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5 INFORMS Transactions on Education 8(2), pp , 2008 INFORMS 93 Figure 11 Sensitivity Report for Modified Model of the assumptions underlying Solver results. Most analytical procedures have built-in assumptions that govern the correctness (or lack thereof) of the technique. Examples include analysis of variance, linear regression, and many other statistical procedures. While such assumptions are often explained in lectures, it is rare that textbook examples and homework problems require students to carefully consider them. A case in point is sensitivity analysis in linear optimization. One crucial assumption in interpreting sensitivity analysis information for changes in model parameters is that all other model parameters are held constant. Thus, it is easy to fall into a trap of ignoring them and blindly crunching through the numbers without due consideration of these assumptions. This is particularly true when using spreadsheet models. Model complexity can confound the situation, as can the manner in which the analysis is taught in the classroom and textbooks. While popular current textbooks note this requirement, the impact of modeling on this assumption is often not clearly addressed. By simply providing examples where the assumptions hold, students can easily ignore them. We provide one example that illustrates this issue along with some suggestions for dealing with it in lectures and assignments. The following is a modified example drawn from an old operations research text (Eck 1976, pp ): A small winery, Walker Wines, buys grapes from local growers and blends the pressings to make two types of wine: shiraz and merlot. It costs \$1.60 to purchase the grapes needed to make a bottle of shiraz, and \$1.40 to purchase the grapes needed to make a bottle of merlot. The contract requires that they provide at least 40% but not more than 70% shiraz. Based on sorne market research it is estimated that the base demand for shiraz is 1,000 bottles, but increases by 5 bottles for each \$l spent on advertising while the base demand for merlot is 2,000 bottles and increases by 8 bottles for each \$l spent on advertising. Production should not exceed demand. Shiraz sells for \$6.25 per bottle while merlot is sold for \$5.25 per bottle. Walker Wines has \$50,000 available to purchase grapes and advertise its products, with an objective of maximizing profit contribution. To formulate this model, let S = number of bottles of shiraz produced, M = number of bottles of merlot produced, A s = dollar amount spent on advertising shiraz, and A m = dollar amount spent on advertising merlot. The objective is to maximize profit (revenue minus costs) = \$6 25S +\$5 25M \$1 60S +\$1 40M +A s +A m = 4 65S M A s A m Constraints are defined as follows. 1. Budget cannot be exceeded: \$1 60S + \$1 40M + A s + A m \$ Contractual requirements must be met: 0 4 S/ S + M 0 7 or, expressed in linear form: 0 6S 0 4M 0 and 0 3S 0 7M Production must not exceed demand: S A s M A m. 4. Nonnegativity. Figure 12 shows a spreadsheet implementation of this model along with the Solver solution. Figure 13 shows the Solver Sensitivity Report. A variety of practical questions can be posed around the Sensitivity Report. For example, suppose that the accountant noticed a small error in computing the profit contribution for shiraz. The cost of shiraz Figure 12 Spreadsheet Implementation of Walker Wines

6 94 INFORMS Transactions on Education 8(2), pp , 2008 INFORMS Figure 13 Walker Wines Solver Sensitivity Report grapes should have been \$1.65 instead of \$1.60. How will this affect the solution? Students will immediately recognize that the unit profit of shiraz drops from \$4.65 to \$4.60. As classical sensitivity analysis is taught, one would examine the allowable decrease associated with the objective coefficient and note that since the change in the profit coefficient is within the allowable decrease of , one would conclude that no change in the optimal solution will result. However, this is not the correct interpretation. If the model is resolved using the new cost parameter, the solution changes dramatically as shown in Figure 14. Because the unit cost is also reflected in the binding budget constraint, the fundamental reason for this result is clearly that any increase in the cost of grapes causes the budget constraint to become infeasible, and thus the solution must be adjusted to maintain feasibility. This can also be illustrated clearly by providing a table showing the actual changes over the range of the parameter changes as shown in Figure 15. Alternatively, it should be noted that the sensitivity range is correct if the change is applied to the selling price of the wine instead of the cost, for example, decreasing the selling price from \$6.25 to \$6.20. In this case, the solution remains the same, although the total profit decreases by \$ The difference between the two scenarios is simple. If the selling price changes, then all other model parameters remain constant the fundamental assumption underlying sensitivity analysis. If the cost changes, then the coefficients in the constraints also change a violation of the assumption. Unless this is clearly understood, most students (and indeed, many instructors), will not recognize the difference. What can be done to mitigate this issue? The simple answer is to just change the value of the parameter in the spreadsheet and resolve. While this will obviously provide the correct solution, it defeats the purpose of sensitivity analysis. The real issue stems from confusing the spreadsheet model with the underlying linear program. One should fully understand the mathematical model first and could easily check whether the cost parameter is included in any of the constraints. However, students often create correct spreadsheet models without correctly formulating the mathematical model. In dealing with the spreadsheet model, one could use Excel s formula auditing capability (see Figure 16). If one selects the cost of Shiraz (cell C5) and applies the Trace Dependent command from the formula auditing menu, we see that the unit cost influences both the unit profit (cell C20) and the bud- Figure 14Solver Solution with New Cost Parameter

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