Example 14.3Football Production at the Pigskin Company

A Multiperiod Production Problem

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Objective

To model and solve Pigskin’s production problem and then to perform interesting sensitivity analyses with SolverTable.

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Background Information

The Pigskin Company produces footballs. Pigskin must decide how many footballs to produce each month. It has decided to use a 6-month planning horizon.

The forecasted demands for the next 6 months are 10,000, 15,000, 30,000, 35,000, 25,000, and 10,000.

Pigskin wants to meet these demands on time, knowing that it currently has 5000 footballs in inventory and it can use a given month’s production to help meet the demand for the month.

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Background Information -- continued During each month there is enough production

capacity to produce up to 30,000 footballs, and there is enough storage capacity to store up to 10,000 footballs at the end of the month.

The forecasted production costs per football for the next 6 months are $12.50, $12.55, $12.70, $12.80, $12.85, and $12.95, respectively.

The holding cost per football held in inventory at the end of the month is figured at 5% of the production cost for that month.

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Background Information -- continued The selling price for football is not considered

relevant to the production decision because it is assumed that all demand will be met exactly when it occurs - at whatever the selling price is.

Therefore, Pigskin wants to determine the production schedule that minimizes the total production and holding costs.

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Solution

In the traditional algebraic formulation, the decision variables are the production quantities for the 6 months, labeled P1 through P6. It is also convenient to let I1 through I6 be the corresponding end-of-the month inventories.

Therefore the obvious constraints on production and storage capacities are Pj < = 300 and Ij < = 100 for each month j, 1 < = j < = 6.

In addition to these constraints, we need “balance” constraints that relate the P’s and I’s.

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Solution -- continued

In any month the inventory from the previous month plus the current production must equal the current demand plus leftover inventory. If Dj is the forecasted demand for month j, then the balance equation for month j is Ij-1 + Pj = Dj + Ij

The first of these constraints for month j=1, uses the known beginning inventory, 50, for all previous inventory (the Ij-1 term).

By putting all the variables on the left and all known values on the right, these balance equations become

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Solution -- continuedP1 - I1 = 100-50

I1 + P2 - I2 = 150

I2 + P3 - I3 = 300

I3 + P4 - I4 = 350

I4 + P5 - I5 = 250

I5 + P6 - I6 = 100

All P’s and I’s must be nonnegative.

Finally, the objective is obvious. It is the sum of unit production costs multiplied by P’s, plus the unit holding costs multiplied by I’s.

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PIGSKIN1.XLS The spreadsheet version of Pigskin’s production problem

is shown here and can be found in this file.

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Developing the Spreadsheet Model The main feature that distinguishes this model from

the product mix model is that some of the constraints, namely the balance equations, are built into the spreadsheet itself by means of formulas. In other words, the only changing cells are the production quantities.

The ending inventories shown in row 21 are determined by the production quantities and equations.

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Developing the Spreadsheet Model -- continued To form the spreadsheet we proceed as follows:

– Inputs. Enter the various inputs in range InitInv (B4), HolPct (B5), B9:G9, ProdCap (B15:G15), Demand (B19:G19), and InvCap (B23:G23).

– Production quantities. Enter any six values in the range Produced (B13:G13) as production quantities.

– On-hand inventory. Enter the formula =B4+B13 in cell B17. This calculates the first month on hand inventory after production. Then enter the “typical” formula =B21+C13 for on-hand inventory after production in month 2 in cell C17 and copy it across row 17.

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Developing the Spreadsheet Model -- continued

– Ending Inventories. Enter the formula =B17-B19 for ending inventory in cell B21 and copy it to the range C21:G21. This formula calculates ending inventory in the current month as on-hand inventory before demand minus the demand in that month.

– Production and holding costs. Enter the formula =B9*B13 in cell B27 and copy it across to cell G27 to calculate the monthly productions costs. The enter the formula =$B$5*B9*B21 in cell B28 and copy it across to cell G28 to calculate the monthly holding costs. Note that these are based on monthly ending inventories. Finally, calculate the cost totals in row 28 by summing.

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Developing the Spreadsheet Model -- continued The logic behind the constraints is now

straightforward. All we have to guarantee is that

– production quantities are nonnegative and do not exceed the production capacities,

– the on-hand inventories after production are at least as large as demands, and

– ending inventories do not exceed storage capacities.

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Using Solver To use Solver, fill out the dialog box as follows and

then click on Solve.

– Model. Fill out the Solver dialog box as shown here. Of course, if we didn’t name ranges, we would refer directly to the corresponding cell addresses.

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Using Solver -- continued

– Options. In the Solver Options dialog box, check the Assume Linear Model and Assume Non-Negative boxes. Note that the latter ensures only that the changing cells will be nonnegative. If we wanted to explicitly constrain the ending inventory cells to be nonnegative, we would have to add an extra constraint in the previous dialog box. However this constraint implicitly guaranteed by the Onhand>=Demand constraint, so it is not needed.

The Solver solution appears on the next slide. This solution is also represented graphically.

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Optimal Solution for Pigskin Production Problem

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Graphical Representation of Optimal Production Schedule

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Interpreting the Solution

We can interpret the solution by comparing production quantities with demands.

– In month 1 Pigskin should produce just enough to meet month 1 demand (taking into account the inventory of 5000).

– In month 2 it should produce 5000 more footballs than month 2 demand, then in month 3 it should produce just enough to meet month 3 demand, still carrying the extra footballs in inventory from month 2 production.

– In month 4 Pigskin should finally use the 5000 footballs, along with the maximum production amount, 30,000, to meet month 4 demand.

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Interpreting the Solution -- continued

– Then in months 5 an 6 it should produce exactly enough to meet these months’ demands.

The total cost is $1,535,563, most of which is production cost.

Could you have guessed that this is the optimal solution? Upon some reflection it makes perfect sense. Because the monthly holding costs are large relative to the differences in monthly production costs, there is little incentive to produce footballs before they are needed to take advantage of a “cheap” production month.

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Interpreting the Solution -- continued Therefore, Solver tells us to produce footballs in

months in which they are needed - when this is possible.

The only exception to the rule is the 20,000 footballs produced during month 2 when only 15,000 are needed. The extra 5000 units produced during month 2 are needed, however, to meet month 3 demand. Thus, month 3 capacity is not available to meet month 4 demand, and 5000 units of month 2 capacity are used to meet month 4 demand.

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Sensitivity Analysis We can again use the Solver/Table add-in to perform a

sensitivity analysis. We illustrate two possibilities.

– First, note that the most inventory we ever carry at the end of a month is 5000 footballs, although storage capacity is 10,000 footballs. Perhaps this is because the holding cost percentage of 5% is fairly large. Would we carry more ending inventory if this holding cost was reduced. Or would we carry less if it was increased. We use Solver/Table to answer these questions. The only input cell is B5 and the single output we keep track of is the maximum ending inventory ever held which we calculate in cell B32 with =MAX(EndInv). As we will see in the output on the next slide, only when the percentage cost is 1% does the holding inventory increase, and even when it reaches 10% we will continue to hold the same.

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Sensitivity of Maximum Ending Inventory to Holding Cost

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Sensitivity Analysis -- continued

– A second possible sensitivity analysis is suggested by the way the optimal production schedule would probably be implemented. The optimal solution to Pigskin’s model specifies the production level for each of the next 6 months. In reality, however, the company might implement the model’s recommendation only for the first month. Then at the beginning of the second month, it will gather new forecasts for the next 6 months (that is, months 2 through 7), solve a new 6 month model, and again implements the model’s recommendation for the first of these months, month2. If it continues in this manner, we say that it is following a 6-month rolling planning horizon.

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Final Question The question then is whether the assumed demand

(really, forecast) toward the end of the planning horizon will have much effect on the optimal production quantity in month 1.

We hope not since these forecasts could be quite inaccurate. The two-way table here shows how the optimal month 1 production quantity varies with the assumed demands in months 5 and 6.

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Final Question -- continued

As we see, if the assumed month 5 and 6 demands remain fairly small, the optimal month 1 production quantity remains at 5000. This is good news.

It means that the optimal production quantity of month 1 is very insensitive to the possibly inaccurate forecasts in month 5 and 6.

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Modeling Issue We assumed that Pigskin is using a 6-month planning

horizon. Why 6 months?

In multiperiod problems such as this, the company has to make forecasts about the future, such as the level of customer demand.

Therefore, the length of the planning horizon is usually the length of time for which the company can make reasonably accurate forecasts.

Here, Pigskin evidently believes that it can forecast up to 6 months from now, so it uses a 6-month planning horizon.