A company produces three types of paint by mixing four different types of raw ma

Question

A company produces three types of paint by mixing four different types of raw materials A,B,C,D that are purchased. The company mixes the raw materials according to the following specifications of the maximum or minimum percentages of raw material in each blend: There is no restriction on the percentage amount of raw material B. The maximum daily availability and cost of raw material is given in the table below: Paint Type Max%A Min%C Max%D Price $/gallon Type 1 50% 20% 15% 8 Type 2 20% 40% 30% 7 Type 3 30% 35% 6 Raw Material Availability (gallons) Cost $/gallon A 4000 0.7 B 6000 0.6 C 3000 0.5 D 6500 0.3 (a) Formulate a linear program to determine the daily production policy that will maximize profit. (b) How would you modify the linear program to determine the daily production policy that will minimize un-utilized raw material.

Explanation / Answer

(a) Linear program is following:

Decision variables: Let Ai, Bi, Ci, and Di be the quantity of raw materials A,B,C,D to be mixed to produce paint type i

Objective: Max (8-0.7)A1+(8-0.6)B1+(8-0.5)C1+(8-0.3)D1+(7-0.7)A2+(7-0.6)B2+(7-0.5)C2+(7-0.3)D2+(6-0.7)A3+(6-0.6)B3+(6-0.5)C3+(6-0.3)D3

s.t.

Constraints:

A1+A2+A3 <= 4000

B1+B2+B3 <= 6000

C1+C2+C3 <= 3000

D1+D2+D3 <= 6500

A1 <= 0.5*(A1+B1+C1+D1)

C1 >= 0.2*(A1+B1+C1+D1)

D1 <= 0.15*(A1+B1+C1+D1)

A2 <= 0.2*(A2+B2+C2+D2)

C2 >= 0.4*(A2+B2+C2+D2)

D2 <= 0.3*(A2+B2+C2+D2)

C3 >= 0.3*(A3+B3+C3+D3)

D3 <= 0.35*(A3+B3+C3+D3)

Ai, Bi, Ci, Di >= 0

(b) In this case, objective function would be modified as under:

Min (4000-A1-A2-A3)+(6000-B1-B2-B3)+(3000-C1-C2-C3)+(6500-D1-D2-D3)

Constraints will remain same as above.

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