Example 15.5Aggregate Planning at SureStep

Aggregate Planning Models

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Objective

To use LP to find the staffing and production schedules that minimize total cost, both without and with backlogging of demand.

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Background Information During the next 4 months the SureStep Shoe Company

must meet (on time) the following demands for pairs of shoes: 3000 in month 1; 5000 in month 2; 2000 in month 3; and 1000 in month 4.

At the beginning of month 1, 500 pairs of shoes are inventory, and SureStep has 100 workers. Each worker is paid $1500 per month and can work up to 160 hours a month before he or she receive overtime. (This amounts to about $9.38 per hour.)

A worker can be forced to work up to 20 hours of overtime per month at an overtime rate of $13 per hour.

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Background Information -- continued It takes about 4 hours of labor and $15 worth of raw

material to produce a pair of shoes.

At the beginning of each month, workers can be hired or fired.

These are fixed costs of $1600 per worker hired and $2000 per worker fired.

At the end of the month, a holding cost of $3 per pair of shoes left in inventory is incurred.

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Background Information -- continued The selling price of shoes will stay constant over this

4-month period, so we don’t need to incorporate it into the model.

Finally, we assume that production in a given month can be used to meet that month’s demand.

SureStep wants to determine a hiring/firing and production schedule that minimizes its total costs over this 4-month period.

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Solution

To model SureStep’s problem we need to keep track of the following:

– number of workers hired, fired and available during each month

– regular-time and overtime hours available and used each month

– production capacity (pair of shoes) and the production quantities each month

– the inventory levels after production and at the end of each month

– the monthly costs and the total costs

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SURESTEP1.XLS

This file contains the setup that can be used to develop the spreadsheet model that appears on the next slide.

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Developing the Spreadsheet Model The spreadsheet model can be developed as follows:

– Inputs. Enter the input data in the range B4:B14 and enter the monthly demands in the Demand range.

– Production, hiring and firing plan. Enter any trial values for the number of pairs of shoes produced each month in the Produced range, the overtime hours used each month in the OTHrs range, the workers hired each month in the Hired range, and the workers fired each month in the Fired range. These four ranges are the changing cells.

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Developing the Spreadsheet Model -- continued

– Workers available each month. In cell B17 enter the initial number of workers available with the formula =B5 Since the number of workers available at the beginning of any other month (before hiring or firing) is equal to the number of workers from the previous month, enter the formula =B20 in cell C17 and copy it to the range D17:E17. Next, calculate the net number of workers on hand after hiring and firing by entering the formula =B17+B18-B19 in cell B20 and copy it across row 20.

– Overtime capacity. Since each available worker can work up to 20 hours of overtime in a month, enter the formula =$B$7*B20 in cell B25 and copy it to the range C25:E25 to calculate the overtime hours capacity for months 2 through 4.

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Developing the Spreadsheet Model -- continued

– Production capacity. Since each worker can work 160 regular-time hours per month, calculate the regular-time hours available in month 1 in cell B22 with the formula =$B$6*B20 and copy it to the range C22:E22 for the other months. Then calculate the total hours available for production in cell B27 with the formula =SUM(B22:B23) and copy it to the range C27:E27 for the months. Finally, since it takes 4 hours of labor to make a pair of shoes, enter the formula =B27/$B$12 in cell B32 and copy it to the range C32:E32.

– Inventories each month. We include two inventory rows, row 34 for inventory after production has occurred and row 37 for ending inventory, after demand has been taken away. To calculate these begin by entering the formula =B4+B30 in cell B34.

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Developing the Spreadsheet Model -- continued

– Inventories each month - continued. Then enter the formula =B37+C30 in cell C34 and copy it to the range D34:E34. This links the inventory in one month to the ending inventory from the previous month. Next, calculate the ending inventory levels by entering the formula =B34-B36 in cell B37 and copying it across row 37.

– Costs. Calculate the various costs in rows 40-45 by multiplying the relevant unit costs in rows 7-14 by the relevant quantities in the body of the spreadsheet. For example, for holding costs enter the formula =$B$14*B37 in cell B45 and copy it to the range C45:E45. Then calculate the row and column totals in column F and row 46.

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Using Solver The Solver dialog box should appear as shown here.

We minimize the total cost, subject to not exceeding the maximum overtime constraint, not exceeding production capacity, and meeting demand on time.

We also include integer constraints on the number of workers hired and fired and the production level of shoes.

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Using Solver -- continued Interestingly, SureStep never hires any workers. It even

fires a worker in month 2 at the same time that it uses 80 hours of overtime. Evidently this is cheaper than hiring and then having to fire when demand decreases.

Note also that SureStep uses all of its production capacity except in month 4. In that month it has more workers than it needs, but there is a big penalty for firing them and no penalty for letting them sit idle.

Finally, the company holds inventory only at the end of month 1. This inventory is then used to help meet month 2 demand.

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Model with Backlogging Allowed In many situations, backlogging is allowed; that is,

customer demand can be met later than it occurs.

We now show how to modify the SureStep model to include the option of backlogging demand.

We assume that at the end of each month a cost of $20 is incurred for each unit of demand (pair of shoes) that remains unsatisfied as the end of the month.

This is easily modeled by allowing a month’s ending inventory to be negative.

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Modifying the Model

To begin, we enter the per unit monthly shortage cost in cell B15.

To take shortages into account, one possibility is to replace the formula currently in cell B45 with =IF(B37>=0,$B$14*B37,-$B$15*B37)

While this accurately computes the holding or shortage cost, Solver is unfortunately unable to deal with IF functions accurately, at least when they involve changing cells (either directly or indirectly).

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Modifying the Model

Alternatively, we could use the equation =MAX($B$14*B37,0) +MAX(0,-$B$15*B37)

instead of the IF expression. However, Solver is also unable to deal appropriately with MAX (or MIN) functions.

We can, however, handles shortages and maintain a linear formulation.

The method is illustrated in the following table.

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SURESTEP2.XLS

This file contains the setup that can be used to develop the modified model.

We again assume that the number of workers hired and fired each month, as well as the number of pairs of shoes produced each month, must be integers.

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Developing the Modified Model

To develop this modified spreadsheet proceed as follows.

– Enter shortage cost. Insert a new row below row 14 and enter the shortage cost per pair of shoes per month in cell B15.

– Rows for amounts held and short. Insert five new rows (which will now be rows 38 through 42) between the Demand and Ending inventory rows. The range B39-E40 will be changing cells. The excess range in row 39 contains the amounts left in inventory (if any), whereas the Shortage range in row 40 contains the shortage (if any), Enter any values in these ranges.

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Developing the Modified Model -- continued

– Ending inventory (positive or negative). Here is the key observation. Let Et be the excess left in inventory at the end of the month t, and let St be the shortage at the end of the month t. Then Et =0 if St is less than or equal to 0 and St=0 if Et is greater than or equal to zero. So if we allow ending inventory It to be negative (meaning that there is a shortage), then for each month we have It = Et - St. To incorporate this into the spreadsheet we enter the formula =B39-B40 in cell B41 and copy it across row 41. We will have to add a constraint to force the two ending inventories to be equal.

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Developing the Modified Model -- continued

– Monthly costs. Insert a new row (which will be row 52) below the holding cost row. Modify the holding cost for month 1 by entering the formula =$B$14*B39 in cell B51 and copy it across row 51. Calculate the shortage cost for month 1 in cell B52 with the formula =$B$15*B40 and copy it across row 52.

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Using Solver for the Modified Model The modifies Solver dialog box appears below. (Not

all the constraints fit in the window, but the new ones do.)

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Using Solver for the Modified Model -- continued Note the changes:

– the Excess and Shortage ranges are additional changing cells,

– the Net range (row 41) is constrained to equal the EndInv range (row 43), and

– the on-hand inventory after production is required only to be at least as great as the demand in month 4, not in all months.

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Comparing Solutions Comparing the two solutions, we see that SureStep

should now fire even more workers in month 3 than before, and it should not use any overtime.

The company again holds inventory in month 1 but it incurs shortages in months 2 and 3.

Of course, the demands for these customers are eventually satisfied by the end of the 4-month period.

The total cost has decreased but only by $2630.

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Comparing Solutions -- continued Note that the optimal cost could not possibly increase.

When we allow more possibilities, as we do when we allow backlogging, the optimal cost can only decrease or stay the same.

Here’s a great opportunity to perform a sensitivity analysis on the unit shortage cost.

In the no-backlogging model, this cost is essentially infinite, whereas it is now relatively small.

The question is how large this cost needs to be before SureStep will not incur any planned shortages.