A PROFIT MAXIMIZATION SINGLE ITEM INVENTORY PROBLEM CONSIDERING DETERIORATION DURING CARRYING FOR PRICE DEPENDENT DEMAND AND PRESERVATION TECHNOLOGY INVESTMENT

. This paper addresses a single item two-level supply chain inventory model considering deterioration during carrying of deteriorating item from a supplier’s warehouse to a retailer’s warehouse as well as deterioration in the retailer’s warehouse. The model assumes preservation technology in the retailer’s warehouse to prevent the rate of deterioration. An upper limit for the preservation technology investment has been set as a constraint to the model. The model maximizes the retailer’s profit per unit time, simultaneously calculated optimal order quantity. A price dependent demand and storage-time dependent holding cost is considered to develop the model. Some theorems are proven to get optimal values of the total cost. A numerical problem is workout as per the developed algorithm and with the help of MATLAB software to study the applicability of our theoretical results


Introduction
Basic inventory models are classified into two types such as Economical Order Quantity (E.O.Q.) and Economical Production Quantity (E.P.Q.).In E.O.Q.model, we optimize order quantity to maximize the total profit of the retailer or minimize total cost.In this regard, first inventory model was introduced by Harris [15].There are many developments on the basic E.O.Q model as of now considering different realistic assumptions which we will discuss in the literature review section.
We know that deterioration plays an important role in an inventory model as if the deterioration rate increases then the profit of a retailer decreases and if the deterioration rate decreases then profit of a retailer increases that is profit varies dis-proportionally with the deterioration rate.So, we can not ignore the deterioration rate in the present study of an inventory model.The deterioration is generally suitable for the items such as raw food items (fruit, vegetables, fishes, meat, eggs, etc.), processed food items, grocery items (salt, sugar, etc.), medical items (Bloods, medicine, vaccine, etc.), radioactive elements, alcohol etc.
This paper is developed considering different types of realistic assumptions.One such realistic assumption is deterioration during carrying.During carrying some quantity of the total order is spoiled because of many reasons.Some of the reasons may be long journey of the carrying vehicle (This type of reason is suitable for radioactive elements, blood, medicine, vaccine, vegetables, fruits, etc.), weather during carrying (This type of reason is suitable for salt, sugar, eggs, fruit, vegetable, meat, fish, etc.), carelessness during loading and unloading (It's the main reason for untrained labor force and this type of reason is suitable for any type of product), etc.
Preservation technology is very much useful to increase products lifetime.Also, during the carrying time, we can not apply preservation technology for more lifetime of the product but in the retailer's warehouse we can apply preservation technology to prevent deterioration.So, deterioration rate of the item during carrying is higher than the deterioration rate in the warehouse.
Another realistic assumption is retailer can sell the defective products in a certain price (which is less then the buying cost) to reduce the loss.This phenomena is used in our model.Retailer can use the equipment (for example -refrigeration equipment) to increase the self life of the item in retailer's warehouse after the item arrives.Thus, retailer has to invest some amount of money to operate the equipment for preservation; which is called preservation technology investment.If preservation technology investment increases then the deterioration rate decreases and if preservation technology investment decreases then deterioration rate increases that is preservation technology investment varies dis-proportionally with the deterioration rates.Also, retailer's can not invest most of the money in preservation technology and hence in this paper we consider an upper bound for the preservation technology investment.
In most cases, tendency of a customer is to buy a product in lesser price.So, if price of an item goes down then demand of the item increases.In this paper we have considered that the demand of an item is depending on the selling price and it varies dis-proportionally with the selling price of an item.Another realistic assumption is that the holding cost varies proportionally to time due to the rent of the warehouse that is if stock time increases then the holding cost increases and if stock time decreases then holding cost decreases.Now we will discuss on the various inventory models developed during the past years in literature review section.

Literature review
Mondal et al. [27] first introduced a model for an ameliorating item with demand of the product dependent on price.Then, Mukhopadhayay et al. [28] proposed an ordering policy on pricing inventory model for deteriorating items.Later, Mukhopadhayay et al. [29] modified their previous problem by introducing the deterioration rate as weibull distribution and price dependent demand.Roy and Chaudhari [33] formulated a model for a deteriorating item and demand depends on price with special sale of the product and later Roy [32] modified his previous problem by incorporating time dependent holding cost.Then Maiti et al. [24] formulated an inventory problem with price dependent demand and stochastic lead time in advance payment system.Next, Sridevi et al. [40] built a price dependent model for deteriorating items with weibull rate of replenishment.Then, Sana [35] formulated an inventory model for perishable items with price-sensitive demand.Maihaimi and Kamalabadi [23] formulated a model with time and price dependent demand for non-instantaneous deteriorating items on jointly time and price.Then, Avinadav et al. [4] introduced a supply chain problem for perishable items.Then, Bhunia and Shaikh [5] formulated an inventory problem for deteriorating items with selling price dependent demand and three-parameter weibull distribution.Next, Ghorieshi et al. [14] formulated an inventory model with price dependent demand and customer returns for non-instantaneous deteriorating items with partial backlogging.Then, Alfares and Ghaitan [2] formulated quantity discount offered to the customer with time dependent holding cost.Then, Jaggi et al. [18] formulated an inventory model for demand dependent on price with credit financing in two storage facilities and non-instantaneous deterioration.Shaikh et al. [36] developed an inventory model with variable demand dependent on price for three-parameter Weibull distributed deteriorating item.Next, Dey et al. [9] formulated an integrated inventory model with price-dependent demand and discrete setup cost reduction.Then, Khanna et al. [20] considered non-linear price-dependent demand for an inventory model with inspection error.In this direction, Gautam et al. [12] developed an inventory model with price-dependent demand for defective items.
An appreciable amount of research paper has been published on inventory control models for deteriorating products.Ghare and Schrader [13] first introduced the concept of deterioration.Then Philip [30] has extended from constant deterioration to three parameters weibull deterioration.Thereafter a lot of researches have been modified by several authors with different types of deterioration.Regarding this context, to reduce deterioration effect Hsu et al. [17] first introduced the preservation technology in his research paper.They introduced an inventory model for deteriorating inventory with preservation technology investment.Next Dye [8] developed a model for non-instantaneous deteriorating item with preservation technology investment.Zhang et al. [43] formulated an inventory model with stock dependent demand and preservation technology investment for deteriorating inventory model.Mishra et al. [26] formulated an inventory model under price and stock dependent demand for deteriorating item with shortage and preservation technology investment.Then Shaikh et al. [39] investigated preservation technology for deteriorating item and time dependent demand with partial backlogging.Li et al. [22] considered preservation technology investment for non-instantaneous deteriorating item and replenishment.Aditi Khanna et al. [21] formulated an inventory model for deteriorating items with stock-dependent demand and time-dependent holding cost.Yongrui and Yu Cao [10] developed an inventory model with stochastic demand with reference price effect for the deteriorating product.Next, Mashud et al. [25] considered a joint pricing inventory model with price-dependent demand and time-dependent deterioration rate under a discount facility system.Also, Asim Paul et al. [3] formulated an inventory model for a deteriorating product with price-sensitive demand and discussed the effect of default risk on optimal credit period in the inventory model.Then, Bashair Ahmad and Lakdere Benkherouf [1] derived an optimal replenishment policy inventory model for the deteriorating product with stock-dependent demand and partial backlogging.Then, Rout et al. [31] formulated a production inventory model for deteriorating items with constant deterioration rate and backlog-dependent demand.After that, Ali Akbar Shaikh et al. [37] developed an inventory model for the deteriorating product with constant deterioration rate and price dependent demand also discussed the decision support system for customers under trade credit policy.Then, Choudhury and Mahata [7] developed an inventory model for growing deteriorating items with price-dependent demand.
Many inventory model developed assuming the holding cost per unit is not constant.Ferguson et al. [11] introduced an inventory model with holding cost as non-linear dependence on the storage time.Next San-Jose et al. [34] developed an E.O.Q.model with the holding cost function having two components: a fixed cost and a variable cost that increases with storage time and partial backlogging.Then Alfares and Ghaithan [2] formulated an inventory and pricing model with time varying holding cost, price dependent demand and quantity discount.Then, Ali-Akbar Shaikh [38] formulated an E.O.Q.model for time dependent holding cost and price discount facility with stock dependent demand.

Research gaps and our contributions
The main highlights of our contribution in this paper are -Table 1 depicts that several researchers considered deterioration and preservation technology in the retailer's warehouse but no one considered deterioration during carrying of an item from supplier warehouse to retailer's warehouse.This model fulfills this gap of research considering different deterioration rate for two intervals.This model can be consider as a generalization of the existing work on deterioration and preservation technology.-Many authors considered that the preservation technology function ( ()) dependent on initial deterioration rate.But during transportation of the items from the supplier warehouse to the retailer's the items has to survive various extreme condition (such as weather during transportation, road condition, etc.) and once the items are arrived at retailer's warehouse the items will be in under preservation technology system, which is a stable system.So, the deterioration rate during carrying and deterioration rate under preservation technology can not be always dependent.So, here we consider the deterioration rate under preservation technology ( ()) is an independent function of initial deterioration rate () and it is define by where  is the sensitivity parameter of investment to the deterioration rate and  is the preservation technology investment per unit time and  () ≪ .-To prove the optimality of the total cost function in this paper we derive two theorems and by using the results 3.2.1 & 3.2.10 from Combini and Martein [6] we prove those theorems analytically.-Here to handle the numerical problem we modify an algorithm from Khan et al. [19] and the modified algorithm is suitable for any similar type of model, which is considered in this paper.

Notation and assumptions
The following assumptions and notation have been used to formulate the mathematical model.

Notation
Tables 2 and 3 describes the notation for the developed model.

Assumptions
Our inventory model is established based on the following assumptions: (1) The model is developed for a single deteriorating item.
(2) We consider the deterioration during carrying.We assume that during carrying the deterioration rate is  (0 <  ≪ 1) is constant.
(3) We assume that the retailer will sell the defective item with a price  ; Which is less than the purchasing cost  per unit item.
(4) Any replacement or repair for the deteriorated products is not considerable during the cycle length  .
(5) We know that if the price of a product is low then demand appears highly in the market.From this observation, we assume that the price is dependent on demand same as Alfares and Ghaithan [2], which is expressed below where  1 is the constant amount demand of the item when selling price  = 0 and  2 is a constant such that In the retailer's warehouse inventory level is gradually decreases because of the demand  and reduced deterioration rate due to the investment on preservation technology is where  is the sensitivity parameter of investment to the deterioration rate and  is the preservation technology investment per unit time and  () ≪ .The relationship between deterioration rate and preservation technology investment parameter are  ()  < 0   where ᾱ is the maximum investment on preservation technology.(8) Similar to Alfares and Ghaithan [2], we consider storage time-dependent holding cost.The holding cost of the products increases linearly concerning the storage time of each unit and it's proportional to the purchase cost  per unit item.Also, the holding cost contained two constant parts, one is the  and another one is ℎ.So, the holding cost function can be expressed as  () =  ( + ℎ) .

Mathematical model
The retailer ordered  units of deteriorating item.Therefore, the inventory level at  = 0 is .During carrying from the supplier's warehouse to retailers warehouse, the stock level gradually declines due to the deterioration rate  and drops at  at the time  =  1 .In the retailer's warehouse inventory level gradually decreases because of the demand () and the reduced deterioration rate due to the investment on preservation technology ( ()).To prevent deterioration in the retailer's warehouse we applied preservation technology (Fig. 1).
Defective product quantity due to carrying is  =  −  , which is given by the equation (5.9).We can sell the defective product with lesser price  .Therefore the revenue by selling defective item is = ( −  ) =  . ( We assumed that cost of preservation technology investment per unit item is . (5) Preservation Technology Cost (  ) =  ( −  1 ).
Therefore the profit function is given by Our target is to find the optimal selling price  * per unit item and the length of the cycle  * to maximize the retailer's profit per unit time   (,  ).

Theoretical results
We formulated the concavity of the objective function for the above problem.To analyze the concavity for the models, we used some results from Combini and Martein [6].Depending on the Theorems 3.2.9 and 3.2.10 in Combini and Martein [6], we know that the function of the form is strictly pseudo-concave if  () is differentiable, non-negative and strictly concave function and  () is positive, convex and differentiable function.
To represent the optimality of our problem by using the above results first, we determined the optimal value of  * then we calculated the optimal value of replenishment  * , which maximizes the retailer's total profit per unit time using the optimal selling price value  * .Theorem 6.1.For a fixed  > 0,   (,  ) is a pseudo-concave function of  .Hence, there exist a unique  (Say  * ) such that   (,  ) attains the maximum value.
Hence, for any fixed  the total profit function   (,  ) is strongly pseudo-concave function of  .So, there exists an unique  * such that   (,  ) attains the maximum value.Theorem 6.2.For any specified value of the cycle length  > 0,   (,  ) is a concave function of .Hence there exists a unique  (Say  * ) such that   (,  ) attains the maximum value.
The necessary condition to find the optimal selling price ( * ) can be found by equating the first order partial derivative of   (,  ) with respect to  of   (,  ) equal to zero.After simplifying, the necessary condition is given by Also, the necessary condition for finding the optimal cycle length ( * ) for a given selling price () can be found by equating the partial derivative   (, )  = 0.So the equation is After getting the value of  * and  * , we can calculate the optimal economic order quantity  * from the equation (5.8).Also we can find the optimal values of the defective item quantity ( * ) and ( * ) (optimal on the demand function.So, It is important to maintain the balance between the higher selling price and the demand of the item. (3) The optimal cycle length ( * ) increases with respect to the expanding values of  1 and  .If the deterioration rate (,  ()) increases then the cycle length decreases.We observed that for the other parameters such as , , ℎ etc. the cycle length is increasing first and then reduces.
(4) The optimal profit   * (,  ) increases if we increase the value of the demand constant  1 that is the optimal profit   * (,  ) varies proportionally with the demand constant  1 and it is shown in the Figure 3. Also, the optimal profit   * (,  ) decreases with respect to the increasing the purchasing cost  and it is shown in the Figure 4.

Conclusion and future research direction
In our paper, we solved a profit maximization problem under several realistic assumptions such as deterioration during carrying, price sensitivity of the demand function, time-dependent holding cost, uses of preservation technology for deteriorating product, etc.Here we consider the deterioration rate due to carrying is constant but after using preservation technology the deterioration rate depends only on the preservation technology investment, also it is an increasing function of preservation technology investment.In addition, the selling price of the product () and the total inventory cycle length ( ) are decision variables in the total profit function of our model.By using the optimal values of the selling price of the product ( ⋆ ) & the total inventory cycle length ( ⋆ ) we calculate the Economic Order Quantity ( ⋆ ), optimal quantity of the defective product due to carrying ( ⋆ ) and the optimal inventory level ( ⋆ ) at the time  1 .Finally, the numerical problem solved by using our modified algorithm as well as the sensitivity analysis of the various parameters, which are involved in the profit function are discussed.The numerical findings and approach of our model are applicable to several types of industries or companies that handling various types of deteriorating product because of realistic assumption of our model and mathematical generalization of various parameters.Therefore from our model and numerical findings the following conclusions can be drawn 1.The optimal cycle length increases concerning the expanding values of  1 .So, the total profit decreases due to production increasing inventory cycle length.Therefore to make the maximum profit, the retailer has to reduce the time taken to bring the deteriorating items from the supplier's warehouse to retailer's warehouse by using various fast transportation modes.2. The parameter  2 has significant impact on increasing  * but it also has a negative influence on the demand function.So, It is important for a retailer to maintain the balance between the higher selling price and the demand of the item by taking various marketing strategies.3. Also, the value of profit function increases with respect to the increasing values of  .So, the retailer can earn more profit by selling the less defective product to the customers at highest possible price.4. The total profit (  ⋆ ) for our model decreases when the value of  (),ℎ, , holding times are enhanced.It indicates that the retailer can earn maximum amount of profit by decresing the values of  () by investing more on preservation technology and also by reducing the values of constant coffecients of the holding cost (ℎ, ) as well as reducing the holding time of the products.5.When the purchasing cost of the product increases(), the total profit decreases but selling price of the product ( ⋆ ) and the optimal cycle length ( ⋆ ) increases.Also, selling price of the product ( ⋆ ) has negative

and 𝜕 2
()  2 < 0. (7) In our model preservation technology investment per unit time is  and it satisfies the condition 0 ≤  ≤ ᾱ

Table 1 .
The comparison between earlier published work and our present work.
* Unit Optimal quantity of defective items due to crrying. * Unit Optimal quantity arrived at retailer's warehouse. Constant the sensitivity parameter of investment to the deterioration rate.ᾱ $/Unit time the maximum investment cost in preservation technology.

Table 4 .
Sensitivity analysis of different parameters.