Question

A company produces and sells 4 types of products – P1, P2, P3, and P4. The table below summarizes the relevant data.

Three types of raw materials – RM1, RM2, and RM3 are required for the manufacturing process. The availability and cost per unit for each raw material are specified under the columns Available and Cost per unit, respectively.

The company is contractually obligated to produce at least a minimum quantity for each product; these are specified in the row Minimum quantity.

The selling price per unit of the products are specified in the row Selling price per unit.

The resource requirements for each product are as follows:

• It takes 2 units of RM1, 4 units of RM2, and 5 units of RM3 to produce each unit of P1.
• It takes 4 units of RM1, 5 units of RM2, and 6 units of RM3 to produce each unit of P2.
• It takes 5 units of RM1, 8 units of RM2, and 10 units of RM3 to produce each unit of P3.
• It takes 2 units of RM1, 3 units of RM2, and 4 units of RM3 to produce each unit of P3.

The fixed cost incurred to produce each unit of product are specified in the row Fixed cost/unit.

The cost of producing each unit of each product is given by the fixed cost plus the cost of the raw material used in producing that unit.

	P1	P2	P3	P4	Available	Cost per unit
RM1	2	4	5	2	18000	$3
RM2	4	5	8	3	30000	$4
RM3	5	6	10	4	36000	$2
Minimum quantity	1000	1000	1000	1000
Fixed cost/unit	$6	$8	$10	$4
Selling price per unit	$50	$60	$90	$40

Formulate this problem as a linear program and obtain the optimal solutions so as to maximize the total profits under the given constraints.

Answer 1

S12 S50 S32 S6 Pl 2000 1000 1000 2500 $10 $40 S26 $4 P4 Max. Profit Available Cost per unit S8 S60 Ptofit Selling price per unit Cost per unit Fixed cost unit S13 S90 S67 S10 P3 $8 P2 70,000 Decisin variable Constrains RM1 RM2 S3 $4 S2 18,000 28,50030000 36,000 36000 2,000 1,000 1,000 2,500 18000 10 10 1000 1000 1000 1000 Minimum quantity P1 Minimum quantity P2 Minimum quantity P3 P4 1 0 0 12 13 14 15 16 17 18

16 17 18 19 20 21 Maximize P 12P+8P +13P +10P Subject to 2P +4P +5P + 2P 18000 4P +5P +8P +3P, 3 28500 5R +6P +10P +4P S 36000 23 26 27 28 29 P 21000 В 1000 В 21000 P 2 1000 31 32 P,R,P, P 20 34