A manufacturing firm is planning a three month production schedule for a new pro

Question

A manufacturing firm is planning a three month production schedule for a new product. From past production records the manager knows that 1,200 units can be produced per month. Also, an additional 400 units can be produced monthly on a overtime basis. The unit costs of items produced on a regular time basis is $15 and $20 on an overtime basis. The demands for the next 3 months are 1000, 1500, and 1800. Excess production may be stored at a cost of $1 per item per month. The manager wants to know the monthly production schedule that will minimize the total cost.
1. Formulate this problem as a regular linear programming model based on the material balance equation.
2. Formulate this problem as a transportation model.

Explanation / Answer

1) LP model

Let

P1, P2, P3 = regular production in 3 months

O1, O2, O3 = overtime production in 3 months

E1, E2, E3 = excess production stored as inventory in 3 months

Objective: Min 15(P1+P2+P3)+20(O1+O2+O3)+1(E1+E2+E3)

s.t.

P1+O1-E1=1000

P2+O2+E1-E2=1500

P3+O3+E2-E3=1800

P1,P2,P3 <= 1200

O1,O2,O3 <= 400

P1,P2,P3,O1,O2,O3,E1,E2,E3 >= 0

2) transportation model

Let

Pij = Regular production in month i consumed in month j , where i,j=1,2,3 and j>=i

Oij = Overtime production in month i consumed in month j , where i,j=1,2,3 and j>=i

Objective: Min 15(P11+P22+P33)+20(O11+O22+O33)+16(P12+P23)+21(O12+O23)+17P13+22O13

s.t.

P11+P12+P13 <= 1200

P22+P23 <= 1200

P33 <= 1200

O11+O12+O13 <= 1200

O22+O23 <= 1200

O33 <= 1200

P11+O11 = 1000

P12+O12+P22+O22 = 1500

P13+O13+P23+O23+P33+O33 = 1800

Pij, Oij >= 0

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