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 Decisions for 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 Demand Forecasting long range short range Long-range strategic planning Production Smoothing Material Planning Operations Scheduling

The Production Smoothing Problem time cumulative Production Cumulative demands backorders Inventory build-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 Alternatives when 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 2 nd and 3 rd 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 continuously adjusting the workforce. Why can’t we find an alternative solution that will minimize the total costs involved in 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 2 nd or 3 rd 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 C P 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 C 0 t = overtime labor rate in time period t ($/hr) Gee, I want to go home. This is like working in a salt mine. Morton Mines

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 C I t = 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 C B t = 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 C S t = cost per unit contracted out during period t

Labor Cost Assume fully burden cost per time period per worker –C W t = cost per worker in period t ($/time period) Cost of hiring includes human relations (HR) cost, training costs, and productivity costs –C H t = cost of hiring one worker in time period t Cost of firing includes terminal pay, HR costs, morale costs –C F 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 P t = number of units produced in month t W t = number of workers in time month t H t = number of workers hired in month t F t = number of workers fired in month t I t = number of units held in inventory at the end of month t B t = number of units backordered at the end of month t S t = number of units contracted out in month t O t = number of overtime hours

The Cost Parameters C P t = production cost (exclusive of labor) ($/unit) C W t = cost per worker in month t ($/mo) C 0 t = overtime labor rate in month t ($/hr) C S t = cost per unit contracted out during month t C B t = unit cost of a shortage or backorder in month t C I t = inventory carrying cost ($/unit per time month) C H t = cost of hiring one worker in month t C F t = cost of firing one worker in month t

The other model parameters T = number of months in planning horizon D t = demand in month t a = labor hours required to produce one unit b = available hours per worker per month f = max overtime hr per worker per month These are great other model parameters.

The Objective Function Production $/unit worker $/month hiring $/worker firing $/worker inventory $/unit/month backorder $/unit contracting $/unit overtime $/hr

The constraints Inventory balance: Worker balance: Production hours: Limit on overtime: Upper bound on contracting:

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

The Mandatory Example The 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 units a month can be contracted out. The company currently has 16 workers. There is to be no outstanding backorders at the end of the planning horizon. A worker is available 145 hours a month. Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sept The Forecast:

The Costs C P t = production cost (exclusive of labor) ($/unit) = $ 40 C W t = cost per worker in month t ($/mo) = $ 5500 mo. 1-6 $ 6000 mo C 0 t = overtime labor rate in month t ($/hr) = $ 55 mo. 1-6 $ 60 mo C S t = cost per unit contracted out during month t = $ 810 C B t = unit cost of a shortage or backorder in month t = $ 900 C I t = inventory carrying cost ($/unit per time month) = $ 50 C H t = cost of hiring one worker in month t = $ 2000 C F t = cost of firing one worker in month t = $ 5000

The Formulation Objective Function MIN 40 P P P P P P P P P P P P W W W W W W W W W W W W O O O O O O O O O O O O S S S S S S S S S S S S I I I I I I I I I I I I B B B B B B B B B B B H H H H H H H H H H H H L L L L L L L L L L L 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 = ) P9 + S9 + I8 - I9 - B8 + B9 = ) P10 + S10 + I9 - I10 - B9 + B10 = ) P11 + S11 + I10 - I11 - B10 + B11 = ) 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 P W1 - O1 = 0 27) 20 P W2 - O2 = 0 28) 20 P W3 - O3 = 0 29) 20 P W4 - O4 = 0 30) 20 P W5 - O5 = 0 31) 20 P W6 - O6 = 0 32) 20 P W7 - O7 = 0 33) 20 P W8 - O8 = 0 34) 20 P W9 - O9 = 0 35) 20 P W10 - O10 = 0 36) 20 P W11 - O11 = 0 37) 20 P 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 = Production = demands: $1,523,233 an increase of $ 80,250 per yr

The Enhancements Bounds on variables: Training: 1-mo training Multiple products: Safety stock:

More Enhanced Enhancements Planned downtime: Let z i = 1 if planned downtime in month i; 0 otherwise Gee how I miss the old factory.

That’s right, More Enhanced Enhancements Production capacity Available workforce 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 , 19, 20, 22, 23, 37, 38