Pricing and Ordering Policies for Quality Unreliable Product with One-Way Substitution

Pricing and Ordering Policies for Quality Unreliable Product with One-way Substitution

Tian Zhiyu

Supervisor:Xu Chen Pan Jingming

Abstract: In order to gain maximum expected profit, the supplierand retailer need to make optimal pricing and ordering policies for substitutable andperishable products with stochastic demand.In this paper we study pricing and ordering policies for quality unreliable product with one-way substitution.We first develop and analysis the retailer’s ordering policy as a Stackelberg follower, then we discuss the supplier’s pricing policy as a Stackelberg leader. We also explore the impact of demand distribution, product’s marginal profit and retail price on the pricing and ordering decision. A series of characters and principles are drawn.

Keywords: Supply chain, Pricing and ordering policies, One-way substitution, Quality unreliable

1Introduction

Customer satisfaction is a key factor for a successful and competitive enterprise and product diversification is a useful tool for the improvement of customer satisfaction. At the same time, product diversification may lead to excess or inadequate of products.Statistics show that in United States stocks inadequate or surplus caused a loss of sales of 25%each year, even more than manufacturing cost[1]; random survey shows that about 8.2% of the customers in the afternoonhave faced a shortage of goods in supermarkets of United States, in the survey period of one month, up to 48% of the commodities at least have one record of shortage[2], which poses a severe challenge for the retailer’s ordering policy.

On the other hand, the survey found that: when consumers facedwith product shortage, only12%~18% abandoned purchase instead of choosing othersubstitutable products,most consumers will turn to choose other size and style of the same brand products[3]. Thus, product diversification will provide more choices for customers when product shortage occurs and the customerswho will be losing originally could be retainedby choosing substitutablegoods. Other related studies show: demand substitutable acts with reality and universality[4, 5]. Thus, the research of substitutable products has important theoretical and practical value.

2 Literature Review

There have substantial studies that range from marketing to operation research andmanagement on the pricing and ordering problems. The marketing literature often focuseson the coordination of pricing decisions in a single period, without production and inventory considerations. The operations literature, on the other hand, has traditionally beenfocused on coordinating production and inventory decisions, assuming that price and demand are given. Elmaghraby and Keskinocak[6]give an extensive literature review of this literature. Now we will concentrate onthose that are related to our study.

Pentico consideredone-way substitutionfor substitutable products, found the best multi-products inventory strategy by dynamic planning[7].Using game theory, Parlar discussedthe products’ substitution effect of two independent decision-makers when their products are in short supply, and found Nash equilibrium solution[8]. Taking a blood bankfor example, Gohstudied inventory system of perishable goods[9]. Chandgeneralizedthe purchase price function of Pentico’s, and assumedthere exist one-way substitution between fixed and accessory demands, derived the optimal dynamic stockportfolio of accessoriesusing dynamic planning [10]. Bassok presented a singlecycledownward substitutionof multi-product inventory model to achieve the optimal orderingpolicy for the maximization of single-cycle profit[11].

Comparing with the above studies, our work has three main differences. First, the quality unreliable problem with one-way substitution is studied. Second, we presented the optimal pricing and ordering policy for the supplier and the retailer. Third, the impact of demand distribution of substitutable product, product’s marginal profit and retail price is studied.

3 Problem Description

We consider a single-period monopoly model with one supplier selling two products: the quality unreliable product (Product 1) and the quality reliable product (Product 2) to one retailer. The retailer faces random and independent demands for each product and the two products have a one-way substitution structure: the quality unreliable product serving as a substitute for the quality unreliable product but not vice versa. In addition the substitution can take place after the demand for Product 2 has been satisfied and if substitution takes place, the retailer charges a lower price than the customer expects to pay, therefore customers always accept the substitute product. Hence, the supplier must find the optimal wholesale price and the retailer must choose the optimal quantity for each product.

The model is depicted in Figure 3-1 and some basic denotation is defined bellow.

ci – unit production cost;pi – unit retail price;wi – unit wholesale price;qi – ordering quantity of theretailer;Di – random market demand for Product i;πr –the retailer’s profit;πs –the supplier’s profit.

Figure 3-1 The pricing and ordering model

We define i=1 denote the quality unreliable product, hereafter Product 1, the density and cumulative demand distribution are, respectively, f(x) and F(X). We also define i=2 denote the quality reliable product, hereafter Product 2, the density and cumulative demand distribution are, respectively, g(x) and G(X). On the basis of above and with some common sense, we can educep1p2, w1w2, c1c2 and pi wi ci. In addition, the supplier and the retailer each has a reservation profit, and they will not participate in the channel if their expected profit are less than that. We assume it is zero for each, so they will choose to participate in the channel on condition that their expected profits are non-negative.

4 Model Formulation and Analysis

4.1 The retailer’s problem

We first consider the retailer’s ordering problem. As a Stackelberg follower, the retailer’s problem is to find optimal ordering policy, given the wholesale price for each product. Given wholesale prices w1and w2, the retailer’s expected profit is:

(4.1)

Proposition 1: For given wholesale prices w1and w2 and any demand densities f(•) and g(•), the retailer’s expected profit Eπrin (4.1) is concave in (q1,q2).

Proof: Taking partial derivatives of Eπrwith respect to q1 andq2, we obtain:

(4.2)

(4.3)

Differentiating the right-hand sides of these expressions with respect to q1 andq2again, we get

Hence,

It follows that Eπr is concave with respect to (q1, q2). Thus the proof is complete.

Hence, there exists a unique optimal ordering quantity set (q1*,q2*) to make the retailer’s expected profit maximization. Setting the right part of equation (4.2) and (4.3) to zero, we can obtain optimal ordering policy of the retailer, i.e., by solving the equation set (4.4) and (4.5), we can obtain the optimal ordering quantity set (q1*,q2*):

Now we study the properties the retailer’s ordering policy.

Proposition2

(1)For any wholesale price w2, the optimal ordering quantity for Product 2 q2* is greater than the optimal ordering quantity from the Newsboy problem q* without substitution.

(2)For any wholesale price w1, the optimal ordering quantity for Product 1 q1* is less than the optimal ordering quantity from the Newsboy problem q* without substitution.

Proof:From equations (4.4) and (4.5), we get

Hence,

The proof is complete. Thus, the retailer will order more Product 2 and less Product 1 when substitution is allowed than without substitution and Product 2 can be used to supply not only its own demand but also the demand of Product 1, i.e., substitution induces the retailer ordering more quality reliable product and less quality unreliable product.

4.2 The supplier’s problem

We now study the supplier’s pricing problem. The supplier has dominant bargaining power as a Stackelberg leader and can correctly anticipate the retailer’s reacting for any wholesale price policies and select the optimal wholesale price policy which maximizes his profit.That is, the supplier faces the retailer’s reaction function q1*(w1, w2) and q2*(w1, w2) for quality unreliable and reliable product, respectively. Hence, the supplier’s profit function is:

(4.6)

From (4.4) and (4.5), we obtain the inverse demand curves as:

(4.7)

(4.8)

Lemma 1

(1)Both w1(q1, q2) and w2(q1, q2) are decreasing in q1 and q2.

(2) w1(q1, q2)=w2(q1, q2) where G(q2)=1.

Proof: (1)Taking partial derivatives of w1(q1, q2) and w2(q1, q2)with respect to q1 andq2, we obtain

(2)We can obtain it from equations (4.7) and (4.8).

Thus, the proof is complete.Hence, in order to sell quantities (q1, q2) products the supplier has to set the wholesale price as equations (4.7) and (4.8) suggest, the wholesale price not only depends on its own quantities but also the other’s. At the retailer’s part, the ordering quantity of one product not only depends on its own wholesale price but also the other’s. That is, in order to sell more units of product, the supplier has to reduce both w1 and w2. Additionally, the supplier can sell Product 2 above its maximal demand level only if he set w2 the same as w1, i.e., the retailer will order more Product 2 than its maximal demand for the purpose of substitution if both the products cost the same.

Substituting (4.7) and (4.8) into (4.6), we obtain the supplier’s profit as:

(4.9)

In order to get more concrete results, we assume the demand distributionsf(•) and g(•) are uniform over (a,b) and (c,d), respectively.

Lemma 2:For a givenq2, πs is quasi-concave with respect to q1;for a givenq1,πs is quasi-concave with respect to q2 where q2∈(0,d). The optimal (q1, q2) lies in the region [a+c-d,b)×[c,d).

Proof:For q2c, πs=(p1-c1)q1+(p2-c2)-p1q1F(q1). Thus, ∂πs/∂q1>0 where q1a;∂2πs/∂q12<0 where a<q1b; ∂πs/∂q1<0 where q1b. Forcq2,∂πs/∂q1>0 where q1a+c-q2;∂2πs/∂q12<0 where a+c-q2q1b; ∂πs/∂q1<0 where q1b. Hence, for a givenq2,πs is quasi-concave in q1. Following the same way, we can proof that πs is quasi-concave with respect to q2 where q2∈(0,d)for a givenq1. We show the result in Figure 4-1:

Figure 4-1 The supplier’s profit as a function of ordering quantity

Thus, the optimal (q1, q2) lies in the region [a+c-d,b)×[c,d). The proof is complete.

From Lemma 2, we can obtain that the supplier’s optimal pricing policy exists, which will lead to the ordering quantity of Product 2 q2 less than it’s maximal demand d even through it could be used for substitution. The supplier’s optimal pricing policy may also lead the ordering quantities of Product 1 q1 less than it’s minimal demand a.

5 Numerical Study

To gain insight how substitutable demands, marginal profit and retail price affect the pricing and ordering policies, we perform a numerical study with uniform distributions.

5.1 The optimal policies

Weuse the following parameter values: p1=15, c1=5, p2=20, c2=10, D1~U(0,100) and D2~U(0,100) to study the optimal pricing and ordering policies and plot πs as a function of (q1, q2) in Figure 5-1:

Figure 5-1 The supplier’s profit as a function of (q1, q2)

We can see that πs is concave in (q1, q2) and there exits optimal (q1, q2) to make the supplier’s profit maximization. Let ∂πs/∂q1=0 and ∂πs/∂q2=0, we get the optimal ordering quantity (q1*, q2*)=(24.26, 34.79). Substitute them into equations (4.7) and (4.8), we get the supplier’s optimal wholesale price w1*=10.45 w2*=16.09 and the supplier’s profit πs*=344.09.

Now we study the impact of some parameters on the optimal pricing and ordering policy. The result is presented in Table 5-1~Table5-3 and some characters and principles are drawn bellow.

Table 5-1 The impact of demand distribution

demand distribution / q1 / q2 / w1 / w2
D1~U(0,100) D2~U(0,100) / 24.26 / 34.79 / 10.45 / 16.09
D1~U(0,100) D2~U(50,100) / 33.33 / 50 / 10.00 / 20
D1~U(50,100) D2~U(0,100) / 29.32 / 40.57 / 18.74 / 18.01

Table 5-2 The impact ofmarginal profit

marginal profit / q1 / q2 / w1 / w2
p1-c1=10, p2-c2=10 / 24.26 / 34.79 / 10.45 / 16.09
p1-c1=12, p2-c2=10 / 25.89 / 35.42 / 11.53 / 16.31
p1-c1=14, p2-c2=10 / 27.08 / 36.08 / 12.62 / 16.55
p1-c1=10, p2-c2=12 / 23.24 / 36.68 / 10.50 / 17.14
p1-c1=10, p2-c2=14 / 22.40 / 38.18 / 10.55 / 18.19

case 1~3: p1=15 c1=5; p1=17 c1=5; p1=19 c1=5 while p2=20 c2=10; case1,4&5: p2=20 c2=10; p2=22 c2=10;p2=24 c2=10 while p1=15 c1=5

From Table 1, we can see that with the increase of the mean and the decrease of the standard deviance of Product 2’s demand, the supplier will set a much higher w2 and a little lower w1,q2 will increase much more than q1. And with the increase of the mean and the decrease of the standard deviance of Product 1’s demand, the supplier will set much higher w1 and a little higher w2, q1 and q2 will increase simultaneously. That is to say, considering the substitution effect, demand distribution has much more impact on the pricing and ordering policies of the quality reliable product.

With the increase of Product 1’s marginal profit, the wholesale price and ordering quantity of Product 1&2 will increase simultaneously. The increase of Product 2’s marginal profit will lead to the increase of the wholesale price of Product 2 and the decrease of the wholesale price and ordering quantity of Product 1. The result is shown in Table 2.

Table 5-3 The impact ofretail prices

retail price / q1 / q2 / w1 / w2
p1=15; p2=20 / 24.26 / 34.79 / 10.45 / 16.09
p1=17; p2=20 / 17.49 / 39.86 / 12.68 / 16.27
p1=19; p2=20 / 9.03 / 48.01 / 15.09 / 16.51
p1=15; p2=22 / 26.33 / 30.56 / 10.35 / 17.95
p1=15; p2=24 / 27.74 / 27.30 / 10.28 / 19.8481

marginal profit is fixed, p1-c1=10, p2-c2=10, change in retail price pi

For fixed marginal profit, we can see from Table 3 that the higher retailer price of Product 2, the higher wholesale price and lower ordering quantity of Product 2. But with the increase of Product 1’s retail price, there will be a increase of the ordering quantity of Product 2 wile the wholesale price of Product 2 increase slightly.

6 Conclusion

In this paper, we studied the impact of substitutability and quality unreliablewith uncertainty of demand on the pricing and ordering policies of the supplier and retailer. We developed the retailer’s ordering model and the supplier’s pricing model considering the quality reliable product can be used as a substitute of the quality unreliable product but not vice verse. We proved the existence of the optimal pricing and ordering policies. We found that the retailer will order less quality unreliable product but more quality reliable product comparing with the newsboy model and the ordering quantity of quality reliable product will never excess its maximal demand unless both products cost the same. We also found that the wholesale price not only depends on its own demand but also the other’s demand and the retailer’s ordering quantity of one product not only depends on its own wholesale price but also the other’s. Numerical studies allow us to find the impact of substitutable demands, marginal profit and retail price on pricing and ordering policies and some characters and principles are drawn.

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The author: Tian Zhiyu, an undergraduate of the management school of UESTC.