Question

Company AAA produces only one product which other manufacturers purchase as a component for their final products. The operations manager wants to plan the production and inventory quantities of the product for the first six months of the next year. The monthly demand is forecasted as follows.

January	February	March	April	May	June
1200	1400	1800	2400	2600	2200

The company has three production options: regular production, overtime production, and subcontracting. Production cost per unit during regular time is $80 per unit. The overtime cost is 50% more than the regular time cost. A unit of product procured from a subcontractor costs $130. The cost to hold one unit of the product from one month to the next is $20.

The regular production capacity is 1,800 units. The minimum production using regular time per month is 1,400 units. Any production beyond 1,800 units in a month is considered as overtime, which cannot be more than 400 units.

The inventory level at the beginning of next year is planned to be 400 units. The minimum inventory at the end of each month is 200 units and the maximum inventory cannot be more than 600 units at the end of each month. The inventory at the end of June must be at least 300 units. a) Formulate the problem as a mathematical (algebraic) model to find the optimal production plan that minimizes the total cost.

Answer 1

(a)

Algebraic Model:

Let,

R_j = Regular production in month-j
O_j = Overtime production in month-j
S_j = Subcontracted units in month-j
E_j = Ending inventory of month-j,

for j=1,2,...,6

Minimize Z = total cost = 80*Σ_j=1(1)6 R_j + 120*Σ_j=1(1)6 O_j + 130*Σ_j=1(1)6 S_j + 20*Σ_j=1(1)6 E_j

Subject to,

R₁ + O₁ + S₁ + 400 - E₁ = 1200
R₂ + O₂ + S₂ + E₁ - E₂ = 1400
R₃ + O₃ + S₃ + E₂ - E₃ = 1800
R₄ + O₄ + S₄ + E₃ - E₄ = 2400
R₅ + O₅ + S₅ + E₄ - E₅ = 2600
R₆ + O₆ + S₆ + E₅ - E₆ = 2200

R_j >= 1400 for j=1,2,...,6
R_j <= 1800 for j=1,2,...,6

O_j <= 400 for j=1,2,...,6

E_j >= 200 for j=1,2,...,5
E₆ >= 300
E_j <= 600 for j=1,2,...,6

R_j, O_j, S_j, E_j >= 0 for j=1,2,...,5