CHAPTER 3 CHAPTER 3 Aggregate PlanningAggregate Planning

McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

Introduction to Aggregate Introduction to Aggregate PlanningPlanning

Goal: To plan gross work force levels and set firm-

wide production plans.

Concept is predicated on the idea of an “aggregate unit” of production. May be actual units, or may be measured in weight (tons of steel), volume (gallons of gasoline), time (worker-hours), or dollars of sales. Can even be a fictitious quantity. (Refer to example in text and in slide below.)

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Overview of the ProblemOverview of the Problem

Suppose that D1, D2, . . . , DT are the forecasts of demand for aggregate units over the planning horizon (T periods.) The problem is to determine both work force levels (Wt) and production levels (Pt ) to minimize total costs over the T period planning horizon.

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Important IssuesImportant Issues Smoothing. Refers to the costs and disruptions that result

from making changes from one period to the next. Bottleneck Planning. Problem of meeting peak demand

because of capacity restrictions. Planning Horizon. Assumed given (T), but what is

“right” value? Rolling horizons and end of horizon effect are both important issues.

Treatment of Demand. Assume demand is known. Ignores uncertainty to focus on the predictable/systematic variations in demand, such as seasonality.

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Relevant CostsRelevant Costs Smoothing Costs

changing size of the work force changing number of units produced

Holding Costs primary component: opportunity cost of investment

Shortage Costs Cost of demand exceeding stock on hand. Why

should shortages be an issue if demand is known?

Other Costs: payroll, overtime, subcontracting.

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Aggregate UnitsAggregate Units

The method is based on notion of aggregate units. They may be

Actual units of production Weight (tons of steel) Volume (gallons of gasoline) Dollars (Value of sales) Fictitious aggregate units

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Example of fictitious aggregate Example of fictitious aggregate units.units.Example 3.1)Example 3.1)

One plant produced 6 models of washing machines:

Model # hrs. Price % sales

A 5532 4.2 285 32

K 4242 4.9 345 21

L 9898 5.1 395 17

L 3800 5.2 425 14

M 2624 5.4 525 10

M 3880 5.8 725 06

Question: How do we define an aggregate unit here?

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Example continuedExample continued

Notice: Price is not necessarily proportional to worker hours (i.e., cost): why?

One method for defining an aggregate unit: requires: .32(4.2) + .21(4.9) + . . . + .06(5.8) = 4.8644 worker hours. Forecasts for demand for aggregate units can be obtained by taking a weighted average (using the same weights) of individual item forecasts.

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Prototype Aggregate Planning Prototype Aggregate Planning ExampleExample(this example is not in the text)(this example is not in the text)

The washing machine plant is interested in determining work force and production levels for the next 8 months. Forecasted demands for Jan-Aug. are: 420, 280, 460, 190, 310, 145, 110, 125. Starting inventory at the end of December is 200 and the firm would like to have 100 units on hand at the end of August. Find monthly production levels.

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Step 1: Determine “net” demand.Step 1: Determine “net” demand.(subtract starting inv. from per. 1 forecast (subtract starting inv. from per. 1 forecast and add ending inv. to per. 8 forecast.)and add ending inv. to per. 8 forecast.)

Month Net Predicted Cum. Net

Demand Demand

1(Jan) 220 220

2(Feb) 280 500

3(Mar) 460 960

4(Apr) 190 1150

5(May) 310 1460

6(June) 145 1605

7(July) 110 1715

8(Aug) 225 1940

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Step 2. Graph Cumulative Net Step 2. Graph Cumulative Net Demand to Find Plans GraphicallyDemand to Find Plans Graphically

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Constant Work Force PlanConstant Work Force Plan

Suppose that we are interested in determining a production plan that doesn’t change the size of the workforce over the planning horizon. How would we do that?

One method: In previous picture, draw a straight line from origin to 1940 units in month 8: The slope of the line is the number of units to produce each month.

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Constant Workforce Plan (zero ending inv)

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Monthly Production = 1940/8 = 242.2 or rounded to Monthly Production = 1940/8 = 242.2 or rounded to 243/month. 243/month. But:But: there are stockouts. there are stockouts.

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How can we have a constant work force How can we have a constant work force plan with no stockouts?plan with no stockouts?

Answer: using the graph, find the straight line that goes through the origin and lies completely above the cumulative net demand curve:

Constant Work Force Plan With No Stockouts

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From the previous graph, we see that cum. net demand From the previous graph, we see that cum. net demand curve is crossed at period 3, so that monthly production curve is crossed at period 3, so that monthly production is 960/3 = 320. Ending inventory each month is found is 960/3 = 320. Ending inventory each month is found from:from:

Month Cum. Net. Dem. Cum. Prod. Invent.

1(Jan) 220 320 100

2(Feb) 500 640 140

3(Mar) 960 960 0

4(Apr.) 1150 1280 130

5(May) 1460 1600 140

6(June) 1605 1920 315

7(July) 1715 2240 525

8(Aug) 1940 2560 620

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ButBut - may not be realistic for several - may not be realistic for several reasons:reasons:

It may not be possible to achieve the production level of 320 unit/mo with an integer number of workers

Since all months do not have the same number of workdays, a constant production level may not translate to the same number of workers each month.

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To overcome these To overcome these shortcomings:shortcomings:

Assume number of workdays per month is given

K factor given (or computed) where

K = # of aggregate units produced by one worker in one day

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Finding KFinding K Suppose that we are told that over a period

of 40 days, the plant had 38 workers who produced 520 units. It follows that:

K= 520/(38*40) = .3421

= average number of units produced by one worker in one day.

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Computing Constant Work Computing Constant Work ForceForce

Assume we are given the following # working days per month: 22, 16, 23, 20, 21, 17, 18, 10. March is still critical month. Cum. net demand thru March = 960. Cum # working days = 22+16+23 = 61. Find 960/61 = 15.7377 units/day implies 15.7377/.3421 = 46 workers required.

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Constant Work Force Production Constant Work Force Production PlanPlan

Mo # wk days Prod. Cum Cum Nt End Inv

Level Prod Dem

Jan 22 346 346 220 126

Feb 16 252 598 500 98

Mar 23 362 960 960 0

Apr 20 315 1275 1150 125

May 21 330 1605 1460 145

Jun 22 346 1951 1605 346

Jul 21 330 2281 1715 566

Aug 22 346 2627 1940 687

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Addition of CostsAddition of Costs Holding Cost (per unit per month): $8.50 Hiring Cost per worker: $800 Firing Cost per worker: $1,250 Payroll Cost: $75/worker/day Shortage Cost: $50 unit short/month

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Cost Evaluation of Constant Work Force PlanCost Evaluation of Constant Work Force Plan

Assume that the work force at end of Dec was 40. Cost to hire 6 workers: 6*800 = $4800 Inventory Cost: accumulate ending inventory:

(126+98+0+. . .+687) = 2093. Add in 100 units netted out in Aug = 2193. Hence Inv. Cost = 2193*8.5=$18,640.50

Payroll cost:

($75/worker/day)(46 workers )(167days) = $576,150 Cost of plan: $576,150 + $18,640.50 + $4800 =

$599,590.50

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Cost Reduction in Constant Work Force PlanCost Reduction in Constant Work Force Plan

In the original cum net demand curve, consider making reductions in the work force one or more times over the

planning horizon to decrease inventory investment.

Plan Modified With Lay Offs in March and May

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Cost Evaluation of Modified Cost Evaluation of Modified PlanPlan

I will not present all the details here. The modified plan calls for reducing the workforce to 36 at start of April and making another reduction to 22 at start of June. The additional cost of layoffs is $30,000, but holding costs are reduced to only $4,250. The total cost of the modified plan is $467,450.

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Zero Inventory Plan (Chase Zero Inventory Plan (Chase Strategy)Strategy)

Here the idea is to change the workforce each month in order to reduce ending inventory to nearly zero by matching the workforce with monthly demand as closely as possible. This is accomplished by computing the # units produced by one worker each month (by multiplying K by #days per mo.) and then taking net demand each month and dividing by this quantity. The resulting ratio is rounded up and possibly adjusted downward.

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I got the following for this problem:I got the following for this problem:

Period Period # hired # hired #fired#fired

1 1 10 Cost of this 10 Cost of this

2 2 20 plan: 20 plan:

3 9 3 9 $555,704.50$555,704.50

4 314 31

5 15 5 15

6 246 24

7 47 4

8 15 8 15

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Optimal Solutions to Aggregate Optimal Solutions to Aggregate Planning Problems Via Linear Planning Problems Via Linear ProgrammingProgramming

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Linear Programming provides a means of solving aggregate planning problems optimally. The LP formulation is fairly complex requiring 8T variables and 3T constraints, where T is the length of the planning horizon. Clearly, this can be a formidable linear program. The LP formulation shows that the modified plan we considered with two months of layoffs is in fact optimal for the prototype problem.

Refer to the latter part of Chapter 3 and the Appendix following the chapter for details.