In a single period model,if shortage cost is four times excess cost,then the optimum service level is:
A)100 percent
B)80 percent
C)60 percent
D)40 percent
E)20 percent

Question

Quizplus · Accepted Answer

In a single period model, the total cost can be represented as the sum of the inventory cost and the shortage cost. Let x represent the service level, then the total cost can be expressed as:

TC = (1-x) * excess cost + x * shortage cost

Given that the shortage cost is four times excess cost, we can write:

TC = (1-x) * excess cost + 4x * excess cost = excess cost * (4x - 3x + 1) = excess cost * (x + 1)

To find the optimum service level that minimizes the total cost, we can take the derivative of the total cost with respect to x and set it equal to zero:

d(TC)/dx = excess cost * (1) = 0
x = 1

However, 100% service level is not realistic or feasible in most situations. So, we need to consider the practical range of service levels. Solving for the practical range of x which makes the 2nd derivative of the total cost positive, we get 0 < x < 0.8.

Substituting x = 0.8, we get:

TC = excess cost * (1.8)

Since we are given that the shortage cost is four times excess cost, we can write:

TC = excess cost * (1.5 * shortage cost)

Equating both expressions for the total cost, we get:

1.8 = 1.5 * (1 - 0.8)
1.8 = 0.3
which is not possible.

Therefore, the optimum service level is at x = 0.8 or 80%.

In a Single Period Model,if Shortage Cost Is Four Times