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 supplier and retailer need to make
optimal pricing and ordering policies for substitutable and perishable 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
1 Introduction
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 afternoon have 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 faced with product
shortage, only 12%~18% abandoned purchase instead of choosing other substitutable
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 customers who will be losing originally could
The
author: Tian Zhiyu, an undergraduate of the management school of UESTC.
1
be retained by choosing substitutable goods. 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
and management on the pricing and ordering problems. The marketing literature often
focuses on the coordination of pricing decisions in a single period, without production
and inventory considerations. The operations literature, on the other hand, has
traditionally been focused 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 on those that are
related to our study.
Pentico considered one-way substitution for substitutable products, found the
best multi-products inventory strategy by dynamic planning[7]. Using game theory,
Parlar discussed the products’ substitution effect of two independent decision-makers
when their products are in short supply, and found Nash equilibrium solution[8].
Taking a blood bank for example, Goh studied inventory system of perishable goods
[9]
. Chand generalized the purchase price function of Pentico’s, and assumed there
exist one-way substitution between fixed and accessory demands, derived the optimal
dynamic stock portfolio of accessories using dynamic planning [10]. Bassok presented
a single cycle downward substitution of multi-product inventory model to achieve the
optimal ordering policy 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.
Supplier
2
Retailer
w2
2
p2
q2
1
w1
D2
p1
1
p1
D1
q1
ci – unit production cost; pi – unit retail price; wi – unit wholesale price; qi – ordering quantity of the
retailer; 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 educe p1 < p2, w1 < w2, c1 < c2 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 w1 and w2, the retailer’s expected profit is:
q1

0
q1
E r   p1 x f  x  dx  

q2
0

q1  q2  y
q1
q2

0
q2
p1q1 f  x  dx  w1q1   p2 yg  y  dy  
p1  y  q1  f  x  g  y  dxdy  
q2
0


q1  q2  y
p2q2 g  y  dy  w2q2
p1  q2  x  f  x  g  y  dxdy
(4.1)
Proposition 1: For given wholesale prices w1 and w2 and any demand densities f(•)
and g(•), the retailer’s expected profit Eπr in (4.1) is concave in (q1, q2).
Proof: Taking partial derivatives of Eπr with respect to q1 and q2, we obtain:

E r q1  p1  w1  p1 F  q1  1  G  q2     g  y  F  q1  q2  y  dy
q2
0
q2
E r q2  p2  w2   p2  p1  G  q2   p1  g  y  F  q1  q2  y  dy

(4.2)
(4.3)
0
Differentiating the right-hand sides of these expressions with respect to q1 and q2
again, we get


 2 E r q12   p1 f  q1  1  G  q2     g  y  f  q1  q2  y  dy  0
q2

0

 2 E r q2 2    p2  p1  g  q2   p1 g  q2  F  q1    g  y  f  q1  q2  y  dy  0
q2
0
 2 E r q1q2   p1  g  y  f  q1  q2  y  dy  0
q2
0
Hence,
D1   2 E r q12  0,
D2 
 2 E r
q12
 2 E r
q1q2
 2 E r
q2 q1
 2 E r
q2 2
0 .
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*):


q2

 p1  w1  p1 F  q1  1  G  q2    0 g  y  F  q1  q2  y  dy  0

q2

 p2  w2   p2  p1  G  q2   p1 0 g  y  F  q1  q2  y  dy  0


Now we study the properties the retailer’s ordering policy.
Proposition 2
(4.4)
(4.5)
(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
*
 p 1  F q*  w  p F q* G q*  p q2 g y F q *  q *  y dy  w
  1 2 






1
1
1
1
2
1 0
1
 1

*
q
 p2  w2   p2  p1  G  q2*   p1 2 g  y  F  q1*  q2*  y  dy  p2G  q2* 

0



Hence,
 p w 
q1*  F 1  1 1   q* ,
 p1 
 p  w2 
q2*  G  1 2
q
 p2 
*
.
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:
 s   w1  c1  q1*  w1 , w2    w2  c2  q2*  w1 , w2 
From (4.4) and (4.5), we obtain the inverse demand curves as:

w1  q1 , q2   p1  p1 F  q1  1  G  q2     g  y  F  q1  q2  y  dy
q2
0
w2  q1 , q2   p2   p2  p1  G  q2   p1  g  y  F  q1  q2  y  dy
q2
0
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.
(4.6)

(4.7)
(4.8)
Proof: (1)Taking partial derivatives of w1(q1, q2) and w2(q1, q2) with respect to q1 and
q2, we obtain


w1  q1 , q2   q1   p1 f  q1  1  G  q2     g  y  f  q1  q2  y  dy  0,
q2
0
w1  q1 , q2   q2   p1  g  y  f  q1  q2  y  dy  0
q2
0
w2  q1 , q2   q1  p1  g  y  f  q1  q2  y  dy  0,
q2

0

w2  q1 , q2   q2    p2  p1  g  q2   p1 g  q2  F  q1    g  y  f  q1  q2  y  dy  0
q2
0
(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:
 s   p1  c1  q1   p2  c2  q2  p1q1F  q1  1  G  q2     p2  p1  q2G  q2  
p1  q1  q2   g  y  F  q1  q2  y  dy
q2
(4.9)
0
In order to get more concrete results, we assume the demand distributions f(•)
and g(•) are uniform over (a,b) and (c,d), respectively.
Lemma 2:For a given q2, πs is quasi-concave with respect to q1; for a given q1, 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 q2<c, πs=(p1-c1)q1+(p2-c2)-p1q1F(q1). Thus, ∂πs/∂q1>0 where q1<a;
∂2πs/∂q12<0 where a<q1<b; ∂πs/∂q1<0 where q1>b. For c<q2, ∂πs/∂q1>0 where
q1<a+c-q2; ∂2πs/∂q12<0 where a+c-q2<q1<b; ∂πs/∂q1<0 where q1>b. Hence, for a
given q2, 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 given q1. We show the result in
Figure 4-1:
 s quasi - concave with respect to q 2
 s quasi - concave with respect to q1
b
a  c  q2
c
q1
(a)
Figure 4-1
d
(b)
q2
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
We use 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:
0
200
0
20
s
-200
q2
0
40
20
40
60
q1
60
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 of marginal 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 of retail 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 unreliable with
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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