Example 4.5

Production Process Models

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Background Information Repco produces three drugs, A, B and C, and can

sell these drugs in unlimited quantities at unit prices $8, $70, and $100, respectively.

Producing a unit of A requires 1 hours of labor.

Producing a unit of B requires 2 hours of labor and 2 units of A.

Producing 1 unit of C requires 3 hours of labor and 1 unit of B.

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Background Information – continued Any product A that is used to produce B cannot be

sold, and any product B that is used to produce C cannot be sold.

A total of 40 hours of labor are available.

Repco wants to use LP to maximize its sales revenue.

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Solution To model Repco’s operation we need to keep track of

the following:

– The number of units produced of each product

– The number of units sold of each product

– The number of units of A and B used to produce other products

– The number of labor hours used

– The revenue received

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Developing the Model The key to developing the spreadsheet model is that

everything that is produced must be used in some way.

Either it must be used as an input to the production of some other product, or it must be sold. Therefore, we have the “balance” equation for each product:

Amount produced = Amount used to produce other products + Amount sold

We will implement this “balance” equation by designing both the amounts produced and the amounts sold as changing cells. Then we will impose a constraint that the above equation must be satisfied.

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REPCO.XLS

This file shows the spreadsheet model for this problem.

The spreadsheet figure on the next slide shows the model.

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4.1 | 4.2 | 4.3 | 4.4 | 4.6 | 4.7

Developing the Model To proceed, carry out the following steps.

– Inputs. Enter the number of labor hours needed to produce a unit of each product in the HrsReqd range and enter the number of units of each product needed to manufacture each other product in the range B7:D9. Then enter the unit selling prices for the products in the UnitPrices range and enter the available labor hours in the HrsAvailable cell.

– Units produced. Enter any trial values for the number of units produced and sold in the UnitsProduced and UnitsSold ranges.

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Developing the Model – continued

– Units used to make other products. In the range G16:I18 calculate the total number of units of each product that are used to produce other products. Begin by calculating the amount of A used to produce A in cell G16 with the formula =B7*B$16 and copy this formula to the range G16:I18 for the other combinations of products. Then calculate the row totals in column J with the SUM function. It is convenient to “transfer” these sums in column J to the B18:D18 range. Use Excel’s TRANSPOSE function, type the formula =TRANSPOSE(UnitsUsedAsInputs) and press Ctrl-Shift-Enter.

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Developing the Model – continued

– Units used total. We want to force the units produced of each product to equal the units used total of each product. We do this in rows 23 and 25. For row 23, enter the formula =B16 in cell B23 and copy it to the range C23:D23. For row 25, enter the formula =B18 + B19 in cell B25 and copy it to the range C25:D25.

– Labor hours used. Calculate the total number of labor hours used in the HrsUsed cell with the formula =SUMPRODUCT(HrsReqd,UnitsProduced).

– Total revenue. Calculate Repco’s revenue from sales in the Revenue cell with the formula =SUMPRODUCT(UnitPrices,UnitsSold).

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Developing the Model – continued Using Solver: To use Solver to maximize Repco’s

revenue, proceed as follows.

– Objective. Select the Revenue cell as the target cell to maximize.

– Changing cells. Select the UnitsProduced and UnitsSold as the changing cells.

– Labor availability constraint. Enter the constraint HrsUsed<=HrsAvailable. This ensures that at most 40 hours of labor are used.

– Product balance constraint. Enter the constraint UnitsProduced=UnitsUsedTotal to ensure that production matches usage for each product.

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Developing the Model – continued

– Specify nonnegativity and optimize. Under SolverOptions, check the nonnegativity box, and use the LP algorithm to obtain the optimal solution shown.

The Solver dialog should appear as shown here.

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Solution We see that Repco obtains a revenue of $700 by

producing 20 units of product A, which are then used to produce 10 units of product B.

All units of product B produced are sold.

Even though product C has the highest selling price, Repco produces non of the product C.

This is because of the large labor equipment for product C.

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Sensitivity Analysis We saw that product C is not produced at all, even

though its selling price is by far the highest.

How high would this selling price have to be to induce Repco to produce any of product C?

We use SolverTable to answer this, using product C selling price as the input variable, letting it vary from $100 to $200 in increments of $10, and keeping track of the total revenue, the units produced of each product, and the units used (row 18) of each product. The results appear on the next slide.

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Sensitivity Analysis – continued As we see, until the product C selling price gets to

$130, Repco uses the same solution as above.

However, when it increases to $130 and beyond, 5.714 units of C are produced.

This in turn requires 5.714 units of product B, which requires 11.429 units of product A, but only product C is actually sold.

Of course Repco would like to produce even more of product C, but the labor hour constraint does not allow it.

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Sensitivity Analysis – continued Therefore, further increases in selling price of product

C have no effect on the solution – other than increasing revenue.

Because available labor imposes an upper limit on the production of product C, even when it is very profitable, it is interesting to see what happens when the selling price of product C and labor hour availability as the two inputs with appropriate values, and selecting the amount produced of product C as the single output.

The results appear on the next slide.

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Sensitivity Analysis – continued This table shows that no product C is produced,

regardless of labor hour availability, until the selling price of C is $130.

The effect of increases in labor hour availability is to let Repco produce more of product C.

Specifically, Repco will produce as much of C as possible, given that 1 unit of B, and hence 2 units of A, are required for each unit of C.

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Sensitivity Analysis – continued Before leaving this example, we provide some further

insight into the sensitivity behavior.

Specifically, why should Repco start producing product C when its unit selling price increases to some value between $120 and $130?

We can provide a straightforward answer to this question because there is a single resource constraint, the labor hour constraint.

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Sensitivity Analysis – continued Consider the production of 1 unit of product B. It

requires 2 labor hours plus 2 units of A, each of which requires 1 labor hour, for a total of 4 labor hours, and it returns $70 in revenue.

Therefore, revenue per labor hours when producing product B is $17.50.

To be eligible as a “winner” product C has to beat this. To beat the $17.50 revenue per labor hour of product B, product C’s unit selling price must be at least $122.50.

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Sensitivity Analysis – continued If its selling price is below this, such as $120, Repco

will sell all products B and no product C.

If its selling price is above this, such as $130, Repco will sell all product C and no product B.

As this analysis illustrates, we can sometimes – but not always – unravel the information obtained by SolverTable.