Journal of Industrial Engineering, University of Tehran, Special Issue, 2011, PP. 103-112 103
Optimal Selling Price, Marketing Expenditure and Order Quantity with Backordering

Samira Mohabbatdar 1 and Maryam Esmaeili *1
1
Department of Engineering, Alzahra University, Tehran, Iran (Received 6 June 2011, Accepted 15 August 2011)

Abstract
Demand is assumed constant in the classical economic order quantity (EOQ) model. However, in the real world, the demand is dependent on many factors such as the selling price, warranty of product and marketing effort. In addition pricing and ordering quantity decisions are interdependent for a seller when demand for the product is price sensitive in the inventory models. These types of models are very popular in the literature as joint pricing and order quantity models. Many researchers consider these models under some conditions such as quantity discount, trade credit and marketing effort. In this paper, we propose a new inventory model for the seller who conducts marketing effort. The marketing effort is the process of performing market research, selling products and/or services to customers and promoting them via advertising to further enhance sales. It is used to identify the customer, to satisfy the customer, and to keep the customer. This process will happen during the planning horizon; therefore the product will be demanded increasingly as time passes. This increasing in the demand leads to the backorder condition in the model. Since the marketing effort as a decision variable is dependent of the time, in this paper, the marketing effort is assumed a linear function of time which has an effect on the demand in addition of price in our model. The model would be included the backorder cost due to raising the shortage of inventory in addition, the purchasing, ordering and holding costs. An algorithm for finding the optimal solution for the selling price, marketing expenditure and the time length of positive stock are obtained when the seller’s profit is maximized. To clarify the model more, numerical examples presented in this paper, including sensitivity analysis of some key parameter- the cost parameters and non-cost parameters- that will compare the obtained results of proposed model.

Keywords: Backorder, Backlog, Inventory control, Marketing effort, Pricing, Shortage

Introduction
The classical economic order quantity (EOQ) model introduced by Harris in 1913 is based on the unreal assumptions such as demand constant. Progressively, the concept of fixing demand is avoided therefore; some new inventory models are emerged (Abad, 1994; Lee, 1993; Lee et al., 1996; Kim and Lee, 1998; Jung and Klein, 2001, 2005). The demand is a function of price over a planning horizon in these models to maximize the firm’s profit. In addition, some other models have been presented by assuming a general demand. In fact; the demand rate is assumed as a convex function of selling price (a linear or nonlinear function of price) under some conditions such as quantity discount or paying for freight (Papachristos and Skouri (2003), Abad (2006), Dye (2007)). Moreover, to avoid fixed demand in EOQ, marketing expenditure would depend on demand over a planning horizon. For instance, Sahidual Islam (2008) has formulated a multi-objective marketing planning inventory model under the limitations of space capacity. The optimal order quantity, marketing expenditure and shortage amount are obtained by applying geometric programming. Similar approaches have also been used in cases where both marketing expenditure and price influence demand (Freeland, 1982; Lee and Kim, 1993, 1998; Esmaeili, 2007).

* Corresponding author: Tel: +98- 21- 88041469 Fax: +98- 21- 88617537 Email: [email protected]
A significant shortcoming of all these models is, considering the marketing expenditure as a decision variable which is independent of the time. However in the real world, during marketing effort, the product observes increasing sales after gaining consumer acceptance. Ignoring the shortages of inventory is another key assumption in the classic EOQ model. Therefore, the concept of backorders (backlogged) captured the mind of inventory modelers to develop the classic EOQ models. Padmanabhan and Vrat (1990) have introduced the backlogging models to represent inventory shortage. They have not considered the necessary conditions for optimal solution, while Chu et al. (2004) used it in his model. In 2005, San Jose et al. have expanded the backlogging models, by considering lost sales penalties in their model. Considering deterministic demand is the common assumption for the above models. However, Zhou et al. (2004) have presented the partial backlogging model under time-varying demand. They emphasize the replenishment costs consist of both fixed and lot size dependent components. They have developed a numerical procedure for determining appropriate lot-sizing policies based on their exploration of the mathematical properties of the model with the sensitivity analysis. Some papers have also obtained optimal stocking policies under backorder and shortage assumption in a supply chain such as Emmett J. Lodree Jr. (2007) and Chun Jen Chung, Hui Ming Wee (2007). One of the common purposes in the mentioned backlogging models is determining the optimal lot size. Although the demand has a significant role in inventory models, they have ignored some factors such as price and marketing expenditure which influence on the demand.
Abad (2008) considers the pricing and lot-sizing model for a product subject to general rate of deterioration and backordering which is more realistic to compare to the previous models. The model is included all three costs-the lost sale, carrying backorders and the shortage cost. However, it is assumed that the demand is a function of one factor, price, in order to avoid the confounding effect of the demand function.
In this paper, we propose a novel model which will enable the sellers in making decisions regarding purchasing, selling price and marketing effort when the backorder occurs. The marketing effort in previous papers is considered static. However, the marketing effort will happen during the planning horizon. The marketing effort is the process of performing market research, selling product and/or services to customers and promoting them via advertising for further enhance sales. It is used to identify, satisfy and keep the customer. Therefore the product will be demanded increasingly as time passes. Unlike most of the models cited above, we use a new approach in including time in marketing effort. In fact, it is considered a linear function of time which has an effect on the demand in addition of price in our model. For that reason, the proposed model can lead to a realistic and distinguished inventory policy in comparison with the previous models. Since the marketing effort influence the demand increasingly as time passes, the seller encounters the shortage cost during the fix planning horizon. The length of the period with positive stock, selling price and marketing expenditure are considered as decision variables and the goal is to determine the optimal solution per period by using an optimization procedure. Logistic costs including purchasing, ordering, holding/carrying, and shortage costs are considered in the proposed model. Note, the order quantity and backorder level can be obtained by the demand over duration of inventory cycle.
The remainder of this paper is organized as follows. Notation, assumptions, decision variables and input parameters are provided in Section 2. A mathematical model and an algorithm for finding the optimal solution are given in Sections 3 and 4. Section 5 presents some computational results including numerical examples. Finally, the paper concludes in Section 6 with some suggestions for future work in this area.

Notation and problem formulation
This section introduces the notation and formulation of our model. Here, we state decision variables, input parameters and assumptions underlying the model.

Decision Variables
P Selling price ($/unit),
M Marketing expenditure per
unit ($/unit),
T1 Duration of the period with positive inventory (T1 ≤ T).

Input Parameters
D(P) Demand rate (units/period),
B(M) Marketing function,
I(t) Net inventory level at time t,
S(t) The shortage level at time t,
π The unit shortage cost per unit of time,
h Holding cost ($/per unit) ($/unit/period) (π < h),
k0 The fixed ordering cost per order ($/order),
Q The order quantity,
b Backorder level,
T Duration of inventory
cycle/cycle time,
K1 Scaling constant for demand function,
α Price elasticity of demand function,
K Scaling constant for
marketing function,
C Unit purchasing cost.

Inventory level

Q-b

Time
T
1

T

Inventory level

Q-b

Time

T

1

T

Figure 1: Inventory pattern over time

Assumptions
The proposed model is based on the following assumptions:
The planning horizon is infinite.
Duration of inventory cycle/cycle time is fixed.
Shortages are permitted and completely backordered.
The product is not perishable.
Demand is a function of price; for
notational simplicity we let D ≡ D(P)

D  K1P;K1  0 (1)

6. It is assumed that marketing effort is an increasing function of marketing expenditure and time. For notational simplicity we let B
≡ B(M) such that

B  K  Mt;0  t T,K  0 (2)

The net inventory system of the seller is illustrated in Fig. 1. It can be described by
the following equation. During t Є (0 T1 ],

dI(t)



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