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This problem is examined because it helps illustrate the role of transaction cost in economic choices, but optimal production run time is an interesting question in its own right. This problem also shows how a nonlinearity, the fixed setup cost of a production run, results in a quantization of the size of a production run. This phenomenon carries over into other areas of economics.

Let Q be the annual rate of sales and N be the number of times per year that the plant is run for a period of time T (in afraction of a year). During the production run a total output of X is achieved. Thus N=Q/X and N is not necessarily an integral number.

Assuming a uniform level of sales throughout the year the inventory increases from 0 at the beginning of the run to (X/T - Q)T = X - QT at the end of the run at time T. The inventory then is used up during the time between the end of the run and the beginning of the next run, which occurs at time (1/N). The average inventory size is (X-QT)/2 during an interval T and (X-QT)/2 during an interval [(1/N)-T]. Thus the average inventory size is (X-QT)/2.

If p is the price per unit the value of the average inventory is p(X-QT)/2. For an interest rate of r the interest cost of the inventory is rp(X-QT)/2. If the setup cost of beginning a production run is S then the setup cost incurred per year is SN=SQ/X.

The level of manufacturing costs depends upon the rate of production x. Let f(x) be the manufacturing cost as a function of the rate of production. Since X=xT, the manufacturing costs can be expressed as Tf(X/T).

The total cost C affected by size of the production run X is then:

C = Tf(x) + rp(xT-QT)/2 + SQ/xT

or, expressed in terms of X and T

C = Tf(X/T) + rp(X-QT)/2 + SQ/X


The choice variables are x and T but choosing x and T to minimumize cost is equivalent to choosing X and T to minimize cost. Minimum cost occurs at X* and T* such that:

∂C/∂X = f'(X*/T*) + rp/2 - SQ/X*2 = 0
∂C/∂T = f(X*/T) - f'(X*/T*)(X*/T*) - rpQ/2 = 0

The second condition implies that x*=X*/T* is such that:

f(x*)-f'(x*)x* = rpQ/2

The minimimum average production cost occurs at the rate of production x where f(x)-f'(x)x = 0, so x* deviates from x on the basis of the level of the interest rate r, the price of the product and the rate of sales Q. The level of x* does not, however, depend upon the level of setup costs.

From x* the value of X* can be found from the first condition; i.e.,

X* = [SQ/(f'(x*)+rp/2)]1/2.

The level of setup costs does affect the size of the optimal production run.

The optimal length of the production run T* is then given by:

T* = X*/x*
= [SQ/(f'(x*)+rp/2)]1/2/x*

The setup cost S affects the optimal production run as well.

If the setup cost is zero then T*=0 and the plant operates essentially continuously and there are no inventories. Thus the optimal operating rate x* would be x, the rate at which average cost is a minimum. This situation would be like an electrical light that switches on and off at such a high frequency that it appears to be lighted continuously.

The above suggests that the problem might be reformulated. First consider the first order conditions for minimizing

C = Tf(x) + rp(x-Q)T/2 + SQ/xT

in terms of x and T rather than X and T.

∂C/∂x = Tf'(x) + rpT/2 - SQ/x2T = 0
∂C/∂T = f(x) + rp(x-Q)/2 - SQ/xT2 = 0


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