## Sum of Maximum Intermediate and Output Buffer States

An overview of the peak buffer states experienced by the intermediate buffers is presented in Figure 2, using the sum of the maximum state of all intermediate buffers. For the case of neural network scheduler this sum is equal to the number of intermediate buffers in the cell (i.e., 77), since the algorithm assigns a target value of one for all intermediate buffers, which cannot be exceeded as the control law forces its output to become zero whenever a buffer state reaches its target value.

When the conventional policies are used, this sum becomes considerably larger and as can be seen from Figure 2, it can be more than five times higher as in the case of CAF. Specifically, while for very low rates, this sum slightly exceeds the limit of 77, a slight increase in the rate results in a considerable increase in the sum of the maximum states for intermediate buffers with most extreme case that of CAF at 0.001 parts per minute.

The NN scheduler controls robustly the maximum possible states of the buffers with respect to raw-material-arrival rates. This is obviously not the case for all conventional schedulers studied that, at higher rates, even force the buffers to reach their capacity limits. Therefore, the proposed scheduler ensures stable manufacturing system operation and minimum capacity requirements for buffers.

Figure 3 presents the sum of maximum output buffers states. The achieved results for both FIFO and NN are identical and equal to the total demand of273 parts for all product types. This sum for the case of the rest of the schedulers, acquires larger values, thus denoting the production of a surplus of parts. This excess of parts originates in the clearing nature ofthe employed policies (i.e., CAF, since CLB is not functioning for rates greater than 0.002).

Figure 6. Average backlogging cost x104

x104

At low rates, all schedulers result in precise achievement of the production goal, due to the fact that all operations, even the clearing ones, empty buffers containing very few parts, causing submachines that output finished products, to process buffers which contain a single part only. Both FIFO & NN emerge as schedulers guaranteeing accurate achievement of production.

WIP and Inventory Costs_

The next two figures represent cost measures for storing parts in buffers, intermediate and output respectively. Let the cost for holding x,. parts at a buffer for Ttime units be considered equal to k, x, T, where k, is the cost for storing a single part in the output buffer of submachine s, , per time unit. Integrating the previous quantity over the entire makespan yields the cost for storing parts in a single buffer. Thus, assuming for simplicity that for all buffers in the system k =1, the cost measure evolves into a plain integral of the respective buffer states. Specifically measures for the Work In Process (WIP) and Inventory costs, are provided by means of the average integral of the intermediate and output buffer states respectively, (i.e. WIP equals to (1/N.) ^='1 1 j" x..dt, while

Inventory cost is (1/No) £ x,,L(0 dt where N.=77, No= 18 are the number of intermediate and output buffers in the cell and T is the production makespan yielded by each scheduling methodology). Due to the low intermediate buffer states guaranteed by the neural network algorithm, a lower WIP is observed, while the rest of the schedulers introduce considerably larger costs.

Figure 7. Average lead time

Thus, the neural network algorithm achieves the most efficient WIP cost control, by keeping intermediate buffer state as small as desired, while guaranteeing system stability. Inventory costs are also considerably smaller than those occurring when employing the conventional scheduling policies due to the overall faster fulfilment of the production demand.

For lower rates, when all schedulers achieve total production almost simultaneously, deviations between the inventory costs yielded by the considered policies, and the one introduced by the NN are not really significant with the exception of the far worst performance of CLB. However, the increase in rates is causing the occurrence of a considerably larger peak in intermediate and output buffer states as well as significant differences in makespan, which sufficientlyjustify, the observed extremely greater WIP & inventory cost for the case of the conventional schedulers, shown in Figures 4 and 5. For higher rates the NN scheduler presents a far more robust-stable behavior when compared to FIFO and CAF. Thus, NN superiority with respect to these popular cost measures is obvious.

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