Computation Section
Subunit Inventory Analysis
 Machine Scheduling

This model is similar to the other models, but it represents a somewhat different system. Here we have a single machine type producing several products. The products correspond to the individual items used for the other optimization models. For most instances of this problem, the individual items have finite replenishment rates, representing the production rates on the machine. The products also have setup times, representing the time required to setup the machine to produce the individual products. Although we allow the constraints used by the other models, the principal constraint for this situation is the machine utilization constraint. If there is only one machine of this type, the total proportion of the times spent in production or setup cannot exceed 1. In our models we allow the number of machines to be variable, so the constraint is changed so that the total utilization of the machine type cannot exceed m where m is the number of machines used.

To create a model, we select Optimize from the menu and click the Machine Scheduling button. The model created is similar to the common cycle model. Note that the Utilization constraint is required for this model. The dialog does not allow it to be unchecked.

To illustrate, we use the same situation as for the common cycle example, but here the replenishment rates are finite. They represent the production rates of the three products using the machine. Of course, the rates must be greater than the demand rate for feasibility.

Reviewing the worksheet below, we note that the first variable is System Cycle Time (Sys CT). This is the cycle time for the machine. We have three variables for the items, N1, N2 and N3, representing the relative cycle times for the three products. The value of 1 means that the item is produced in every cycle. The value of 2 means that the item is produced every other cycle. Higher values have similar meanings. The relative cycle times must be integer. For this model we have one additional variable, Number of Resources. This is the number of machines used for the system. For some situations it may be impossible to meet the production requirements with a single machine. This variable allows the solution to use an integer number of machines, greater than 1.

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

The worksheet is shown below with the optimum solution. The optimum solution uses 2 machines. The Solver could find no feasible solution when the number of machines was limited to 1.


The constraints are the same as for the common cycle model except the Utilization constraint. This constraint requires an added row of data in row 36, the setup time, assumed to be one week for all three products. The computed column shown in row 37 is the proportion of time used for production, that is the (demand rate)/(replenishment rate). The total is shown in cell H37. This value is transferred to cell H21 in the row for the utilization constraint. The remainder of the coefficients in row 21 are computed as the

prop. of time for item setup = (item setup time)/((system cycle time)*(relative number of cycles)

Cell P21 holds the constant -1. The term enters the utilization formula as the negative of the number of resources. The sum of the contents of the range H21... K21 plus the product of -1 and the number of resources must not exceed 0. The relationship is imposed by the utilization constraint.

Models of this kind are also available when shortages are allowed.

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