Linear programming is an optimization process. In production scheduling linear programming is used to optimise production plan for the linear objective function with some constraints. These programming models are able to handle large numbers and constraints and ultimately to reach the feasible solution. There are various ways of solving a linear programming problem like graphical method, simplex method and for huge number of data Microsoft Excel Solver. These types of mathematical tools are used mainly in determining production quantities, staffing, inventory etc of production process. For example, for planning the production of a product family without using backorders. Each worker produces 5000 units per month if used productively on regular time or overtime. Other possible options are subcontracting and overtime production. Let Dₓ=demand as product units in month x (presumed known, not a variable)
Wₓ=workers on hand at the start of month x, Hₓ= hires at the start of the month x
Lₓ= Lay-offs at the start of the month x, Iₓ = inventory as product units at the end of month x
Sₓ= subcontracted production as product units in month x, Oₓ= overtime production in month x
Then for each month the constraints are Wₓ=Wₓ₋₁ +Hₓ -Lₓ
Iₓ = Iₓ₋₁ +5000 Wₓ + Oₓ + Sₓ -Dₓ
Oₓ≤ 0.15 (5000 Wₓ)
So, there are six variables where Dₓ is the decision variable and three constraints for each month . If the production process covers 12 months then there will be 72 decision variables and 36 constraints. So the objective function for maximizing profits and minimizing cost will be
TC=Ʃⁿᵢ₌₁(c₁Wₓ+c₂Hₓ+c₃Lₓ+c₄Sₓ+c₅Oₓ) where n = 12 months of a year.From the above example, we can see that to get the feasible solution the huge data requires computer to solve. But with this solution the managers or other management of a company can the production schedule and implement them.
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Operation Management by Lee J. Krajewski
