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• Parameters: found in or computed from the data

• Decision variables: unknowns to be determined

• Objective function: the bottom line

• Constraints: Satisfy demands and state relationships among variables

• Single Product model here; can be generalized

Aggregate Production Planning by Linear Programming:

Terminology

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• dt = amount of product demanded in period t

• pt = productivity per worker in period t

• Lt = unit labor cost per worker in period t

• ht = unit hiring cost per worker in period t

• ft = unit firing cost per worker in period t

Parameters: find these in the data!

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• ct = inventory holding cost per unit per period in period t

• at = backorder cost per item per period in period t

Parameters: find these in the data!

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• wt = number of workers employed in period t

• ut = number of workers hired between periods t-1 and t

• vt = number of workers fired between periods t-1 and t

• it = amount of product in inventory at the end of period t

• bt = amount backordered at the end of period t

Decision Variables: find values by solving the

model

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• io = initial inventory level

• wo = initial workforce level

Initial values: “fixed variables”; find in

data

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The LP Model

Minimize t (Ltwt + htut + ftvt + ctit + atbt)

s.t.

ut - vt = wt – wt-1 for each period t : workforce change

ptwt + it-1 - it + bt – bt-1 = dt for each period t: demand balance

wt, ut, vt, it, bt 0 for each time period t: nonnegativity

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LP Example: based Ch 13 problems

Period DemandPeriod 1 6000Period 2 4800Period 3 7840Period 4 5200Period 5 6560Period 6 3600

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LP Example: based on Ch 13 problems

Cost DataRegular time labor cost per hour $8.00Overtime Labor Cost per hour $12.00Subcontracting cost per unit $60.00Holding Cost per unit per period $10.00Back order cost per unit per period $100.00Hiring Cost per employee $500.00Firing Cost per employee $1,000.00Capacity DataBeginning Workforce 210Beginning inventory 400Labor Standard per unit 6Regular-time available per period 160overtime available per period 32subcontracting maximum per period 1000subcontracting minimum per period 500

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The LP Model

Minimize t (128wt + 500ut + 1000vt + 10it + 100bt)

s.t.

u1 – v1 = w1 – w0 : workforce change, period 1,…

26.67wt + io – i1 + b1 – bo = dt: demand balance, period 1…

wt, ut, vt, it, bt 0 for each time period t: nonnegativity