Example 3.1a Sensitivity Analysis and Solver Table Add-in

| 3.2 | Question n Check how sensitive the optimal profit and the optimal product mix are to 1.Changes in the number of labor hours available 2.The cost per ounce of metal. n Then check how sensitive the optimal profit is to simultaneous changes in the hourly labor cost and the total labor hours available.

| 3.2 | Solution n To begin this solution we assume that the product mix model has been formulated and optimized and that the Solver/Table add-in has been installed n The solution to question 1 is shown in the following table.

| 3.2 | Solution – continued n We obtained this output we use the Data/SolverTable menu item, select one-way table in the first dialog box, and fill in the second dialog box as shown here. n When we click OK, the Solver solves a separate optimization problem for each of the 11 rows of the table and then reports the requested outputs in the table.

| 3.2 | Solution – continued n Note that the SolverTable enters comments (indicated by small red triangles) in several cells to help you interpret the output. n There are several ways to interpret the output from this sensitivity analysis.

| 3.2 | Solution – continued n First, we can look at columns B-E to see how product mix changes as more labor hours become available. n Second, we can see how extra labor hours add to the total profit. We show this numerically in column G, where each value is the increase in profit from the previous row. n As column G indicates, it is worthwhile to obtain extra labor hours, even though we have to pay for them, because profit increases.

| 3.2 | Solution – continued n However, the increase in profit per extra labor hour, called the shadow price of labor hours, is not constant. n The line chart below illustrates how the shadow price decreases as more labor hours are already owned.

| 3.2 | Solution – continued n The answer to sensitivity question 2 is similar and appears in this table. We used the SolverTable exactly as before; only the input cell and input values differ.

| 3.2 | Solution – continued n Note how the optimal product mix remains unchanged for a cost of metal in the $.30 to $.70 range. Within this range, the only thing that changes is the profit, and it decreases only because metal gets more expensive. n Outside of this range, however, we change the product mix (and obtain less profit). n Intuitively, once metal becomes expensive enough, products that use metal most heavily become less attractive. They will be produced at lower levels or dropped from the mix altogether.

| 3.2 | Solution – continued n Finally, we answer sensitivity question 3 with a two- way table, as shown on the next slide. n Now the values of the two inputs, hourly labor cost and labor hours available are listed along the top and left-hand side, and the address of the single output, total profit is placed at the upper left cell of the table. n To produce this table we fill in the SolverTable’s second dialog box as shown on the slide after the next.

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| 3.2 | Solution – continued n We can see how total profit decreases in each row as the hourly labor cost increases and how it increases in each column as the available labor increases. n We can also chart the profits in a table as shown on the next slide. n It is always possible to run a sensitivity analysis by changing inputs directly in the spreadsheet model and rerunning Solver. The advantages of SolverTable are that it enables us to perform systematic sensitivity analysis for any selected inputs and outputs, and it keeps track of the results in a table.

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