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Channel Coordination with
Price-Quality Sensitive Demand and
Concave Transportation Cost
(Working paper of Choi, Lei, Wang, and CX Fan, 2004)
We analyze the impact of channel coordination on supply chain profitability for a single-product supply process involving a supplier, a buyer and a transporter with concave transportation cost functions. The buyer purchases the product from the supplier and then sells in the market. The market demand is assumed to be sensitive to the buyer’s selling price and the supplier’s (or manufacturer’s) product quality. We establish the dominance relationship between the profitability achieved by a totally centralized coordination and the sum of individual partner’s maximum profitability in a decentralized business environment. The effect of transporter’s coordination is analyzed. Policies to jointly optimize the buyer’s market selling price, the supplier’s quality level, and the shipping quantities handled by the transporter are proposed. Empirical observations that show the improvement on supply chain profitability by using the optimal policies assuming a market demand function that is decreasingly convex to the buyer’s (retailer’s) selling price and increasingly linear to the supplier’s (manufacturer’s) product quality are reported.
1. Introduction
As supply chain management captures the attention of more and more top level executives, companies are changing their ways of interacting, collaborating, sharing the information, and making decisions (O’Reilly, 2002). One important category of business decisions influenced by such changes is channel coordination policies that guide collaborative operations of partners involved in a supply chain process. Effective supply chain management necessitates a strong collaboration of participating partners and a solid implementation of strategies to optimize the total profitability of the chain as a whole entity. Among numerous examples of practical needs for supply chain coordination is the operational problems handled by British Airway Catering (BAC). BAC is responsible for delivering about 44 millions of meals per year prepared by hundreds of third party
catering partners. Just in terms of London Kitchen alone, about 250 tons of chicken and 73 tons of eggs are prepared for BAC on annual basis. Together with finished meals, BAC also manages to purchase many other non-food items such as crockery, glassware, plastics, blankets, non-perishable dry foods, and others. Every time a jumbo jet takes off, about 40,000 items are pulled through the BAC supply chain. Any lack of coordination and integration can result in a serious problem in British Airway operations, either a delay in meeting passenger needs, overstock in expensive airport inventories, huge cost due to perishable foods, and loss in profitability (Christopher, 1998).
Buyer-supplier coordination has received a significant attention of researchers during the past two decades. Primary results in this regard can be found in the work by Goyal (1976, 1988), Monahan (1984), Lee and Rosenblatt (1986), Banerjee (1986), Goyal and Gupta (1989), Benjamin(1990), Anupindi and Akella (1993), Kohli and Park (1994), Lau and Lau(1994), Weng (1995,1997), and several of others, etc. These initial work have produced many valuable insights into the development of mechanisms to ensure the implementation of coordination policies in practice, and have built a foundation for later studies in the area. There have also been several important extensions recently. One is by Corbett and Groote (2000). They studied the optimal quantity discount policies under asymmetric information about the buyer’s cost structure, and compared the policy to the situation where the supplier has full information. Cheung and Lee (2002) analyzed the joint impact of shipment coordination strategy (between buyer and supplier) and stock rebalancing strategy to maximize the joint profit of supplier and buyer. The work is a very first one to study the joint effect of the two major supply chain strategies. Chen, Federgruen and Zheng (2001) generalized existing channel coordination models with identical buyers and developed effective mechanisms for managing a distribution process involving one supplier and multiple non-identical buyers. There have also been a number of papers published for the channel coordination with stochastic demand. A comprehensive review for this area can be found in the work of Cachon (2001).
In section 2 of the paper, we develop the profitability functions for individual partners involved in a supply chain process. We prove that the yearly profit of the transporter can be improved, even in a decentralized/independent business environment, if his/her fixed and variable operation cost can be incorporated into the buyer’s consideration on ordering quantities. We also discuss the conditions required to achieve this improvement. In section 3, we analyze joint optimal policies for all the three partners that maximize the overall (total) profitability of a supply chain. We show that the willingness of the transporter to coordinate and to share the profitability with other partners have a significant impact on total supply chain profitability. In section 4, we derive optimal selling prices, quality level, and shipping quantity under a decentralized, and a totally centralized, business environment, respectively, assuming the market demand is a decreasing convex function of buyer’s selling price and a linear increasing function of the supplier’s product quality. In section 5, we report our empirical observations on the impact of transporter’s cost parameters on economic ordering quantities and supply chain profitability under different coordination environment. Finally in section 6, we discuss future research extensions.
2. Models for individual partner’s yearly profitability
Our analysis is based on a supply process with a supplier (a manufacturer or a purchaser), a buyer (a retailer or a distributor who directly faces the market demand), and a third party logistics partner or a transporter who transports the shipment from the supplier to the buyer. We assume that the operation costs, including set up cost per order and the holding cost (per unit per year) incurred to all the partners are known. In addition, we assume the market demand to the product is deterministic, continuous, decreases as buyer’s (retailer’s) market selling price x increases, and increases as supplier’s product quality q improves (or say increases as the supplier’s cost on quality improvement increases). We shall use D ( x,q ) to denote this continuous price-quality sensitive demand.
Notations used in our analysis are summarized as follows:
x : The buyer’s (or the retailer’s) unit selling price to the market; p : The buyer’s unit purchasing price ( p < x ) from the supplier; q : The product quality level (0< q< 1), and q (^) ⇒1 if the product is of top quality in the respective market; c : The supplier’s variable cost of manufacturing( c < p ); u : The supplier’s budget on quality for each unit of product, u = ϕ( q );
T ( p ) : The transporter’s unit shipping (or mileage) cost, as an increasing function of p ; g : Shipping rate quoted by the transporter, g > T ( p ) ; k : The transporter’s profit per unit of shipment, as a constant component of g ; Sb : The buyer’s fixed ordering cost per order (e.g., fixed cost per truck); Ss : The supplier’s fixed processing and set up cost per order; St : The transporter’s fixed cost per order; Hb : The buyer’s unit holding cost per year; Hs : The supplier’s unit holding cost per year; Ht : The transporter’s unit holding cost for inventory-in-transit per year; Q : The order size (or the shipping quantity) per order; D ( x,q ): The annual market demand, as a function of x and q.
We assume that the supply process is a free-on-board(FOB)-destination process which requires the supplier to pay for the shipping cost. This implies that the supplier’s selling price p includes a variable production cost c , a unit product quality cost u (as one of the decision variables of the model), the shipping rate g , and the profit that the supplier may want to charge from his/her sales to the buyer, or c + u + g < p < x ,where x stands
for the buyer (or retailer)’s market selling price. Given the assumptions, the yearly profitability of the supplier, the transporter and the buyer can be represented as
Supplier: Π (^) s ( p , q )=( p − c − u − g ) D ( x , q )− SsD ( x , q ) Q − HsQ 2 (1)
Transporter: Π (^) t ( g )=( g − T ( p )) D ( x , q )− StD ( x , q ) Q − HtQ 2 (2)
Buyer: Π (^) b ( x )=( x − p ) D ( x , q )− SbD ( x , q ) Q − HbQ 2 (3)
Π (^) b ( x | Qb )=( x − p ) D ( x , q )−[ 2 SbHbD ( x , q )]^12. (3a)
For any given supplier’s price p and quality level q , the buyer’s interest is to choose his/her optimal market selling price x * b^ ( p , q )that together with
Qb ( p , q ) = 2 SbD ( xb *( p , q ), q ) H b
maximize (3a). Given x * (^) b^ ( p , q )and Q (^) b ( p , q ), transporter’s yearly profit becomes
Π (^) t ( g )=( g − T ( p )) D ( xb *^ ( p , q ), q )−( St Sb + Ht Hb )[ SbHbD ( xb *( p , q ), q ) 2 ]^12 (2a)
and the supplier’s yearly profit becomes
( , ) ( ) ( *^ ( , ), ) ( )[ SH D ( x *( p , q ), q ) 2 ]^12 H
H
S
p q p c u g Dx pq q S b b b b
s b
Π (^) s = − − − b − s + (1a)
Since the supplier has a complete information about the buyer’s Q and , he/she
can always optimize the values of p and q to maximize
b x^ * b^ ( p , q ) Π (^) s upon a given demand function
D(x,q). The maximum supply chain profit that may be possibly achieved under such an independent (decentralized) business environment can now be represented as (see Choi, Lei and Wang, 2002, for details)
*^12
2 [ 1 ( )/ 2 ][ ( ( , ), ) 2 ]
SS S HHH SH Dx pq q
x pq c u T p Dx pq q s b t s b t b b b
s t b b b − + + +^ +
Note that (4) remains unchanged regardless the contract between the supplier and the buyer is based on a FOB-destination agreement (the supplier’s expense on shipping cost) or a FOB-origin agreement (the buyer’s expense on shipping cost).
Example 1. Consider an independent business environment with Sb =$200, St =$600,
Ss =$4000, Hb =$25, Hs =$20, Ht =20, d = 2 × 1010 , c =$40. Suppose D ( x , q )= dq x^2 ,
a =10%, k =$8, pb *^ =$ 126 , qb *^ = 0. 874 , g = k + ap + St Q + HtQ [ 2 D ( x , q )]and
(we shall introduce the formulas in Section 4 to compute that
maximize the supplier’s, and the buyer’s, yearly profit in an independent business environment, respectively), then we have
xb *^ =$ 252 p * b^ , q * b , x * b
Π * b xb *^ − p * b ) D ( xb *^ , qb *)− [ 2 SbHb D ( xb *^ , q * b )]^12 =34.694× 106 , Π * t g − apb * ) D ( xb *^ , q * b )− ( St S 2 ] 12 = 2.
,
, ) ) ( 6
q * b^ − ub * Dx
52. 189 × 106.
15.289 10
pb − c − g pb
Π * t^ +Π* b = $
Π * s
Π * s
Qt ( p , q )
21 [( S^ tHb SbH^ t^ )^12 +^ (^ Sb ( St Sb Ht ) 12 + (
= ( (^) b + Ht Hb )[ SbHbD ( x * b^ , qb *) 206 × 106 , = ( (^) b *^ , q * b )−( Ss Sb + Hs Hb )[ SbHbD ( xb *, qb *) 2 ]^12
and +
Now, consider the following results:
Lemma 1 : With the buyer’s economic ordering quantity, Q , and the transporter’s
profit/cost structure (2a), the transporter’s yearly profit is never higher than the maximum (or the transporter’s fixed plus holding cost is never lower than the minimum) that can be
achieved by the transporter’s own ETQ,
b (^ p , q^ )
Qt ( p , q )= 2 StD ( xb *^ ( p , q ), q ) H t.
Proof : We prove the correctness of the claim by showing that the transporter’s operation (holding plus fixed processing) cost under Q is no less than that under Q With
, the transporter’s operation cost is
b t (^ p , q ). 2 S (^) t HtD ( xb *^ ( p , q ), q ) and with ,
this costs becomes
Qb ( p , q )
H (^) t StHb ) 12 ] 2 StHtD ( x * b ( p , q ), q )
Since , the
claim holds. g
Hb SbHt StHb ) 12 ≥ 2 ( StHb SbHt )^14 ( SbHt StHb )^14 = 2
Lemma 2 : With the buyer’s economic ordering quantity Q , the supplier’s yearly
profit is never higher than the maximum (or the supplier’s fixed plus holding cost is
b (^ p , q^ )
The discussion above is sufficient to lead to a conclusion that, in an independent business environment, each individual partner makes a maximum yearly profit that is no more than that can be achieved by his/her own economic ordering quantity, regardless the supply chain ordering quantity is decided by the buyer or the supplier. This conclusion also implies that the sum of maximum yearly profit achieved by individual partners in an independent business environment, Π (^) s +Π t +Π b , is a lower bound of that may possibly
be achieved by a centralized channel coordination, as we will discuss in Section 3.
As a motivation to our analysis on centralized channel coordination in next section, the following result shows that the transporter’s yearly profitability (2a) may be improved even in an independent/decentralized business environment if the buyer incorporates, in his/her decisions on ordering quantity, the operation costs incurred to all the other partners in the supply chain. Let S (^) J = Ss + Sb + St , H (^) J = Hs + Hb + Ht ,
and QJ = 2 S (^) JD ( x , q )/ HJ.
Lemma 5 : In an independent business environment, for any given p , q , g and x , if
b
s t b
s t H
H H
S
S + S ≥ + and J
J b
b t
t H
S
H
S
H
S (^) ≥ ⋅ , then ( | ) ( | ) Π (^) t g QJ ≥Π t g Qb , where
( | ) ( ( )) ( , ) ( )[ S H D ( x , q ) 2 ]^12 H
H
S
g Q g T p Dxq S Π (^) t J = − − Jt + Jt J J ,
and ( g | Q ) ( g T ( p )) D ( x , q ) ( SS HH )[ SbHbD ( x , q ) 2 ]^12 b
t b
Π (^) t b = − − t +.
Proof : The claim Π (^) t ( g | QJ )≥Π t ( g | Qb )holds if and only if
( + )^2 −( + )^2 J J ≥ 0. J
t J b b t b
t b
t (^) S H H
H
S
S H S
H
H
S
S
Since
[ ] [ ]
( ) ( ) [ ] [ ]
2 2
2 2 2 2 2 2
b
b J t J J
J b t b
bt bt b b Jt Jt J J tb b tb b tJ J tJ J
H
S
H
H S
S
H
S
S H
SS HH S H SS HH S H SS H HH S SS H HH S
b
s t b
s t H
H H
S
S + S ≥ + and J
J b
b t
t H
S
H
S
H
S (^) ≥ ⋅ , we have
[ ][ ( ) ( ) ] 0.
[ ] [ ]
2 2
2 2
btJ bt J b s t s t^ b
b
b J t J J
J b t b
SSS HHH H S S H H S
H
S
H
H S
S
H
S
S H
g
Among the two conditions, b
s t b
s t H
H H
S
S + S ≥ + and J
J b
b t
t H
S
H
S
H
S (^) ≥ ⋅ , we see that
b
s t b
s t H
H H
S
S + S ≥ + can be easily justified since S (^) t + Ss >> Sb , Ht ≈ Hs ≈ Hb hold for
most applications. However, condition J
J b
b H
S
H
S ⋅
t
t H
S (^) ≥ holds only for the situations
where the fixed cost of the transporter is relative higher than the fixed cost of the buyer.
In particular, we can show that the condition J
J b
b t
t H
S
H
≥ S^ ⋅
H
S (^) holds whenever
S (^) b ≤ (^) m^1 +^1 St , and S (^) t ≥ (^) m^1 ( Sb + Ss ), where parameter m ≥ 1 can be any positive integer.
Figure 1 below, as an example, shows the improvement on transporter’s yearly profit when the buyer’s ordering quantity changes from Q to Q. In this example, the
transporter’s yearly profitability increased by [(0.65-0.48)/0.48] % or 35% after the buyer changes his/her ordering quantity from Q to Q. In practice, this change in the buyer’s
decision, if can be implemented, could offer a strong incentive to the transporter which in turn may lead to a better coordination and better logistics services.
b J
b J
J J J H Q = 2 S D ( x , q ) (6)
which maximizes the total supply chain profitability (5) at
Π (^) J ( x , p , q | QJ )=( x − c − u − T ( p )) D ( x , q )−[ 2 SJHJD ( x , q )]^12 (5a)
for any given supplier’s p, q , transporter’s g , and buyer’s price x.
Theorem 1. (Dominance relationship) For any given p , q , g and x , the total supply chain profitability under QJ dominates the sum of individual partners’ maximum yearly profit in an independent business environment using ordering quantity Qb .That is
Π (^) J ( x , p , q | QJ )≥Π s +Π t +Π b.
Proof : From (1a), (2a) and (3a), we have
2 [ 1 ( )/ 2 ][ ( , ) 2 ]^12
SS S HHH SHDx q
x c q T p Dx q s b t s b t b b
s t b
− + + +^ +
ϕ
Since
2 [ 1 ( )( ) ]^12 [ ( , ) 2 ]^12
SS S S SSHH H HHH SHDx q
x c q T p Dx q
x pq Q
s b t s tb bs t s b t b b
J J
− + + + + + +^ +
ϕ
and
[ 1 +( + + + )/ 2 ]^2 −[ 1 +^ + +( + )(^ + )+ + ]≥ 0 b
s t b b
s t s t b
s t b
s t b
s t H
H H
SH
S S H H
S
S S
H
H H
S
S S ,
the claim holds. g
In a very similar way, we can prove the correctness of the following result.
Theorem 2 : (Generalized dominance relationship) For any given p , q , g and x , we have Π (^) J ( x , p , q | QJ )≥Π s +Π t +Π b
regardless the ordering quantity Q for the independent business environment is decided by which partner, or say regardless Q = Qs , Qb , orQt.
Theorem 3 : The total supply chain profitability achieved under joint ordering quantity Q (^) J , Π* J^ ( p ), increases as the supplier’s unit selling price p decreases.
Proof : Assume that p 1 (^) < p 2 , then
1 * 1 1 * 1 * 1 * 1 * 1 12
1 * * * 1 ( ( ) ( ) ( )) ( ( ), ( )) [ 2 ( ( ), ( ))]
x p c u p T p Dx p q p S H Dx p q p
p x q p J J J J J J J J
J J J J = − − − −
2 * 2 2 * 2 * 2 * 2 * 2 12
2 * * * 2 ( ( ) ( ) ( )) ( ( ), ( )) [ 2 ( ( ), ( ))]
x p c u p T p Dx p q p S H Dx p q p
p x q p J J J J J J J J
J J J J = − − − −
( )
( ( ) ( ) ( )) ( ( ), ( )) [ 2 ( ( ), ( ))]
( ) ( ( ) ( ) ( )) ( ( ), ( )) [ 2 ( ( ), ( ))]
p
x p c u p T p D x p q p S H Dx p q p
p x p c u p T p D x p q p SH Dx p q p
J
J J J J J J J J
J J J J J J J J J
= Π
− − − −
Π ≥ − − − −
g
Theorem 3 reveals the importance of supplier’s coordination. If the supplier is willing not to charge a profit from his/her sales to the buyer, then the total supply chain profitability will be higher than otherwise they may achieve. This observation also leads to the following result.
Corollary 1 : In a joint channel coordination environment, for any given g , the supplier’s optimal unit selling price p * J are given by Π (^) s ( p * J^ )= 0 or
p * J = c + u * J + g + Ss / QJ + HsQJ /[ 2 D ( x * J , q * J )]
In addition, we have
Theorem 4 : The total supply chain profitability achieved under joint ordering quantity
Q (^) J , Π* J^ ( g ), increases as the transporter’s shipping rate g decreases.
Proof: Assume that g 1 (^) < g 2 , then
( ) / /[ 2 ( , )],with 0
/ /[ 2 ( , )]
J J t J t J J J J
J J J s J s J J J g T p S Q HQ Dx q k
p c u g S Q HQ Dx q (7)
In next section, we shall focus our analysis on the optimal operation policies when the market demand is a continuous decreasing convex function of buyer’s selling price x and a linear increasing function of supplier’s product quality q , and when the shipping rate g is a continuous function of demand D ( x,q ).
4. The impact of channel coordination under D ( x , q )= x^ dq 2
In this section, we develop optimal operation policies, in terms of selling prices, product quality and ordering quantity, that together maximize the total profitability of a supply chain, assuming that the market demand has a structure of D , where parameter d >> 1 stands for the market factor. Furthermore, we assume that the supplier’s product quality q can be modeled as a continuous increasing function of u ,
( x , q )= dq / x^2
1 q = e − u^ < q < where u stands for the dollar amount the supplier invests to ensure the quality of each unit of product. We assume the transporter’s shipping rate g to be
g = k + T ( p )+ St Q + HtQ [ 2 D ( x , q )], T ( p )= ap
or ] 2 ( , )
[
SHDxq
g = k + ap + St^ H + HtS
where k is a constant representing transporter’s profit charge, ap stands for the unit shipping operation cost, and S and H are dependent of the ordering quantity Q.
4.1. Optimal policies under a decentralized environment
In an independent/decentralized business environment, the buyer applies and executes an order quantity that maximizes his/her own profitability for any given p and q.
That is Qb = 2 S (^) bD ( x , q ) Hb and
Π b ( x )=( x − p ) D ( x , q )− SbD ( x , q ) Q − HbQ 2
u u b b u
b b dxe dpx e dSxH e
x S H dq x x p dq 2 21 1 1 2
1
2 2 [ 2 ]
= − − − −^ −
Then, the following results hold.
Theorem 6. For an independent business environment where the market demand is
approximated by ( x , q )= xdq 2 ,
b
D the buyer’s optimal selling price that maximizes his/her
yearly profitability Π is
p dq
S H
x b b
b ⋅ −
for any given supplier’s selling price p and product quality level q.
Proof. Let 2 [^2 ]^20
1 2
(^112) 3
1 =− 2 + + = Π (^) b −^ u − u b b e − u x
e dS H x
e dp x
d dx
d (^) , we have
p dq
x S H
de
S H
p de dS H e
x dpe b b b u
u b b u b b
u b ⋅ −
−
− −
−
1 2
[ 2 ]
(^21) 1 12 1
1
From the analysis above, we can also see that when the supplier has a top quality in the market (i.e., q and when the supply chain’s market share is sufficiently high
(i.e., , then the buyer’s optimal selling price can be approximated by
d >> Sb Hb )
(^122112)
(^122112)
(^12211212)
(^122112)
( 1 ) [ 1 ( ) ( ) )[ 2 ]
( 1 )( ( 2 ) ) [( ) ( ) )[ 2 ]
u s t b s t b b b
u b b
u b b s t b s t b b b
u b b
ad e a S S S H H H SH
c u k d e SH
a d e SH S S S H H H SH
p c u k d e SH
−
−
−
−
Now let ∂∂Π us^ = 0 and then solve for p again
[( ) ( ) )[ 2 ] 4 0
( )( (^2 ) )
[ ( 2 ) ]
2
12 12 21
2 2 2 12 21
12 21 1 2
(^1221122)
−
− − −
pu S S S H H H SH d e
u d e pSH p c u k ap d e puSH d e u s t b s t b b b
s u b b u b b u
this gives
21 12 12
21 12 12 21 122 2 21 2 ( 1 )( ( 2 ) ) [( ) ( ) ][ 2 ]
2 ( )( ( 2 ) ) 2 [ ( 2 ) ]
u b b s t b s t b b b
u b b u b b u a de dSH S S S H H H dSH
p c u k de dSH d e SH ue − − − + + +
−
− − (10)
Equations (9) and (10) together define the following relationship
( 1 )[ ( )]^22 [ 1 ( ) ( ) ][ 2 ]^120 2 1 − a u − u − c + k e − u − u − a + Ss + St Sb + Hs + Ht Hb SbHb d =
Since d >> Sb Hb , the relationship above can be approximated by
u^2 − u −( c + k )= 0
or u * (^) b^ = 1 +^1 + 24 ( c^ + k ) (11)
Equation (11) implies that when c >>1, u (^) b *^ ≈ c + k , and qb e c + k
−
. g
Results (8a), (8b), and (11) provide guidelines for the optimal operational policies that the buyer and the supplier should follow, for a given shipping rate g , to maximize their own yearly profitability in an independent/decentralized business environment. In
next section, we shall switch our analysis to the optimal operational policies in a totally coordinated environment.
4.2 Optimal policies under a centralized (total coordinated) environment
When all the partners (the supplier, the buyer, and the transporter) are willing to coordinate their operations to optimize the total supply chain profitability, the optimal
joint ordering quantity becomes QJ =( 2 SJD ( x , q ) HJ )^12 which maximizes the supply
chain yearly profitability at
u u J J u
J J J dxe c u ap xd e dSxH e
x S H dq x x c u ap dq
2
1 12 1 2
1
2 2
( ) [^2 ]
= − − + + − −^ −
Let ( )^2 [^2 ]^20
1 2
(^112) 3
1 ∂ =−^2 + + + + =
∂Π (^) J −^ u − u J J e − u x e dS^ H x e c u ap^ d x
d x , we obtain
u
u J J u J J
u J
de
S H
c u ap de dS H e
x c u apde (^21)
1 (^12) 1
1
1 2
[ 2 ]
−
− −
−
−
This leads to the following result (proof skipped).
Theorem 8. If the market demand can be approximated by D ( x , q )= x^ dq 2 , then the
optimal market selling price that maximizes the joint supply chain profitability (12) is defined by
dq
S H
x c u ap J J
J 2
*^2 ( )
= +^ + (13)
for any given supplier’s decision on product quality u and q.
