Diversification in economic order quantities (EOQ) model

Citation

Parikh, M., Shah, N.H. and Shah, Y.K. (1999), "Diversification in economic order quantities (EOQ) model", Kybernetes, Vol. 28 No. 3. https://doi.org/10.1108/k.1999.06728caf.001

Diversification in economic order quantities (EOQ) model

Diversification in economic order quantities (EOQ) model

Keywords Cost minimization, Cybernetics, Inventory, Suppliers

Abstract In this paper an economic order quantities (EOQ) model is considered under the assumption that there exist two suppliers (sources) who ship amount-ordered quantities. Since they may charge different prices, diversification may be advantageous rather than using a single supplier. We analyze this problem for EOQ models under two different situations: when orders are simultaneously placed from the two suppliers; and when orders are placed one after the other from the two suppliers. Analytic results for the optimal order quantities and the minimum cost are obtained for the EOQ model. The paper concludes with a numerical example.

1. Introduction

In recent years, one crucial assumption made in these studies concerns the number of suppliers involved. It is always assumed that there is only one supplier (source). Only recently, Gerchack and Parlar (1990) have modelled the tradeoffs involving two suppliers in the economic order quantities (EOQ) context. They compared the associated cost with a single source with lower order cost and they have found conditions under which diversification is optimal.

In portfolio theory, it is well-known that risk-averse investors should diversify their investments (Levy and Sarnat, 1984). In our problem, we will show, the objective function to minimize is convex for a wide range of parameter values and this would suggest that a similar pattern of diversification might emerge. An analysis of the resulting diversification issues may suggest that it could be beneficial to order smaller quantities from two suppliers, rather than one large quantity from a single supplier.

In this communication, we develop the results by making the very general assumption that the unit prices charged by two suppliers and the unit holding costs incurred for items purchased from the two suppliers are different. We developed the total cost expression as a function of the order quantities from each supplier when both the orders are placed simultaneously and when orders are placed one after the other from the two suppliers.

Our model is based on the model which determines the optimal order quantity by minimizing the sum of inventory holding cost and ordering costs. In the development of the model, we will assume that lead time is zero. The benefits of diversification are achieved through possible reduction in the total annual costs which include ordering, holding and purchase costs. We conclude with a discussion of a numerical example for each situation and a comparison of the single source versus diversification.

2. Diversification in the EOQ model

There exist two suppliers to a customer. The customer needs to determine the amount that it should order from each supplier every period such that his holding and ordering costs will be minimum.

In this section, we assume that the annual demand is R units and we may order from two independent sources in which case K is the total ordering cost when both sources are used. Q_i > 0, i = 1, 2 units are ordered from source i. It is also assumed that shortages are not allowed and the unit price charged by source i is C_i and inventory holding charge fraction I/annum for both the sources.

Situation I: when orders are simultaneously placed from the two suppliers

When Q_i units are ordered from each supplier the total amount ordered is Q_1I + Q_2I which becomes the initial inventory at the beginning of each cycle. The cycle length is (Q_1I+ Q_2I)/R.

Referring to Figure 1, the total cost incurred per time unit is

(1)

Minimizing TC₁ (Q_1I, Q_2I) subject to Q_1I > 0 and Q_2I > 0 (since diversification is in effect, i.e. both sources are being used) would give us the optimal quantity to order from each source. To minimize TC_I, we perform partial differentiations and obtain the following:

Figure 1.Inventory level in the EOQ model

(2)

(3)

Assuming (2) to be quadratic in Q_2I, we get

Substituting this value of Q_1I in equation (3), it can be solved for Q_2I using Newton-Raphson's method.

It can be shown that it is not possible to obtain the explicit form of the minimum value of the objective function TC^*_I in terms of all the parameter values (Parikh et al., 1997):

(4)

To see whether diversification is worthwhile, one needs to compare TC^*_I with TC^*_i; the minimum cost incurred at source i = 1, 2 is used exclusively. Letting K_i to be order cost of source i is used, we have optimal order quantity Q^*_i as

(5)

The corresponding minimum cost is

(6)

A comparison of (4) and (6) reveals that the optimal solution to the diversification problem will depend on which of TC^*_i, TC₁ or TC₂ is the smallest, given the parameters R, K, K_i, and C_i. In section 3, we will discuss numerical examples where the comparison is made.

Situation II: when orders are placed one after the other to the two suppliers

Arguing, as above, in this situation, the total cost of an inventory system per time unit is given by

(7)

To minimize TC_II, we perform partial differentiation and obtain the following:

(8)

and

(9)

solving equation (8) for Q_1II (i.e. assuming it to be quadratic in Q_1II) we get,

(10)

Substituting value of Q_1II from (10) in (9), the final expression of (9) can be solved for optimum Q_2II using Newton-Raphson's method and hence optimum

(11)

can be found out using (7).

Again, here, to see whether this diversification is worthwhile we compare TC^*_II with TC^*_i, i = 1,2 (as defined in situation I, equation (6)).

Comparing (4) and (11) indicates that

Thus, it may be advantageous to place a second order once the first order gets depleted.

The above comparison is also discussed by a numerical example in section 3.

3. Numerical example

We will use this example to determine the optimal order quantities and total cost for both situations.

1. 1.

   Situation 1: when orders are simultaneously placed from the two suppliers.

   Situation 2: when orders are placed one after the other to the two suppliers.

We use the following parameter values:

1. 1.

   Order cost, K = $25.00/order

   Demand rate, R = 1,000 units per year

   Unit price charged by source I, C₁ = $20.00 per unit

   Unit price charged by source II, C₂ = $30.00 per unit

   Inventory carrying charge fraction, I = $10 per cent per annum.

We assume that the second source charge more, (C₂ > C₁).

Using these data, the optimal order quantities and the annual cost are found by the procedure described in the last section.

In Table I:

We suppose that the order can be placed at one source. If the order costs at individual suppliers are for example K₁ = $13.00 and K₂ = $12.00 as above, we obtain

Comparing these TC^*_i ­ values with the TC^*_I and TC^*_II obtained when two suppliers are used (K = $25.00), we see that it would be preferable to use only supplier I since TC^*₁ < TC^*_I < TC^*₂.

It is interesting to note when the values of K₁ and K₂ are interchanged, i.e. for K₁ = $12.00, K₂ = $13.00 the optimal solution is to order from supplier I, since in this case we get

	Situation - I	Situation - II
Q*1	909.91	788.63
Q*2	7.94	0.73
TC*	TC*I = 30,400.01	TC*II = 30,362.14
Cycle time	0.92	0.79

Table I.Cycle times for bothsituations

4. Summary and discussion

There is often the situation where a customer has to decide how much he/she should order from a set of suppliers. Most work on EOQ models do not address this scenario. The common assumption in those models is that all quantities are ordered from the same supplier. We provide an extension of this model to two suppliers and determine optimal order quantities.

In this paper, we have provided an analysis of the diversification situation in the EOQ model under two different situations. The two situations we modelled are: both the orders are placed simultaneously; and when second order is placed, one by one, once the first is depleted. We obtained an analytical solution for the optimal order quantities from two sources and the minimum average cost in terms of all parameter values for the EOQ model.

For both situations, we provided numerical examples and compared the diversification results with using a single supplier.

An extension of this presented study would be the analysis of the diversification case with more than two sources. We think that it may be a very difficult problem due to the complicated nature of the solution for even the two sources case discussed here. A feasible way to analyze these problems may be by providing heuristics.

Manisha ParikhDepartment of Statistics,Gujarat University, Ahmedabad ­ 380000,Gujarat, IndiaE-mail: bibahsl@vsin.ernet.inFax: 0091-079-6638497

Nita H. Shah Department of Mathematics,Gujarat University, Ahmedabad ­ 380000,Gujarat, IndiaE-mail: bibahsl@vsin.ernet.inFax: 0091-079-6638497

Y.K. ShahDepartment of Statistics,Gujarat University, Ahmedabad ­ 380000,Gujarat, IndiaE-mail: bibahsl@vsin.ernet.inFax: 0091-079-6638497