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Chapter 3Chapter 3

Aggregate PlanningAggregate Planning

McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.

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Introduction to Aggregate PlanningIntroduction to Aggregate Planning

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

wide production plans.wide production plans.

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

3-3The Hierarchy of The Hierarchy of Production Planning DecisionsProduction Planning Decisions

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Overview of the ProblemOverview of the Problem

Suppose that DSuppose that D11, D, D22, . . . , D, . . . , DTT are the are the

forecasts of demand for aggregate units forecasts of demand for aggregate units over the planning horizon (T periods.) The over the planning horizon (T periods.) The problem is to determine both work force problem is to determine both work force levels (Wlevels (Wtt) and production levels (P) and production levels (Ptt ) to ) to

minimize total costs over the T period minimize total costs over the T period planning horizon. planning horizon.

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Important IssuesImportant Issues

Smoothing.Smoothing. Refers to the costs and disruptions that Refers to the costs and disruptions that result from making changes from one period to the next.result from making changes from one period to the next.

Bottleneck PlanningBottleneck Planning. Problem of meeting peak demand . Problem of meeting peak demand because of capacity restrictions. because of capacity restrictions.

Planning HorizonPlanning Horizon. Assumed given (T), but what is . Assumed given (T), but what is “right” value? Rolling horizons and end of horizon “right” value? Rolling horizons and end of horizon effect are both important issues. effect are both important issues.

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

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Relevant CostsRelevant Costs

Smoothing CostsSmoothing Costs– changing size of the work forcechanging size of the work force– changing number of units producedchanging number of units produced

Holding CostsHolding Costs– primary component: opportunity cost of investmentprimary component: opportunity cost of investment

Shortage CostsShortage Costs– Cost of demand exceeding stock on hand. Why Cost of demand exceeding stock on hand. Why

should shortages be an issue if demand is known? should shortages be an issue if demand is known?

Other Costs:Other Costs: payroll, overtime, subcontracting. payroll, overtime, subcontracting.

3-7Cost of Changing Cost of Changing the Size of the Workforcethe Size of the Workforce

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Holding and Back-Order CostsHolding and Back-Order Costs

Back-orders Positive inventory

Slope = CP

Slope = Ci

$ C

ost

Inventory

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Aggregate UnitsAggregate Units

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

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

3-10Example of fictitious aggregate units.Example of fictitious aggregate units.(Example 3.1)(Example 3.1)

One plant produced 6 models of washing machines:One plant produced 6 models of washing machines:

Model # hrs. Price Model # hrs. Price % sales % sales

A 5532A 5532 4.24.2 285285 3232

K 4242K 4242 4.94.9 345345 2121

L 9898L 9898 5.15.1 395395 1717

L 3800L 3800 5.25.2 425425 1414

M 2624M 2624 5.45.4 525525 1010

M 3880M 3880 5.85.8 725725 0606

Question: How do we define an aggregate unit here?Question: How do we define an aggregate unit here?

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Example continuedExample continued

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

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

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

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

3-13Step 1: Determine “net” demand.Step 1: Determine “net” demand.(subtract starting inv. from per. 1 forecast and add ending inv. (subtract starting inv. from per. 1 forecast and add ending inv. to per. 8 forecast.)to per. 8 forecast.)

MonthMonth Net PredictedNet Predicted Cum. NetCum. Net DaysDays

Demand Demand Demand Demand

1(Jan)1(Jan) 220220 220220 2222

2(Feb)2(Feb) 280280 500500 1616

3(Mar)3(Mar) 460460 960960 2323

4(Apr)4(Apr) 190190 1150 1150 2020

5(May)5(May) 310310 14601460 2121

6(June)6(June) 145145 16051605 1717

7(July)7(July) 110110 17151715 1818

8(Aug)8(Aug) 225225 19401940 1010

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

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Cum Net Dem

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Constant Work Force PlanConstant Work Force Plan

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

One method: In previous picture, draw a straight One method: In previous picture, draw a straight line from origin to 1940 units in month 8: The line from origin to 1940 units in month 8: The slope of the line is the number of units to slope of the line is the number of units to produce each month.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 plan How can we have a constant work force plan with no stockouts?with no stockouts?

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

Constant Work Force Plan With No Stockouts

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3-19From the previous graph, we see that cum. net demand curve From the previous graph, we see that cum. net demand curve is crossed at period 3, so that monthly production is 960/3 = is crossed at period 3, so that monthly production is 960/3 = 320. Ending inventory each month is found from:320. Ending inventory each month is found from:

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

1(Jan)1(Jan) 220220 320 320 100 100

2(Feb)2(Feb) 500 640 140500 640 140

3(Mar)3(Mar) 960 960 0960 960 0

4(Apr.) 1150 1280 1304(Apr.) 1150 1280 130

5(May)5(May) 1460 1600 140 1460 1600 140

6(June)6(June) 1605 1920 315 1605 1920 315

7(July)7(July) 1715 2240 525 1715 2240 525

8(Aug)8(Aug) 1940 2560 620 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 It may not be possible to achieve the production level of 320 unit/moproduction level of 320 unit/monthnth with an with an integer number of workersinteger number of workers

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

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To overcome these shortcomings:To overcome these shortcomings:

Assume number of workdays per month is Assume number of workdays per month is givengiven

K factor given (or computed) where K factor given (or computed) where

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

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Finding KFinding K

Suppose that we are told that over a period Suppose that we are told that over a period of 40 days, the plant had 38 workers who of 40 days, the plant had 38 workers who produced 520 units. It follows that:produced 520 units. It follows that:

K= 520/(38*40) = K= 520/(38*40) = 00.3421 .3421

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

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Computing Constant Work ForceComputing Constant Work Force

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

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Constant Work Force Production PlanConstant Work Force Production Plan

MoMo # wk days Prod. Cum Cum N # wk days Prod. Cum Cum Neet End Invt End Inv

Level Prod DemLevel Prod Dem

Jan 22 346 346 220 126Jan 22 346 346 220 126

Feb 16 252 598 500 98Feb 16 252 598 500 98

Mar 23 362 960 960 0Mar 23 362 960 960 0

Apr 20 315 1275 1150 125Apr 20 315 1275 1150 125

May 21 330 1605 1460 145May 21 330 1605 1460 145

Jun 22 346 1951 1605 346Jun 22 346 1951 1605 346

Jul 21 330 2281 1715 566Jul 21 330 2281 1715 566

Aug 22 346 2627 1940 687Aug 22 346 2627 1940 687

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Addition of CostsAddition of Costs

Holding Cost (per unit per month): $8.50Holding Cost (per unit per month): $8.50 Hiring Cost per worker: $800Hiring Cost per worker: $800 Firing Cost per worker: $1,250Firing Cost per worker: $1,250 Payroll Cost: $75/worker/dayPayroll Cost: $75/worker/day Shortage Cost: $50 unit short/monthShortage 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.Assume that the work force at end of Dec was 40. Cost to hire 6 workers: 6*800 = $4800Cost to hire 6 workers: 6*800 = $4800 Inventory Cost: accumulate ending inventory: Inventory Cost: accumulate ending inventory:

(126+98+0+. . .+687) = 2093. Add in 100 units netted (126+98+0+. . .+687) = 2093. Add in 100 units netted out in Aug = 2193. Hence Inv. Cost = out in Aug = 2193. Hence Inv. Cost = 2193*8.5=$18,640.502193*8.5=$18,640.50

Payroll cost: Payroll cost:

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

$599,590.50$599,590.50 ~ $600K ~ $600K

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

planning horizon to decrease inventory investmentplanning horizon to decrease inventory investment..

Plan Modified With Lay Offs in March and May

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Cost Evaluation of Modified PlanCost Evaluation of Modified Plan

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

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

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

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

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