Download 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
Xu Chen
School of Management, University of Electronic Science and Technology, P.R.China, 610054
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 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 OR/OM 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 OR/OM 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, product’s marginal
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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 1 and some basic denotation is defined bellow.
Supplier
2
Retailer
w2
2
p2
q2
1
D2
p1
w1
1
p1
D1
q1
Figure 1: The pricing and ordering model
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. We
define i=1 denote Product 1, the density and cumulative demand distribution are, respectively, f(x) and
F(X). We also define i=2 denote Product 2, the density and cumulative demand distribution are,
respectively, g(x) and G(X). On the basis of above, 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, which we assume is zero for each,
so they will choose to participate in the channel if only 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
Eπ r = ∫ p1 x f (x )dx + ∫
+∫
q2
0
∫
q1 + q2 − y
q1
q2
+∞
0
q2
p1 q1 f ( x )dx − w1 q1 + ∫ p2 yg ( y )dy + ∫
p1 ( y − q1 ) f (x )g ( y )dxdy + ∫
q2
0
∫
+∞
q1 + q2 − y
p2 q 2 g ( y )dy − w2 q 2
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 with respect to (q1, q2).
Proof: Taking partial derivatives of Eπr with respect to q1 and q2, we obtain:
q
(4.2 )
∂Eπ r ∂q1 = p1 − w1 − p1  F (q1 )(1 − G (q2 )) + ∫ g ( y )F (q1 + q2 − y )dy 
0
2


∂Eπ r ∂q 2 = p 2 − w 2 − ( p 2 − p1 )G (q 2 ) − p1 ∫ g ( y )F (q1 + q 2 − y )dy
(4.3)
Differentiating the right-hand sides of these expressions with respect to q1 and q2 again, we get
q
2
∂ 2 Eπ r ∂q1 = − p1  f (q1 )(1 − G (q 2 )) + ∫ g ( y ) f (q1 + q 2 − y )dy  ≤ 0
0
q2
0
2

∂ E π r ∂q 2
2
2

q2

= −( p 2 − p1 )g (q 2 ) − p1  g (q 2 )F (q1 ) + ∫ g ( y ) f (q1 + q 2 − y )dy  ≤ 0
0


∂ 2 Eπ r ∂q1 ∂q 2 = − p1 ∫ g ( y ) f (q1 + q 2 − y )dy ≤ 0
q2
0
Hence,
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∂ 2 Eπ r
2
D1 = ∂ 2 Eπ r ∂q1 ≤ 0 ,
2
D2 =
∂q1
∂ 2 Eπ r
∂q 2 ∂q1
∂ 2 Eπ r
∂q1∂q 2
≥0 .
∂ 2 Eπ r
∂q 2
2
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 + q 2 − y )dy  = 0 (4.4)

q
 p 2 − w2 − ( p 2 − p1 )G (q 2 ) − p1 2 g ( y )F (q1 + q 2 − y )dy = 0 (4.5)
∫0

Now we study the properties the retailer’s ordering policy.
Proposition 2
(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 q g ( y )F (q * + q * − y )dy ≥ w
1
1
1
1
2
1 ∫0
1
2
1
 1
*
2

q*2
 p2 − w2 − ( p 2 − p1 )G q *2 = p1 ∫ g ( y )F q1* + q *2 − y dy ≤ p 2 G q2*

0
( )
(
)
( )
Hence,
 p − w1 
 = q *′ ,
q1* ≤ F −1  1
 p1 
 p − w2 
 = q * .
q2* ≥ G −1  2
 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:
π s = (w1 − c1 )q1* (w1 , w2 ) + (w2 − c2 )q2* (w1 , w2 )
(4.6)
From (4.4) and (4.5), we obtain the inverse demand curves as:
q
w1 (q1 , q2 ) = p1 − p1  F (q1 )(1 − G (q2 )) + ∫ g ( y )F (q1 + q 2 − y )dy  (4.7 )
0
2


w2 (q1 , q 2 ) = p 2 − ( p 2 − p1 )G (q 2 ) − p1 ∫ g ( y )F (q1 + q 2 − y )dy
(4.8)
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.
Proof: (1)Taking partial derivatives of w1(q1, q2) and w2(q1, q2) with respect to q1 and q2, we obtain
q
q
∂w1 (q1 , q 2 ) ∂ q1 = − p1  f (q1 )(1 − G (q2 )) + ∫ g ( y ) f (q1 + q2 − y )dy  ≤ 0,
∂w1 (q1 , q2 ) ∂ q 2 = − p1 ∫ g ( y ) f (q1 + q2 − y )dy ≤ 0
0
0
2

2

q2
q2
∂w2 (q1 , q2 ) ∂ q1 = p1 ∫ g ( y ) f (q1 + q 2 − y )dy ≤ 0, ∂w2 (q1 , q2 ) ∂ q2 = −( p2 − p1 )g (q 2 ) − p1  g (q 2 )F (q1 ) + ∫ g ( y ) f (q1 + q 2 − y )dy  ≤ 0
0
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 w as equations (4.7) and (4.8) suggest, the wholesale price not only depends on its own quantities
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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, assume market demand is finite, 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:
π s = ( p1 − c1 )q1 + ( p2 − c2 )q2 − p1q1 F (q1 )(1 − G (q2 )) − ( p2 − p1 )q2G (q2 ) − p1 (q1 + q2 )∫0 g ( y )F (q1 + q2 − y )dy
q2
(4.9 )
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 2:
π s quasi - concave with respect to q 2
π s quasi - concave with respect to q1
b
a + c − q2
c
(a)
q1
d
(b)
q2
Figure 2 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 3:
0
200
ps
0
20
-200
q2
0
40
20
40
60
q1
60
Figure 2 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
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result is presented in Table 1~Table 3 and some characters and principles are drawn bellow.
Table 1 The impact of demand distribution
demand distribution
q1
q2
D1~U(0,100) D2~U(0,100)
24.26
34.79
D1~U(0,100) D2~U(50,100)
33.33
50.00
D1~U(50,100) D2~U(0,100)
29.32
40.57
Table 2 The impact of marginal profit
marginal profit
q1
q2
p1-c1=10, p2-c2=10
24.26
34.79
p1-c1=12, p2-c2=10
25.89
35.42
p1-c1=14, p2-c2=10
27.08
36.08
p1-c1=10, p2-c2=12
23.24
36.68
p1-c1=10, p2-c2=14
22.40
38.18
w1
10.45
10.00
18.74
w2
16.09
20.00
18.01
w1
10.45
11.53
12.62
10.50
10.55
w2
16.09
16.31
16.55
17.14
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.
retail price
p1=15; p2=20
p1=17; p2=20
p1=19; p2=20
p1=15; p2=22
p1=15; p2=24
Table 3 The impact of retail prices
q1
q2
24.26
34.79
17.49
39.86
9.03
48.01
26.33
30.56
27.74
27.30
w1
10.45
12.68
15.09
10.35
10.28
w2
16.09
16.27
16.51
17.95
19.85
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 while 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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