Production Smoothing

An insightful look at the intricacies of tactical level production planning

An engineering management graduate busily determining an optimal production schedule.

“Let all things be done decently and in order.”I Corinthians

Production Goals

• On-time deliveries with no backorders

• Minimal work in process (inventories)

• Short lead-times

• Maximum use of resources

• Least possible cost

CEO

But aren’t these goals

conflicting?

Management Tactical Decisionsfor the next year or two…

• Production rates• Staffing

– recruiting & training new workers– Laying-off workers

• Procurement– Supplier contracts and orders

• Outsourcing (long lead-times)• Budgeting for expenditures

Too many difficult

decisions to be made.

Production Planning

medium range

Dem

and

For

ecas

ting

long range

short range

Long-range strategic planning

ProductionSmoothing

Material Planning

OperationsScheduling

The Production Smoothing Problem

time

cumulativeProduction

Cumulativedemands

backorders

Inventorybuild-up

Production Smoothing Terms

• Smoothing is concerned with the costs resulting from changes in workforce and production levels.

• Planning horizon is the number of time periods (often months) for which the workforce, inventory, and production levels are to be determined.

Management Alternativeswhen product demands vary over the planning horizon

• Build inventories during months of slack demand in anticipation of higher demand rates later.

• Carry backorders or tolerate lost sales during month of peak demand.• Use of 2nd and 3rd shifts (work force implications)• Use overtime in peak months or under-time in slack months to vary

output while work force and facilities remain constant.• Use subcontracting in peak months• Vary product mix in attempt to maintain constant production.• Planned downtime• Vary capacity through changes in plant and equipment (not always an

option in the short term)• Control demand by promotional campaigns, discount sales,

counterbalancing products

Extreme Solutions.

Alternative 1: Maintain constant production thereby stabilizing the workforce while observing increasing inventory levels or backorders and potential lost sales.

Alternative 2: Vary production to meet demands thereby minimizing inventories and backorders while continuouslyadjusting the workforce.

Why can’t we find an alternative solutionthat will minimize the total costs involvedin changing production levels, inventories,workforce sizes, and backorders?

The relevant costs

• Production costs• Inventory carrying costs• Shortage costs (Backorder and lost sales)• Contracting out or procurement costs• Costs of increasing or decreasing workforce• Costs of overtime • costs of 2nd or 3rd shift operation• Cost of changing production rates• Cost of under-utilization of workforce (lost

opportunity costs)

Production Costs

• Increasing with level of production– assumed linear based upon an average learning

curve cost

• Labor cost not included• At some point a fixed cost for expanded

capability would be incurred• May bound production levels• CP

t = dollars per unit produced in time period t

Overtime Cost• Assumed linear cost per hour of overtime

– at some point worker efficiency will drop

• Normally set an upper bound on overtime hours

• C0t = overtime labor rate in time period t

($/hr) Gee, I want to go home. This is like

working in a salt mine.MortonMines

Inventory Costs

• Assume a dollar cost per item carried over from one time period to the next

• Costs may include storage cost, obsolescence cost, pilferage cost, insurance cost, and investment cost

• CIt = inventory carrying cost ($/unit per

time period)

Shortage Costs

• Assume a dollar cost per item short at the end of the time period

• Items short may be backordered or may result in lost sales

• Costs may include additional administrative costs, lost revenue, goodwill cost

• CBt = unit cost of a shortage or backorder in

period t

Outsourcing Cost

• Assumed proportional to the number of units outsourced

• Assumed to cost more per unit than producing in-house– otherwise outsource all units if possible

• May have lower and upper bounds

• CSt = cost per unit contracted out during period t

Labor Cost• Assume fully burden cost per time period per

worker– CW

t = cost per worker in period t ($/time period)

• Cost of hiring includes human relations (HR) cost, training costs, and productivity costs– CH

t = cost of hiring one worker in time period t• Cost of firing includes terminal pay, HR costs,

morale costs– CF

t = cost of firing one worker in time period t• Underutilization of Work Force

– incur fully burden wages– excess capacity

The Assumptions

• Single product • Costs are linear• backorders or lost sales are permitted• demands are variable but deterministic and known• infinite planning horizon (rolling horizon)• inventory carrying costs are incurred for inventory

carried over from one month to the next• overtime is permitted

The Decision Variables

Pt = number of units produced in month tWt = number of workers in time month tHt = number of workers hired in month tFt = number of workers fired in month tIt = number of units held in inventory at the end of month tBt = number of units backordered at the end of month tSt = number of units contracted out in month tOt = number of overtime hours

The Cost Parameters

CPt = production cost (exclusive of labor) ($/unit)

CWt = cost per worker in month t ($/mo)

C0t = overtime labor rate in month t ($/hr)

CSt = cost per unit contracted out during month t

CBt = unit cost of a shortage or backorder in month t

CIt = inventory carrying cost ($/unit per time month)

CHt = cost of hiring one worker in month t

CFt = cost of firing one worker in month t

The other model parameters

T = number of months in planning horizonDt = demand in month ta = labor hours required to produce one unitb = available hours per worker per monthf = max overtime hr per worker per month

These are great other model parameters.

The Objective Function

1

(

)

TP W H F It t t t t t t t t t

t

B S Ot t t t t t

Min z C P C W C H C F C I

C B C S C O

Production$/unit

worker$/month

hiring$/worker

firing$/worker

inventory$/unit/month

backorder$/unit

contracting$/unit

overtime$/hr

The constraints

I B I B P S D t Tt t t t t t t 1 1 1 2, ,...,

Inventory balance:

1t t t tW W H F Worker balance:

a P bW Ot t t Production hours:

O f Wt tLimit on overtime:

S Ut tS

Upper bound on contracting:

Gee! This looks like a really terrific model.But an example would surely help here.

The Mandatory ExampleThe Factory

The Factory manufactures things. A thing has a variable monthly demand. Demand forecast for the next 12 month is shown below. It takes 20 labor hours to produce one thing. No more than 30 unitsa month can be contracted out. The company currently has 16 workers. There is to be no outstanding backorders at the end of theplanning horizon. A worker is available 145 hours a month.

Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sept120 180 220 100 90 110 100 120 130 150 170 190

The Forecast:

The Costs

CPt = production cost (exclusive of labor) ($/unit) = $ 40

CWt = cost per worker in month t ($/mo) = $ 5500 mo. 1-6

$ 6000 mo. 7-12C0

t = overtime labor rate in month t ($/hr) = $ 55 mo. 1-6$ 60 mo. 7-12

CSt = cost per unit contracted out during month t = $ 810

CBt = unit cost of a shortage or backorder in month t = $ 900

CIt = inventory carrying cost ($/unit per time month) = $ 50

CHt = cost of hiring one worker in month t = $ 2000

CFt = cost of firing one worker in month t = $ 5000

The Formulation Objective Function

MIN 40 P1 + 40 P2 + 40 P3 + 40 P4 + 40 P5 + 40 P6 + 40 P7 + 40 P8 + 40 P9 + 40 P10 + 40 P11 + 40 P12 + 5500 W1 + 5500 W2 + 5500 W3 + 5500 W4 + 5500 W5 + 5500 W6 + 6000 W7 + 6000 W8 + 6000 W9 + 6000 W10 + 6000 W11 + 6000 W12 + 55 O1 + 55 O2 + 55 O3 + 55 O4 + 55 O5 + 55 O6 + 60 O7 + 60 O8 + 60 O9 + 60 O10 + 60 O11 + 60 O12 + 810 S1 + 810 S2 + 810 S3 + 810 S4 + 810 S5 + 810 S6 + 810 S7 + 810 S8 + 810 S9 + 810 S10 + 810 S11 + 810 S12 + 50 I1 + 50 I2 + 50 I3 + 50 I4 + 50 I5 + 50 I6 + 50 I7 + 50 I8 + 50 I9 + 50 I10 + 50 I11 + 50 I12 + 900 B1 + 900 B2 + 900 B3 + 900 B4 + 900 B5 + 900 B6 + 900 B7 + 900 B8 + 900 B9 + 900 B10 + 900 B11 + 2000 H1 + 2000 H2 + 2000 H3 + 2000 H4 + 2000 H5 + 2000 H6 + 2000 H7 + 2000 H8 + 2000 H9 + 2000 H10 + 2000 H11 + 2000 H12 + 5000 L1 + 5000 L2 + 5000 L3 + 5000 L4 + 5000 L5 + 5000 L6 + 5000 L7 + 5000 L8 + 5000 L9 + 5000 L10 + 5000 L11 + 5000 L12

The Formulation Inventory Balance Constraints

2) P1 + S1 - I1 + B1 = 120 3) P2 + S2 + I1 - I2 - B1 + B2 = 180 4) P3 + S3 + I2 - I3 - B2 + B3 = 220 5) P4 + S4 + I3 - I4 - B3 + B4 = 100 6) P5 + S5 + I4 - I5 - B4 + B5 = 90 7) P6 + S6 + I5 - I6 - B5 + B6 = 110 8) P7 + S7 + I6 - I7 - B6 + B7 = 100 9) P8 + S8 + I7 - I8 - B7 + B8 = 120 10) P9 + S9 + I8 - I9 - B8 + B9 = 130 11) P10 + S10 + I9 - I10 - B9 + B10 = 150 12) P11 + S11 + I10 - I11 - B10 + B11 = 170 13) P12 + S12 + I11 - I12 - B11 = 190

The Formulation Worker Balance Constraints

14) W1 - H1 + L1 = 16 15) - W1 + W2 - H2 + L2 = 0 16) - W2 + W3 - H3 + L3 = 0 17) - W3 + W4 - H4 + L4 = 0 18) - W4 + W5 - H5 + L5 = 0 19) - W5 + W6 - H6 + L6 = 0 20) - W6 + W7 - H7 + L7 = 0 21) - W7 + W8 - H8 + L8 = 0 22) - W8 + W9 - H9 + L9 = 0 23) - W9 + W10 - H10 + L10 = 0 24) - W10 + W11 - H11 + L11 = 0 25) - W11 + W12 - H12 + L12 = 0

The Formulation Production Constraints

26) 20 P1 - 145 W1 - O1 = 0 27) 20 P2 - 145 W2 - O2 = 0 28) 20 P3 - 145 W3 - O3 = 0 29) 20 P4 - 145 W4 - O4 = 0 30) 20 P5 - 145 W5 - O5 = 0 31) 20 P6 - 145 W6 - O6 = 0 32) 20 P7 - 145 W7 - O7 = 0 33) 20 P8 - 145 W8 - O8 = 0 34) 20 P9 - 145 W9 - O9 = 0 35) 20 P10 - 145 W10 - O10 = 0 36) 20 P11 - 145 W11 - O11 = 0 37) 20 P12 - 145 W12 - O12 = 0

The Formulation Overtime Constraints

38) - 40 W1 + O1 <= 0 39) - 40 W2 + O2 <= 0 40) - 40 W3 + O3 <= 0 41) - 40 W4 + O4 <= 0 42) - 40 W5 + O5 <= 0 43) - 40 W6 + O6 <= 0 44) - 40 W7 + O7 <= 0 45) - 40 W8 + O8 <= 0 46) - 40 W9 + O9 <= 0 47) - 40 W10 + O10 <= 0 48) - 40 W11 + O11 <= 0 49) - 40 W12 + O12 <= 0

The Formulation Upper Bound (Contract Out) Constraints

50) S1 <= 30 51) S2 <= 30 52) S3 <= 30 53) S4 <= 30 54) S5 <= 30 55) S6 <= 30 56) S7 <= 30 57) S8 <= 30 58) S9 <= 30 59) S10 <= 30 60) S11 <= 30 61) S12 <= 30

… and then the plant manager hired an OR specialist to find the optimum production and worker levels.

The Solution

Cost = $1,442,983

Constant production: $1,463,711 an increase of $20,728 per yr production = 132.5

Production = demands: $1,523,233 an increase of $ 80,250 per yr

Month Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sept CostUnits Produced (P) 131.3 131.3 167.5 115.6 115.6 115.6 115.6 115.6 115.6 115.6 115.6 115.6 $58,800Workers (Wt) 18.1 18.1 18.1 15.9 15.9 15.9 15.9 15.9 15.9 15.9 15.9 15.9 $1,135,518Overtime hours (O) 0.0 0.0 724.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 $39,832Nbr Hired (H) 2.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 $4,211Nbr Fired (L) 0.0 0.0 0.0 2.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 $10,833Contracted Out (S) 30.0 30.0 30.0 0.0 0.0 0.0 0.0 0.0 30.0 30.0 30.0 30.0 $170,100Ending Inventory (I) 41.3 22.5 0.0 15.6 41.1 46.7 62.2 57.8 73.3 68.9 44.4 0.0 $23,689Shortages (B) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 $0

Cost = $1,442,983

The Enhancements

B B H H L L gW P Pt tU

t tU

t tU

t t tU ; ; ;

Bounds on variables:

W W H Lt t t t 1 1

Training:

1-mo training

1

1

[

( )]

TW H F Ot t t t t t t t

t

nP I B Sit it it it it it it it

i

Min z C W C H C F C O

C P C I C B C S

Multiple products:

I It tLSafety stock:

More Enhanced Enhancements

Planned downtime: Let zi = 1 if planned downtime in month i; 0 otherwise

1

(1 )

1

(1 )

(1 )

Ut t t

T

tt

t t

t t

P P z

z

H M z

F M z

Gee how I miss the old

factory.

That’s right, More Enhanced Enhancements

t PP U

1.9t t t tW W H F

Production capacity

Available workforcet WW U

1 1 .95t t t t t t tI B I B P S D

5% reject rate

What about attrition. Every month ten percent of us leave

this cotton picking job!

Production Smoothing by LP

• Strengths– optimizes rather

than satisfying (e.g. spreadsheets)

– flexibility– use of dual solution

and sensitivity analysis

• Weaknesses– assumption of

deterministic demand

– use of linear costs functions

– single measure of effectiveness

– data requirements

The End

Production smoothing - a rolling horizon model.

Rolling horizon. What could that possibly mean??

Homework: Chapter 3 - 18, 19, 20, 22, 23, 37, 38