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	$0.98	$1.06
PT-300	$1.34	$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 $0.21 per unit, and the cost of shipping from the Mexico plant is $0.18 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.
	Let P1 = number of PT-100 battery packs produced at the Philippines plant P2 = number of PT-200 battery packs produced at the Philippines plant P3 = number of PT-300 battery packs produced at the Philippines plant M1 = number of PT-100 battery packs produced at the Mexico plant M2 = number of PT-200 battery packs produced at the Mexico plant M3 = number of PT-300 battery packs produced at the Mexico plant
	Min P1 + P2 + P3 + M1 + M2 + M3 s.t. P1 + P2 + P3 + M1 + M2 + M3 = Production PT-100 P1 + P2 + P3 + M1 + M2 + M3 = Production PT-200 P1 + P2 + P3 + M1 + M2 + M3 = Production PT-300 P1 + P2 + P3 + M1 + M2 + M3 ≤ Capacity Phi PT-100 & 200 P1 + P2 + P3 + M1 + M2 + M3 ≤ Capacity Mex PT-100 & 200 P1 + P2 + P3 + M1 + M2 + M3 ≤ Capacity Phi PT-300 P1 + P2 + P3 + M1 + M2 + M3 ≤ Capacity Mex PT-300 P1, P2, P3, M1, M2, M3 ≥ 0
(b)	Solve the linear program developed in part (a), to determine the optimal production plan.
	Qty Produced Phillipines Mexico PT-100 PT-200 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 $  / unit.

Answer 1

Determine the Production + Shipping cost per unit for each plant as follows:

Plant	Philippines	Mexico
Product	Production	Production + Transportations Cost	Production	Production + Transportation Cost
PT-100	$0.95	0.95+0.21 = $1.16	$0.98	0.98+0.18 = $1.16
PT-200	$0.98	$1.19	$1.06	$1.24
PT-300	$1.34	$1.55	$1.15	$1.33

Formulation:

Objective function: Objective to minimze the total production and shipping cost

Min     $1.16P1 + $1.19P2 + $1.55P3 + $1.16M1 + $1.24M2 + $1.33M3                           

s.t.                                                                                                                                 

1*P1 + 0*P2 + 0*P3 + 1*M1 + 0*M2 + 0*M3 =   200,000         Production PT-100

0*P1 + 1*P2 + 0*P3 + 0*M1 + 1*M2 + 0*M3 =   100,000         Production PT-200

0*P1 + 0*P2 + 1*P3 + 0*M1 + 0*M2 + 1*M3 =   150,000         Production PT-300

1*P1 + 1*P2 + 0*P3 + 0*M1 + 0*M2 + 0*M3 <= 175,000        Capacity Phi PT-100 & 200

0*P1 + 0*P2 + 0*P3 + 1*M1 + 1*M2 + 0*M3 <= 160,000        Capacity Mex PT-100 & 200

0*P1 + 0*P2 + 1*P3 + 0*M1 + 0*M2 + 0*M3 <= 75,000          Capacity Phi PT-300

0*P1 + 0*P2 + 0*P3 + 0*M1 + 0*M2 + 1*M3 <= 100,000        Capacity Mex PT-300

P1, P2, P3, M1, M2, M3 ≥ 0                    

Part b:

Excel Model and solver solution:

Optimal Solution:

Qty Produced
	Phillipines	Mexico
PT-100	75,000	125,000
PT-200	100,000	None
PT-300	50,000	100,000

Total cost= $560,500

Part C.

Sensitivity report:

According to sensitivity analysis the PT-100 is produced in Philippines plant, thus its reduced cost is already zero, thus it not need to reduce the cost of this plant

ANS: 0

Part D.

PT 200 is not produced in Mexico and the reduced cost of M2 is 0.05, the production and shipping cost should be reduced by 0.05 so that variable M2 is considered in optimal so;ution.

ANS: atleast by $0.05/unit