the budget is very tight and if the non-negativity constraints are relaxed negative order quantities may be obtained, and therefore some products have to be deleted from the original list. We show how the values of the thresholds that divide the regions are computed and extend the previous methods, when necessary, to cover each of the three-solution's domains in order to determine the optimum order quantity for the various products. Numerical examples are given to illustrate the application of the developed procedures.
The newsboy problem, also known as the single period stochastic inventory model, is found to be a suitable tool for decision-making regarding stocking issues in today's supply chains. Motivated by the interest of the community, in an earlier publication entitled “Exact, approximate, and generic iterative models for the multi-product newsboy problem with budget constraint’’ (Abdel-Malek et al., 2004), we developed models to determine the optimum lot size for the capacitated situation. The model formulation used is as follows: equation(1) View the MathML sourceMinimizeE=∑τ=1Ncτxτ+hτ∫0xτ(xτ-Dτ)fτ(Dτ)dDτ+vτ∫xτ∞(Dτ-xτ)fτ(Dτ)dDτ, Turn MathJax on subject to equation(2) View the MathML source∑τ=1Ncτxτ⩽BG, Turn MathJax on where EE is the expected cost; NN the total number of items; ττ the item index; cτcτ the cost per unit of product ττ; xτxτ the amount to be ordered of item ττ which is a decision variable; hτhτ the cost incurred per item ττ for leftovers at the end of the specified period; DτDτ the random variable of item's ττ demand; fτ(Dτ)fτ(Dτ) probability density function of demand for item ττ; vτvτ the cost of revenue loss per unit of product ττ, and BGBG the available budget. This model is based on the classical one that was developed originally by Hadley and Whitin (1963). The solution is obtained using the Lagrangian method and it is as follows: equation(3) View the MathML sourceF(xτ**)=vτ-(1+λ)cτvτ+hτ, Turn MathJax on where View the MathML sourcexτ** is the optimal solution under budget constraint; F(·)F(·) the cumulative distribution function (CDF), and λλ the Lagrangian multiplier. As can be seen, this approach first ignores the budget constraint and finds the optimum values of the lot size for each product. Then, these values are plugged into the budget to see if they satisfy its constraint, otherwise the inequality constraint is set to equality and the Lagrangian approach is used. It should be also noted that Hadley and Whitin's model relaxes the non-negativity constraints of the order quantities. In fact most of the existing Lagrangian based models regarding the capacitated newsboy problem do not pay too much attention to the lower bounds of the order quantities (non-negativity constraints), see for example, Abdel-Malek et al. (2004), Ben-Daya and Abdul (1993), Erlebacher (2000), Gallego and Moon (1993), Khouja (1999), Moon and Silver (2000), and Vairaktarakis (2000). One should mention that if, when tackling a three-product problem, the non-negativity constraints are not relaxed and Kuhn–Tucker conditions are applied, the number of nonlinear equations to be solved simultaneously is more than 20. This could be one of the reasons that most existing models relax the lower bounds to make the problem tractable. Nevertheless, in doing so and as Lau and Lau, 1995 and Lau and Lau, 1996 were among the first to observe, this could lead to infeasible order quantities (negative) for some of the considered products. While modeling a case study for a large bakery, Lau and Lau experienced this phenomenon and offered a conceptual explanation for its occurrence. To address the aforementioned situation, we extend the previous paper by Abdel-Malek et al. to efficiently solve the capacitated multi-product newsboy problem (CMPNP) and to help the decision-maker in recognizing the implications of the available budget. Hence, the decision-maker can avoid infeasible (negative) order quantities by deleting products from further consideration when the constraint is too tight. Additionally, this extension provides a means to conduct sensitivity analysis, when necessary, for increasing the budget to include desired products if the initial amount does not allow for their consideration. Our taxonomy in this article is as follows. After this introduction, in Section 2, we present the problem and give some insights into its solution space. Section 3 follows where a general approach for solving the CMPNP and the necessary proofs are shown. Afterwards, Section 4 shows numerical examples to illustrate the application of these methods. Finally, Section 5 presents the conclusion of this paper.
نتیجه گیری انگلیسی
The extension in this paper has been propelled by the earlier observation of Lau and Lau (1996). In their article, they noted that if the budget is tight, the existing solutions methods, which are mostly Lagrangian based, could lead to negative optimum order quantities. Our study shows that the negativity of order quantities phenomenon stems from the fact that most of the existing models relax the lower bounds of the order quantities. To address this problem, the solution space for this model is analyzed. Our study reveals insights leading to efficient solution methods that depend on the value of the available budget and the parameters governing the problem. It is found that the solution space can be divided into three distinct regions by two thresholds that have been defined in the paper. Formulae that are based on the available budget, products cost, and demand density functions are developed to determine the values of these thresholds in order to define these regions. The first region is where the budget is large enough to order the optimum quantity of each item without exceeding the allocated amount of money. In this case the classical model by Hadley and Whitin (1963) can be directly applied to obtain the optimum order quantities of each item while relaxing the budget constraint. In the second region, the budget constraint is binding. Therefore, the Lagrangian approach while relaxing the lower bound can be followed as basis to solve the problem. Depending on the type of probability density function of each product demand, one can choose from the solution models, exact, approximate or GIM (Generic Iterative Method) that are developed in Abdel-Malek et al. (2004) to obtain the lot size of each product. The third region, where the main contribution of this work lies, addresses the tight budget issue. That is when the budget is not enough to order all the products. Based on duality theory, our approach starts by deleting, in ascending order, products with lower marginal utilities at their lower bounds, until the remaining ones can fit within the available budget. Then, the non-negativity constraints are relaxed and one can apply one of the solution methods followed in the second region. It should be mentioned that the approach proposed here relaxes the integrality constraints on the number of items ordered from each product. In practical sense, this is almost optimal when the number of items from each product is large, more than 10 (numerical experiments showed that the total costs are almost identical in most cases when rounding off the resulting solution). Nevertheless, if the number of items is small, the yielded solution can be an excellent starting point for Branch and Bound Approach since the objective function is convex. In addition to its applicability to large number of products that could have different probability density functions, the proposed methodology can be used for post optimality analysis, where the decision-maker can decide on increasing the available budget for more customer satisfaction or even tighten it more depending on the marginal utilities of each product. This should be helpful in the analysis of a firm's inventory policies. Finally, the model introduced in this paper complements existing Lagrangian based models. After defining the degree of tightness of the budget and deleting those products with low marginal utility, if any, one can apply other models that are already available in the literature. Also, the models here can be helpful as a first cut solution for more complex structures of the newsboy problem such as those developed in Moon and Silver (2000) and Silver and Moon (2001).