Optimal Sales Discount Decision Model with Fixed Ordered Quantities

1.1 Overview of results

A major finding in this paper is that the sellers can increase sales by sales discounts, to realize maximize profits according to the demand rate and enable the category of goods inventory always in a relatively reasonable level. This paper shows that sales discounts can be determined under the fixed order quantity by taking demands as a “bridge”. It breaks the assumption that sellers is always in the optimal state.

1.2 Literature review

The assumption that sellers are always in the optimal state plays a crucial role in most researches relevant to the EOQ model. Then, the key question is how the suppliers can make biggest profit by their own price discounts strategy^[12]. This assumption is questionable. Michael Porter’s diamond model has used the concept of bargaining power. It reveals the contradictions between suppliers and sellers comprehensively. In addition, there are other challenges to the traditional EOQ model, such as time-varying demand and cost issues (i.e., deterministic problems of discrete-time non-homogeneous inventory system in finite stage); adjustment supplementary strategic issues; supplementary costs subtle changes issue; the demand rate issues and so on.

Goyal et al.^[13] pointed out that “Many researchers have assumed that the sales price and the purchase cost is the same, this is a disadvantage for the development of various types of EOQ model”. They also designed a kind of algorithm for sellers to determine the optimal prices and quantities simultaneously.

In fact, the issue on combination of inventory control and pricing^[14-15] is a classical one. A single-stage model is a starting point of this issue, which simply determines a unique price and economic order quantity. Subsequently, some scholars discuss the cost and uncertainty demand, equilibrium conditions under single-stage, and the increased or decreased marginal cost. A preliminary analysis of joint decision on the multi-stage storage/pricing model was also made^[16-17]. After that, Karlin and Carr^[18] assumed that the sales price could be set only at the beginning of the stage; demand is random and is a function of price; the highest price in finite and infinite period is the one where demand is zero. They also extended the model from single-stage to multi-stage. Other scholars have studied joint decision problem of stochastic demand/pricing^[19-26] in the single-stage. For example: Literature [21] considered simultaneous pricing and procurement decisions associated with a one-period pure inventory model under deterministic or probabilistic demand. It investigated the necessary and sufficient conditions for an (σ,Σ) type policy to be optimal for the determination of the procurement quantity. How the corresponding optimal price could be obtained was also shown in this literature. Literature [24] explained the phenomenon of price rigidity (or price smoothing) as the outcome of the optimal inventory policy of a multi-period profit-maximizing firm under demand and output uncertainties. Price smoothing might be manifested in two forms. First, price changes might be moderated with respect to those implied by the demand function; and second, the firm might choose to restrict price fluctuations by establishing upper and/or lower bounds on prices. It showed that the extent of the asymmetry in price smoothing depends on the relationship between the inventory holding cost and the backlog penalty cost. The model accommodated a wide range of price behaviour as observed in empirical studies on the issue. The point of Literature [25] is that inventory adjustment attenuates downward pressure on price when realized demand is low because firms accumulate inventory holdovers, speculating that demand will be stronger in the succeeding period. When realized demand is high the firm draws down its inventories until a “stock-out” occurs and price rises to clear the market. Literature [26] addressed the simultaneous determination of pricing and inventory replenishment strategies in the face of demand uncertainty. More specifically, the single item, periodic review model was analyzed. Demands in consecutive periods were independent, but their distributions depend on the item’s price in accordance with general stochastic demand functions. The price charged in any given period can be specified dynamically as a function of the state of the system. A replenishment order may be placed at the beginning of some or all of the periods. Stock-outs are fully backlogged. Both finite and infinite horizon models, with the objective of maximizing total expected discounted profit or its time average value were addressed, assuming that prices can either be adjusted arbitrarily (upward or downward) or that they can only be decreased. The structure of an optimal combined pricing and inventory strategy for all of the above types of models were characterized. An efficient value iteration method to compute these optimal strategies was also developed. Finally, an extensive numerical study that characterizes various qualitative properties of the optimal strategies and corresponding optimal profit values was made.

Early studies mainly focus on joint decision problem of dynamic storage/pricing under deterministic demand. Thomas^[27], Kunreuther and Schrage^[28] studied the joint decision problem of deterministic dynamic demand/pricing model in discrete-time based on the research of Wagner-Whitin^[29]. Rajan, et al.^[30] extended discrete-time to the continuous time. He assumed the demand rate was a function of the price and time, and the price changed continuously in a complementary cycle, and is a function of time elapsed. The article discussed optimal pricing and storage strategies under the premise of assuming the supplement was in equal time intervals. Moreover, Elmaghraby and Keskinocak^[31] reviewed the research progress and practical application about the inventory decisions and dynamic pricing.

Our contributions to the literature and the management practice are: first, to proposing objective function based on sellers’ maximize profits, and extending EOQ model to an optimal sales discounts decision model. Second, we also contribute by analyzing sellers’ optimal price in an optimal sales discounts decision model. Third, proposed and proved propositions in this paper have certain practical significance.

2.1 Basic definitions and assumptions

• The seller is a non-production retailer; does not provide non-physical products, such as services, no non-operating income.

• Business profits only determined by the amount of sales and the total cost.

• The total annual cost of inventories only constituted by the unit cost (C_h) and annual ordering cost (C₀).

• The seller takes profit maximization as the goal.

• The cost parameters are greater than zero.

2.2 Sales discount decision problem of single commodity with fixed order quantity in a single-stage

First consider sales discount decision problem of single commodity with fixed order quantity in a single-stage. The target for the seller is to pursue the total annual maximum profit. Model assumptions are as follows,

• Demand is a linear function of the price i.e. D = –kp + b, where D denotes the annual demand for the product, denotes the sales price of the goods; denote the slope and intercept of the linear function of demand respectively (k > 0, b > 0).

• “Stock-out” is not allowed.

• The procurement quantity Q > 0.

Total annual corporate profit will be obtained from the following general formula:

The total annual cost is the same as it in the classical EOQ model. Formula (2) can be obtained from (1),

2.3 Sales discount decision problem of multi-category goods with fixed order quantity in a single-stage

In a sales discount decision problem of single commodity with fixed order quantity in a single-stage, we assume that order quantity is fixed. The assumption is still valid in this section p_i(i = 1,2,···). D_i denotes the price of commodities, i.e. D_i = –k_i∗ p_i + b_i.

Total annual profit is given by,

2.4 Sales discount decision problem of single commodity with fixed order quantity in a multi-stage

Assumed further that

• In finite time, the time interval is (0, H). Sales period in terms of years, starting point is not fixed.

• Fixed time of replenishment and fixed replenishment quantity constraints.

• Fixed replenishment costs in the various stages.

• The sales prices unchanged, in addition to the price of holdovers in various stages, p_h = d_s* p_i, which d_s is sales discount rate, p_h denotes the price of holdovers, p_i denotes the price of inventory.

• The backlog in the previous stage is taken as inventory in a later stage, namely Q_i = Q_h + Q_r, which Q_i denotes quantities of inventory, Q_h denotes quantities of holdovers, Q_r denotes quantities of replenishment.

• Demand functions are the same at each stage.

In a certain sales cycle (t₀, t₀ + 1), then

Where in, D_t = –k(t)p + b(t).

2.5 Sales discount decision problem of multi-category goods with fixed order quantity in a multi-stage

In section 2.3, the assumption of three products a little too strict, companies might be interested in this classification suspicious. In this section, when we expand the problem to the one with multi-product in multi-stage. The original mark p to p_i(i = 1,2, · · ·) for i class product prices will be revised as follows, first kind of demand for class product is denoted by

D_i(D_i = –k_i(t)p_i+ b_i(t)).

3.1 Problem about optimal price

Due to the good position in the bargain, some brands suppliers developed relatively harsh ordering conditions. The following parts is a problem about optimal price. Suppose a garment supplier requires the following conditions that its brand vendors must attend a required ordering each year, order quantity shall not be less than 300 pieces of items; A vendor ordering cost is 350 yuan, the average holding cost of goods is 60 yuan. Goods demand curve is . If the vendor has ordered 300 pieces of items, how much is the optimal price for the vendor which can maximize profits? Solution: Use Theorem 1, can solve for

Maximum profit

3.2 Optimal sales discount for holdovers

Continue 3.1 problems; Suppose the vendor operates have six categories of goods; The total order quantity is 300 pieces of items. The vendor has holdovers in this year due to the fixed order quantity per year. How much is the sales discounts of the holdovers in the next year, which can ensure the vendor to make maximum profits? Specific data is in Table 1.

According to data in Table 1, we can be found that there are backlogs in category B; the current backlogs are 39 pieces of items. According to Theorem 5, put the above data into equation (15). The following result can be obtained:

At this point,

This result is of great significant; the results of this discussion will be launched in the next section.

4.1 Research implication of optimal sales discounts decision model

Competition in the market can be seen as a game among the various economic agents. In a game played by suppliers and sellers, in most cases they will ‘fight’ (i.e. bargain), and seldom cooperate. Determination of sales discounts can be seen as sellers compromise to suppliers by seeking the optimal strategy with downstream customers. However, the determination of sales discount has long been considered to be a marketing problem, discount strategies often comes from an intuition of managers. This resulted in the certain operating risks. This study will prevent from managers’ intuition. It improves decision-making process, and enables the process scientific by theoretical analysis and modeling.

As most of the non-linear model can be linearization, this study can be extended to optimal pricing or sales discounts research questions with respect to a function with complex demand. In this way, this paper will play a role in simplifying the practical problems. A complex pricing and demand data to fit the contents of the study can be processed by system identification model. The author will be discussed in detail elsewhere.

4.2 Managerial implication of optimal sales discounts decision model

According to the trend of observed data in section 3.2, we may believe that the price of goods in category B is prefer to be fixed because of a higher demand in the next period. The result in section 3.2 was interesting. The optimal sales discounts obtained, turned out to be less than 60% of the regular price. It seems to be unreasonable. But numerous managers in the actual operations are often doing like that. Do not forget, the supplier requires same procurement quantity of the vendors in the next period. That will make further holdovers of such goods, unless the demand in next period becomes larger. The possible gloomy result is very serious. In addition, it can also be found that in sales discounts optimization process, if the demand to other commodities is higher than that in previous period, the sales discount of holdovers can be smaller. Otherwise you will need a larger sales discount to enhance the sale and maximize profits.

This model has good openness, which can incorporate into various types of goods demand forecasts. To obtain the true and simplify demand function will be the next work should be done. Correct prediction to future demand will benefit to making a more favorable sales discount for the vendors. The model will help sales managers to make more scientific and effective decision-making in terms of the expanding aspects.

5.1 Single category of sales discounts decision problem

This paper revised the assumption — vendors are always in the optimal state, in the existing literature of inventory system decision problems. First, we assumed that sellers only have one commodity; sellers can get maximum profit in a single stage by finding the optimal sales price. We prove that in a sales discount decision system of single commodity with fixed order quantity in a single-stage, when the annual ordering cost is constant, the optimal price depends only on the slope and intercept of the demand function. Intercept and the optimal pricing are proportional inverse; the slope and the optimal pricing are positively proportional. On this basis, sales discount decision problems single commodity in multi-stage is discussed. We further inference that the optimal sales discount is proportional to the elasticity of demand.

5.2 Multi-category sales discount decision problem

This paper draws on static pricing and dynamic pricing two classical concepts, starting from the actual situation, establishes a sales discount decision model with multi-category goods. With respect to sales discount decision system of multi-category goods in single-stage, we found basic conditions of optimal sales price for the holdovers in multi-category goods. By simplifying the optimal sales price calculated to specific value, a practical significance was given. With respect to sales discount decision system of multi-category goods in multi-stage, we first proposed a lemma, “There must be a point in time which only one category goods is backlog among multi-category goods in finite time (0, H)”. Subsequently we proposed and proved conditions for sales discount of holdovers in a sales discount decision system of multi-category goods with fixed order quantity in a multi-stage. This model is more in line with China’s market reality. It plays an important role in helping the sellers make their sales discount decision.

5.3 Possible expansion direction in the future

This paper presents and proves the conditions for the determination of the optimal sales discount, fully accounting of Michael Porter’s diamond model. It will cause some difficulties for non-academic readers, since a number of elements in practice have been simplified to several important factors. In addition, the determination of the demand function does not source from the actual data directly. Although some examples of real data have been given, overall, how to do the exact setting of sales discount is still a question should be faced. Many related work to serve the sellers in reality should be done for the next stage-oriented (i.e. demand unknown stage) model.

In subsequent studies, the author are willing to conduct research work in demand forecasts, decision-making with the model, the integration of work, and a lot of the actual interaction between data and model validation, and model of the software with other scholars. We hope to realize intelligent data analysis and decision-making in reality.