Question1 A semiconductor company is facing the problem of determining the amoun

Question

Question1 A semiconductor company is facing the problem of determining the amount of capital expenditure to reduce setup costs. This company has yearly demand rate of 3000 units for a specific type of chip. The annual production rate is 6000 units. Holding one unit of this chip in the inventory for a year has a cost of $20. The setup cost K is dependent on the capital expenditure according to the following relation K-200+1000e 0,003s) where S is the capital cost incurred every year. a) Compute the optimal values of the capital expenditure S and the lot size Q that jointly minimize the average annual costs. b) What is the average annual cost at the optimal solution?

Explanation / Answer

T = Total Cost

T = (D/Q) x K + (Q/2).(P - D).H/P + S

Given, D = 3000; P = 6000; H = 20; and K = 200+1000.exp(-0.0003S)

T = (3000/Q) x (200+1000.exp(-0.0003S)) + 10Q + S

T'Q (partial derivative w.r.t. Q) = (-1/Q^2) x 3000 x (200+1000.exp(-0.0003S)) + 10

T'S (partial derivative w.r.t. S) = (3000/Q) x 1000 x (-0.0003) x exp(-0.0003S) + 1

We equate both of the above to get the minima

(-1/Q^2) x 3000 x (200+1000.exp(-0.0003S)) + 10 = 0
(3000/Q) x 1000 x (-0.0003) x exp(-0.0003S) + 1 = 0

Solving these two, we will get 10*(Q^2) - (Q/0.0003) - 600000Q = 0 or Q = 462

And correspondingly exp(-0.0003S) = -1/((3000/462) x 1000 x (-0.0003)) or, S = $2216

Total cost = (3000/462) x (200+1000.exp(-0.0003*2216)) + 10*245 + 2216 = $11475

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