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.
(a)
Algebraic Model:
Let,
Rj = Regular production in month-j
Oj = Overtime production in month-j
Sj = Subcontracted units in month-j
Ej = Ending inventory of month-j,
for j=1,2,...,6
Minimize Z = total cost = 80*Σj=1(1)6 Rj + 120*Σj=1(1)6 Oj + 130*Σj=1(1)6 Sj + 20*Σj=1(1)6 Ej
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
Rj >= 1400 for j=1,2,...,6
Rj <= 1800 for j=1,2,...,6
Oj <= 400 for j=1,2,...,6
Ej >= 200 for j=1,2,...,5
E6 >= 300
Ej <= 600 for j=1,2,...,6
Rj, Oj, Sj, Ej >= 0 for j=1,2,...,5
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