OPTIMAL ORDERING POLICY FOR SUBSTITUTABLE AND COMPLEMENTARY PRODUCTS UNDER QUADRATIC DEMAND

In this article, two mutually substitutable products are considered in the inventory. Out of which one of the product is assumed to be a bundle of two complementary components. The inventory of both products are depleted by its demand, which is assumed to be quadratic. In case of stock out, partial substitution is carried out by the other product to fulfil the demand of the customers with a rate of substitution. The unmet demand is assumed to be lost. The proposed model is formulated and a solution procedure is suggested to obtain the optimal ordering quantity. Sensitivity analysis has been carried out extensively for all the parameters in the model in order to analyse the behaviour of the model. We observe that, there is a substantial improvement in the total cost with substitution compared to that of total cost without substitution.


INTRODUCTION
Inventory management decides the performance of the company or firm. It involves majorly in planning and controlling the level of the stocks in the warehouse by determining two important aspects in the management, how much to order and when to order. The improper planning of the inventory affects the profit of the company. In addition, the customer's satisfaction and maintenance of the goodwill of the company rely on the management of the inventory. We often notice in super markets where two complementary products are bundle together as single product for utilization. For example, tooth paste and toothbrush, hardware and software, fountain pen and inkbottle, pencil and eraser. Usually a single component usage will not be sufficient to get the complete utility of the whole product, such products are called complementary products.
Often we come across in our daily life, where a consumer choose between two different products like different brands of hand sanitizer, different brands of face mask, different brand of mobile phone, different brand of AC, different brand of laptops, different brand of coffee , shampoo, bath soap etc. These products are called substitutable products, as one product is an alternate to another product. In general, when two products serves the need in all aspect like quality, price etc, even after interchanging are said to be substitutes. The advantage of product substitution is that it saves the purchasing time of the customers, when their preferred product is out of stock, customers are willing to accept the substitute as they does not wish to visit another store. Substitutable products benefits the company by minimizing its total cost. In addition, it further improves the good will of the company. The effectiveness of the system is established by the product substitution.
However, in real life not all customers willing to accept the substitute product, such demands are assumed lost.
Many articles describes the substitutable product and complementary products which are as follows, Benkherouf et al [1] developed an inventory model for substitutable products with time varying demand and the planning horizon is assumed finite. Drenzner et al [2] proposed an inventory model with two substitutable products under EOQ model with joint replenishment policy. Durga, B. K., & Chandrasekaran, E [3] considered an inventory model consisting of two 1851 OPTIMAL ORDERING POLICY substitutable products having quadratic demand. This model also considers the cost of substitution of the item. Goyal [4] developed an inventory model with the option of substitution. Gurnani and Drezner [5] extended the work of Drenzner et al [2] for multiple products. Krommyda et al [6] considered stock dependant demand and partial substitution. Liu et al [7] considered one-way substitution with or without conversion cost for perishable products. Maddah et al [8] considered an EOQ model with multiple products with partial substitution. Mishra [9] also developed an inventory model for substitutable and deteriorating products with cost of substitution. Mishra and Shanker [10] proposed a model for substitutable products with cost of substitution.
Mukhopadhyay, A., & Goswami [12] developed an EOQ inventory model with one-way substitution for imperfect quality of products. Salameh et al [13] extended the work of Drenzer et al [2] with partial substitution where only a fraction of customers is interested to buy the substitute item. Stavrulaki [14] developed a study of products, which are substitutable based on the level of inventory.. Zhang et al [17] considered correlated demand between products and partial backordering under EOQ framework. Zhang et al [18] developed EOQ model with substitution and partial backordering.
Yue et al [16] developed a Bertrand model of pricing of complementary goods under information asymmetry in which he considered two complementary goods as mixed bundle given by two different company. Mukhopadhyay et al [11] developed an optimum pricing strategy for complementary products by applying a game theoretical approach. Yan and Bandyopadhyay [15] developed a model on complementary products with bundle pricing in which he combine highly complementary products and change a relatively lower price.
It can be seen that not many structured work on substitutable and complementary products considered together in the literature. In this article, two mutually substitutable product is considered, where one of the product consists of two complementary components. The inventory depletes by its demand, which is assumed to be quadratic. The objective of this model is to find the optimal ordering policy. The products are assumed to be similar in quality and price. When one of the products becomes out of stock, the demand of it is partially met from the inventory of the other product.
The proposed model is described in detail in the subsequent sections. In Section 2 the assumptions and notations used in the model is presented. In section 3 the proposed model is mathematically formulated. In section 4 a solution procedure is suggested to obtain the optimal ordering quantity by showing that the total cost function is pseudo convex. Extensive sensitivity analysis has been done for all the parameter in the model to test the performance of the inventory system and presented in section 5. Finally in section 6 conclusion and future work is stated.

ASSUMPTIONS AND NOTATIONS
The assumptions made in the proposed model are listed below.
• The proposed inventory model consists of two products, which are substitutable.
• Product 2 is a bundle of two complementary products  and  .
• The demand of the product is quadratic.
• The demand of the product is partially substituted by the other product when stocks out with a rate of substitution. The unmet demand is assumed to be lost incurring a shortage cost.
• Both products are ordered jointly in each ordering cycle.
• The ordering cost and holding cost is constant.
• Lead-time is zero and instant replenishment.
The following notations are used throughout the article. 12 , kk Ordering cost of Product 1 and 2 12 , hh Holding cost of Product 1 and 2 1 2 , DD Demand rate of Product 1 and 2 12 , aa T .The demand of product 1(both component and  ) and product 2 are denoted by 1 D and 2 D respectively which is assumed to be quadratic. Two products are assumed to be substitutable products that is, they are similar in quality and price. Therefore, when one product becomes out of stock, the demand of the product is partially fulfilled from the inventory of the other product. Let 1  and 2  be the rate of substitution of product 1 and product 2 respectively. That is, when product 1 runs out of stock, customer's demand is satisfied by product 1. Whereas, when product 2 is unavailable, the customer will wait for the packaging of complementary components  and  , together to be consumed as product and get satisfied with the substitution.
Since the substitution is partially done, the unmet demand of the products is assumed to be lost.
Two cases is considered for this model based on the inventory cycle of the products, ie. When 1 2 TT  and 1 2 TT  .

Fig 1
The inventory level for case (i) when In this case, product 1 stocks out first. The future demand of this product is met from the leftover inventory of the product 2. The total cost in this case consists of the ordering cost of product 1 which is composed of two complementary components and  , the ordering cost of product 2 , holding cost of product 1 and 2. Also, the unmet demand of product 1 which is assumed to be lost is included in the shortage cost.
Therefore, sum of all the above components comprises of the total cost involved in case (i), which is given by . Therefore, the total cost equation is rewritten as

SOLUTION PROCEDURE
In this section, we propose a solution procedure to obtain the optimal ordering quantities of the inventory model. We state the following theorem.

Proof: Appendix B
We now present an algorithm to find the optimal order quantity for the proposed model.

Algorithm:
Step1 Initialize the parametric values of the inventory model.
Step 2 Solve the following constrained nonlinear optimization problem and obtain the optimal values. Step 3 Compare the optimal values obtained in step 2 and hence the required optimal ordering quantity.

Sensitivity Analysis
In this section, numerical illustration of the proposed model is presented. Initialize the parameter of the model with the following.         Table 7 Summary of Sensitivity analysis

CONCLUSION
An inventory system with two mutually substitutable products has been considered, where one of the product is composed of two complementary components  and  . When one product becomes out of stock, the future demand of the product is met partially from the leftover inventory of the other product. This model is applicable to similar items consisting of complementary products such as, different sim cards and mobile phone, different laptop with graphic software, coffee and milk, different brands of toothpaste and brush etc,. The pseudo convexity of the total cost function for two different cases has been derived. Analysis of this proposed model shows that substitution between products saves in the total cost. Sensitivity analysis is carried out extensively to validate the proposed model. The summary of the results is presented in Table 7 .When the substitution rates increases, the percentage improvement in the total cost of substitution over without substitution decreases. Whereas when there is an increment in ordering cost and holding cost, the percentage improvement in the total cost of substitution over without substitution also increases. This article can be further extended by considering, cost of substitution, both products as complementary products, deterioration of items, also one can extend the work by considering multiple product.

Parameter Variation TCU(WS) TCU(WOS) % improvement
Ordering cost ( ) 12 , kk  Since the Hessian matrix is positive semi-definite, ( ) 22 , TC q Q is a convex function. We make use of the fact that, ratio of the positive convex function over a linear function is pseudo convex.
Hence the Proof.

APPENDIX B
Proof of Theorem 2: Similar to Proof of Theorem 1

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.