An Analysis of the Multiproduct Newsboy Problem with a Budget Constraint

Carol HuayungaNavjot SinghIE 440Professor MalekApril 28, 2014

I. Introduction

Many people have long faced budget constraint problems in real life. A budget constraint problem is when a consumer and or a producer are restricted in purchasing/making goods and services due to a certain income. In consequences, a single period stochastic inventory model has been created and used throughout many years in fiscal and operational management fields with the ultimate goal to reduce costs and increase profits during the process of obtaining and selling products. Then, this newsboy problem is an ideal tool in order to find solutions to stocking issues in todays supply chain problems. Therefore, in the article, Exact, approximate, and generic iterative model for the multiproduct newsboy problem with budget constraint (Abdel Malek 2004) , which was inspired by Lau and Lau (Eur Operational Research 1996), gives an exact answer to the problem where the demand probability density function is uniform. In other words, all the demands are equally likely and near optimal solution when the demand is distributed otherwise. Then, the approach for the solution will add the space being divided into three regions. These three regions are later than marked by two unique thresholds.

Furthermore out of the three regions, the first region consists of where the budget is very large and the solution stays exactly like the unconstrained problem. Secondly, the second region is where the budget is not very large but is medium with the constraint binding. In other words, the newsboy can order the products using Lagrangian approach. Lastly, the third region consists of were the budget is very small and the negativity constraints will lead to negative ordered quantities. This shows that some products need to be removed from the native list. By reading further it will profoundly explain how the values in the three regions are computed in order to find the optimal order quantity for the products. To start with the looking solution, the following equation set by Hadley and Whitin (1963) shows the expected minimum cost.

With the budget restriction:

Where:

E= Expected Cost.N= Total number of items.= The item index.=The cost per unit of product .=The amount to be ordered of item (variable decision).=The inventory cost. =The scholastic demand of the item.= Probability demand function of the demand.= Cost of revenue or Loss per unit.=The budget.

Then to can find the probability that estimates the optimum lot size, it is necessary to use the Lagrangian approach. In consequence, the solution for the equation (1) is.

Where:=Cumulative distribution function.= The Lagrangian multiply.=Optimal solution under budget constraint.

II. The Problem

At the beginning the analysis decision-market does not consider the budget constraints as part of newsboy issue. Later, this factor will be included on the next studies. Hadley and Whitins (1963) obtained the equation that minimizes the expected cost of order. To find the optimum lot size order, they used the Lagrangian approach that relaxes the non-negative constraint. However, Lau and Lau (1995,1996) figured out that using this approach gave negative orders. On the other hand, if an analysis the lower bounds are not relaxing and the Kuhn-Tucker condition is applied, the newsvendor problem results infeasible too. The reason is because the number of equations that are obtained results more than 20. Then, to obtain an optimum order this case should be studied. The fig (1) shows the case when exist an available budget and if the Lagrangian approach is applied, the optimum lot size are obtained.

Fig 1.(a)Solution space where non-negativity constraint can be relaxed.

The opposite case is represent in the fig (2). There, if we have a tight budget and the Lagrangian approach is used to find the desired number of orders, some of those items will have negative values.

Fig 2. (b) A solution space where non-negativity constraint must be considered (relaxing the non-negativity constraints may lead to negative order quantities).

Then, as this paper is based in An analysis of the multi-product newsboy problem with a budget constraint (Abel Malek et al., 2005), the problem will be analyzed different cases according the budget tightness. Finally, the selection of optimum order will be possible to find using a convenient approach.

III. Approach

As it was mentioned before, the newsvendor problem will be analyzed depending the available budget. Therefore, the space that represent this budget will be divided in tree regions as the fig (3) shows.

Fig (3). Thresholds for the budget ranges.

These three regions are separated for two thresholds that will permit to get the desired budget classification. The two thresholds are merely equations that take factor such as the cost of unit, number of units, cost of inventory of units and the demand probability distribution. Then, three regions are:

Region1:

This region represents the case where is sufficient budget. Therefore, if the newsvendor opts to order certain number of item according the of the first equation (1), he will obtain the optimum order. This is feasible because there is not budget restriction in this area. Then solution in this part should be:

: The optimum number of quantities to order.

Region2:

However, in this region the budget is little tight. Then, the newsboy has to consider the budget restrictions. Therefore, to get the solution for this second region, the non-negativity should be relaxed using the Lagrangian approach. In this way, the optimum order lot size will be reached. Also, in Abdel-Maleks paper, a determined solution could get using the demand probability density function. The solution is represented by this equation:

Region3:

In this case, the degrees of tightness of budget are more than the last region. Therefore, if the newsboy chose to use the Lagrangian approach to get his desired number of order, he will get negative values. The explanation is because the lower bounds are not being considered. The solution starts deleting the items that have lower marginal utility. In this way, the cost function value keeps low until it can match with the constraint budget. After this, it is possible use the Lagrangian approach and the optimum lot size are obtained.

IV. Numerical Example: Next we will follow a precise numerical example in order to make things more lucid. From the given article we are looking at Example 2 (4.2) and we see that we are given a 17-item situation with a budget constraint Bg=$2500. The table below (table 2) will show the data and demand normally distributed.

Therefore, our next step learning from earlier of this article will be to defining the two thresholds. The two thresholds are BG(1) and BG(2). Next, we will find the first threshold and in order to do that we will use equation (4) from the article: This is where we obtain BG (1)> BG-> $21988 >$2500.Next we will find the next threshold by using equation (5) and equation (6) from the article:

The following table (table 3) will help us find N

For BG(2) j and j=1, N can be found using the table above.BG(2) > BG ($18807 > $2500) since BG(2) is larger we will use Range 3 from the 3 degree budget tightness method.

Since BG(2) is from the items list, we need to reduce the list to make the problem feasible by taking out item 9 (lowest marginal utility) and go from lowest to highest in ascending order. We will start with the following:, And substituting its value in the following equation: We get: which is larger than the available budget in which at this point items 9 and 15 have been deleted. By keep repeating this procedure at this rate, until we get . When ($2163