Agg Chapter03

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Chapter 3

Aggregate Planning

McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hil l Companies, Inc. All r ights reserved.



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Aggregate Planning Strategies

Should inventories be used to absorb changes indemand during planning period?

Should changes be accomodated by varying thesize of the workforce?

Should part-timers be used, or should overtimeand idle time absorb fluctuations?

Should subcontractors be used on fluctuating

orders so a stable workforce can be maintained? Should prices or other factors be changed to

influence demand?



3

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,

» May be measured in weight (tons of steel), volume

(gallons of gasoline), time (worker-hours), or dollars ofsales.

» Can even be a fictitious quantity. (Refer to example intext and in slide below.)





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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. Whyshould shortages be an issue if demand is known?

Other Costs: payroll, overtime, subcontracting.



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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 fordemand 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. Forecasteddemands for Jan-Aug. are: 420, 280, 460,

190, 310, 145, 110, 125. Starting inventory

at the end of December is 200 and the firmwould 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

Demand Demand

1(Jan) 220 2202(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 Demand

to Find Plans Graphically

0

200

400

600

800

1000

1200

1400

1600

1800

2000

1 2 3 4 5 6 7 8

Cum Net Dem



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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 planninghorizon. How would we do that?

One method: In previous picture, draw astraight 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)

0

500

1000

1500

2000

1 2 3 4 5 6 7 8

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 1403(Mar) 960 960 0

4(Apr.) 1150 1280 130

5(May) 1460 1600 1406(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/mo with an

integer number of workers

Since all months do not have the same

number of workdays, a constant productionlevel may not translate to the same number

of workers each month.



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To 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) = .3421

= average number of units produced by

one worker in one day.



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Computing Constant Work Force

Assume we are given the following # working

days per month: 22, 16, 23, 20, 21, 22, 21,

22. March is still the critical month. Cum.net demand thru March = 960. Cum #

working days = 22+16+23 = 61. Find

960/61 = 15.7377 units/day implies15.7377/.3421 = 46 workers required.



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Why again did we pick on March?

Examining the graph we see that that was

the “Trigger point” where our constant

production line intersected the cumulativedemand line assuring NO STOCKOUTS!

Can we “prove” this is best?



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Tabulate Days/Production per Worker Vs.

Demand to find minimum numbers

Month # Work Days #Units/worker Forecast Demand net Min # Workers C. Net Demand C.Units/Worker

Min #

Workers

Jan 22.00 7.53 220.00 29.23 220.00 7.53 29.23

Feb 16.00 5.47 280.00 51.15 500.00 13.00 38.46

Mar 23.00 7.87 460.00 58.46 960.00 20.87 46.00

Apr 20.00 6.84 190.00 27.77 1150.00 27.71 41.50

May 21.00 7.18 310.00 43.15 1460.00 34.89 41.84

Jun 22.00 7.53 145.00 19.27 1605.00 42.42 37.84

Jul 21.00 7.18 110.00 15.31 1715.00 49.60 34.57

Aug 22.00 7.53 225.00 29.90 1940.00 57.13 33.96



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What should we look at?

Cumulative Demand says March needs

most workers – but will mean building

inventories in Jan + Feb If we keep this number of workers we will

continue to build inventory through the rest

of the plan



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Constant Work Force Production Plan

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 145Jun 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

A Al i i ll d h “Ch



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An Alternative is called the “Chase

Plan”

Here, we hire and fire workers to keep

inventory low

We would employ only the number ofworkers needed each month to meet

demand

Examining our chart (earlier) we need:» Jan: 30; Feb: 51; Mar: 59; Apr: 27; May: 43

» Jun: 20; Jul: 15; Aug: 30

A Al i i ll d h “Ch



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Plan”

So we hire or Fire (lay-off) monthly» Jan (starts with 40 workers): Fire 10 (cost $8000)

» Feb: Hire 21 (cost $16800)

» Mar: Hire 8 (cost $6400)

» Apr: Fire 31 (cost $38750)

» May: Hire 15 (cost $12000)

» Jun: Fire 23 (cost $28750)

» Jul: Fire 5 (cost $6250)» Aug: Hire 15 (cost $12000)

Total Personnel Costs: $128950

A Alt ti i ll d th “Ch



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Plan”

Inventory cost is essentially 165*8.5 =$1402.50

Employment costs: $428325 Chase Plan Total: $558677.50

Betters the “Constant Workforce Plan” by:

» 599590.50 – 558677.50 = 40913 But will this be good for your image?

Can we find a better plan?

Cost Reduction in Constant Work Force Plan



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Cost Reduction in Constant Work Force Plan

& Chase 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

0

500

1000

1500

2000

1 2 3 4 5 6 7 8



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Cost 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 makinganother 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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Optimal Solutions to Aggregate Planning

Problems Via Linear Programming

Linear Programming provides a means of solvingaggregate planning problems optimally. The LPformulation is fairly complex requiring 8Tvariables and 3T constraints, where T is the lengthof the planning horizon. Clearly, this can be aformidable linear program. The LP formulationshows that the modified plan we considered withtwo months of layoffs is in fact optimal for the

prototype problem.

E l i th O ti l (L P )



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Exploring the Optimal (L.P.)

Approach

We need an Objective Function for cost of the aggregate

plan (target minimization):

– Here the ci‟s are cost for hiring, firing, inventory, production,

etc – HT and FT are number of workers hired and fired

– IT, PT, OT, ST AND UT are numbers inventoried, produced on

regular time, overtime, by „sub-contract‟ or idle worker hours

respectively

1

T

H H F F I T R R o T u T S T

t

c N c N c I c P c O c U c S




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Approach

This objective Function would be subject to

a series of constraints (one for each period)

– Number of Worker Constraints:

– Inventory Constraints:

– Production Constraints:

1t t t tW W H F

1t t t t t I I P S D

Where: k*n is the number of units produced by a workert

on regular time during a period

t t t t t P k n W O U




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Approach

Assuming we allow no idle time and will produceonly on regular time

» No overtime or subcontracting

We would have:» 9 worker variables (W0 to Waug)

» 8 Hire Variable

» 8 Fire Variables

» 9 Inventory Variables (I0 to Iaug)» 8 Production Variables

» 8 „Demands‟

» And 1 complicated Objective function




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Approach

Lets try Excel!

This is a toughee!

Lots of variables and lots of constraints –

but work is straight forward!



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Disaggregation

Aggregate plans were built to optimal staffinglevels for “families” or groups of products

Disaggregation is a means to build specific“Master Production Schedules”

Typically by breaking down the aggregatingweights to individual parts – or working onschedules of these families as optimal

Later leads to values similar to EOQ which wewill explore in Chapter 4!

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