Discussion Paper

No. 2018-77 | October 17, 2018
An EOQ Model for multiple products with varying degrees of substitutability


In this paper, the authors present an EOQ model with substitutions between products and a dynamic inventory replenishment policy. The key assumption is that many products in the market are substitutable at different levels, and that, in most cases, a customer who discovers that a desired product is unavailable will choose to consume a product with similar attributes or functionality, rather than not purchase at all. Therefore, given a firm that stocks multiple substitutable products, the authors assume that a stock out of one product has a direct impact on other products’ demand. The main purpose of the model is to enable inventory managers to develop ordering policies that ensures that, in the event that a specific product runs out and cannot be replenished due to unforeseen circumstances, the consequent increase in demand for related products will not cause further stock out incidents. To this end, the authors introduce a dependency factor, a variable that indicates the level of dependency, or correlation, between one product and another. The dependencies among the various products offered by the firm are embedded into the EOQ formula and assumptions, enabling managers to update their ordering schedules as needed. This approach has the potential to generate more practical and realistic purchasing and inventory optimization policies.

JEL Classification:

C44, C51, C61, L60


  • Downloads: 1610


Cite As

Leonid Eksler, Roei Aviram, Amir Elalouf, and Aakash Kamble (2018). An EOQ Model for multiple products with varying degrees of substitutability. Economics Discussion Papers, No 2018-77, Kiel Institute for the World Economy. http://www.economics-ejournal.org/economics/discussionpapers/2018-77

Comments and Questions

Anonymous - Referee Report 1
December 03, 2018 - 08:34

This paper discusses an inventory ordering policy when considering dependency factor, a variable that indicates the level of dependency between one product and its substitutions. The basic EOQ model with lead time is modified to optimize the inventory outcomes. I suggest a major revision as explained below.

1. One ...[more]

... major issue with this paper is that the authors need to justify why EOQ model still apply to the scenario described in the paper. The goal of EOQ is to minimize the sum of ordering cost and holding cost. However, as illustrated in Section 4, the authors seem to maximize revenue using their inventory ordering policy. Also, when considering the penalty cost for shortage, why EOQ model is the best option?
2. Another major issue with the algorithm is that the authors did not specify how to choose the substitutable products when one item is out of stock. The numerical study in Section 4 assumed only two products in the group, which obviously avoided the discussion of substitutable product selection. If item A is out of stock and customers can switch to products B or C, then which products between B and C should have higher ROP? The answer was merely discussed in this paper although the authors claimed that the proposed solution is for “multiple substitutable items within an inventory system”.
3. Last but not least, I believe the paper needs to compare its ordering policy with some other existing policy in order to demonstrate that the proposed policy achieves a much better result. Can this policy generate lower cost (or higher revenue) that the cited paper? Comparing the proposed policy with the basic EOQ provided very limited insights.

Besides the three major issues listed above, there are some minor comments.
1. The algorithm needs to specify when the ROP will be adjusted: when one product is out of stock or when its supply from the upstream is delayed.
2. Is eta_{i,j}=eta_{j,i}? If not, then the definition of eta_{i,j} (defined as Strength of the correlation between product 𝑖 and product 𝑗} is unclear. Also, I am confused with the definition of consumption gap.
3. Figure 1 lacked some important information. Does i start from 1? If so, does j start from i+1? DF in Figure 1 is not predefined.
4. Why eta is the same in Section 3.3?

Please correct the following typos or errors:
a. On page 7, in figure 1, “i+1” should be “product i+1” and “product I to product j” should be “product i to product j”.
b. On page 8, “P_i belongs to G_i” should be “P_i belongs to G”?
c. On page 9, “By finding CGij, we can now compute the correlation strength Ƞ𝑖𝑖 as follows:” should be deleted.

Aakash Kamble - Reply to Referee Report 1
January 03, 2019 - 08:37

see attached file

Anonymous - Invited Reader Comment
December 03, 2018 - 08:51

The authors argue that by correctly accounting for the transfer of demand from an out-of-stock item to substitutable items, the expected order date of the latter must be corrected.

The argument is true, if very simple. In short, if a shop sells two kinds of orange juice and ...[more]

... one runs out of stock, the stock of the latter will depleted faster and one can predict when this will happen, which is straightforward to compute assuming constant consumption rates.

The added layer here is that of a dependency matrix, which allows the above argument to be generalized to any number of items and any number of substitutable items. The whole question is then how this matrix is determined from data? Only a vague statement is provided. Several open questions remain:

1. Why should coefficients of the dependency matrix be 1 or 0 only?
1.1 I am not sure that correlations are adequate here. The authors probably mean dependencies.
2. What are the economic consequences of wrong positives, wrong negatives in the inference of the dependency matrix?
3. Could the authors offer a dynamic example?
4. Is there data available?
5. Why are the consumption rates constant?
6. Are we sure that the whole demand of an out-of-stock product fully propagates to other products?

In short, provided that this generalization does not pre-exist (I am not in a position to have an opinion on this point), I think that this discussion paper is a necessary but not sufficient contribution, and that this way of thinking should be extended much further to show non-trivial dynamical properties.

Aakash Kamble - Reply to Invited Reader Comment
January 03, 2019 - 08:38

see attached file