Computation Section
Subunit Inventory Analysis
 Common Cycle Time

The common cycle time problem is similar to the multi-item problem except that here we restrict the cycle times of the items to either be the same or be integral multiples of a common cycle time. This kind of system would be used when it is more convenient to order replenishments for several items at a time. Again we select Optimize from the menu, but this time we select the Common Cycle button. We do not repeat features of the model that are same as the multi-item case.

For the example we use a similar inventory system as we used to illustrate the multi-item case. For this system, however, there is an setup cost to place an order for the entire system. There may also be additional setup costs for each item.

Reviewing the worksheet below, note that in addition to the three columns for the individual items, there is a fourth column for the system. The only data included in this column is the system setup cost.

The variables of the math-programming model for the common cycle situation are quite different than the variables for the multi-item system. The principal variable is System Cycle Time (Sys CT), measured in weeks for the example. The variables labeled N1, N2 and N3 are the relative order cycles for the items. A value of 1 means that the item is ordered in every cycle of the system. A value of 2 means that the item is ordered every other cycle. These variables are restricted to integer values. The default upper and lower bounds for the integer variables are both 1, meaning that every item is ordered in every cycle. We have changed the upper bound to 10 to allow variability above 1. The worksheet is shown below with the optimum solution.

The yellow cells in column H through K hold nonlinear expressions of the decision variables that define the investment, size, residence time and order frequency for the system. The profit is a separable nonlinear function of the decision variables.


The constraints are the same as for the multi-item system except the Order Frequency constraint. For this model we compute this value as 1/(system cycle time). The individual items do not contribute to the order frequency.

This model can also be constructed for finite replenishment rates and/or shortages. We do not illustrate these options here. In every case the models are nonlinear. They also include integer variables. When the latter are fixed at 1, the model has only one decision variable, the system cycle time. That problem is easily handled by the Solver algorithm. When the integer values are allowed vary, the result is an integer/nonlinear programming model. These problems are often quite difficult to solve. Again, we recommend finding a number of solutions using different initial values for the decision variables.

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Operations Management / Industrial Engineering
by Paul A. Jensen
Copyright 2004 - All rights reserved