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Master Production Scheduling

Material Requirements Planning

Order Scheduling

Process Planning

Strategic Capacity Planning

Aggregate Planning

Long-range

Intermediate-range

Short-range

Hierarchy of Planning Problems

Ch 11 - 22© 1998 by Prentice-Hall IncRussell/Taylor Oper Mgt 2/e

Hierarchical Planning ProcessItems

Product lines or families

Individual products

Components

Manufacturing operations

Resource level

Plants

Individual machines

Critical work centers

Production Planning Capacity Planning

Resource Requirements Plan

Rough-Cut Capacity Plan

Capacity Requirements Plan

Input/Output Control

Aggregate Production Plan

Master Production Schedule

Material Requirements Plan

Shop Floor Schedule

All work centers

Forecast of AggregateDemand For t Period

Planning Horizon

Forecast of AggregateDemand For t Period

Planning Horizon

Aggregate Production Plan Determination of Aggregate Production &

Work Force Levels for t Period Planning Horizon

Aggregate Production Plan Determination of Aggregate Production &

Work Force Levels for t Period Planning Horizon

Master ProductionSchedule

Aggregate Planning Translating annual or quarterly business plans

into broad labor and output plans for the intermediate term (6 to 18 months).

Objective: to minimize the cost of resources required to meet demand over that period.

Optimal combination of production rate, workforce level, and inventory on hand is sought throughout the intermediate term.


Inputs and Outputs to Aggregate Production Planning

AggregateProductionPlanning

CompanyPolicies

FinancialConstraints

StrategicObjectives

Units or dollarssubcontracted,backordered, or

lost

CapacityConstraints

Size ofWorkforce

Productionper month

(in units or $)

InventoryLevels

DemandForecasts


Aggregate Production Planning (APP)

Matches market demand to company resources

Plans production 6 months to 12 months in advance

Expresses demand, resources, and capacity in general terms

Develops a strategy for economically meeting demand

Establishes a companywide game plan for allocating resources

Goal: To plan gross work force and production levels and set firm-wide production plans.

We try to determine aggregate production targets and the levels of resources required to achieve these production goals given the demand projection for the intermediate term.

Namely given demand as D1, D2,…,DT

Find the number of workers employed in each period

Find the number of aggregate units to be produced in each period

Concept is based on the idea of an “aggregate unit” of production. Aggregate Planning requires aggregation of different goods or services.

They may be Actual units of production if items are similar Weight (tons of steel) Volume (gallons of gasoline) Dollar value (Value of sales) Fictitious aggregate units

Aggregation Unit - Example Six different models of washing machines

Model Number Number of Worker-Hours Required to Produce

Selling Price

Percentage of total sales

A5532 4.2 $285 32

K4242 4.9 345 21

L9898 5.1 395 17

L3800 5.2 425 14

M2624 5.4 525 10

M3880 5.8 725 6

Can we use dollars as an aggregation unit? Notice: Price is not necessarily proportional to worker hours (i.e., cost): why?Use a fictitious washing machine as an aggregation unit which requires(.32)(4.2)+(.21)(4.9)+(.17)(5.1)+(.14)(5.2)+(.10)(5.4)+(.06)(5.8)=4.856 hours of labor timeForecasts for demand for aggregate units can be obtained by taking a weighted average (using the same weights) of individual item forecasts.

Different levels of aggregation

Items: Final products to be delivered to the customer Families: A group of items that a share common

manufacturing setup cost Types: Groups of families with production quantities

that are determined by a single aggregate production plan

Above may not always work, the aggregation method should be consistent with the firm’s organizational structure, product line, planning needs and availability of forecast and other data.

Production Rate

What is the best production curve in seasonal business

Demand

Time


Strategies for Meeting Demand1. Use inventory to absorb fluctuations in demand

(level production)

2. Hire and fire workers to match demand (chase demand)

3. Maintain resources for high demand levels

4. Increase or decrease working hours (over & undertime)

5. Subcontract work to other firms

6. Use part-time workers

7. Provide the service or product at a later time period (backordering)


Strategy Details

Level production - produce at constant rate & use inventory as needed to meet demand

Chase demand - change workforce levels so that production matches demand

Maintaining resources for high demand levels -ensures high levels of customer service

Overtime & undertime - common when demand fluctuations are not extreme


Strategy Details

Subcontracting - useful if supplier meets quality & time requirements

Part-time workers - feasible for unskilled jobs or if labor pool exists

Backordering - only works if customer is willing to wait for product/services


Level Production

Time

Production

Demand

Units


Chase Demand

Time

Units

Production

Demand


Aggregate Planning for Services

1. Most services can’t be inventoried

2. Demand for services is difficult to predict

3. Capacity is also difficult to predict

4. Service capacity must be provided at the appropriate place and time

5. Labor is usually the most constraining resource for services

Overview of the Problem

Suppose that D1, D2, . . . , DT are the forecasts of demand for aggregate units over the planning horizon (T periods.)

The problem is to determine both work force levels (Wt) and production levels (Pt ) to minimize total costs over the T period planning horizon.

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.

Costs in Aggregate Planning

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.

Regular time costs Overtime or subcontracting costs Idle time costs

Smoothing costs

Holding and Back-Order Costs

Back-orders Positive inventory

Slope = CP

Slope = Ci

$ C

ost

Inventory

Prototype Problem Forecast demands over the next six

months for disk drives.Month Forecast

January 1280

February 640

March 900

April 1200

May 2000

June 1400

*Inventory at the end of December expected to be 500*Company requires ending inventory of 600 at the end of June*Initial workforce is 300

Cost of hiring one worker = CH = $500Cost of firing one worker = CF = $1000Inventory holding cost per unit, per month = CI = $80

Prototype Problem Plant manager observed that over 22

working days, with the workforce level of 76, the firm produced 245 disk drives.

K= # of aggregate units produced per worker per day = 245 / (22 * 76) = 0.14653

Month Net Predicted Demand

Net Predicted Cumulative Demand

January 780 780

February 640 1420

March 900 2320

April 1200 3520

May 2000 5520

June 2000 7520

Zero Inventory Plan (Chase)

Month Number of working days

Number of units produced per worker

Predicted net demand

Minimum number of workers required

January 20 2.931 780 267

February

24 3.517 640 182

March 18 2.638 900 342

April 26 3.810 1200 315

May 22 3.224 2000 621

June 15 2.198 2000 910

Zero Inventory Plan (Chase)Month Number

of workers

Number hired

Number fired

Number of units per worker

Number of units produced

Cumulative production

Cumulative demand

Ending Inventory

January 267 33 2.931 783 783 780 3February

182 85 3.517 640 1423 1420 3

March 342 160 2.638 902 2325 2320 5April 315 27 3.810 1200 3525 3520 5May 621 306 3.224 2002 5527 5520 7June 910 289 2.198 2000 7527 7520 7Total 755 145 30

Hiring costs = 755 * 500 = $377,500Firing costs = 145 * 1000 = $145,000Holding costs = (600+30) * 80 = $50,400Total cost = $ 572,900

Constant workforce plan (Level)

Month Cumulative net demand

Cumulative number of units produced per worker

Ratio (rounded up)

January 780 2.931 267

February 1420 6.448 221

March 2320 9.086 256

April 3520 12.896 273

May 5520 16.120 343

June 7520 18.318 411

Constant workforce plan (Level)Month Number of

units per worker

Number of units produced

Cumulative production

Cumulative demand

Ending Inventory

January 2.931 1205 1205 780 425February 3.517 1445 2650 1420 1230March 2.638 1084 3734 2320 1414April 3.810 1566 5300 3520 1780May 3.224 1325 6625 5520 1105June 2.198 903 7528 7520 8Total 5962

Hiring costs = (411-300) * 500 = $55,500 Holding costs = (5962+600) * 80 = $524,960 Total cost = $ 580,460

Mixed Strategies

Other scenarios

0

1000

2000

3000

4000

5000

6000

7000

0 1 2 3 4 5 6

Month

Cu

mu

lati

ve n

um

ber

of

un

its

Section 1

Comparing strategies

comparing strategies

0

100

200

300

400

500

600

700

Chase Level Mixed

strategy

tota

l co

st

(th

ou

san

ds)

Hiring/firing costsInventory costs

Mathematical programming formulations

Cost definitions and input dataCH = Cost of hiring one workerCF = Cost of firing one workerCI = Cost of holding one unit of stockCR = Cost of producing one unit in regular timeCO = Cost of producing one unit in over timeCU = Idle cost per unit of productionCS = Cost to subcontract one unit of productionnt = Number of production days in period tK = Number of aggregate units produced by one worker in one

dayI0 = Initial inventoryW0= Initial workforceDt = Forecast of demand in period t

Problem variablesWt = Workforce level in period t

Pt = Number of units produced in period t

It = Inventory level at the end of period t

Ht = Number of workers hired in period t

Ft = Number of workers fired in period t

Ot = Overtime production in units

Ut = Worker idle time in units

St = Number of units subcontracted from outside

Constraints Conservation of workforce

Wt = Wt-1+ Ht – Ft, for all t=1,..,T Inventory balance

It = It-1+Pt+St-Dt, for all t=1,..,T Relationship between production and

workforcePt = K nt Wt + Ot - Ut, for all t=1,..,T

Constraints Initial levels for inventory and workforce

I0=I0, W0= W0

Ending levels for inventory and workforceIT=IT, WT= WT

Non-negativity constraintsWt, Pt, It, Ht, Ft, Ot, Ut, St >=0 for all t=1,..,T

Linear program

TtPWSUOIFH

TtDSPII

TtUOWKnP

TtFHWW

ScUcOcPcIcFcHc

tttttttt

ttttt

ttttt

tttt

tStUtOtRtItF

T

ttH

1 allfor 0,,,,,,,

1 allfor

1 allfor

1 allfor

Subject to

)( Minimize

1

1

1

Properties of the optimal solution

If cF>=0 and cH>=0, there could be either hiring or firing in one period (if at all), not both. i.e., HtFt=0, for all t=1,..,T

If cO>=0 and cI>=0, there could be either overtime or idle time in one period (if at all), not both. i.e.,OtUt=0, for all t=1,..,T

IF THESE CONSTRAINTS ARE INCLUDED INTO THE MODEL, IT BECOMES NON-LINEAR. FROM OPTIMALITY CONDITIONS, NOT NECESSARY TO INCLUDE INTO THE MODEL.EXTREME POINTS OF THE FEASIBLE REGION SATISFY THESE CONDITIONS

Properties of the optimal solution

Not necessarily integers- fractional numbers may not make sense for some variables, eg # of people hired/fired.

Requires an integer-linear programming model, imposing that variables are integers. Hovewer this is computationally difficult.

Solving as LP and then rounding may be possible. However, rounding must be performed carefully( may lead to infeasibilities).

Round to the next larger integer!

Extensions Adding buffers for uncertainty

It >= Bt, for all t=1,..,T

Bt= Buffer stock for period t defined in advance

Storage constraints for inventoryIt <= Vt, for all t=1,..,T

Vt=Storage capacity in period t

Extensions Limits on overtime

Ot<=Mt, for all t=1,..,T

Mt: maximum overtime in period t Capacity constraints on production

Pt<=Ct, for all t=1,..,T

Ct: capacity in period t

Allowing for backorders: Inventory level is the difference of +

and - inventory It=I+t-I-t , all non-negative for all periods

Holding cost: charged for + inventory Stockout cost: charged against –

inventory

Backorders and inventory In any period, there are either backorders or

positive inventory (if at all), but not both. i.e.,

TtII tt 1 allfor 0

Allowing for backorders

TtPWSUOIIFH

TtIII

TtDSPII

TtUOWKnP

TtFHWW

ScUcOcPcIcIcFcHc

ttttttttt

ttt

ttttt

ttttt

tttt

tStUtOtRtPtItF

T

ttH

1 allfor 0,,,,,,,,

1 allfor

1 allfor

1 allfor

1 allfor

Subject to

)( Minimize

1

1

1

Convex piecewise-linear costs

Linearization

TtPWSUOIFHH

TtDSPII

TtUOWKnP

TtHH

TtHHH

TtFHWW

ScUcOcPcIcFcHcHc

ttttttttt

ttttt

ttttt

t

ttt

tttt

tStUtOtRtItF

T

ttHtH

1 allfor 0,,,,,,,,

1 allfor

1 allfor

1 allfor *

1 allfor

1 allfor

Subject to

)( Minimize

21

1

1

21

1

12211

# of workers hired upto H*

Example-1 The Paris Paint Company is in the process of planning labor force

requirements and production levels for the next 4 quarters. The marketing department has provided production with the following forecasts of demand for Paris Paint over the next year.

Quarter Demand forecast in (1000s of gallons)

1 380

2 630

3 220

4 160

Assume that there are currently 280 employees with the company. Employees are hired for at least one full quarter. Hiring costs amount to $1,200 per employee and firing costs are $2,500 per employee. Inventory costs are $1 per gallon per quarter. It is estimated that one worker produced 1,000 gallons of paint each quarter. Assume that Paris currently has 80,000 gallons of paint in inventory and would like to end the year with an inventory of at least 20,000 gallons.Formulate the problem as an LP.

Linear Program-1

TtPWIFH

DDDD

WI

I

TtDPII

WP

TtFHWW

IFH

ttttt

tttt

tt

tttt

tt

T

tt

1 allfor 0,,,,

160000,220000,630000,380000

280,80000

20000

1 allfor

1000

1 allfor

Subject to

)25001200( Minimize

4321

00

4

1

1

1

Example 2 Sun Microsystems is the producer of computer workstations. For the year 2003,

they estimate their quarterly demand for high-end workstations to be the following:

Q1 5000

Q2 5000

Q3 9000

Q4 8000

Sun Microsystems focuses on innovation and design rather than manufacturing. Therefore, Sun developed strategic relationships with contract manufacturers Solectron and Celestica; Solectron promising 6000, Celestica promising 2000 units of maximum manufacturing capacity per quarter for Sun.Solectron is the preferred vendor for Sun and charges $1000 for each high end workstation that it manufactures. Sun may chose to use any amount of the capacity that Solectron promised without paying any penalties. Celestica, on the other hand, is the secondary vendor for Sun. It charges $1100 for each high-end workstation. Celestica requires that Sun declares the Q1 subcontracted amount in advance. Also, Sun can increase the subcontracted amount each quarter only if it pays $100 per unit of increase. In addition, Sun can never reduce the amount it subcontracted from Celestica from quarter to quarter.Inventory holding costs are $50 per quarter per unit.How much should Sun subcontract from Solectron and Celestica each quarter?

Linear Program-2

0,

1 allfor 0,,,,

8000,9000,5000,5000

0

1 allfor 2000

1 allfor 6000

1 allfor

1 allfor

1 allfor

Subject to

)5010011001000( Minimize

200

221

4321

0

2

1

1

21,22

21

221

1

SI

TtCSSSI

DDDD

I

TtS

TtS

TtDSII

TtCSS

TtSSS

ICSS

ttttt

t

t

tttt

ttt

ttt

ttt

T

tt

Example-3 The personal department of the A&M Corporation wants to

know how many workers will be needed each month for the next 4 month production period. The demands would be 1250, 1100, 950, 900 in months July, August, September, October.

The inventory on hand at the end of June was 500 units. The company wants to maintain a minimum inventory of 300 units each month. Each unit requires 5 employee hours. There are 20 working days each month, and each employee works an 8-hour day. The workforce at the end of June was 35 workers.

The workers can also work overtime, but overtime cannot exceed 30% of the regular time in each month. Overtime costs an additional $20 per unit produced.

Hiring costs $2500 per employee, firing costs $4000 per employee, payroll costs $3000 per employee per month, and inventory holding cost is $100.

Formulate the problem as an LP and solve

Linear Program-3

TtOPWIFH

DDDD

WI

TtI

TtDPII

TtWO

TtWP

TtFHWW

OIWFH

tttttt

t

tttt

tt

tt

tttt

tttt

T

tt

1 allfor 0,,,,,

900,950,1100,1250

35,500

1 allfor 300

1 allfor

1 allfor 6.9

1 allfor O 32

1 allfor

Subject to

)20100300040002500( Minimize

4321

00

1

t

1

1

Example 4- Component availability constraints Seagate is a manufacturer of hard drives for personal and

enterprise use. The enterprise division is trying to plan its production for the first six months in 2003. The forecast for these 6 months are given below

January 12000

February 14000

March 15000

April 16000

May 16500

June 16000

Seagate does not have any constraints on workforce or equipment for manufacturing hard drives. However, the enterprise hard drive requires two counts of a specific chip (application specific integrated circuit – ASIC) which is sourced from Texas Instruments. TI’s capacity is fixed for the first 4 months and it can allocate only a portion of its capacity to Seagate. This amounts to 26000 such chips in each of the first 4 months in 2003. Seagate may also choose to use subcontractors for manufacturing the hard drives which would cost an additional (on top of its own manufacturing) $50 per drive.If the inventory holding costs are $20 per unit per month for the hard drives and $5 per unit per month for the ASICs, find the optimal production plan for the hard drives and optimal purchase plan for the hard drives and ASICs.

Linear Program-4

61 allfor 0,,,,

16000,16500,1600015000,14000,12000

0

41 allfor 26000

61 allfor 2

61 allfor

Subject to

)50520( Minimize

554

321

00

1

1

6

1t

tSPPII

DDDDDD

II

tP

tPPII

tDSPII

SII

tc

th

tct

ht

ch

ct

ht

ct

ct

ct

tth

tht

ht

tc

th

t

Production Planning problems with concave costs

Why concave costs? Economies of scale Setup costs associated with

production, subcontracting, overtime, alternate resources

Cannot be modeled as linear programs

May be more difficult to solve

Modeling fixed (setup) costs Assume there is a fixed cost associated with

subcontracting. No idle time or overtime allowed.

largely sufficient

)1,0(

1 allfor 0,,,,,

1 allfor

1 allfor

1 allfor

Subject to

)( Minimize

1

1

1

M

y

TtPWSIFH

MyS

TtDSPII

TtWKnP

TtFHWW

yKScPcIcFcHc

t

tttttt

tt

ttttt

ttt

tttt

tStStRtItF

T

ttH

Example 5- Component availability constraints Seagate is a manufacturer of hard drives for personal and

enterprise use. The enterprise division is trying to plan its production for the first six months in 2003. The forecast for these 6 months are given below

January 12000

February 14000

March 15000

April 16000

May 16500

June 16000Seagate does not have any constraints on workforce or equipment for manufacturing hard drives. However, the enterprise hard drive requires two counts of a specific chip (application specific integrated circuit – ASIC) which is sourced from Texas Instruments. TI’s capacity is fixed for the first 4 months and it can allocate only a portion of its capacity to Seagate. This amounts to 26000 such chips in each of the first 4 months in 2003. Seagate may also choose to use subcontractors for manufacturing the hard drives which would cost an additional (on top of its own manufacturing) $10 per drive. In addition, there is a fixed cost of $20,000 working with a subcontractor in any month.If the inventory holding costs are $20 per unit per month for the hard drives and $5 per unit per month for the ASICs, find the optimal production plan for the hard drives and optimal purchase plan for the hard drives and ASICs.

Mixed Integer Program

61 allfor 0,,,,

16000,16500,1600015000,14000,12000

0

61 allfor )1,0(

61 allfor 89500

41 allfor 26000

61 allfor 2

61 allfor

Subject to

)2000010520( Minimize

554

321

00

1

1

6

1t

tSPPII

DDDDDD

II

ty

tyS

tP

tPPII

tDSPII

ySII

tc

th

tct

ht

ch

t

tt

ct

ht

ct

ct

ct

tth

tht

ht

ttc

th

t

From Aggregate Plan to Master Production Schedule

The result of the aggregate plan is the production quantities, inventory levels and required resources at the aggregate level in the mid term.

The firms are expected to act (acquire workforce & resources, contact suppliers) based on the aggregate plan.

Such actions will be input (constraints) to lower level (i.e. more detailed and shorter term) decisions in the company

Consistency between the aggregate plan and the master production schedule is desired, but not always possible

Initial master production schedule dictated by the demand plan (forecast) may not always be feasible.

Disaggregation into MPS X*=optimal # of aggregatee units to

be produced for a product group , found by LP solution

This aggregate qty should be disaggregated into individual products within this product group

Disaggregation into MPS Min Kj j/Yj , j=1.....J

s.t. Yj=X* j=1.....J

aj<=Yj<=bj Kj= setup for item j within the familyj = annual demand for item j

aj,bj: lower/upper bounds for item jKj j/Yj: average annual setupcost for item j

Inputs to Master Production Schedule

Forecast by end item (sometimes also by customer class or distribution channel) usually weekly sometimes monthly

For each item, manufacturing/distribution process flow

For each stage of manufacturing/distribution process flow

Consumption rate of critical components Consumption rate of critical resources

Availability for critical components and resources

Lead times (for time phasing) Prioritization and allocation schemes for constrained

situations

MPS

Links tactical and operational planning stages Engine that drives MRP, CRP, purchasing and shop

floor execution systems Anticipated build schedule for end-products or

major product options A statement of production, not demand Sales forecast is an input but MPS is not a forecast Always stated in terms of part numbers for which

BOM exists After constructing the MPS we make a rough-cut

capacity planning in terms of critical work centers

Creating MPS Decide on appropriate planning horizon Estimate requirements for each period in

terms of end-products in the planning horizon Account for any backorders Compare requirements against actual and

projected inventory balance Build the MPS Use rough-cut capacity planning to evaluate

feasibility If not, revise as appropriate

Output: a matrix in which rows showing individual products, columns showing time periods

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