Session 09 Production Planning and Control

Aggregate Planning..

D 0 8 5 4 Supply Chain : Manufacturing and Warehousing

Bina Nusantara University 2

Aggregate Planning• Aggregate planning, which might also be called

Macro Production Planning,address the problem of deciding :– how many employees the firm should retain,– and for a manufacturing firm,

• The quantity and the mix of products to be produced.• Macro production planning is not limited to manufacturing

firms.• Macro production planning strategies are a fundamental

part of the firms’s overall business strategy.

Bina Nusantara University 3Source : Production and Operations Analysis 4th Edition, Steven Nahmias McGraw Hill International Edition

Aggregate Planning• The methodology of aggregate planning in this topic

requires the assumptioassumption that – Demand is deterministic , or known in advance.

• This assumption is made to simplify the analysis and allow us to focus on the systematic or predictable changes in the demand pattern, rather than on the unsystematic or random changes.

• The goal of the analysis is to determine the number of workers that should be employed each period and the number of aggregate units that should be produced each period.

Bina Nusantara University 4Source : Production and Operations Analysis 4th Edition, Steven Nahmias McGraw Hill International Edition

Aggregate Planning• The objective is to minimize costs of production, payroll,

holding and changing size of the workforce. The cost of making changes are generally referred to as smoothing costs.

• Most of the aggregate planning models discuss in this topic assume that all costs are linear functions.

This means that the cost of hiring an additional worker is the same as the cost of hiring the previous worker,andthe cost of holding and additional unit of inventory is the same as the cost of holding the previous unit of inventory.

Bina Nusantara University 5Source : Production and Operations Analysis 4th Edition, Steven Nahmias McGraw Hill International Edition

Hierarchy of Hierarchy of Production Production DecisionsDecisions

Long-range Capacity PlanningLong-range Capacity Planning

Planning HorizonPlanning Horizon

Aggregate planningAggregate planning: : Intermediate-range Intermediate-range capacity planning, usually covering 2 to 12 capacity planning, usually covering 2 to 12 months.months.

Shortrange

Intermediate range

Long range

Now 2 months 1 Year

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Aggregate Planning StrategiesAggregate Planning Strategies• Should Should inventoriesinventories be used to absorb changes in demand be used to absorb changes in demand

during planning period?during planning period?• Should demand changes be accommodated by varying the Should demand changes be accommodated by varying the

size of the size of the workforceworkforce??• Should part-timers be used, or should Should part-timers be used, or should overtime overtime and/or and/or

machine idle time machine idle time be used to absorb fluctuations?be used to absorb fluctuations?• Should Should subcontractorssubcontractors be used on fluctuating orders so a be used on fluctuating orders so a

stable workforce can be maintained?stable workforce can be maintained?• Should prices or other factors be changed to influence Should prices or other factors be changed to influence

demand?demand?

Why Aggregate Planning Is NecessaryWhy Aggregate Planning Is Necessary

• Fully load facilities and minimize overloading and Fully load facilities and minimize overloading and underloadingunderloading

• Make sure enough capacity available to satisfy Make sure enough capacity available to satisfy expected demandexpected demand

• Plan for the orderly and systematic change of Plan for the orderly and systematic change of production capacity to meet the peaks and production capacity to meet the peaks and valleys of expected customer demandvalleys of expected customer demand

• Get the most output for the amount of resources Get the most output for the amount of resources availableavailable

Aggregation Method Suggested by Hax and Aggregation Method Suggested by Hax and MealMeal

• They suggest grouping products into three categories: They suggest grouping products into three categories: • items, families, and types. items, families, and types. • Items are the finest level in the product structure and Items are the finest level in the product structure and

correspond to individual stock-keeping units. For example, correspond to individual stock-keeping units. For example, a firm selling refrigerators would distinguish white from a firm selling refrigerators would distinguish white from almond in the same refrigerator as different items. almond in the same refrigerator as different items.

• A family in this context would be refrigerators in general. A family in this context would be refrigerators in general. • Types are natural groupings of families; kitchen appliances Types are natural groupings of families; kitchen appliances

might be one type.might be one type.

Steps in Aggregate PlanningSteps in Aggregate Planning

• Prepare the sales forecast (Note that all producting Prepare the sales forecast (Note that all producting planning activities begin with sales forecast)planning activities begin with sales forecast)

• Total all the individual product or service forecasts into one Total all the individual product or service forecasts into one aggregate demand (if not homogeneous use labor-hours, aggregate demand (if not homogeneous use labor-hours, machine-hours or sales dollars)machine-hours or sales dollars)

• Transform the aggregate demand into worker, material and Transform the aggregate demand into worker, material and machine requirementsmachine requirements

• Develop alternative capacity plans Develop alternative capacity plans • Select a capacity plan which satisfies aggregate demand Select a capacity plan which satisfies aggregate demand

and best meets the objectives of the organization.and best meets the objectives of the organization.

Overview of the ProblemOverview of the Problem

Suppose that DSuppose that D11, D, D22, . . . , D, . . . , DTT are the forecasts of are the forecasts of demand for aggregate units over the planning demand for aggregate units over the planning horizon (T periods.) The problem is to determine horizon (T periods.) The problem is to determine both work force levels (Wboth work force levels (Wtt) and production levels ) and production levels (P(Ptt ) to minimize total costs over the T period ) to minimize total costs over the T period planning horizon. planning horizon.

Important IssuesImportant Issues in in Aggregate PlanningAggregate Planning

• Smoothing.Smoothing. Refers to the costs and disruptions Refers to the costs and disruptions that result from making changes that result from making changes in production in production and workforce levels and workforce levels from one period to the from one period to the nextnext (cost of hiring and firing workers) (cost of hiring and firing workers)..

• Bottleneck PlanningBottleneck Planning. Problem of . Problem of not not meeting meeting the the peak demand because of capacity peak demand because of capacity restrictions. A bottleneck occurs when the restrictions. A bottleneck occurs when the capacity of the productive system is insufficient capacity of the productive system is insufficient to meet a sudden surge in the demand. to meet a sudden surge in the demand. Bottlenecks can also occur in a particular part Bottlenecks can also occur in a particular part of the productive system due to the breakdown of the productive system due to the breakdown of a key piece of equipment or the shortage of of a key piece of equipment or the shortage of a critical resource.a critical resource.

Important IssuesImportant Issues in in Aggregate PlanningAggregate Planning

• Planning HorizonPlanning Horizon. The planning horizon is the number . The planning horizon is the number of periods of demand forecast used to generate the of periods of demand forecast used to generate the aggregate plan. If the horizon is too short, there may aggregate plan. If the horizon is too short, there may be insufficient time to build inventories to meet be insufficient time to build inventories to meet future demand surges and if it is too long the future demand surges and if it is too long the reliability of the demand forecasts is likely to be low.reliability of the demand forecasts is likely to be low. (ın practice, rolling schedules are used)(ın practice, rolling schedules are used)

• TTreatment of Demandreatment of Demand. Assume demand is known. . Assume demand is known. Ignores uncertainty to focus on the Ignores uncertainty to focus on the prpredictable/systematic variations in demand, such as edictable/systematic variations in demand, such as seasonality.seasonality.

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

in inventoryin inventory• Shortage CostsShortage Costs

– Cost of demand exceeding stock on hand. Cost of demand exceeding stock on hand.

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

Cost of Changing Cost of Changing the Size of the Workforcethe Size of the Workforce

Fig. 3-2

Holding and Back-Order CostsHolding and Back-Order Costs

Fig. 3-3

Back-orders Positive inventory

Slope = CP

Slope = Ci$

Cos

t

Inventory

Aggregate UnitsAggregate Units

The method is based on notion of aggregate units. The method is based on notion of aggregate units. They may beThey 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(See example 3.1)(See example 3.1)

Example 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?

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. 4.8644 worker hours. This approach for this This approach for this example is reasonable since products produced example is reasonable since products produced are similar. When products produced are are similar. When products produced are heterogeneous, a natural aggregate unit is sales heterogeneous, a natural aggregate unit is sales dollars.dollars.

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

Step 1: Determine “net” demand.Step 1: Determine “net” demand.(subtract starting inv(subtract starting inventoryentory from per from periodiod 1 forecast and add 1 forecast and add

ending invending inventoryentory to per to periodiod 8 forecast.) 8 forecast.)

MonthMonth Net PredictedNet Predicted Cum. NetCum. Net

Demand DemandDemand Demand

1(Jan)1(Jan) 220220 220220

2(Feb)2(Feb) 280280 500500

3(Mar)3(Mar) 460460 960960

4(Apr)4(Apr) 190190 1150 1150

5(May)5(May) 310310 14601460

6(June)6(June) 145145 16051605

7(July)7(July) 110110 17151715

8(Aug)8(Aug) 225225 19401940

Step 2. Graph Cumulative Net Demand to Find Plans Step 2. Graph Cumulative Net Demand to Find Plans GraphicallyGraphically

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Cum Net Dem

Basic StrategiesBasic Strategies• Constant Workforce (Level Capacity)Constant Workforce (Level Capacity) strategy: strategy:

– Maintaining a steady rate of regular-time output while Maintaining a steady rate of regular-time output while meeting variations in demand by a combination of meeting variations in demand by a combination of options.options.

• Zero InventoryZero Inventory (Matching Demand)(Matching Demand)strategy: strategy:

– Matching capacity to demand; the planned output for Matching capacity to demand; the planned output for a period is set at the expected demand for that a period is set at the expected demand for that period.period.

Constant WorkforceConstant Workforce Approach Approach

• AdvantagesAdvantages

– Stable output rates and workforceStable output rates and workforce

• DisadvantagesDisadvantages

– Greater inventory costsGreater inventory costs

– Increased overtime and idle timeIncreased overtime and idle time

– Resource utilizations vary over timeResource utilizations vary over time

Zero InventoryZero Inventory Approach Approach

• AdvantagesAdvantages

– Investment in inventory is lowInvestment in inventory is low

– Labor utilization iLabor utilization iss high high

• DisadvantagesDisadvantages

– The cost of adjusting output rates and/or workforce The cost of adjusting output rates and/or workforce levelslevels

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 produce slope of the line is the number of units to produce each month.each month.

Constant Workforce Plan (zero ending inv)

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

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

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

Constant Work Force Plan With No Stockouts

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

month is found from: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

ButBut - may not be realistic for several reasons: - may not be realistic for several reasons:• It may not be possible to achieve the production It may not be possible to achieve the production

level of 320 unit/mo with an integer number of level of 320 unit/mo with an integer number of workersworkers

• Since all months do not have the same number of Since all months do not have the same number of workdays, a constant production level may not workdays, a constant production level may not translate to the same number of workers each translate to the same number of workers each month.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 given (reasonable!)given (reasonable!)

• Compute a “K factor” given by:Compute a “K factor” given by:K = number of aggregate units produced by K = number of aggregate units produced by one worker one worker

in one dayin one day

Finding KFinding K

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

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

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

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Computing Constant Work Force -- RealisticallyComputing Constant Work Force -- Realistically

• Assume we are given the following # working days per month: Assume we are given the following # working days per month: 22, 16, 23, 20, 21, 22, 21, 22. 22, 16, 23, 20, 21, 22, 21, 22. – March is still the critical month. March is still the critical month.

• Cum. net demand thru March = 960. Cum. net demand thru March = 960. • Cum # working days = 22+16+23 = 61. Cum # working days = 22+16+23 = 61. • We find that:We find that:

– 960/61 = 15.7377 units/day 960/61 = 15.7377 units/day – 15.7377/.3421 = 46 workers required15.7377/.3421 = 46 workers required– Actually 46.003 – here we truncate because we are set to build Actually 46.003 – here we truncate because we are set to build

inventory so the low number should work (check for March stock inventory so the low number should work (check for March stock out) – however we must use care and typically ‘round up’ any out) – however we must use care and typically ‘round up’ any fractional worker calculations thus building more inventoryfractional worker calculations thus building more inventory

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Why again did we pick on March?Why again did we pick on March?

• Examining the graph we see that that was the Examining the graph we see that that was the “Trigger point” where our constant production “Trigger point” where our constant production line intersected the cumulative demand line line intersected the cumulative demand line assuring NO STOCKOUTS!assuring NO STOCKOUTS!

• Can we “prove” this is best?Can we “prove” this is best?

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Tabulate Days/Production Per Worker Vs. Demand To Find Tabulate Days/Production Per Worker Vs. Demand To Find Minimum NumbersMinimum 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?What Should We Look At?

• Cumulative Demand says March needs most Cumulative Demand says March needs most workers – but will mean building inventories in Jan workers – but will mean building inventories in Jan + Feb to fulfill the greater March demand+ Feb to fulfill the greater March demand

• If we keep this number of workers we will If we keep this number of workers we will continue to build inventory through the rest of the continue to build inventory through the rest of the plan!plan!

Constant Work Force Production PlanConstant Work Force Production Plan

MoMo # wk days Prod. Cum Cum Nt End Inv # wk days Prod. Cum Cum Nt 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

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

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

• Assume that the work force at Assume that the work force at the the end of Dec was 40.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 out (126+98+0+. . .+687) = 2093. Add in 100 units netted out in Aug = 2193. Hence Inv. Cost = 2193*8.5=$18,640.50in Aug = 2193. Hence Inv. Cost = 2193*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

Cost Reduction in Constant Work Force PlanCost Reduction in Constant Work Force Plan(Mixed Strategy)(Mixed Strategy)

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

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

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An Alternative is called the “Chase Plan”An Alternative is called the “Chase Plan”

• Here, we hire and fire (layoff) workers to keep Here, we hire and fire (layoff) workers to keep inventory low!inventory low!

• We would employ only the number of workers We would employ only the number of workers needed each month to meet demandneeded each month to meet demand

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

43 Jun: 20; Jul: 15; Aug: 3043 Jun: 20; Jul: 15; Aug: 30• Found by: (monthly demand) Found by: (monthly demand) (monthly (monthly

pr. /worker)pr. /worker)

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An Alternative is called the “Chase Plan”An Alternative is called the “Chase Plan”

• So we hire or Fire (lay-off) monthlySo we hire or Fire (lay-off) monthly• Jan (starts with 40 workers): Fire 10 (cost $8000)Jan (starts with 40 workers): Fire 10 (cost $8000)• Feb: Hire 21 (cost $16800)Feb: Hire 21 (cost $16800)• Mar: Hire 8 (cost $6400)Mar: Hire 8 (cost $6400)• Apr: Fire 31 (cost $38750)Apr: Fire 31 (cost $38750)• May: Hire 15 (cost $12000)May: Hire 15 (cost $12000)• Jun: Fire 23 (cost $28750)Jun: Fire 23 (cost $28750)• Jul: Fire 5 (cost $6250)Jul: Fire 5 (cost $6250)• Aug: Hire 15 (cost $12000)Aug: Hire 15 (cost $12000)

• Total Personnel Costs: $128950Total Personnel Costs: $128950

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

Period Period # hired # hired #fired#fired

1 1 10 10

2 2 2 211

3 3 88

4 314 31

5 15 5 15

6 246 24

7 47 4

8 15 8 15

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An Alternative is called the “Chase Plan”An Alternative is called the “Chase Plan”

• Inventory cost is essentially 165*8.5 = $1402.50Inventory cost is essentially 165*8.5 = $1402.50• Employment costs: $428325Employment costs: $428325• Chase Plan Total: $558677.50Chase Plan Total: $558677.50• Betters the “Constant Workforce Plan” by:Betters the “Constant Workforce Plan” by:

• 599590.50 – 558677.50 = 40913599590.50 – 558677.50 = 40913• But will this be good for your image?But will this be good for your image?• Can we find a better plan?Can we find a better plan?

Disaggregating The Aggregate PlanDisaggregating The Aggregate Plan

• Disaggregation of aggregate plans mean Disaggregation of aggregate plans mean converting an converting an aggregate plan to a detailed master production schedule aggregate plan to a detailed master production schedule for each individual itemfor each individual item (remember the hierarchical product (remember the hierarchical product structure given earlier: items, families, types)structure given earlier: items, families, types)..

• Keep in mind that uKeep in mind that unless the results of the aggregate plan nless the results of the aggregate plan can be linked to the master production schedule, the can be linked to the master production schedule, the aggregate planning methodology could have little value.aggregate planning methodology could have little value.

Aggregate Plan to Master ScheduleAggregate Plan to Master Schedule

AggregatePlanning

Disaggregation

MasterSchedule

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 formulation is fairly complex requiring 8T decision decision variablesvariables(1.workforce level,(1.workforce level, 2. production level, 3. 2. production level, 3. inventory level, 4. # of workers hired, 5. # of workres inventory level, 4. # of workers hired, 5. # of workres fired, 6. overtime production, 7. idletime, 8. fired, 6. overtime production, 7. idletime, 8. subcontracting) subcontracting) and 3T constraintsand 3T constraints (1. workforce, 2. (1. workforce, 2. production, 3. inventory)production, 3. inventory), where T is the length of the , where T is the length of the planning horizon. planning horizon. (See section 3.5, pg.125)(See section 3.5, pg.125)

Optimal Solutions to Aggregate Planning Problems Optimal Solutions to Aggregate Planning Problems Via Linear ProgrammingVia Linear Programming

Clearly, this can be a formidable linear program. Clearly, this can be a formidable linear program. The LP formulation shows that the modified plan The LP formulation shows that the modified plan we considered with two months of layoffs is in we considered with two months of layoffs is in fact optimal for the prototype problem.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.

Exploring the Optimal (L.P.) ApproachExploring the Optimal (L.P.) Approach

• We need an Objective Function for cost of the aggregate plan We need an Objective Function for cost of the aggregate plan (target is to minimize costs):(target is to minimize costs):

– Here the cHere the cii’s are cost for hiring, firing, inventory, production, etc’s are cost for hiring, firing, inventory, production, etc

– HHTT and F and FTT are number of workers hired and fired are number of workers hired and fired

– IITT, P, PTT, O, OTT, S, STT AND U AND UTT are numbers units inventoried, produced on regular are numbers units inventoried, produced on regular time, on overtime, by ‘sub-contract’ or the number of units that could time, on overtime, by ‘sub-contract’ or the number of units that could be produced on idled worker hours respectivelybe produced on idled worker hours respectively

1

T

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

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

Exploring the Optimal (L.P.) ApproachExploring the Optimal (L.P.) Approach

• This objective Function would be subject to a series of constraints This objective Function would be subject to a series of constraints (one of each type for each period)(one of each type for each period)

• ‘‘Number of Workers’ Constraints:Number of Workers’ Constraints:

• Inventory Constraints:Inventory Constraints:

• Production Constraints:Production Constraints:

1t t t tW W H F

1t t t t tI I P S D

t t t t tP k n W O U Where: nt * k is the number of units

produced by a worker in a given period of nt days

Real Constraint Equation (rewritten for L.P.):Real Constraint Equation (rewritten for L.P.):• Employee Constraints:Employee Constraints:

• Inventory Constraints:Inventory Constraints:

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1 0 1 1

0

0

Specifically:

t t t tW W H F

W W H F

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specifically:

t t t t tP I I S D

P I I S D

Real Constraint Equations (rewritten for L.P.):Real Constraint Equations (rewritten for L.P.):• Production Constraints:Production Constraints:

1 1 1 1 1

0

0

specifically:

t t t t tP k n W O U

P k n W O U

Real Constraint Equations (rewritten for L.P.):Real Constraint Equations (rewritten for L.P.):

• Finally, we need constraints defining:Finally, we need constraints defining:– Initial Workforce sizeInitial Workforce size– Starting InventoryStarting Inventory– Final Desired InventoryFinal Desired Inventory– And, of course, the general constraint forcing all And, of course, the general constraint forcing all

variables to be variables to be 0 0