"A simulation analysis of ordering policies under inflationary conditions": a critique.

In a recent article in this journal Mehra and Amini[1] presented an inflation-adjusted economic order quantity (EOQ) model. They found that order quantifies were stable over a range of inflation rates. This is inconsistent with their analysis which inferred that order sizes should increase under inflationary conditions. Mehra and Amini[1] used a discrete formulation identical to one in Mehra et al.[2] with exactly the same variable names, equations and presentation format. A reference to the earlier work was needed since roughly 40 per cent of Mehra and Amini's article was taken directly from it. The lack of a reference is misleading in its implications that the whole of the article is original to Mehra and Amini.

The stability that Mehra and Amini found is a consequence of their assumption that the number of set-ups each year must be an integer. This is an arbitrary constraint and is not necessary. In a discrete model it is possible and sensible to allow "fractional" numbers of set-ups per year. For example, one could find that the optimal order cycle is to order every five months. This is a discrete period. Such a possibility is excluded by their assumption about setups. Mehra and Amini do not explicitly recognize that constraining the number of set-ups to integer values imposes a one-year planning horizon. The assumption of integer numbers of set-ups (1,2,3, ...) limits the order cycles possible (12 months, six months, four months, ...). The excluded periods may be optimal solutions. There is no good reason why an arbitrary one-year horizon should be placed on this model. The possibility of longer or shorter periods of inflation makes this approach questionable. The objective should be to optimize over the whole inflationary period rather than in annual blocks. The formulation is sub-optimal. Mehra and Amini's model could be recast with its parameters for demand, inflation and holding costs adjusted to reflect longer or shorter periods. This would yield a more suitable formulation but their approach would still be inappropriate. Their "simulation" model provides rough answers where existing analytical models give exact answers. Although Mehra and Amini disdain continuous formulations, these models provide more effective solutions.

Mehra and Amini simply state that costs increase by various proportions as inflation increases. This is a trivial analysis. The critical question is: if the unadjusted EOQ is inappropriate, what cost penalties then follow from using it? The costs of the unadjusted EOQ should be higher than those from adjusted models if it is not appropriate. Mehra and Amini do not provide evidence of the relative merits of their approach.

Mehra and Amini assert that continuous models are inappropriate without substantiating their views. It is widely recognized that continuous models are approximations for more complex discrete models which, in theory, more accurately describe the situation. I am not aware that anyone has found non-trivial cost penalties resulting from applying continuous formulations. Silver and Peterson[3] describe a continuous formulation of an inflation-adjusted EOQ model and present a readily usable result. The standard EOQ can be multiplied by a factor: (1/(1 - i/K)) (1/2) where "K" is the holding rate and "i" the inflation rate to yield an adjusted order quantity. This approach may be compared to Mehra and Amini[1] by using an example from Mehra et al.[2]. The set-up costs are US\$4,000, the holding costs are 30 per cent, the initial unit cost is US\$40, with a demand for 12,000 units per year and an inflation rate of 12 per cent. Under these circumstances the optimal order quantity is 3,689. Using the same parameter values and solving for the optimum order quantity using Silver and Peterson's[3] formulation gives a result of 3,651. The difference in quantities is about 1 per cent. The difference in costs is practically invisible, US\$56,554 for Silver and Peterson[3] and US\$56,553 for Mehra et al.[2].

A comprehensive set of comparisons is needed to establish how well Mehra and Amini's[1] approach works. This may be seen in Table I. The benefits of a broad comparison are obvious. It shows the relative merits of discrete and continuous approaches, and compares both to an unadjusted approach. The key is a consistent comparison of costs under inflationary conditions across all policies. I have used Mehra and Amini's formula for total cost under inflationary conditions for all cost calculations. Table I is based on an evaluation of the first row of Mehra and Amini's Table I. It also includes costs for their approach, and the order quantities and costs for the unadjusted EOQ. and for Silver and Peterson's continuous formulation. If the unadjusted EOQ were used when inflation was 12 per cent its costs would be about 5 per cent greater than those in the adjusted models. This confirms Mehra and Amini's[1], and others' views. The comparison of Mehra and Amini's discrete approach with Silver and Peterson's[3] formulation is the key result. Mehra and Amini's[1] model is consistently more costly. The differences are small though, as would be expected if continuous formulations were good approximations to discrete phenomena. An analysis of other examples from their Tables I to VI provided comparable results. In many instances there was a cost benefit of less than 1 per cent from any inflation adjustment, and the worst was about 5 per cent as shown in Table I. It might be possible to contrive a scenario in which their model performs better against the EOQ and alternative formulations of inflation models. Nevertheless, if the cases presented are to be taken as ideal or typical applications its performance is unimpressive.

[TABULAR DATA FOR TABLE I OMITTED]

Mehra and Amini's[1] finding that inflation affects the optimal lot size simply duplicates Mehra et al.'s[2] conclusions, using their model. The approach taken implicitly imposes planning horizon limits. This unnecessary limitation affects their results and subverts the "interesting and revealing" finding of their work. If set-ups are restricted to integers the model becomes inflexible and it is no surprise then that order quantities are stable. Their disagreement with using continuous formulations may be soundly based on purely theoretical considerations, but it can be shown to have no practical effect on model performance. The model used is complex in spite of Mehra and Amini's assertion that it can be easily programmed. Silver and Peterson's[3] model is easier to understand and use, and its performance seems equal to the optimal discrete model of Mehra et al.[2]; and it betters the simulation approach of Mehra and Amini[1].

References

1. Mehra, S. and Amini, M.M., "A simulation analysis of ordering policies under inflationary conditions", International Journal of Operations & Production Management, Vol. 14 No. 10, 1994, pp. 72-83.

2. Mehra, S., Agrawal, S.P. and Rajagoplan, M., "Some comments on the validity of EOQ formula under inflationary conditions", Decision Sciences, Vol. 22 No. 1, 1991, pp. 206-12.

3. Silver, E.A. and Peterson, R., Derision Systems for Inventory Management and Production Planning, 2nd ed., John Wiley & Sons, New York, NY, 1985.
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