Chapter 3 Aggregate Planning

www.izmirekonomi.edu.trAsst. Asst. Prof. Dr. Prof. Dr. Mahmut Ali Mahmut Ali GÖKÇE, Izmir University of EconomicsGÖKÇE, Izmir University of Economics

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Chapter 3

Aggregate Planning

www.izmirekonomi.edu.tr

Asst. Asst. Prof. Dr. Prof. Dr. Mahmut Ali Mahmut Ali GÖKÇE, Izmir University of EconomicsGÖKÇE, Izmir University of Economics

FallFall, 2007, 20072 of 31

Introduction to Aggregate Planning 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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The Hierarchy of Production Planning Decisions



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Overview of the ProblemSuppose 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 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 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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Cost of Changing the Size of the Workforce



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Holding and Back-Order Costs

Back-orders Positive inventory

Slope = CP

Slope = Ci

$ C

ost

Inventory



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Aggregate 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 units.(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 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 Example(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.(subtract starting inv. from per. 1 forecast and add ending inv. to per. 8 forecast.)

Month Net Predicted Cum. Net Days

Demand Demand

1(Jan) 220 220 22

2(Feb) 280 500 16

3(Mar) 460 960 23

4(Apr) 190 1150 20

5(May) 310 1460 21

6(June) 145 1605 17

7(July) 110 1715 18

8(Aug) 225 1940 10



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Step 2. Graph Cumulative Net Demand to Find Plans Graphically

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500

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1500

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2500

1 2 3 4 5 6 7 8



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



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How can we have a constant work force 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

0

500

1000

1500

2000

2500

3000

1 2 3 4 5 6 7 8



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From the previous graph, we see that cum. net demand curve is crossed at period 3, so that monthly production is 960/3 = 320. Ending inventory each month is found 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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But - may not be realistic for several reasons:

It may not be possible to achieve the production level of 320 unit/month 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.



FallFall, 2007, 200721 of 31To overcome these

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 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) = 0.3421

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



FallFall, 2007, 200723 of 31Computing Constant Work

Force Assume we are given the following # of

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



FallFall, 2007, 200724 of 31Constant Work Force

Production Plan

Mo # wk days Prod. Cum Cum Net 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 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 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 ~ $600K



FallFall, 2007, 200727 of 31Cost 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.

162017801940



FallFall, 2007, 200728 of 31Cost Evaluation of Modified

Plan 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 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 month) 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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How to disaggregate? Have to disaggregate a aggregate plan to a

master production schedule (MPS) MPS will then be used to plan for material

requirements of the production and detailed scheduling



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Optimal Solutions to Aggregate Planning Problems Via Linear Programming 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.

Chapter 3 Aggregate Planning

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