1AGGREGATE PLANNING

Production Planning and Production Planning and ControlControl

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Haeryip SihombingFakulti Kejuruteraan Pembuatan

Universiti Teknologi Malaysia Melaka

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Chapter OutlineChapter Outline

I.I. IntroductionIntroduction

II.II.The Concept of AggregationThe Concept of Aggregation

III.III. An Overview of Production-Planning ActivitiesAn Overview of Production-Planning Activities

IV.IV. Framework for Aggregate Production PlanningFramework for Aggregate Production Planning

V.V. Techniques for Aggregate Production Techniques for Aggregate Production PlanningPlanning

VI.VI. Aggregate Planning in Service CompaniesAggregate Planning in Service Companies

VII.VII. Implementing Aggregate Production Plans - Implementing Aggregate Production Plans - Managerial IssuesManagerial Issues

VIII.VIII. Hierarchical Production PlanningHierarchical Production Planning

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Aggregate production Aggregate production planningplanning is medium-term is medium-term

capacity planning over a two to capacity planning over a two to eighteen month planning eighteen month planning

horizon. It involves determining horizon. It involves determining the lowest-cost method of the lowest-cost method of

providing the adjustable capacity providing the adjustable capacity for meeting production for meeting production

requirementsrequirements..

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Capacity Decisions HierarchyCapacity Decisions Hierarchy

LinkagesFacilitiesPlanningAggregatePlanning

Scheduling

Time FrameFacilities Planning

Aggregate PlanningScheduling

Time

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Aggregation Aggregation refers to the idea refers to the idea of focusing on overall capacity, of focusing on overall capacity, rather than individual products rather than individual products

or services.or services.

Aggregation is done according to:Aggregation is done according to: ProductsProducts LaborLabor TimeTime

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Production PlanningProduction Planning

Long Range PlanningLong Range Planning Strategic planning (1-5 years)Strategic planning (1-5 years)

Medium Range PlanningMedium Range Planning Employment, output, and inventory Employment, output, and inventory

levels (2-18 months)levels (2-18 months) Short Range PlanningShort Range Planning

Job scheduling, machine loading, and Job scheduling, machine loading, and job sequencing (0-2 months)job sequencing (0-2 months)

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Aggregate production planning Aggregate production planning involves managing...involves managing...

Work force levelsWork force levels - the number - the number of workers required for production.of workers required for production.

Production ratesProduction rates - the number - the number of units produced per time period.of units produced per time period.

Inventory levelsInventory levels - the balance of - the balance of unused units carried forward from unused units carried forward from the previous period.the previous period.

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Common objectives of production Common objectives of production planning...planning...

MINIMIZE:MINIMIZE:cost, inventory levels, changes in work force levels, use of overtime, use of subcontracting, changes in production rates, changes in production rates, plant/personnel idle time

MAXIMIZE:MAXIMIZE:profits, customer service

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Methods of Influencing DemandMethods of Influencing Demand

Price IncentivesPrice Incentives ReservationsReservations BacklogsBacklogs Complementary Products or ServicesComplementary Products or Services Advertising/promotionAdvertising/promotion

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Methods of Influencing SupplyMethods of Influencing Supply

Hiring/firing workersHiring/firing workers Overtime/slack timeOvertime/slack time Part time/temporary laborPart time/temporary labor SubcontractingSubcontracting Cooperative arrangementsCooperative arrangements InventoriesInventories

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Aggregate Production Planning Aggregate Production Planning Variable CostsVariable Costs

Hiring/firing costsHiring/firing costs Overtime/slack time costsOvertime/slack time costs Part time/temporary labor costsPart time/temporary labor costs Subcontracting costsSubcontracting costs Cooperative arrangements costsCooperative arrangements costs Inventory carrying costsInventory carrying costs Backorder or stock out costsBackorder or stock out costs

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Aggregate Production Planning Aggregate Production Planning StrategiesStrategies

Chase strategyChase strategy production rates or work force levels are production rates or work force levels are

adjusted to match demand requirements adjusted to match demand requirements over planning horizonover planning horizon

Level strategyLevel strategy constant production rate or work force constant production rate or work force

level is maintained over planning horizonlevel is maintained over planning horizon Mixed strategyMixed strategy

both inventory level changes and work both inventory level changes and work force level changes occurforce level changes occur

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Aggregate Production Planning Aggregate Production Planning TechniquesTechniques

Trial-and-error methodTrial-and-error method Mathematical techniquesMathematical techniques

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Trial-and-Error MethodTrial-and-Error Method

Examples of alternative strategies:Examples of alternative strategies: Vary work force levelsVary work force levels Level work force, vary inventories Level work force, vary inventories

and backordersand backorders Level work force, use Level work force, use

subcontractingsubcontracting Level work force, use overtime and Level work force, use overtime and

subcontractingsubcontracting

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Mathematical TechniquesMathematical Techniques

- Linear Decision Rule - Linear Decision Rule

- Mgmt. Coefficient - Mgmt. Coefficient ModelsModels

- Parametric Prod. - Parametric Prod. Planning Planning

- Search Decision - Search Decision RuleRule

- Production-- Production-Switching Heuristic Switching Heuristic

- Linear Programming- Linear Programming

- Transportation - Transportation MethodMethod

- Goal Programming- Goal Programming

- Mixed Integer - Mixed Integer Programming Programming

- Simulation Models- Simulation Models

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Managerial Issues Managerial Issues in Aggregate Production Planningin Aggregate Production Planning

1. APP should be tailored to the particular company and situation.

2. APP may be constrained by union contracts or company policies.

3. Mathematical techniques will likely have to be balanced with managerial judgment and experience.

4. A tendency to blur the distinction between production planning and production scheduling.

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Aggregate Planning in ServicesAggregate Planning in Services

For service companies, For service companies, aggregate planning results in aggregate planning results in staffing plans that call for staffing plans that call for changing the number of changing the number of employees or subcontracting.employees or subcontracting.

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The ENDThe END

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Production Planning Production Planning EnvironmentEnvironment

Competitor’sBehavior

Raw MaterialAvailability

Market Demand

Planningfor

Production

ExternalCapacity

(outsourcing)

EconomicConditions

CurrentPhysicalCapacity

CurrentInventory

CurrentWork Force

RequiredProductionActivities

““Long-range plan” (3-10 years) updated Long-range plan” (3-10 years) updated yearlyyearly

Inputs: aggregate forecasts (units) and current Inputs: aggregate forecasts (units) and current plant capacity (hours)plant capacity (hours)

Decision: build new plant, expand an existing Decision: build new plant, expand an existing plant, create new product line, expand, contract, plant, create new product line, expand, contract, or delete existing product linesor delete existing product lines

Level of detail: Very AggregatedLevel of detail: Very Aggregated Degree of uncertainty: HighDegree of uncertainty: High

Planning ProductionPlanning Production

““Intermediate-range plan” (6 month – 2 Intermediate-range plan” (6 month – 2 years) updated quarterlyyears) updated quarterly

Inputs: aggregate capacity and product Inputs: aggregate capacity and product decisions from the long-term plan, units are decisions from the long-term plan, units are aggregated by product line or family and plant aggregated by product line or family and plant departmentdepartment

Decision: changes in work force, additional Decision: changes in work force, additional machines, subcontracting, overtimemachines, subcontracting, overtime

Level of detail: AggregatedLevel of detail: Aggregated Degree of uncertainty: MediumDegree of uncertainty: Medium

Planning ProductionPlanning Production

““Short-range plan” (1 week – 6 month) Short-range plan” (1 week – 6 month) updated daily or weeklyupdated daily or weekly

Inputs: decisions from the intermediate-term plan, Inputs: decisions from the intermediate-term plan, units are aggregated by particular product and units are aggregated by particular product and capacity – available hours on a particular machine, capacity – available hours on a particular machine, short range forecast, inventory levels, work force short range forecast, inventory levels, work force levels, processeslevels, processes

Decision: overtime and undertime, possibility of Decision: overtime and undertime, possibility of not fulfilling all demand, subcontracting, delivery not fulfilling all demand, subcontracting, delivery dates for suppliers, product qualitydates for suppliers, product quality

Level of detail: Very DetailedLevel of detail: Very Detailed Degree of uncertainty: LowDegree of uncertainty: Low

Planning ProductionPlanning Production

Production Planning ExampleProduction Planning Example Small company makes one product – plastic cases to hold CD’s.Small company makes one product – plastic cases to hold CD’s. Two different types of mold are used to produce top & bottom.Two different types of mold are used to produce top & bottom. Two halves are manually put together, placed in the boxes & shipped.Two halves are manually put together, placed in the boxes & shipped. The injection molding machines can make 550 pieces per hour.The injection molding machines can make 550 pieces per hour. A worker can finish 55 cases in 1 hour (10 workers / machine)A worker can finish 55 cases in 1 hour (10 workers / machine) Forecasts of demand: 80,000 cases per month for next year Forecasts of demand: 80,000 cases per month for next year at 4 at 4

weeks/month the demand should be 20,000 cases per week.weeks/month the demand should be 20,000 cases per week. Company runs 5 out of 7 days per week: 4,000 cases per day needed.Company runs 5 out of 7 days per week: 4,000 cases per day needed. Each worker can not work more than 8 hours per dayEach worker can not work more than 8 hours per day 4,000/8 = 500 pieces per hour have to be produced.4,000/8 = 500 pieces per hour have to be produced.

Plan: 1 machine, 10 workers, 8 hours/day, 5 days/weekPlan: 1 machine, 10 workers, 8 hours/day, 5 days/week

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The Hierarchy of The Hierarchy of Production Planning DecisionsProduction Planning Decisions

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Goal: To plan gross work force levels and Goal: To plan gross work force levels and set firm-wide production plans. set firm-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 ” of production. May be actual units, or may be measured in weight actual units, or may be measured in weight (tons of steel), volume (gallons of gasoline), (tons of steel), volume (gallons of gasoline), time (worker-hours), or dollars of sales. Can time (worker-hours), or dollars of sales. Can even be a fictitious quantity. (Refer to even be a fictitious quantity. (Refer to example in text and in slide below.)example in text and in slide below.)

Introduction to Aggregate Introduction to Aggregate PlanningPlanning

Introduction to Aggregate Introduction to Aggregate PlanningPlanning

Constant production rate can be satisfied with Constant production rate can be satisfied with constant capacity. constant capacity.

Work force is constant, production rate slightly less Work force is constant, production rate slightly less that capacity of people & machines: good utilization that capacity of people & machines: good utilization without overloading the facilities.without overloading the facilities.

Raw material usage is also constant.Raw material usage is also constant. If supplier and customers are also close, frequent If supplier and customers are also close, frequent

deliveries of raw material and finished goods will deliveries of raw material and finished goods will keep inventory low.keep inventory low.

How realistic is this example?How realistic is this example? Strategies to cope with fluctuating demand?Strategies to cope with fluctuating demand?

-- change the demand -- produce at constant rate anyway-- vary the production rate -- use combination of above strategies

Introduction to Aggregate Introduction to Aggregate Planning:Planning:

Influencing DemandInfluencing Demand

Do not satisfy demand during peak Do not satisfy demand during peak periodsperiods

Capacity < Peak demand , constant production rateCapacity < Peak demand , constant production rate Loss of some salesLoss of some sales

Japanese car manufacturers often take this stanceJapanese car manufacturers often take this stance Determine percentage of the market shareDetermine percentage of the market share Constant production is set at this levelConstant production is set at this level Sales personal expected to sell produced amountSales personal expected to sell produced amount

Ease of planning must be compared to lost revenueEase of planning must be compared to lost revenue

Introduction to Aggregate Introduction to Aggregate Planning:Planning:

Influencing DemandInfluencing Demand

Shift demand from peak periods to non-peak Shift demand from peak periods to non-peak periods / create new demand for non-peak periods / create new demand for non-peak periodsperiods

Creating new demand can be done through advertising Creating new demand can be done through advertising or incentive programs (automobile industry: rebates; or incentive programs (automobile industry: rebates; telephone company’s – differential pricing system)telephone company’s – differential pricing system)

Smoothing demandSmoothing demand

Introduction to Aggregate Introduction to Aggregate Planning:Planning:

Influencing DemandInfluencing Demand

Produce several products with peak Produce several products with peak demand in different periodsdemand in different periods

Products should be similar, so that Products should be similar, so that manufacturing them is not too differentmanufacturing them is not too different

Snowmobiles and jetskis – same engines, Snowmobiles and jetskis – same engines, similar body worksimilar body work

Lawn-mowers – snowblowers; baseball – Lawn-mowers – snowblowers; baseball – football equipmentfootball equipment

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Medium Range Planning: Medium Range Planning: Aggregate Production PlanningAggregate Production Planning

Establish production rates by major product Establish production rates by major product groupsgroups by labor hours required or units of productionby labor hours required or units of production

Attempt to determine monthly work force Attempt to determine monthly work force size and inventory levels that minimizes size and inventory levels that minimizes production related production related costscosts over the planning over the planning period (for 6-24 month)period (for 6-24 month)

Relevant Costs InvolvedRelevant Costs Involved

Regular time costsRegular time costs Costs of producing a unit of output during regular working Costs of producing a unit of output during regular working

hours, including direct and indirect labor, material, hours, including direct and indirect labor, material, manufacturing expensesmanufacturing expenses

Overtime costsOvertime costs Costs associated with using manpower beyond normal working Costs associated with using manpower beyond normal working

hourshours Production-rate change costsProduction-rate change costs

Costs incurred in substantially altering the production rateCosts incurred in substantially altering the production rate Inventory associated costsInventory associated costs

Out of pocket costs associated with carrying inventoryOut of pocket costs associated with carrying inventory Costs of insufficient capacity in the short runCosts of insufficient capacity in the short run

Costs incurred as a result of backordering, lost sales revenue, Costs incurred as a result of backordering, lost sales revenue, loss of goodwill + costs of actions initiated to prevent loss of goodwill + costs of actions initiated to prevent shortagesshortages

Control system costsControl system costs Costs of acquiring the data for analytical decision, Costs of acquiring the data for analytical decision,

computational effort and implementation costscomputational effort and implementation costs

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

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 result from making changes from one period to the next.next.

Bottleneck PlanningBottleneck Planning. Problem of meeting peak . Problem of meeting peak demand because of capacity restrictions. demand 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 predictable/systematic variations in demand, such as seasonality.as 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 primary component: opportunity cost of

investmentinvestment 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.

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Cost 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

Aggregate UnitsAggregate Units

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

Example of fictitious aggregate Example of fictitious aggregate units.units.

(Example.1)(Example.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 55324.24.2 285285 3232

K 4242K 42424.94.9 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 Notice: Price is not necessarily proportional to worker hours (i.e., cost): why?to 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) + . . . requires: .32(4.2) + .21(4.9) + . . . + .06(5.8) = 4.8644 worker hours. Forecasts + .06(5.8) = 4.8644 worker hours. Forecasts for demand for aggregate units can be for demand for aggregate units can be obtained by taking a weighted average obtained by taking a weighted average (using the same weights) of individual item (using the same weights) of individual item forecasts. forecasts.

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

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

add ending inv. to per. 8 forecast.)add ending inv. 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 Step 2. Graph Cumulative Net Demand to Find Plans GraphicallyDemand to Find Plans Graphically

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Cum Net Dem

Constant Work Force PlanConstant Work Force Plan

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

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

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

Constant Workforce Plan (zero ending inv)

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

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

Constant Work Force Plan With No Stockouts

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

Ending inventory each month is found from: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

ButBut - may not be realistic for - may not be realistic for several reasons:several 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 with an integer number of workersan integer number of workers

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

To overcome these shortcomings:To overcome these shortcomings:

Assume number of workdays per month Assume number of workdays per month is givenis given

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

Finding KFinding K

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

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, working days per month: 22, 16, 23, 20, 21, 17, 18, 10. March is still critical month. 21, 17, 18, 10. March is still critical month. Cum. net demand thru March = 960. Cum Cum. net demand thru March = 960. Cum ## of of 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.

Constant Work Force Production Constant Work Force Production PlanPlan

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

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 Cost Evaluation of Constant Work Force PlanPlan

Assume that the work force at end of Dec was Assume that the work force at end of Dec was 40.40.

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 (126+98+0+. . .+687) = 2093. Add in 100 units netted out in Aug = 2193. Hence Inv. Cost = netted 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) = ($75/worker/day)(46 workers )(167days) = $576,150$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

Cost Reduction in Constant Work Force Cost Reduction in Constant Work Force PlanPlan

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

inventory investmentinventory 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 workforce to 36 at start of April and making another reduction to 22 at start making another reduction to 22 at start of June. The additional cost of layoffs is of June. The additional cost of layoffs is $30,000, but holding costs are reduced $30,000, but holding costs are reduced to only $4,250. The total cost of the to only $4,250. The total cost of the modified plan is $467,450. modified plan is $467,450.

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

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

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

Aggregate UnitsAggregate Units

The method is based on notion of The method is based on notion of aggregate units. 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

Overview of the ProblemOverview of the Problem

DD11, D, D22, . . . , D, . . . , DTT - the forecasts of demand for aggregate - the forecasts of demand for aggregate units over the planning horizon (T periods) units over the planning horizon (T periods)

Determine: Determine: WWtt - work force levels - work force levels

PPtt - production levels - production levels

IItt – inventory levels – inventory levels

HHtt – number of workers hired in this period – number of workers hired in this period

FFtt – number of workers fired in this period – number of workers fired in this period

OOtt – overtime production in units – overtime production in units

UUtt – “undertime”, worker idle time in units – “undertime”, worker idle time in units

SStt – number of units subcontracted from – number of units subcontracted from outsideoutside

to minimize total costs over the T period planning horizonto minimize total costs over the T period planning horizon

Example of fictitious aggregate Example of fictitious aggregate units:units:

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

Model # hrs.Model # hrs. Price % sales Price % salesA 5532A 5532 4.24.2 285285 3232K 4242K 42424.94.9 345345 2121L 9898L 9898 5.15.1 395395 1717L 3800L 3800 5.25.2 425425 1414M 2624M 2624 5.45.4 525525 1010M 3880M 3880 5.85.8 725725 0606

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

Price/Price/#hours#hours

67.8667.8670.4170.4177.4577.4581.7381.7397.2297.22125.0125.0

Example (continued)Example (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:

0.32(4.2) + 0.21(4.9) + 0.17(5.1) + 0.14(5.2) 0.32(4.2) + 0.21(4.9) + 0.17(5.1) + 0.14(5.2) + 0.10(5.4) + 0.06(5.8) = 4.856 worker hours+ 0.10(5.4) + 0.06(5.8) = 4.856 worker hours

Forecasts for demand for aggregate units can Forecasts for demand for aggregate units can be obtained by taking a weighted average be obtained by taking a weighted average (using the same weights) of individual item (using the same weights) of individual item forecasts. forecasts.

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 monthsfor the next 8 months

Forecasted demands for Jan-Aug. are: Forecasted demands for Jan-Aug. are:

420, 280, 460, 190, 310, 145, 110, 125420, 280, 460, 190, 310, 145, 110, 125

Starting inventory at the end of December is Starting inventory at the end of December is 200 and the firm would like to have 100 units 200 and the firm would like to have 100 units on hand at the end of Auguston hand at the end of August

Find monthly production levelsFind monthly production levels

Example Example (continued)(continued)

Step 1: Determine “net” demand.Step 1: Determine “net” demand.(subtract starting inventory from period 1 forecast and (subtract starting inventory from period 1 forecast and

add ending inventory to period 8 forecast)add ending inventory to period 8 forecast)

Month ForecastedMonth Forecasted Net PredictedNet Predicted Cum. NetCum. Net Demand Demand Demand Demand Demand Demand1(Jan)1(Jan) 420420 420-200=220420-200=220 220 2202(Feb)2(Feb) 280 280 280280 500 5003(Mar)3(Mar) 460 460 460460 960 9604(Apr)4(Apr) 190 190 190190 1150 11505(May)5(May)310 310 310310 1460 14606(June)6(June) 145 145 145145 1605 16057(July)7(July) 110 110 110110 1715 17158(Aug)8(Aug) 125 125 125+100=225125+100=225 1940 1940 Starting inventory - 200 and final inventory - Starting inventory - 200 and final inventory -

100 units100 units

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

Determine a production plan that doesn’t change the size of the workforce over the planning horizon. What to do?

Draw a straight line from first point 220 to 1940 units in month 8: The slope of the line is the number of units to produce each month.

Monthly Production =

= 1940 / 8 = 242.5 (rounded to 243/month)

Any shortfalls in this solution?Any shortfalls in this solution?

Constant Workforce Plan (zero ending inv)

0

500

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1 2 3 4 5 6 7 8

Demand is backlogged

Constant work force plan with no 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?

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

From the previous graph, we see that cum. net From the previous graph, we see that cum. net demand curve is crossed at period 3, so that monthly demand curve is crossed at period 3, so that monthly

production is 960/3 = 320. Ending inventory each production is 960/3 = 320. Ending inventory each month is found from:month is found from:

Constant work force plan with no stockouts

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However…However…

This solution may not be realistic for several This solution 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/mo with an production level of 320 unit/mo 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 monthof workers each month

Some thoughts: Some thoughts: Final inventory is 620 units, not 100 unitsFinal inventory is 620 units, not 100 units Cost of carrying inventory in each periodCost of carrying inventory in each period

Production Strategies:Production Strategies:

Constant production rate with Zero Constant production rate with Zero inventoryinventory stockoutsstockouts carrying inventorycarrying inventory

Constant production rate with no stockoutsConstant production rate with no stockouts carrying inventorycarrying inventory extra inventory at the period Textra inventory at the period T

Mixed strategyMixed strategy few changes in the workforce allowedfew changes in the workforce allowed more flexibilitymore flexibility lower costslower costs

Example #2Example #2 (based on example #1)(based on example #1)

The plant has 38 workers who produced 630 The plant has 38 workers who produced 630 units in a period of 40 daysunits in a period of 40 days

K= 630/(38*40) = 0.414 K= 630/(38*40) = 0.414 average number average number of units produced by one worker in one dayof units produced by one worker in one day

Assume we are given the following # Assume we are given the following # working days per month: working days per month:

jan 22jan 22 apr apr 2020 juljul 1818feb 16feb 16 may may 2121 augaug 1010mar 23 mar 23 junjun 1717

Constant Work Force Production Plan:Constant Work Force Production Plan:

38 workers, K= .414 38 workers, K= .414

Month # wk Month # wk Prod. Cum Cum Nt End Inv Prod. Cum Cum Nt End Inv days Dem Level Prod Demdays Dem Level Prod Dem Jan 22 220 346 346 220 126Jan 22 220 346 346 220 126 Feb 16 280 252 598 500 98Feb 16 280 252 598 500 98 Mar 23 460 362 960 960 0Mar 23 460 362 960 960 0 Apr 20 190 315 1275 1150 125Apr 20 190 315 1275 1150 125 May 21 310 330 1605 1460 145May 21 310 330 1605 1460 145 Jun 22 145 346 1951 1605 346Jun 22 145 346 1951 1605 346 Jul 21 110 330 2281 1715 566Jul 21 110 330 2281 1715 566 Aug 22 125 346 2627 1940 687Aug 22 125 346 2627 1940 687 +100 +100

Addition of CostsAddition of Costs

Holding Cost (per unit per month): $ 8.50Holding Cost (per unit per month): $ 8.50 Hiring Cost per worker: $ 800.00Hiring Cost per worker: $ 800.00 Firing Cost per worker: $ 1,250.00Firing Cost per worker: $ 1,250.00 Payroll Cost ( per worker/day): $ 75.00Payroll Cost ( per worker/day): $ 75.00 Shortage Cost (unit short/month): $ 50.00Shortage Cost (unit short/month): $ 50.00

Cost Cost EvaluationEvaluation of Constant Work of Constant Work Force PlanForce Plan

Assume that the work force at end of Dec was 32Assume that the work force at end of Dec was 32 Cost to hire 6 workers: 6*800 = $4,800Cost to hire 6 workers: 6*800 = $4,800 Inventory Cost Inventory Cost accumulate ending inventory: accumulate ending inventory:

(126+98+0+125+145+346+567+687) = 2,095(126+98+0+125+145+346+567+687) = 2,095

(100 units at the end of the august in included in 687 (100 units at the end of the august in included in 687 units inventory) units inventory)

Hence Inventory Cost = 2095*8.5=$17,809.37Hence Inventory Cost = 2095*8.5=$17,809.37 Payroll cost: Payroll cost:

($75/worker/day)(38 workers )(167days) = $475,950($75/worker/day)(38 workers )(167days) = $475,950 Cost of plan: Cost of plan:

$475,950 + $17,809.37 + $4800 = $475,950 + $17,809.37 + $4800 = $498,559.37$498,559.37

Cost Reduction in Constant Work Force Cost Reduction in Constant Work Force PlanPlan

In the original cum net demand curve, consider In the original cum net demand curve, consider making reductions in the work force one or more making reductions in the work force one or more times over the planning horizon to decrease inventory times over the planning horizon to decrease inventory investment.investment.

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Cum Dem

Cost Evaluation of Modified Plan with Cost Evaluation of Modified Plan with One Workforce Adjustment:One Workforce Adjustment:

The modified plan calls forThe modified plan calls for hiring 6 workers in Jan (to 38)hiring 6 workers in Jan (to 38) reducing the workforce to 23 (from 38) at start of Aprilreducing the workforce to 23 (from 38) at start of April

cost of hiring iscost of hiring is $ 4,800.00 $ 4,800.00 $ 4,800.00 $ 4,800.00 cost of layoffs is $ cost of layoffs is $ 18,750.0018,750.00 $ 0.00 $ 0.00 payroll cost is payroll cost is $ 356,700.00 $ $ 356,700.00 $ 475,950.00475,950.00 holding costs are $ 2,528.93 $ holding costs are $ 2,528.93 $ 17,809.3717,809.37 shortage costs are $ shortage costs are $ 7,770.407,770.40 $ 0.00 $ 0.00 The total cost of the modified plan is $ 390,548.33The total cost of the modified plan is $ 390,548.33 Original plan had cost of Original plan had cost of $ 498,559.37$ 498,559.37

Cost Evaluation of Modified Plan with Cost Evaluation of Modified Plan with Two Workforce Adjustment:Two Workforce Adjustment:

The modified plan calls for The modified plan calls for hiring 6 workers in Januaryhiring 6 workers in January firing 8 workers at start of Aprilfiring 8 workers at start of April firing 12 workers at start of Junefiring 12 workers at start of June

TwoTwo One One None None cost of hiring is $ 4,800.00 $ 4,800.00 $ 4,800.00cost of hiring is $ 4,800.00 $ 4,800.00 $ 4,800.00 cost of layoffs is cost of layoffs is $ 25,000.00$ 25,000.00 $ $ 18,750.0018,750.00 $ 0.00 $ 0.00 payroll cost is $ 353,850.00 payroll cost is $ 353,850.00 $ 356,700.00$ 356,700.00 $ 475,950.00$ 475,950.00 holding costs are holding costs are $ 3,452.87$ 3,452.87 $ 2,528.93 $ 2,528.93 $ 17,809.37$ 17,809.37 shortage costs are $ 0.00 shortage costs are $ 0.00 $ 7,770.40$ 7,770.40 $ 0.00 $ 0.00 The total cost : $ 387,102.87 The total cost : $ 387,102.87 $ 390,548.33$ 390,548.33 $ 498,559.37$ 498,559.37

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Cum Dem

Constant Work Force Production Plan:Constant Work Force Production Plan: 38 workers, K= .414 38 workers, K= .414

Month # wk Month # wk Prod. Cum Cum Nt End Inv Prod. Cum Cum Nt End Inv days Dem Level Prod Demdays Dem Level Prod Dem Jan Jan 22 22 220 346 346 220 346 346 220220 126 126 Feb Feb 1616 280 252 598 280 252 598 500500 98 98 Mar Mar 2323 460 362 960 460 362 960 960960 0 0 Apr Apr 2020 190 315 1275 190 315 1275 11501150 125 125 May May 2121 310 330 1605 310 330 1605 14601460 145 145 Jun Jun 2222 145 346 1951 145 346 1951 16051605 346 346 Jul Jul 2121 110 330 2281 110 330 2281 17151715 566 566 Aug Aug 2222 125 346 2627 125 346 2627 19401940 687 687 +100 +100

Cost Reduction in Constant Work Force Cost Reduction in Constant Work Force PlanPlan

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

Idea: Idea:

change the workforce each month in order to match change the workforce each month in order to match the workforce with monthly demand as closely as the workforce with monthly demand as closely as possiblepossible

This is accomplished by computing the # units This is accomplished by computing the # units produced by one worker each month (by multiplying produced by one worker each month (by multiplying K by #days per month)K by #days per month)

Then take net demand each month and dividing by Then take net demand each month and dividing by this quantity. The resulting ratio is rounded up and this quantity. The resulting ratio is rounded up and possibly adjusted downward.possibly adjusted downward.

At the end of December there are 32 workers

Period # hired #fired

1 7 Cost of this

2 17 plan:

3 6 $461,732.08

4 25

5 13

6 20

7 4

8 13

83

Hybrid Hybrid StrategiesStrategies

Use a combination of options:Use a combination of options: Build-up inventory ahead of rising demand & use backorders Build-up inventory ahead of rising demand & use backorders

to level extreme peaksto level extreme peaks Finished goods inventories: Anticipate demandFinished goods inventories: Anticipate demand Back orders & lost sales: Delay delivery or allow demand Back orders & lost sales: Delay delivery or allow demand

to go unfilledto go unfilled Shift demand to off-peak times: Proactive marketingShift demand to off-peak times: Proactive marketing

Overtime: Overtime: Short-term optionShort-term option Pay workers a premium to work longer hoursPay workers a premium to work longer hours

84

Hybrid Hybrid StrategiesStrategies

Undertime:Undertime: Short-term option Short-term option Slow the production rate or send workers home early Slow the production rate or send workers home early

(lowers labor productivity, but doesn’t tie up capital in (lowers labor productivity, but doesn’t tie up capital in finished good inventories)finished good inventories)

Reassign workers to preventive maintenance during lullsReassign workers to preventive maintenance during lulls

Subcontracting:Subcontracting: Medium-term option Medium-term option Subcontract production or hire temporary workers to cover Subcontract production or hire temporary workers to cover

short-term peaksshort-term peaks

Hire & fire workers:Hire & fire workers: Long-term option Long-term option Change the size of the workforceChange the size of the workforce Layoff or furlough workers during lullsLayoff or furlough workers during lulls

85

Another APP ExampleAnother APP Example

__________________________________________________Hiring cost = $100 per workerHiring cost = $100 per worker

Firing cost = $500 per workerFiring cost = $500 per worker

Inventory carrying cost = $0.50 per pound per Inventory carrying cost = $0.50 per pound per quarterquarter

Production per employee = 1,000 pounds per quarterProduction per employee = 1,000 pounds per quarter

Beginning work force = 100 workersBeginning work force = 100 workers

Quarter Sales Forecast (lb)

Spring 80,000

Summer 50,000

Fall 120,000

Winter 150,000

86

Level Production StrategyLevel Production Strategy

SalesSales ProductionProduction

QuarterQuarter Forecast Forecast PlanPlan InventoryInventory

SpringSpring 80,00080,000 100,000100,000 20,00020,000

SummerSummer 50,00050,000 100,000100,000 70,00070,000

FallFall 120,000120,000 100,000100,000 50,00050,000

WinterWinter 150,000150,000 100,000100,000 00

400,000400,000 140,000140,000

Cost = 140,000 pounds x 0.50 per pound = $70,000Cost = 140,000 pounds x 0.50 per pound = $70,000

87

Chase Demand Strategy (Zero Inventory)Chase Demand Strategy (Zero Inventory)

SalesSales ProductionProductionWorkersWorkers WorkersWorkers WorkersWorkers

QuarterQuarter ForecastForecast PlanPlan NeededNeeded HiredHired FiredFired

SpringSpring 80,00080,000 80,00080,000 8080 -- 2020

SummerSummer 50,00050,000 50,00050,000 5050 -- 3030

FallFall 120,000120,000 120,000120,000 120120 7070 --

WinterWinter 150,000150,000 150,000150,000 150150 3030 --

100100 5050

Cost Cost = (100 workers hired x $100) + (50 workers fired x = (100 workers hired x $100) + (50 workers fired x $500)$500)

= $10,000 + 25,000 = $35,000 = $10,000 + 25,000 = $35,000

Hiring cost = $100 per worker; Firing cost = $500 per workerInventory carrying cost = $0.50 per pound per quarterProduction per employee = 1,000 pounds per quarterBeginning work force = 100 workers

88

APP By Linear ProgrammingAPP By Linear ProgrammingMin Z = $100 (HMin Z = $100 (H11 + H + H22 + H + H33 + H + H44) + $500 (F) + $500 (F11 + F + F22 + F + F33 + F + F44)+ $0.50 (I)+ $0.50 (I11 + I + I22

+ I+ I33 + I + I44))

Subject toSubject to

PP11 - I - I11 = 80,000 = 80,000 (1)(1) DemandDemand

II11 + P + P22 - I - I22 = 50,000 = 50,000 (2)(2) constraintsconstraints

II22 + P + P33 - I - I33 = 120,000 = 120,000 (3)(3)

II33 + P + P44 - I - I44 = 150,000 = 150,000 (4)(4)

PP11 - 1,000 W - 1,000 W11 = 0 = 0 (5)(5) ProductionProduction

PP22 - 1,000 W - 1,000 W22 = 0 = 0 (6)(6) constraintsconstraints

PP33 - 1,000 W - 1,000 W33 = 0 = 0 (7)(7)

PP44 - 1,000 W - 1,000 W44 = 0 = 0 (8)(8)

WW11 - H - H11 + F + F11 = 100 = 100 (9)(9) Work force Work force

WW22 - W - W11 - H - H22 + F + F22 = 0 = 0 (10)(10) constraints constraints

WW33 - W - W22 - H - H33 + F + F33 = 0 = 0 (11)(11)

WW44 - W - W33 - H - H44 + F + F44 = 0 = 0 (12)(12)

whereHt = # hired for period tFt = # fired for period tIt = inventory at end of period tWt = workforce at period tPt = # units produced at period t

89

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

ProgrammingProgramming

DDtt – the forecasts of demand for aggregate units for period t, t = – the forecasts of demand for aggregate units for period t, t = 1 … T1 … T

nntt– number of units that can be made by one worker in period t– number of units that can be made by one worker in period tCCtt

PP – cost to produce one unit in period t– cost to produce one unit in period tCCtt

WW – cost of one worker in period t– cost of one worker in period tCCtt

HH – cost to hire one worker in period t– cost to hire one worker in period tCCtt

LL – cost to layoff one worker in period t– cost to layoff one worker in period tCCtt

II – cost to hold one unit in inventory in period t– cost to hold one unit in inventory in period tCCtt

BB – cost to backorder one unit in period t– cost to backorder one unit in period t

WWtt– number of workers available in period t – number of workers available in period t PPtt – number of units produced in period t– number of units produced in period tIItt – number of units held in the inventory at the end of period t– number of units held in the inventory at the end of period tHHtt – number of workers hired in period t– number of workers hired in period tFFtt – number of workers fired in period t– number of workers fired in period t

known info

decision variables

90

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

ProgrammingProgrammingLP:LP:

s.t constraints s.t constraints

All variables are continuously divisible – is it All variables are continuously divisible – is it a problem?a problem?

Solution: Solution: Produce 214.5 of aggregated Produce 214.5 of aggregated unitsunits

Hire 56.38 workersHire 56.38 workers

IP:IP:s.t constraints*s.t constraints*

Some variables are continuously divisible, Some variables are continuously divisible, some are real number only – problem?some are real number only – problem?

xfor minmax

xfor minmax

91

Linear Programming: Linear Programming: Objective Function and ConstraintsObjective Function and Constraints

T

tt

Btt

Itt

Ltt

Htt

Wtt

Pt BCICLCHCWCPC

1

min

TtWnP ttt ,,2,1,

TtLHWW tttt ,,2,1,1

TtDPBIBI tttttt ,,2,1,11

TtILHWP ttttt ,,2,1,0,,,,

s.t.

productionconstraint

labourconstraint

inventoryconstraint

production salary hiring layoffs inventory backlogs

• Number of constraints is 3T, number of unknown is 5T• W0, I0, B0 – initial workforce, initial inventory/backlog

92

Linear Programming: Product Mix Linear Programming: Product Mix PlanningPlanning

Multiple products processed on various workstationMultiple products processed on various workstation

ii – an index of product, i = 1, … , m– an index of product, i = 1, … , mjj – an index of workstation, j = 1, … , n– an index of workstation, j = 1, … , ntt – an index of period, t = 1, … , T– an index of period, t = 1, … , TDDitit– the maximum demand for product i for period t– the maximum demand for product i for period tdditit – the minimum sales allows of product i for period t– the minimum sales allows of product i for period taaijij – time required on workstation j to produce one unit of – time required on workstation j to produce one unit of

product iproduct iccjtjt – capacity of workstation j in period t in the same units as a– capacity of workstation j in period t in the same units as aijij

rrii – net profit from one unit of product i – net profit from one unit of product i hhii – cost to hold one unit of product i for one period in the – cost to hold one unit of product i for one period in the

inventoryinventory

XXitit – amount of product i produced in period t– amount of product i produced in period tSSitit – amount of product i sold in period t– amount of product i sold in period tIIitit – number of units of product i held in the inventory at the – number of units of product i held in the inventory at the

end of period tend of period t

93

Linear Programming: Product Mixed Linear Programming: Product Mixed Planning Planning

Objective Function and ConstraintsObjective Function and Constraints

s.t.

salesconstraint

capacityconstraint

inventoryconstraint

profit inventory

This model can be used to obtain information on demand feasibility bottleneck location product mix

T

t

m

iitiiti IhSr

1 1

max

tiallforDSd ititit ,,

tjallforcXa jt

m

iitij ,,

1

tiallforSXII itititit ,,1

tiallforISX ititit ,,0,,

Product Mix PlanningProduct Mix Planning

Demand feasibilityDemand feasibility Determine if the set of demands is capacity-feasibleDetermine if the set of demands is capacity-feasible If SIf Sitit=D=Ditit then demand is feasible, otherwise demand is then demand is feasible, otherwise demand is

infeasibleinfeasible If “could not find a feasible solution”, then lower bound dIf “could not find a feasible solution”, then lower bound d itit is is

too high for a given capacitytoo high for a given capacity

Bottleneck locationsBottleneck locations Constraints restrict production on each workstation in each Constraints restrict production on each workstation in each

periodperiod Observe binding constraints to determine which workstations Observe binding constraints to determine which workstations

limit capacitylimit capacity Consistently binding workstation is a “bottleneck”Consistently binding workstation is a “bottleneck” Require close management attentionRequire close management attention

Product mixProduct mix If capacity is an issue, then model will try to maximize revenue If capacity is an issue, then model will try to maximize revenue

by utilizing products with high net profitby utilizing products with high net profit

tjallforcXa jt

m

iitij ,,

1

95

Homework AssignmentHomework Assignment

Read “Production & Operations Read “Production & Operations Analysis” by S.Nahmias chapter 3, Analysis” by S.Nahmias chapter 3, sections 1 – 4sections 1 – 4

Problems:Problems: 3.53.5 3.9 – 3.113.9 – 3.11 3.14 – 3.163.14 – 3.16

The ENDThe END