Original Article

AN INVENTORY MODEL ESTABLISHING ECONOMIC SUSTAINABILITY FOR DEVELOPING EFFICIENT MARKDOWN POLICIES WITH PRODUCT DETERIORATION CONSIDERATIONS

 

 

INTRODUCTION                                                      

The deteriorating inventory model, that has multi-variable demand rate and utilizes reduction policy in boosting the profit, is being studied. Inflation will also rise in this paper; as a result of that, the purchasing power per unit of money will reduce. In this paper, we also assume that the holding costs will be time dependent. The reduction policy can also increase the accumulated amount of profit yet the shortest of the reduction cases is the case of time and price dependence. One of them is a numerical example. Lastly, sensitivity analysis has also been prepared in consideration to some important parameters.

The most important process to control is inventory control, which entails stocks, improved services and other storage space. It involves the planning of the sales and the stock-outs, the maximization of the inventory profit and the removal of the dead stock piling. There are increased challenges faced by the companies when dealing with goods that are becoming deteriorated. The definition of deterioration includes depreciation, destruction, wear and tear of the products as well as obsolescence of the products. Business school scholars have conducted gigantic research on the notion and the majority of its facets were examined. The model developed by Ghare and Schrader (1963) is a first time economic order quantity where the planning time constant rate of demand is finite and the rate of deterioration is constant. In Goyal and Giri (2001), the new tendencies of the inventory modelling growth were taken into account. The system of supply chain where items were deteriorated by reverse logistics was formed partially and congested Singh and Sharma (2018).

It is known that the classical inventory model is characterised by a fixed holding cost. As a matter of fact, material goods require and holding cost might be time dependent. The time would also feature prominently on the inventory systems and as such we do believe that holding costs are also time sensitive. Trying to create the inventory model, Jaggi (2014) created the price-sensitive demand. It factored in functions of holding expenses that were time-related. A model of the quantity of production was suggested based on the speculations by Tayal et al. (2015) in terms of variable holding cost as an outline of non-instantaneous deteriorating products. Singh and Rana have discussed EO model in which the time dependent holding cost and multi-variable demand were addressed Singh and Rana (2020). At inflation profit search, they obtained maximum profit. Singh and Rana Singh and Rana (2020) examined the demand, which is satisfied by a new and recycle product. Inflation was also introduced to them as an efficient element of new product as well as old product.

Retailers have embraced the use of markdown policies to sell (delicate) things. The good consumers buy on distorted times and price of sale. Widyadana and Wee (2007) used depreciating inventory model which involved price-sensitive demand, and price-sensitive use reduction policy to maximize the profit. Wang et al. (2016) have derived the most appropriate markdown policy that can be enforced on the perishable food that the consumer perceives based on the prices, Nagare and Utia Nagare and Dutta (2018) have taken the single-period sensitivity.

The determinants of the demand of a commodity in the competitive market are seen to be extremely numerous due to its competitive nature. The multiplex demand of a commodity is one of such factors. Multi-variable demand of commodity is significant to the demand of the product. In Omar and Zulkipli (2014), the demand was assumed to be deterministic and in a positive relationship with the display marker of the items. Singhal and Singh (2017) examined a chain of commodities supplying system which is broken in terms of time and quality and various needs in the market.

 

ASSUMPTIONS AND NOTATION

Assumptions

·        The demand rate is deterministic and is a function of both price and on-hand inventory level:

 

, where and with

 

·        The holding cost is time-dependent where:

 

 

·        The deterioration is time-dependent where 0<a<1

·        There is a salvage value on the deteriorated units.

·        All items are mandatory to be sold.

·        Inflation and time-value of the currency are considered.

·        The deduction value is applied only once in a planning period, and the reduction value is known.

·      The production time is relative to the cycle time where ..

·      Reduction time varies between , which is equivalent to .

 

Notations

·        I(t): Inventory level at time t

·        a: Deterioration rate

·        p: Constant production rate

·        r: Inflation rate

·        γ: Reduction rate

·        ϵ: Increase price rate

·        1: Production percentage

·        λ: Reduction percentage

·        m: Initial price

·        O: Unit ordering cost

·        : Salvage value related to deteriorating units during cycle time where 0<β<1

·        : Unit production cost

 

Model Formulation

The production and supply start instantaneously, and the production ends at a time with the inventory level reached. We assume there is no deterioration during the production uptime. In the interval , inventory level declines due to demand and deterioration. At the time , a markdown is offered to grow the demand rate. The position of the inventory at any time over a period is governed by the following differential equation:

                                     

                                                                                                                                                  (1)

With γ=1(no markdown) and initial boundary condition ,

                                                                                  

                                                                                                                         (2)

 

With γ=1(no markdown) and boundary condition ,

                                                    

                                                                                            (3)

 

With boundary condition .

Solution of the above differential equations are given by:

 

                                                                                                                                         (4)

                                                                                  

                                                                                                             (5)

 

                                                                                       (6)

 

Where, using the boundary condition and continuity condition, the inventory level

 

                                                                                                                                                            (7)

                                                                                                                

                                                                                                                                             (8)

 

Ordering Cost = O/T

 

Production Cost =

 

Holding cost = 

 

Deterioration cost = 

 

Sales Revenue cost = 

Annual profit = Sales revenue cost – Holding cost– Ordering cost – Production cost– Deterioration cost

 

 

Note that Annual Profit is a function of t1, t2 and t3. We optimize the AP function by following Srivastava and Gupta (2013) procedure where we rewrite,

 

Srivastava and Gupta (2013) only varies T in order to find their optimal solution. However, in our case we are able to find a better solution by varying T and . Substitute t1, t2 and t3 into equation (9), then we obtain new equation w.r.t T and .

 

Solution Procedure

This sector determines the optimum values of T and λ which maximize the total profit AP(T, λ). The necessary condition for maximizing the AP are:

 

 

Also satisfied with the following conditions:

 

 

Numerical Example: In this section, a numerical example is deliberately given to illustrate the model. The following criteria are given below, which are used in the examples.

 

 

With these values, the solutions of the system were found as follows:

 

 

 

SENSITIVITY ANALYSIS”

In order to achieve more awareness on the issue of cost, all the parameters of +30 percent to -30 percent are run in a sensitivity analysis. Table 1 shows the result. Based on the sensitivity analysis, it is possible to conclude the following insights:

 

OBSERVATIONS

·        In Table 1 it is resolute on the hypothesis that it is the decrease in the parameter of the cost of holding vcan, which in truth seeks to reduce the total cost of the organism. The greater the increase of the v fades with increase of the cycle time T and the larger the inventory levels Q1,Qincrease with increase of the cycle time T, thus, the less the v, the less the markdown percentage 8.

·        As shown in Table 1, increasing the parameter 80 of the markdown percentage +30 to markdown percentage -10 decreases the cycle time T, annual profit, and negative changes in the inventory level Q 1,Q 2 and negative variation in the inventory level 1Q 2Q level with a negative change in the markdown percentage. Nevertheless, 0.20 -20% and -30% is the unclear answer.

·        As a fact is known, parameter C pdecreasing can lead towards the working well to decrease the annual income of the system in table 1. It is necessary to mention that there are no changes in other parameters because of C-pdecreases.

·        It is clear that the reduction of the cost deterioration Cdwould bring T,Q 1,Q 2 and maximal annual profit of the system, as shown in table 1. But it does decline by a small percentage in the percent of markdown 4by the fall of Cd.

·        As shown in Table 1, all the parameters are on the rise as the value of alpha decreases. The rdecreases, but in table 1, optimum annual profit is also better and the markdown percentage of the system λ gets better. Also, cycle time Tand extent of stock Q decreases or increases marginally. In this regard, the lower the quality of inventory Q2, the lower Q2 of inventory.

 

CONCLUSION

This paper presents a lifetime deteriorating inventory model that has a multi-variable demand rate. When the inventory and profit want to be maximum, then reduction policy would be used. As this paper shows, the monetary expansion in the business world is a good thing because it is evident in the sensitivity analysis where effect of monetary expansion is directly pointed to be the optimal markdown time and optimal cost. Cost is taken to be a variable function because to bring the study closer to reality which increases as time progresses. It is also possible to determine that policy makers must be very keen to make the reduction rate dependent. The proposed model can be generalized in cases of partial backorder, stochastic demand and the other one is permitable delay in payment.

  

ACKNOWLEDGMENTS

None.

 

REFERENCES

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Goyal, S. K., and Giri, B. C. (2001). Recent Trends in Modeling of Deteriorating Inventory. European Journal of Operational Research, 134(1), 1–16. https://doi.org/10.1016/S0377-2217%2800%2900248-4

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Singh, S. R., and Rana, K. (2020). Optimal Refill Policy for New Product and Take Back Quantity of Used Product with Deteriorating Items Under Inflation and Lead Time. In Strategic System Assurance and Business Analytics. https://doi.org/10.1007/978-981-15-3647-2

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