Question

Photon Technologies, Inc., a manufacturer of batteries for mobile phones, signed a contract with a large electronics manufacturer to produce three models of lithium-ion battery packs for a new line of phones. The contract calls for the following
Battery Pack Production Quantity
PT-100 200,000
PT-200 100,000
PT-300 150,000
Photon Technologies can manufacture the battery packs at manufacturing plants located in the Philippines and Mexico. The unit cost of the battery packs differs at the two plants because of differences in production equipment and wage rates. The unit costs for each battery pack at each manufacturing plant are as follows:
Plant
Product Philippines Mexico
PT-100 0.95 0.98
PT-200 1.34 0.98
PT-300 1.06 1.15
The PT-100 and PT-200 battery packs are produced using similar production equipment available at both plants. However, each plant has a limited capacity for the total number of PT-100 and PT-200 battery packs produced. The combined PT-100 and PT-200 production capacities are 175,000 units at the Philippines plant and 160,000 units at the Mexico plant. The PT-300 production capacities are 75,000 units at the Philippines plant and 100,000 units at the Mexico plant. The cost of shipping from the Philippines plant is $0.15 per unit, and the cost of shipping from the Mexico plant is 0.08 per unit.
(a) Develop a linear program that Photon Technologies can use to determine how many units of each battery pack to produce at each plant to minimize the total production and shipping cost associated with the new contract.
(b) Solve the linear program developed in part (a), to determine the optimal production plan. Qty Produced Phillipines Mexico PT-100 16000c 40000 PT-200 100000 0 5000 1000 PT-300 Total Cost-$
(c) Use sensitivity analysis to determine how much the production and/or shipping cost per unit would have to change to produce additional units of the PT-100 in the Philippines plant. If required, round your answer to two decimal digits At least $ / unit.
(d) Use sensitivity analysis to determine how much the production and/or shipping cost per unit would have to change to produce additional units of the PT-200 in the Mexico plant. If required, round your answer to two decimal digits. At least $ .05/ unit.

Answer 1

Answer:

Check the explanation

Explanation:

a) Linear program model:

Decision variables: Let

P1 = Number of PT-100 products produced at Philippines plant

P2 = Number of PT-200 products produced at Philippines plant

P1 = Number of PT-300 products produced at Philippines plant

M1 = Number of PT-100 products produced at Mexico plant

M2 = Number of PT-200 products produced at Mexico plant

M3 = Number of PT-300 products produced at Mexico plant

Objective: Min (0.95+0.15)P1 + (0.98+0.15)P2 + (1.34+0.15)P3 + (0.98+0.08)M1 + (1.06+0.08)M2 + (1.15+0.08)M3

or,

Min 1.10P1 + 1.13P2 + 1.49P3 + 1.06M1 + 1.14M2 + 1.23M3

s.t.

P1 + M1 ≥ 200,000

P2 + M2 ≥ 100,000

P3 + M3 ≥ 150,000

P1 + P2 ≤ 175,000

M1 + M2 ≤ 160,000

P3 ≤ 75,000

M3 ≤ 100,000

P1, P2, P3, M1, M2, M3 ≥ 0

(b) Solution of the linear program using Excel Solver can be seen in the first attached image below.

Formula: H2 =SUMPRODUCT(B2:G2,$B$11:$G$11)   copy to H2:H9

Optimal Solution:

Decision Variable              Value

P1                                     40000

P2                                     100000

P3                                     50000

M1                                     160000

M2                                     0

M3                                     100000

Total production and shipping cost = $ 524,100

Sensitivity report can be seen in the second attached image below.

Referring to above sensitivity analysis,

(c) Allowable decrease in objective coefficient of P1 is 0.04 therefore production and/or shipping cost per unit has to decrease by $ 0.04 to produce additional units of PT-100 in Philippines plant.

(d) Allowable decrease in objective coefficient of M2 is 0.05 therefore production and/or shipping cost per unit have to be decreased by $ 0.05 to produce additional units of PT-200 in Mexico plant.