Pricing policy for declining demand using item preservation technology
 Uttam Kumar Khedlekar^{1}Email author,
 Diwakar Shukla^{1} and
 Anubhav Namdeo^{1}
Received: 5 July 2016
Accepted: 1 November 2016
Published: 11 November 2016
Abstract
We have designed an inventory model for seasonal products in which deterioration can be controlled by item preservation technology investment. Demand for the product is considered price sensitive and decreases linearly. This study has shown that the profit is a concave function of optimal selling price, replenishment time and preservation cost parameter. We simultaneously determined the optimal selling price of the product, the replenishment cycle and the cost of item preservation technology. Additionally, this study has shown that there exists an optimal selling price and optimal preservation investment to maximize the profit for every business setup. Finally, the model is illustrated by numerical examples and sensitive analysis of the optimal solution with respect to major parameters.
Keywords
Mathematics subject classification
Background
Retailers and manufacturers identify the importance of pricing policy in a competitive environment. Adequate pricing and marketing policies may boost the company’s bottomline in such competition. Products such as fashion apparel, cosmetics, and winter wears become obsolete with the passing of time. Therefore, a pricing policy is required to ensure sale of the entire stock before entering the next cycle. Mostly, in a diminishing market, the manufacturers put their efforts into uplifting sales by reducing the price through media and pricing policy when demand declines. Fergany and Wakeel (2006) considered a probabilistic lost sales inventory system where order cost is a function of order quantity and lead time demand follows the normal distribution by using Lagrangian method. Chiu et al. (2016) developed two extended economic manufacturing quantity model to examine the production process, end product delivery and intrasupply chains.
Items that have a decreasing in quality or quantity over time are known as deteriorating items, and this process is called deterioration. One of the major assumptions of the inventory system is that the stock of goods can be indefinite, however some types of products either deteriorate or become obsolete with time, and the storage of such goods becomes difficult. This process can generally be seems in the case of commonly used goods such as fruits, vegetables, meat, perfumes, alcohols, gasoline, radioactive substances, photographic films and electronic components where deterioration is usually observed during their normal storage period. Banu and Mondal (2016) considered an economic order quantity model for deteriorating items where the demand function is linked with the customers’ credit period and the duration of policy of such credit period is exponential in nature.
Blackburn and Scudder (2009) examined warehouse temperature according to warehouse capacity constraints. Fergany (2016) proposed a probabilistic multiitem, singlesource inventory model with varying mixture shortage costs under two restrictions. Kouki et al. (2013) reveal that a continuous temperature control policy can be very efficient for inventory management. Widyadana and Wee (2010) devised an EPQ model for deteriorating items by considering stochastic demand. In this model, lost sales will occur when the machine unavailability time is longer than the nonproduction time. Khedlekar et al. (2014) developed a production inventory model for deteriorating items with production disruption and analyzed the system under different situations. Chandel and Khedlekar (2013) designed an integrated inventory model to optimize the total expenditure for warehouse setup. Shukla et al. (2012) considered an optimal selling price for optimal profit in a certain business setup and concluded that if demand for products is less pricesensitive, optimal profit will be greater but permit less pricesetting. Khedlekar and Shukla (2012) developed a dynamic pricing model for products with logarithmic decline pricesensitive demand and found that β is the most significant parameter affecting optimal profit and the respective number of price settings.
Chen and Zhang (2010) developed a threeechelon supply chain system consisting of suppliers, manufacturers and customers under demand disruptions by using a jumpdiffusion model. Furthermore, an improved analytical hierarchy process (AHP) studied for selection of the best suppliers based on quantitative factors such as optimal longterm total cost. The objective was to minimize the total cost under different demand disruption scenarios. Banerjee and Roy (2010) formulated a multiobjective inventory model for both exponential and uniform lead time where demand was taken, and it was found that fuzzy optimization obtained better results.
Roy and Chaudhuri (2011) introduced an economic production lotsized model where the production rate depends on the stock and selling price per unit. In this model, deterioration is assumed as a constant fraction allowing no shortages. Balkhi and Bakry (2009) devised a dynamic inventory model of deteriorating items in which each of the production, demand and deterioration rates were assumed to be general functions of time. Both inflation and time value of money were incorporated, and the optimal stopping and restarting production times in any cycle could be determined.
Giri et al. (2003) extended the economic lotscheduling problem where the production follows a normal distribution. Sarkar and Moon (2011) designed a classical EPQ model with stochastic demand under the effect of inflation. The model is described by considering a general distribution function. Chang et al. (2010) determined an EOQ model for deteriorating items by assuming that demand rate depends not only on the ondisplay stock level but also on the selling price per unit as well as the shelf or display space. They formulated two types of mathematical models to manifest the extended EOQ models for maximizing profits and derived the algorithms to find the optimal solution. The recent development of some EOQ models [Shukla and Khedlekar (2015), He and He (2010) and Zang et al. (2014)] has been important in the current competitive market. Chang and Dye (1999) modeled a business with a constant deterioration rate and timevarying demand with the assumption that the shortages were partially backlogged and the backlogging rate was a decreasing function of the waiting time for the next replenishment. A continuous production control inventory model for deteriorating items with shortages was developed by Samanta and Roy (2004), they assumed demand and production rates are constant and the deterioration of an item follows an exponential distribution. Hence, this model is applicable to food items, drugs, pharmaceuticals, etc. Dye and Hsieh (2012) extended Hsu et al.’s (2010) model by incorporating a timevarying deterioration rate. He and Huang (2013) devised a model for deteriorating seasonal products whose deterioration rate can be controlled by investing in preservation efforts. This model suggests preservation technology investment and pricing strategies for deteriorating seasonal products. Zang et al. (2014) designed an inventory model wherein demand is dependent on selling price and time and deterioration can be controlled by preservation technology. There are numerous studies on inventory models for deteriorating items under different conditions, such as Goswami and Chaudhuri (1991), Alamri and Balkhi (2007), Khedlekar et al. (2012), Chung and Huang (2007), Ouyang et al. (2005), and Kumar and Sharma (2012).
The study of a deteriorating inventory model is of extreme importance for the smooth and efficient running of any business organization. Deterioration may occur when goods decay, damage, spoil, or evaporate, or are obsolete or pilfered, all of which lead to loss of utility or loss of marginal value of commodities. The deterioration rate is assumed to be constant at the beginning of inventory modeling, but it varies according to time, weather, season, introduction of new technologies or new generation, such as grease, oil, petrol, alcohol, pharmaceutical substances, medicine, iron products, vegetables, perfumes, computer apparels, mobiles, and cosmetics. We apply preservation technology to protect the product from such types of deterioration, but this effort adds an additional cost to the total cost, known as the preservation technology investment cost.
The assumption of this study is that if we invest an optimal preservation cost u, the reduced deterioration rate becomes \(\theta \left( {1  f\left( u \right)} \right)\), which maximizes the total profit TP. Similarly, the demand rate is generally taken to be constant, but demand is never constant in reality. Since demand is affected by time, season and weather, it is a very sensitive factor related to any business management. Market price is highly related to market demand, so another assumption is that demand is considered timedependent and linearly related to market price. Thus, here we assume the demand \(D\left( {p,t} \right) = \alpha  at  \beta p\) is price sensitive, time dependent and linearly declining. Since the time horizon is taken as infinite in this study, we determined the optimal time for replenishment.
The main goal of this study is to determine the optimal selling price, the optimal length of the replenishment cycle and the optimal preservation technology investment simultaneously, such that the total profit per unit time is maximized.
The rest of the paper is organized as follows: In section “Assumption and notation”, the assumption and notations are presented. In section “Mathematical model”, a mathematical model to maximize the total profit per unit time is established, and three propositions are developed. In section “Numerical example”, two numerical examples are given. In section “Discussions”, the conclusion is presented to summarize the outputs.
Assumption and notation
 p :

Market price per unit, where p > c
 h :

Inventory holding cost unit per unit time
 T :

Length of the replenishment cycle, where the end of the cycle inventory is zero
 I(t) :

Inventory level at time t
 Q :

Order quantity per cycle
 K :

Replenishment cost per order
 c :

Purchasing cost per unit
 D(p,t):

Demand rate is a function of both price and time
 θ :

Deterioration rate, where 0 ≤ θ ≤ 1
 u :

Preservation technology investment parameter per unit time to reduce the deterioration rate
 f(u):

Proportion of reduced deterioration rate, where 0 ≤ f(u) ≤ 1
 TP(T, p, u):

Total profit per unit time
Mathematical model
Sales revenue
Purchasing cost
Inventory holding cost
Preservation cost
Replenishment cost
Proposition 1
There exists a unique p* that maximizes the profit function \(TP\left( {T,\,p,\,u} \right)\) for fixed T and optimal u.
Proof
Thus, p* is the optimal market price that maximizes the profit function for fixed T and optimal u.
Proposition 2
The profit function \(TP(T,\,p,\,u)\) is concave in the replenishment cycle T.
Proof
The first and second partial derivatives of the profit function \(TP(T,\,p,\,u)\) with respect to T are as follows, where \({\text{x }} = \, \theta \left( {1  f(u)} \right)\)
Hence, the total profit function is concave in T. Thus, there exists a unique optimal replenishment time T ^{ * } that maximizes \(TP(T,\,p,\,u)\), and the optimal T ^{ * } can be obtained by solving \(\frac{\partial TP(T,\,p,\,u)}{\partial T} = 0\).
Proposition 3
For any given feasible p and T, there exists a unique optimal preservation technology investment u ^{*} that maximizes TP(p,T,u).
Proof
Since f′(u) > 0 and f ″(u) < 0, it is clear from the above equation that \(\frac{{\partial^{2} TP(p,T,u)}}{{\partial u{}^{2}}} < 0\).
Hence, the total profit is a concave function of the preservation cost.
Numerical example
Example 1
In this example, we have considered that the reduced deterioration rateis \(f(u) = 1  e^{{  \gamma {\text{ u}}}}\), \(\gamma > 0\). The parametric values of the inventory system are as follows: K = $10 per order, c = $10 per unit, h = $0.5 per unit per month, u = $5 per unit time, θ = 0.01, α = 500, β = 0.5, γ = 0.05, and ε = 0.0001. Then, the optimal selling price per unit is p* = $12.6834, the optimal replenishment time T ^{*} = 0.712, the total profit per unit time \(TP(T^{ * } ,\,p^{ * } ,\,u^{ * } )\) = $416 and the order quantity Q ^{*} = 26.
Sensitivity analysis with respect to the major parameters
Input parameters  Output parameters  −40%  −20%  0%  20%  40% 

α  p  8.77  10.72  12.69  14.67  16.65 
Q  17  22  26  30  33  
T  1.050  0.830  0.712  0.633  0.576  
TP  103  239  416  633  891  
β  p  19.34  15.18  12.69  11.03  9.85 
Q  28  27  26  25  24  
T  0.67  0.69  0.712  0.74  0.77  
TP  765  549  416  325  258  
γ  p  12.685  12.684  12.6833  12.683  12.682 
Q  26  26  26  26  26  
T  0.712  0.712  0.712  0.711  0.711  
TP  416.06  416.03  416  415.97  415.90  
h  p  12.65  12.67  12.68  12.70  12.72 
Q  34  29  26  24  22  
T  0.910  0.792  0.712  0.654  0.606  
TP  420.24  418.18  416  413.79  411.70 
Discussions
Based on the results in Table 1, we observe the following facts:
The increasing initial demand α, the optimal total profit per unit time \(TP(T^{ * } ,\,p^{ * } ,\,u^{ * } )\), the optimal selling price p*, and optimal order quantity Q ^{ * } increase while the optimal replenishment time T ^{ * }decreases. As a result, we have to maintain a high initial demand by ordering more quantity per replenishment cycle and shortening the replenishment cycle. Moreover, if the scaling factor α is getting low, then the enterprise terminates the order.
When the price sensitivity parameter β increases, the optimal \(TP(T^{ * } ,\,p^{ * } ,\,u^{ * } )\), p* and Q ^{ * } decreases while the optimal length of replenishment cycle T ^{ * } increases. Thus, demand declines with a higher market price. To maintain this demand, we must reduce the optimal selling price.
When increasing value of parameter γ, the total profit per unit time \(TP(T^{ * } ,\,p^{ * } ,\,u^{ * } )\) and the optimal selling price p*decrease marginally. Therefore, this parameter does not change the model output.
When the holding cost per unit time h increases, the optimal selling price p* increases, while the optimal order quantity Q ^{ * }, the optimal replenishment cycle T ^{ * }and the total profit \(TP(T^{ * } ,\,p^{ * } ,\,u^{ * } )\) decrease. The minimum holding cost leads to maximizing the total profit.
Example 2
Sensitivity analysis with respect to θ
θ  T  P  TP  Q 

0.01  0.712  12.6833  416.00  26.10 
0.02  0.708  12.6879  416.20  26.00 
0.03  0.701  12.6914  416.30  25.75 
0.04  0.691  12.6945  416.33  25.50 
0.05  0.682  12.6974  416.36  25.16 
0.06  0.773  12.7002  415.38  24.87 
0.07  0.664  12.7028  414.40  24.57 
0.08  0.656  12.705  413.39  24.31 
0.09  0.648  12.708  412.38  24.05 
0.10  0.639  12.711  411.38  23.75 
Discussions
Conclusion
The concept of item preservation technology is developed for pricesensitive demand. This paper concludes that there is a unique optimal selling price, optimal length of replenishment cycle and optimal item preservation technology investment for obtaining optimal profit. Numerical examples are provided to illustrate the proposed model. Sensitivity analysis is provided with respect to some key parameters to manage the system. Ordering of small lots is advised for retailers to reduce deterioration. The incorporation of preservation technology investment may significantly reduce the deterioration. Sensitivity analysis reveals that if initial demand increases or decreases, then we must adapt by ordering either more quantities per cycle or by reducing orders accordingly. Higher market price affects the market demands. We need to keep market demand progressive by reducing the selling price accordingly. Moreover, to maximize the model output, we need to keep the numerical value for parameter γ as low as possible. There is a need for balance between holding costs and preservation technology investment costs in order to obtain maximum profit.
One can extend this model for stochastic demand and variable holding costs. The proposed model can also be designed in a fuzzy environment. The theory can also be applied to growing and deckling markets separately with variable deterioration rates.
Declarations
Authors’ contributions
All authors have contributed to the manuscript equally. We all discussed the outcomes of the proposed model and approved the final manuscript. All authors read and approved the final manuscript.
Acknowledgements
The authors express their gratitude to the editor and the reviewer for their valuable suggestions and comments, which have contributed to the improvement of this paper.
Competing interests
We, the authors, declare that we have no competing interests in this regard.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Alamri AA, Balkhi ZT (2007) The effects of learning and forgetting on the optimal production lot size for deteriorating items with time varying demand and deterioration rates. Int J Prod Econ 107:125–138View ArticleGoogle Scholar
 Balkhi ZT, Bakry ASH (2009) A general and dynamic production lot size inventory model. Int J Math Models Methods Appl Sci 3(3):187–195Google Scholar
 Banerjee S, Roy TK (2010) Solution of single and multi objective stochastic inventory models with fuzzy cost components by intuitionist fuzzy optimization technique. Adv Oper Res 2010:765278MATHGoogle Scholar
 Banu A, Mondal SK (2016) Analysis of credit linked demand in an inventory model with varying ordering cost. Springerplus 5:926View ArticlePubMedPubMed CentralGoogle Scholar
 Blackburn J, Scudder G (2009) Supply chain strategies for perishable products: the case of fresh produce. Prod Oper Manag 18(2):129–137View ArticleGoogle Scholar
 Chandel RPS, Khedlekar UK (2013) A new inventory model with multiple warehouses. Int Res J Pure Algebra 3(5):192–200Google Scholar
 Chang HJ, Dye CY (1999) An EOQ model for deteriorating items with time varying demand and partial backlogging. J Oper Res Soc 50:1176–1182View ArticleMATHGoogle Scholar
 Chang CT, Chen YJ, Tsai TR, Wu SJ (2010) Inventory model with stock and price dependent demand for deteriorating items based on limited shelf space. Yugosl J Oper Res 20(1):55–69MathSciNetView ArticleMATHGoogle Scholar
 Chen X, Zhang J (2010) Production control and supplier selection under demand disruptions. Int J Ind Eng Manag 3(3):421–446Google Scholar
 Chiu SW, Chen SW, Chiu YSP, Li TW (2016) Producerretailer integrated EMQ system with machine breakdown, rework failures, and a discontinuous inventory issuing policy. Springerplus 5:1351View ArticleGoogle Scholar
 Chung KJ, Huang TS (2007) The optimal retailer’s ordering policies for deteriorating items with limited storage capacity under trade credit financing. Int J Prod Econ 106:127–145View ArticleGoogle Scholar
 Dye CY, Hsieh TP (2012) An optimal replenishment policy for deteriorating items with effective investment in preservation technology. Eur J Oper Res 218:106–112MathSciNetView ArticleMATHGoogle Scholar
 Fergany HA (2016) Probabilistic multiitem inventory model with varying mixture shortage cost under restrictions. Springerplus 5:1351View ArticlePubMedPubMed CentralGoogle Scholar
 Fergany HA, Wakeel ELMF (2006) Constrained probabilistic lost sales inventory system with normal distribution and varying order cost. J Math Stat 2(1):363–366MathSciNetView ArticleMATHGoogle Scholar
 Giri BC, Moon I, Yun WY (2003) Scheduling economic lot sizes in deteriorating production systems. Nav Res Logist 50(6):650–661MathSciNetView ArticleMATHGoogle Scholar
 Goswami A, Chaudhuri KS (1991) An EOQ model for deteriorating items with shortage and linear trend in demand. J Oper Res Soc 42:1105–1110View ArticleMATHGoogle Scholar
 He Y, He J (2010) A production model for deteriorating inventory items with production disruptions. Discrete Dyn Nat Soc 2010:1–14MathSciNetMATHGoogle Scholar
 He Y, Huang H (2013) Optimizing inventory and pricing policy for seasonal deteriorating products with preservation technology investment. J Ind Eng 2013:1–7MathSciNetGoogle Scholar
 Hsu PH, Wee HM, Teng HM (2010) Preservation technology investment for deteriorating inventory. Int J Prod Econ 124(2):388–394Google Scholar
 Khedlekar UK, Shukla D (2012) Dynamic pricing model with logarithmic demand. Opsearch 50(1):1–13MathSciNetView ArticleGoogle Scholar
 Khedlekar UK, Shukla D, Chandel RPS (2012) Managerial efficiency with disrupted production system. Int J Oper Res 9(3):141–150MathSciNetGoogle Scholar
 Khedlekar UK, Shukla D, Chandel RPS (2014) Computational study for disrupted production system with time dependent demand. J Sci Ind Res 73(5):294–301Google Scholar
 Kouki C, Sahin E, Jemai Z, Dallery Y (2013) Assessing the impact of perishability and the use of time temperature technologies on inventory management. Int J Prod Econ 143(1):72–85View ArticleGoogle Scholar
 Kumar R, Sharma SK (2012) An M/M/1/N queuing model with retention of reneged customers and balking. Am J Oper Res 2(1):1–5MathSciNetView ArticleGoogle Scholar
 Ouyang LY, Chang CT, Teng JT (2005) An EOQ model for deteriorating items under trade credits. J Oper Res Soc 56:719–726View ArticleMATHGoogle Scholar
 Roy T, Chaudhuri KS (2011) An EPLS model for a variable production rate with stockprice sensitive demand and deterioration. Yugosl J Oper Res 21(2):1–12MATHGoogle Scholar
 Samanta GP, Roy A (2004) A production inventory model with deteriorating items and shortages. Yugosl J Oper Res 14(2):219–230MathSciNetView ArticleMATHGoogle Scholar
 Sarkar B, Moon IK (2011) An EPQ model with inflation in an imperfect production system. Appl Math Comput 217:6159–6167MathSciNetMATHGoogle Scholar
 Shukla D, Khedlekar UK (2015) Inventory model for convertible item with deterioration. Commun Stat Theory Methods 54(4):1137–1147MathSciNetMATHGoogle Scholar
 Shukla D, Khedlekar UK, Chandel RPS, Bhagwat S (2012) Simulation of inventory policy for product with price and timedependent demand for deteriorating items. Int J Model Simul Sci Comput 3(1):1–30View ArticleGoogle Scholar
 Widyadana GA, Wee HM (2010) Production inventory models for deteriorating items with stochastic machine unavailability time, lost sales and pricedependent demand. J Tek Ind 12(2):61–68MathSciNetGoogle Scholar
 Zang J, Bai Z, Tang W (2014) Optimal pricing policy for deteriorating items with preservation technology investment. J Ind Manag Optim 10(4):2014MathSciNetGoogle Scholar