# Simultaneous pricing and inventory decisions for substitute and complementary items with nonlinear holding cost

• M. A. Edalatpour
• S. M. J. Mirzapour Al-e-Hashem
Production Management

## Abstract

In practice, the holding cost in inventory models can be considered as linear or nonlinear functions. Nevertheless, nonlinear holding cost rarely attracted the attention of the researchers, and to our knowledge, it is novel in the presence of complementary or substitute items. It can also be interpreted as a way to include the perishability of products. Except for the perishability, nonlinear holding cost can be employed for other purposes like cold supply chain. In this paper, a new pricing method based on a multi-product inventory model is presented for complementary and substitute items. This paper aims to find the optimum value of the replenishment cycle and prices for the products supplied together (complementary) or interchangeable (substitute), such that the total cost of the inventory system is minimized. Demand is assumed as a price sensitive function while the proposed model is trying to obtain the optimal values of prices and the replenishment cycle simultaneously.

## Keywords

Pricing Complementary Substitute Nonlinear holding cost Multi-product Inventory

i

Product index

k

Model type index

## Decision variables

$$T$$

Order cycle

$$D_{i}$$

Demand of product i

$$p_{i}$$

Price of product i

$$Q_{i}$$

Order quantity of product i

$$R_{k}$$

Revenue function

$$k$$

Item type

## Parameters

$$A_{i}$$

Fixed cost of ordering of product i

$$K_{i}$$

Total purchasing cost of product i

$$\bar{K}_{i}$$

Average purchasing cost per ordering cycle ($$T$$)

$$a_{i}$$

Base demand (demand for zero price)

$$s$$

Price sensitivity of the demand

$$h_{i}$$

Holding cost of product i per each unit of time

$$c_{i}$$

Purchase cost of product i

$$\theta_{i}$$

Ratio of complementarity between product $$i$$ and its complement products $$\left( {0 \le \theta_{i} \le 1} \right)$$

$$\mu_{i}$$

Ratio of substitution between product $$i$$ and its substitute products ($$0 \le \mu_{i} \le 1)$$

$$\varphi$$

Deterioration rate

## References

1. 1.
Alfares HK, Ghaithan A (2016) Inventory and pricing model with price-dependent demand, time-varying holding cost, and quantity discounts. Comput Ind Eng 64:170–177Google Scholar
2. 2.
Avinadav T, Herbona A, Spiegel U (2013) Optimal inventory policy for a perishable item with demand function sensitive to price and time. Int J Prod Econ 144(2):497–506Google Scholar
3. 3.
Chandra S (2017) An inventory model with ramp type demand, time varying holding cost and price discount on backorders. Uncertain Supply Chain Manag 5(1):51–58Google Scholar
4. 4.
Chen X, Hu P (2012) Joint pricing and inventory management with deterministic demand and costly price adjustment. Oper Res Lett 40:385–389
5. 5.
Cohen Morris A (1977) Joint pricing and ordering policy for exponentially decaying inventory with known demand. Naval Res Logist Q 24(2):257–268
6. 6.
Dobson G, Pinker EJ, Yildiz O (2017) An EOQ model for perishable goods with age-dependent demand rate. Eur J Oper Res.
7. 7.
Ferguson M, Jayaraman V, Souza GC (2007) Note: an application of the EOQ model with nonlinear holding cost to inventory management of perishables. Eur J Oper Res 180(1):485–490
8. 8.
Fujiwara O, Perera ULJSR (1993) EOQ models for continuously deteriorating products using linear and exponential penalty costs. Eur J Oper Res 70:104–114
9. 9.
Giri BC, Chauduri KS (1998) Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost. Eur J Oper Res 105:467–474
10. 10.
Gupta S, Loulou R (1998) Process innovation, product differentiation, and channel structure: strategic incentives in a duopoly. Market Sci 17(4):301–316Google Scholar
11. 11.
Hsieh Tsu-Pang, Dye Chung-Yuan (2010) Pricing and lot-sizing policies for deteriorating items with partial backlogging under inflation. Expert Syst Appl 37(10):7234–7242Google Scholar
12. 12.
Kang S, Kim I (1983) A study on the price and production level of the deteriorating inventory system. Int J Prod Res 21(6):899–908
13. 13.
Maity K, Maiti Manoranjan (2009) Optimal inventory policies for deteriorating complementary and substitute items. Int J Syst Sci 40(3):267–276
14. 14.
Mokhtari H (2018) Economic order quantity for joint complementary and substitute items. Math Comput Simul 154:34–47Google Scholar
15. 15.
Mukhopadhyay Samar K, Yue Xiaohang, Zhu Xiaowei (2011) A Stackelberg model of pricing of complementary goods under information asymmetry. Int J Prod Econ 134(2):424–433Google Scholar
16. 16.
Pando V, San-José LA, García-Laguna J, Sicilia J (2013) An economic lot-size model with non-linear holding cost hinging on time and quantity. Int J Prod Econ 145(1):294–303Google Scholar
17. 17.
Rastogi M, Singh S, Kushwah P, Tayal S (2017) An EOQ model with variable holding cost and partial backlogging under credit limit policy and cash discount. Uncertain Supply Chain Manag 5(1):27–42Google Scholar
18. 18.
San-José LA, Sicilia J, García-Laguna J (2015) Analysis of an EOQ inventory model with partial backordering and non-linear unit holding cost. Omega 54:147–157Google Scholar
19. 19.
Sazvar Z, Baboli A, Jokar MRA (2013) A replenishment policy for perishable products with non-linear holding cost under stochastic supply lead time. Int J Adv Manuf Technol 64:1087–1098Google Scholar
20. 20.
Shavandi Hassan, Mahlooji Hashem, Nosratian Nasim Ekram (2012) A constrained multi-product pricing and inventory control problem. Appl Soft Comput 12(8):2454–2461Google Scholar
21. 21.
Tiwari S, Cardenas-Barron LE, Goh M, Shaikh AA (2018) Joint pricing and inventory model for deteriorating items with expiration dates and partial backlogging under two-level partial trade credits in supply chain. Int J Prod Econ. Google Scholar
22. 22.
Tripathi RP, Tomar SS (2015) Optimal order policy for deteriorating items with time-dependent demand in response to temporary price discount linked to order quantity. Int J Math Anal 9(23):1095–1109Google Scholar
23. 23.
Wei J, Zhao J, Li Y (2013) Pricing decisions for complementary products with firms’ different market powers. Eur J Oper Res 224(3):507–519
24. 24.
Weiss H (1982) Economic order quantity models with nonlinear holding costs. Eur J Oper Res 9:56–60
25. 25.
Whitin TM (1995) Inventory control and price theory. Manag Sci 2(1):61–68Google Scholar
26. 26.
Yan R, Bandyopadhyay S (2011) The profit benefits of bundle pricing of complementary products. J Retail Consum Serv 18(4):355–361Google Scholar
27. 27.
Yue Xiaohang, Mukhopadhyay Samar K, Zhu Xiaowei (2006) A Bertrand model of pricing of complementary goods under information asymmetry. J Business Res 59(10):1182–1192Google Scholar