FORMULATE ONLY 2 A product can be made in three sizes: Small, Medium, and Large.
ID: 3119355 • Letter: F
Question
FORMULATE ONLY
2 A product can be made in three sizes: Small, Medium, and Large. The respective profit/unit are $10, $15, and $20. The products are made in three centers (A, B, and C), each having the following capacities for making the product (regardless of the size and combination of sizes):
200 units/day, 300 units/day, and 500 units/day. To make the above products, cooling water and electricity are needed as follows.
Water (gallons/unit) Electricity (Kwatts/unit)
Small 10 gallons 45 KW
Medium 14 60 KW
Large 18 65 KW
Center A has 8000 gallons of Water and 10,000 KW of electricity.
Center B has 5000 gallons of Water and 15,000 KW of electricity.
Center C has 10,000 gallons of Water and 15,000 KW of electricity.
Daily demand of each of the products is as follows:
Small 500 units
Medium 700 units
Large 800 units
To maintain an overall balance, the company policy is to ensure that the fraction (scheduled production/center’s capacity) remains consistent for all three centers.
Explanation / Answer
Solution
Let xij = production of product i in centre j, i = 1, 2, 3 for Products Small, Medium and Large respectively; j = 1, 2, 3 for Centres A, B, C respectively.
Then formulation of LPP is:
Max Z(total profit) = 10(x11 + x12 + x13) + 15(x21 + x22 + x23) + 20(x31 + x32 + x33)
Subject to
(x11 + x12 + x13) 500 ……. Demand constraint for Product 1
(x21 + x22 + x23) 700 ……. Demand constraint for Product 2
(x31 + x32 + x33) 800 ……. Demand constraint for Product 3
(x11 + x21 + x31) 200 ……. Capacity constraint for Centre 1
(x12 + x22 + x32) 300 ……. Capacity constraint for Centre 2
(x13 + x23 + x33) 500 ……. Capacity constraint for Centre 3
(10x11 + 14x21 + 18x31) 8000 ……. Water constraint for Centre 1
(10x12 + 14x22 + 18x32) 5000 ……. Water constraint for Centre 2
(10x13 + 14x23 + 18x33) 10000 ……. Water constraint for Centre 3
(45x11 + 60x21 + 65x31) 10000 ……. Electricity constraint for Centre 1
(45x12 + 60x22 + 65x32) 15000 ……. Electricity constraint for Centre 2
(45x13 + 60x23 + 65x33) 15000 ……. Electricity constraint for Centre 3
{(x11 + x21 + x31)/200} = {(x12 + x22 + x32)/300} = {(x13 + x23 + x33)}/500
[Production Balancing constraint]
xij 0 for all i, j = 1, 2, 3…… Non-negativity condition.
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