IMPACT OF SUBSTITUTION COST ON OPTIMAL ORDER QUANTITY FOR SUBSTITUTABLE AND COMPLEMENTARY PRODUCTS WITH QUANTITY DISCOUNT UNDER QUADRATIC DEMAND

In this article, impact of substitution cost on obtaining optimal ordering quantities for substitutable and complementary products has been studied. Product 1 and 2 are mutually substitutable and each product comprises of two complementary components. The phenomenon of quantity discount is considered in this model. When a product becomes out of stock, partial substitution has been carried out to fulfil the demand of the other product. The demand of both products is assumed to be quadratic. Mathematical model is formulated and a solution a procedure is suggested to obtain the optimal ordering quantities. Optimal total cost for all possible cases has been discussed in this paper. Numerical example is presented and sensitivity analysis is carried out extensively to prove the viability of the proposed inventory model. Substantial improvement in the optimal total cost with substitution over without substitution is observed. 5630 B. KOMALA DURGA, E. CHANDRASEKARAN


INTRODUCTION
In supermarkets it is common to see, the unavailability of the customer's desired product. In this situation, the behavior of the customers that is very commonly seen is willingness to purchase a substitute product whenever their desired product is out of stock. Some customers may switch to another store to check for the availability of their desired product. Many researchers, study the former behavior of the customers. The demand of a product is met by the other mutually substitutable product. Two products which could serve the same purpose as the other product does is called as substitutable products. Moreover, they are identical in means of similarity and utility in the eyes of the customer. For example, two different soft drinks like Pepsi and coke, tea and coffee, butter and margarine, printed book and kindle etc. In an inventory system, when one product becomes out of stock, the demand of it is fulfilled from the inventory of other product.
Substitution can be done in two ways, complete substitution and partial substitution. Most of the researchers have developed an inventory system consisting of two or more substitutable product and identified that only partial substitution of demand of the products bring forth enormous profit compared to complete substitution. Whenever, substitution takes place, an additional cost is incurred called as substitution cost. Complementary product is a product, which is composed of two or more components, packed together to be utilized for customers. For example, pencils and notebooks, burger and burger buns, DVD and DVD player, mobile phones and sim cards etc. The utility of the products will not be completed without both the components. For instance, the DVD is useless without the DVD player, a customer must buy a DVD player in order to make use of the DVD. Many examples are available for substitutable product composed of complementary components. Different brands of tennis balls and different brands of tennis rackets, different brand of mobile phones with different sim cards, different brands of toothpaste and different brands of tooth brush etc. Quantity discount is offered to the retailer in order to encourage them to buy 5631 IMPACT OF SUBSTITUTION COST ON OPTIMAL ORDER QUANTITY products in bulk. The retailer also gets benefited as the unit price of the quantity is decreased when ordered in large quantities. This phenomenon of quantity discount is applied in the proposed model.

Many researchers have contributed in area of inventory model with product substitution,
Benkherouf et al [1] developed an inventory model with product substitution and found an optimal order schedule minimizing the total cost. Whereas, Chen, Y. et al [2] studied an inventory model by considering partial substitution with shortages allowed, results indicated that partial substitution decreases the total expected cost. Most of the articles considered the EOQ framework and extended the idea of substitution of products, which are mutually substitutable. Drezner et al [3] considered an EOQ model with two substitutable products and showed that partial substitution or no substitution may be optimal because of the non-linearity of the total cost equation. The demand of the product is assumed to be quadratic in the articles by Durga, B. K., & Chandrasekaran, E. [4] in which they developed a model with two substitutable product under quadratic demand. Also, Durga, B. K., & Chandrasekaran, E. [5] showed the impact of substitution when obtaining the optimal ordering policy for an inventory system consisting of two substitutable products. Very few researchers have considered articleas on both substitutable and complementary products in EOQ framework. In particular, Durga, B. K., & Chandrasekaran, E. [6] developed a model with two products which are mutually substitutable and complementary products under quadratic demand.
Edalatpour et al [7] presented a model based on multi product consisting of complementary and substitutable items. In this paper demand is assumed as a function of price sensitive. The inventory model is also considered in game theory approach by some of the researchers such as Giri et al [8], in which they considered two substitutable product and one complementary product and solved using game theoretical approach. Most articles study about the stock -out based substitution, like Goyal, S. K. [9] proposed a model with substitution between products and Gurnani, H., & Drezner, Z. [10] extended his work and considered a model with multiple products and determined the optimal ordering policy and substitutable quantities. Krommyda et al [11] studied about stock out based demand and product substitution to maximize the profit function. Maddah et al [12] formulated a joint replenishment model in EOQ framework for multiple products under product substitution. Mcgillivray, R., & Silver, E. [13] investigated a generalized inventory model with effect of product substitution by considering unit variable cost and shortage cost. The feature of quantity discount is considered byMishra, V. K. et al [14] in his article where he studied the aspect of substitution of product, quantity discount and substitution cost. Yet another important work on substitution and complementary product is done by Mokhtari, H. [15] in which he extended the classical approach of EOQ by considering the situation of some product being complement with others. Some of the other contribution in this area is as follows, Pan et al [16] shows that the visibility of the products stock level brings positive impact and increases the demand for a product. Salameh et al [17] combines the works on substitution and joint replenishment model under EOQ framework. Taleizadeh et al [18] discussed the effect of deterioration in the combination of complementary and substitutable products. Tang et al [19] in their model adopted fixed and variable pricing strategy to obtain the optimal ordering quantity. Transchel, S [20] examined a stochastic inventory model to determine the demand function in the stock-out based substitution. Wei et al [21] two complementary products is studied is under game theoretical approach. Yue, X. et al [22] complementary goods as a mixed bundle is studied.
In this article, we consider an inventory system with two products. Product 1 consists of two complementary components and product 2 also comprises of two complementary components.
The inventory is depleted by its own demand. The demand of the products are assumed to be quadratic. When a product becomes out of stock, the demand of it is partially met from the inventory of product 2 with an additional cost of substitution. The unmet demand is assumed to be lost. The phenomenon of Quantity discount is utilized. The main objective of this model is to find an optimal ordering policy for the proposed model.
The rest of the article is organized as follows, Section 2 points to the notation and assumptions used throughout the model. Section 3 describes the formulation of the proposed model. • The inventory system consists of two products. Each product consists of two complementary components, which are mutually substitutable.
• The inventory of the products depletes by its own demand, which is assumed to be quadratic.
• During the stock out, partial substitution between products takes place with a rate of substitution and a substitution cost associated with it.
• Lead time is zero and replenishment is instantaneous.
• The distributor offer the following discount based on quantity to the retailer is given below.

FORMULATION OF MODEL
Consider an inventory system consisting of two products which are mutually substitutable. Product  and 2  with utilization rates 3 a and 4 a respectively. The demand rates of the products are assumed to be quadratic. When product 1 depletes, the further demand of it is met 5635 IMPACT OF SUBSTITUTION COST ON OPTIMAL ORDER QUANTITY from the inventory of product 2 partially, the unmet demand is assumed to be lost and vice-versa.
Whenever substitution takes place, there incurs an additional cost called substitution cost. In this model we have considered partial substitution. The rate in which substitution takes places is knowns as rate of substitution. 1  is the fraction of demand met from the inventory of product 2 to product 1. Similarly 2  is the fraction of demand met from the inventory of product 1 to product 2. Three scenarios has been discussed based on the time interval of the products. i.e.

Scenario 1
Consider a situation where the product 1 inventory is completed first. The future demand is partially substituted from the inventory of product 2 and the unmet demand is assumed to be lost.
In this case the cycle time of product 1 is less than that of product 2 i.e. 12 TT  . The total cost equation in this situation comprises of the following cost that is listed below.
The total cost is given by 11

Scenario 2
Consider the case where 12 TT  , product 2 depletes faster than product 1. Further demand of product 2 is met from the inventory of the product 1, the unmet demand is assumed to be lost. The total cost comprises of the following cost.   In this case we consider that there is no substitution taking place. Then the total cost equation includes the ordering cost and the holding cost of the products alone.

SOLUTION PROCEDURE
In this section a method to obtain an optimal ordering quantity for Scenario 1 and 2 is recommended. By proving 1 TCU and 2 TCU as pseudo-convex function, will guarantee a unique optimal policy.
The pseudo convexity of the total cost per unit time has been established below. The following algorithm helps to find a unique optimal ordering quantities as the total cost per unit time is pseudo convex.

Algorithm to obtain optimal order quantities under quantity discount
Step 1 Initialize all the parameters Step 2 Solve the optimization problem with the given constraint Step 3 Obtain the optimal ordering quantities ( ) ** 13 , qq and hence the optimal total cost.
Step 4 If Step 3 is not feasible, take different combinations of order quantity and then go to Step 2.
Step 5 Repeat Step 4 until appropriate order quantities are found.
Step 6 Stop the algorithm 5641 IMPACT OF SUBSTITUTION COST ON OPTIMAL ORDER QUANTITY

SENSITIVITY ANALYSIS
In this section, numerical example with sensitivity analysis has been done extensively to illustrate the behavior of the proposed model. We initialize the parametric values as follows: 1  Similarly, solving Step 2 (ii) the optimal ordering quantities are obtained as 1

CONCLUSION
An inventory system consisting of two mutually substitutable products where, each product consists of two complementary components is considered in this model. The inventory of both products is depleted by its own demand, which is assumed to be quadratic. When one product becomes out of stock, the demand of the other product is met from the inventory of the second product with a rate of substitution. In addition, substitution cost is incurred whenever substitution takes place. The unmet demand is assumed to be lost. The phenomenon of quantity discount is considered in this paper. The proposed model is applicable for products such as different brands of mobile and sim cards with different network, different brands of DVD players and different DVDs, etc. Three scenario has been discussed for the proposed model. A solution procedure is suggested in this paper to obtain the optimal total ordering quantities. Numerical example is provided to see the viability of the model. Sensitivity analysis has been extensively carried out to illustrate the percentage improvement in optimal total inventory cost with substitution over without substitution.
Further extension can be done by considering more than two product. The phenomenon of deterioration can be applied. In addition, it can be extended in the direction of probability by introducing, stochastic demand etc. Now, we apply the result which states that "ratio of a positive convex function over a linear function is pseudo convex". Hence, 1 TCU is pseudo convex.