Inventory models for deteriorating items with discounted selling price and stock demand Freddy Andrés Pérez ( fa.perez10@uniandes.edu.co) Universidad de los Andes, Departamento de Ingeniería Industrial Fidel Torres ( ftorres@uniandes.edu.co) Universidad de los Andes, Departamento de Ingeniería Industrial Abstract New families of inventory models are developed with pre- and post-deterioration discounts when demand is initially stock and then becomes constant. Shortages are allowed and partially backordered depending on the waiting time for the next replenishment. Furthermore, the time value of money is considered over a fixed planning horizon. Keywords: Inventory, Deteriorating, Inflation Introduction Deteriorating inventory models accounting for partial back-ordering and financial aspects such as pricing strategies, inflation and time value of money are well covered in the current literature; however, there are some characteristics in the modeling of inventory systems with deteriorating goods hardly studied. For instance, incorporating a known price discount is very common and corresponds very well with the real world situations for deteriorating items, but a few authors consider two deterioration phases associated to two subsequent price discounts. In this sense, several authors over time have investigated different pricing strategies for deteriorating inventory when shortages are allowed and partially backordered depending on the waiting time (Abad 2001, Abad 2003, Dye 2012, Dye and Hsieh 2011, 2013, Dye et al. 2007, Ghosh et al. 2011, Maihami and Abadi 2012, Maihami and Nakhai Kamalabadi 2012, Shavandi et al. 2012, Soni and Patel 2012, Teng et al. 2007, Valliathal and Uthayakumar 2010), as well as inventory systems dealing with time value of money and shortage (Dye and Hsieh 2011, Berk et al. 2009, Mirzazadeh et al. 2009, Abdul and Murata 2011, Chauhan and Kumari 2011, Gaur et al. 2011, Jaggi et al. 2011, Mishra et al. 2011, Valliathal and Uthayakumar 2011, Jain and Aggarwal 2012, Yang 2012, Ali et al. 2013, Bhunia et al. 2013, Taleizadeh and Nematollahi 2013, Yang and Chang 2013). Most of the studies with exception of a few (Dye and Hsieh 2011, Berk et al. 2009, Valliathal and Uthayakumar 2011) do not endeavor to unify pricing decisions and deterioration research streams with studies where shortages are allowed, and/or the time value of money is taken into account. Berk et al. (2009) considered the presence of price-sensitive renewal demand processes to maximize the discounted expected profit by determining the selling 1
prices dynamically for items with fixed lifetime, but shortages are not considered. Dye and Hsieh (2011) developed an inventory model under inflation for deteriorating items with priceand stock- demand and limited capacity. Meanwhile, Valliathal and Uthayakumar (2011) presented a production lot-sizing model for deteriorating item under inflation by considering a coordinated pricing and production decisions as well as price decision disregarding the production cost. The major assumptions used in the existing works with pricing decision and partially backordered depending on the waiting time have been summarized in Table 1. Table 1- Summary of related literature for inventory models with partially backlogging Author(s) and year Deterioration Demand Time Pricing decision Time value of per inventory horizon money cycle (Abad 2003) (Ghosh et al. 2011) (Dye 2012) (Maihami and Abadi 2012) (Maihami and Nakhai Kamalabadi 2012) (Valliathal and Uthayakumar 2010) (Dye and Hsieh 2011) (Dye and Hsieh 2013) (Soni and Patel 2012) (Shavandi et al. 2012) (Teng et al. 2007) (Abad 2001) (Dye et al. 2007) (Valliathal and Uthayakumar 2011) Present paper Time (noninstantaneously (noninstantaneously (noninstantaneously (noninstantaneously Time Time Time (noninstantaneously Time and price Time and price Time and price Time and price Stock level and price No Finite Two-phase pricing No Infinite Equals No Infinite Equals No Infinite Equals Yes Finite Dynamic pricing Price No Infinite Two-phase pricing Prices No Finite One pricing decision Per item Price and time Stock and constant Yes infinite Equals Yes Finite Two-phase price reduction As a consequence, according to the present authors knowledge, no one has attempted to introduce a temporary price reduction on selling price before the start of deterioration as well as a discount on selling price as the deterioration starts in order to boost the inventory depletion rate when inventory systems are dealing with both waiting time backlog rate and deteriorating inventory accounting for inflation and time value of money. 2
So, the current paper represents above issue in detail extending the model in Panda et al. (2009). In what remains, this paper is organized in the following order. First, it formulates the basic mathematical model with pre- and post-deterioration discounts on selling price. Second, the special cases from the formulation of the basic model are submitted. Third, a numerical example is presented to exemplify the model developed. And finally, the conclusion of this study is given. Mathematical Model In this paper, a deterministic inventory model with pre- and post-deterioration discounts on selling price in order to investigate how much discount on selling price may be given to maximize the profit over a finite horizon H is considered. The total time horizon H has been divided into m equal parts of length (i.e., ). At the beginning of the replenishment cycle the inventory level raises to. As time progresses it decreases due to instantaneous stock demand up to the time. After deterioration starts and the inventory level decreases for deterioration and constant demand until it reaches zero level at. Ultimately, backorders are accumulated from time to. So, the evolution of inventory level occurs according to the following system of linear differential equations = (1) ( ) (2) (3), (4) Where a > 0 is the initial demand rate in of stock level, b > 0 is the stock sensitive demand parameter, I(t) is the instantaneous inventory level at time t,, is the effect of discount over demand,, is the effect of discounted selling price over demand, is the deterioration rate, and is the fraction rate of backordered items at time t with. Note that the fraction of shortages backordered is a decreasing function of time t, and it reflects the fact that less waiting time implies more backordered items. Also note that and are determined from prior knowledge of the seller depending on the influence caused by the reduction rate of selling price and the waiting time over demand rate in each case respectively. Solving the differential equations with the initial and boundary conditions we get (5) ( ) (6) = [ ] (7) [ ] (8) Now, at the point t = τ we have from Equations (6) and (7) 3
( ( ) ) (9) The present value of the holding cost and disposal cost during the entire horizon H are [ ] (10) Where and are the respective holding and disposal costs per unit and m denotes the number of replenishment periods during the time horizon H. The remaining costs during the entire time horizon H are found as follow: Let c the per unit purchase cost of the items. Because shortage during the last cycle is replenished at time, and shortage during the first replenishment cycle should be backordered during the next replenishment cycle, the present worth of the purchasing cost is given by { [ ] } [ ( ) ] (11) Let be the shortage cost for backordered items. The present value of the total shortage cost during the entire time horizon H is equal to [ ] [ ( ) [ ]] (12) Defining as the per unit shortage cost for lost items, the present value of the total shortage cost of the items being lost in the entire time horizon H is given by [ [ ] ] [ ( ) ] (13) Defining s as the per unit selling price of the items. The present worth of the total sales revenue in the entire horizon H can be found by equation (14). 4
[ ( ) ( ) [ ]] (14) cost Thus, the net present value of the system during the entire horizon H (including set up ) is (15) On integration and simplification of the relevant costs, the net profit value during the entire horizon H becomes (16). Where, PC, SC, LC are equal to equations (11), (12), (13) respectively (, ) [ ] (16) Where: ; [ ] ( ) ( ) * ( ) + Now, considering the constraints in which the pre- and post-deterioration discounts on selling price is given in such a way that the discounted selling price is not less that the unit cost of the product, the objective here is to determine the optimal values of and which maximize the function constrained by ; and. Note that it s very difficult to derive the results analytically. Thus, some of the numerical algorithms for constrained nonlinear optimization must be applied to derive the optimal values of, and m. There are several methods to cope with constraint optimization problem numerically. But here it s uses The Differential Evolution Algorithm (Price et al. 2005) to derive the optimal values of the decision variables. 5
Some special cases Let be the above model described. The following situations can be easily deduced from equations (9) and (16) as follow: Model with only post-deterioration discount on unit selling price In this case only discount on selling price will be given as soon as the deterioration starts. So, the order quantity is obtained from equation (9) by Substituting and as ( ( ) ) (17) Also, from equation (16) the net present value of the system is found as * + [ ( ) ] (18) [ ( ) ] Therefore, the goal here is to find the optimal values of and to maximize the function constrained by and. Where PC, SC, LC are equal to equations (11), (12), (13) respectively, bearing in mind that in (11) now. Model for no discount on unit selling price In this model there is no pre-deterioration as well as no post deterioration discount on unit selling price. So, the order quantity is found substituting, from Equation (9) as ( ( ) ) (19) And from equation (16) the net present value becomes (20). Where PC, SC, LC are equal to equations (11), (12), (13) respectively, taking into account that in (11) now [ ] [ ( ) ] (20) [ ( ) ] 6
So, the goal in (20) is to find the optimal values of, and to maximize the function constrained by. Model for instant deterioration with post deterioration discount In this case pre-deterioration discount is no option because the items start to deteriorate as soon as it s received in the stock. So, from equations (9) the order quantity can be found by substituting as (21) Similarly, from equation (16) net present worth becomes (22). Where PC, SC, LC are equal to equations (11), (12), (13) respectively, considering that in (11) now. * + * + (22) Hence, the aim here is to determine the optimal values of and which maximize the function constrained by and Model with instant deterioration and without discount (, ) The initial inventory level can be obtained from equations (21) by substituting as (23) So, from equation (22) the net present value becomes (24). Where PC, SC, LC are equal to equations (11), (12), (13) respectively with in (11). Thus, we have to determine and from which the function (24) is maximized and meet * + (24) Model for fixed life time products with pre deterioration discount In this scenario there is only pre-deterioration discount on selling price and cycle length must be less than life time of items. So the order quantity (25) and net present worth (26) for fixed life 7
time products where as follows can be found from equation (9) and (16), by letting ( ) (25) * ( ) ( )+ [ ( ) ] (26) ( ) * ( ) + Thus, the aim is to determine and to maximize (26) constrained by, Where PC, SC, LC are equal to (11),(12),(13) respectively, bearing in mind that in (11) now. Model for fixed life time products with no discount Letting, the initial inventory level and the net present value becomes (27) and (28) from (25) and (26) respectively. So, the objective here is to found and to maximize (28) constrained by. Where PC, SC, LC are equals to (11),(12),(13) respectively, taking in account that in (11) now. (27) [ ( ) ] (28) Numerical example In order to illustrate the proposed model, an example has been solved with the following parameter values: ;. As was mentioned it used The Differential Evolution Algorithm (Price et al. 2005) to derive the optimal values of the decision variables. For model with both pre- and post-deterioration discounts, the predeterioration discount on unit selling price is 0.59 and discount starts at time and continued to.then the product start to deteriorates. During this time in order to boost inventory depletion rate, 59% discount is provided for remaining time of replenishment cycle. The net present worth is 3994. The optimal order quantity is 2494 and the cycle length is 1.75. These results and the others subcases are listed in Table 1. 8
As can be noted, the net present value in model is greater than model and model, but there are combinations in the parameters where it can be different. So, a more deep analysis is needed to know when it is useful for managers each models in order to maximize the profit when the time value of money is taken into consideration. Table 2- Optimal values of the decision variables, order quantity and profit of the models Models 0.59 0.59 0.856602 1.75 1.75 2494.22 3994.08-0 - 0.777778 0.777778 75.4906 3615.75 - - - 1.75 1.75 266.17 1973.32-0.59-4.56344 7 2395.04 168 828 - - - 0.7 0.7 63.6662 91 409.9 0.59-0.340424 1.3 1.4 4070.68 828 408 - - - 1.3 1.4 177.221 1569.13 Conclusion Panda et al. (2009) presented a practical inventory model with pre- and post-deterioration discounts on selling price. The mathematical model is developed in order to investigate how much discount on selling price may be given to maximize the profit per unit time when demand is both stock and constant, but the time value of money and shortage are not considered. In this paper, that model has been extended to make it a little more applicable when inventory systems are dealing with the inventory replenishment problem for partial backlogging and time value of money. Although this research represent an important contribution of existing inventory models for deteriorating items with temporary price discounts, the model developed here can be further improved by employing more elaborate inventory models. Potential topics for notable future research include the incorporation of stochastic demand rate, multi-echelon inventory and multiitem approaches. References Abad, P.L. 2001. Optimal price and order size for a reseller under partial backordering. Computers & Operations Research 28(1): 53 65. Abad, P.L. 2003. Optimal pricing and lot-sizing under conditions of perishability, finite production and partial backordering and lost sale. European Journal of Operational Research 144(3 ): 677 685. Abdul, I. and Murata, A. 2011. Optimal production strategy for deteriorating items with varying demand pattern under inflation. International journal of industrial engineering computations 2(3): 449-466. Ali, S.S., Madaan, J., Chan, F.T.S. and Kannan, S. 2013. Inventory management of perishable products: a time decay linked logistic approach. International Journal of Production Research 51(13): 3864-3879. Berk, E., Gürler, Ü. and Yıldırım, G. 2009. On pricing of perishable assets with menu costs. International Journal of Production Economics 121(2): 678-699. Bhunia, A.K., Shaikh, A.A. and Gupta, R.K. 2013. A study on two-warehouse partially backlogged deteriorating inventory models under inflation via particle swarm optimisation. International Journal of Systems Science. DOI:10.1080/00207721.2013.807385: 1-15. Chauhan, S.S. and Kumari, R. 2011. Optimal Policy for Deteriorating Inventory with Permissible Delay in Payments Under Inflation. International Transactions in Applied Sciences 3(4): 615-636. 9
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