Download Efficiency of Independent Downstream Firm Could Counteract

Efficiency of Independent Downstream Firm Could Counteract
Coordination Difficulties
Richard K. Cho
Department of Management Sciences, University of Waterloo
Waterloo, Ontario, N2L 3G1, Canada
[email protected]
Yigal Gerchak*
Department of Industrial Engineering, Tel-Aviv University
Ramat-Aviv, Tel Aviv 69978, Israel
FAX: 972-3-640-7669, [email protected]
October, 2001
Abstract
Outsourcing and decentralization are popular in large part due to the recognition that
independent agents are more efficient and effective than corresponding divisions of vertically
integrated firms. Yet supply chain coordination models implicitly assume that cost and other
parameters are unaffected by decentralization, thus rendering the integrated firm a systemoptimal benchmark. A case in point are independent downstream firms vs. corresponding
divisions of integrated firms, where the former's production or operating costs are likely to be
lower. We examine the effect of reduction in these costs on lot sizes, wholesale prices and
profits. We endow Lariviere and Porteus' (2001) "selling to the newsvendor" and Pasternack's
(1985) "optimal pricing and return policies" models with downstream (possibly non-linear)
operating costs and examine the consequences. We also explore a downstream firm's optimal
investments in reducing its operating costs within a coordinated decentralized system.
*
Corresponding author.
Efficiency of Independent Downstream Firm Could Counteract
Coordination Difficulties
1.
Introduction
The emerging literature on coordination in decentralized supply chains (e.g. Cachon 1999,
Lariviere 1999) treats the downstream firm (e.g., a retailer) as a costless operation, other than the
cost of mismatches between demand realization and product availability (Lariviere and Porteus
2001; "L&P"). In practice, of course, retailers do incur various operating costs - rent, utilities,
labor etc. - many of which are essentially independent of the amount actually sold; they depend
more on the amount ordered, as that dictates the scale of the operation. Moreover, the
downstream firm may actually perform some finishing or assembly operations on the product,
some of which need to take place before demand is realized. As such, it would be of interest to
examine how the presence of these costs affects coordination issues and profit split between
manufacturer and retailer or supplier and buyer (we shall use the "manufacturer-retailer" and
"operating costs" terminology for concreteness). Note that while for the retailer the operating
costs play a similar role to that of the wholesale price, the manufacturer does not receive them,
so the scenario is more general than that discussed by L&P and Lariviere (1999). Of particular
interest to us is the effect of retailer's increased efficiency, one of the manifestations of which are
reduction in operating costs, on scale and profits.
A key motivation for the recently widespread phenomena of outsourcing and
decentralization is that independent specialized firms are more efficient and effective than
corresponding divisions of a large corporation. Independent retailers can forecast their demands
better, and thus replenish more wisely, deploy the appropriate sales effort to increase demand,
1
and be more diligent in controlling their operating costs. Independent suppliers can produce
higher quality at lower cost. Assembly plants can be more efficient if their owners/managers
focus solely on the assembly operations. But it is well known that decentralization brings about
tricky coordination problems. These problems have been extensively discussed in the emerging
literature on coordination in decentralized supply chains, and various mechanisms for mitigating
the adverse impacts have been proposed. But an implicit assumption in this literature is that the
cost, quality and other parameters pertaining to each agent are unaffected by (de)centralization.
A centralized/integrated firm is then, by definition, the most efficient arrangement, which serves
as benchmark for decentralized systems. But if divisions of the centralized firm are less efficient
than spun-off firms, that has to be traded off against the costs imposed by coordination
difficulties in decentralized chains. We consider these tradeoffs within several stylized
manufacturer-retailer models with uncertain demand.
This paper uses the framework of L&P's "Selling to the Newsvendor: An Analysis of PriceOnly Contracts" to examine the effect of operating costs on order quantities, wholesale price, and
expected profits of both retailer and manufacturer. We first review L&P’s model, and then
generalize it to include retailer's operating costs, non-linear production costs and individual
rationality for both parties. Basic properties are shown to remain unchanged and additional ones
established. It is proved that as retailer's efficiency improves, the optimal quantity increases, and
the retailer's and manufacturer's profits improve. While wholesale price also increases in
retailer's efficiency for constant manufacturing cost, it might not be so for general non-linear
manufacturing cost. We then generalize the operating costs to non-linear, and show that the
quantities increase in these costs for convex operating costs, but may decrease for highly
concave costs. We then examine a scenario with returns (buybacks). Pasternack (1985; see also
2
Lariviere 1999) proved that judicious use of a return policy by a manufacturer can coordinate the
channel. We show that with a linear production cost and a full return policy the ratio of buyback
price to wholesale price declines as retailer's efficiency increases.
Until this point the operating cost were assumed to be exogenous, and thus did not constitute
a decision variable. Such operating cost may, however, be a function of investments made and
as such be one of the important decision variables. The paper thus goes on to scenarios with
controllable operating cost. We show that the retailer has less incentive to invest in reducing
operating cost in a decentralized system than in an integrated system.
2.
Literature Review
There are several streams of literature that analyze coordination problems for a two-member
vertical channel. For each stream, the environmental setting is different, especially in regard to
demand pattern.
Jeuland and Shugan (1983) studied two vertically related firms with deterministic demand.
They considered a retailer's efficiency as affecting the retail price and thus the demand. A
quantity discounts policy was introduced to coordinate the channel and reach global optimality.
Corbett and de Groote (2000) analyzed a single supplier and single buyer model with EOQ
(Economic Order Quantity)-type ordering. The demand is deterministic and known to both
parties. The main variable of the system is buyer's holding cost, which is similar to what we call
retailer's operating cost. They investigated the effect of buyer's holding cost, which was either
known or unknown to the supplier, on the profit of both parties, and developed a quantity
discount policy to coordinate the system. When the buyer's holding cost was unknown to the
3
supplier, they assumed a distribution with a decreasing reverse hazard rate. Corbett and Tang
(1999) analyzed channel coordination with price-sensitive demand and where buyer's internal
cost is unknown to the supplier. They found that cost efficiency of the buyer raises the wholesale
price, or reduces the subsidy offered by a supplier to a retailer.
Pasternack (1985) uses the newsvendor-type demand with a single manufacturer and
retailer, and the main research issue is a channel coordination using a return policy. He shows
that if the wholesale price paid by a retailer to the manufacturer and the credit for a returned unit
are related by a particular linear relation the channel will be coordinated. An increase in credit
for returned products actually worsens the retailer's profit because it raises the wholesale price
and maintains the same order quantity. Lariviere (1999) and L&P (2001) investigated the optimal
wholesale price and resulting quantity in the two-member vertical supply chain and a wholesaleprice-driven contract. We will discuss their research in detail in the next section. Lariviere (1999)
developed contracts for coordinating a decentralized supply chain such as quantity discount and
quantity flexibility. Atkinson (1979) introduced an incentive scheme based on the difference
between an owner's request and a manager's actual decision. His scheme encourages the manager
to develop better knowledge of end-user demand, especially when demand is uncertain.
Most research using newsvendor-type demands has not included a retailer or buyer
efficiency factor and hence an integrated or centralized system is always preferred. While a
return policy is considered one of the best coordinating tool for decentralized system, it may
discourage effort of retailers to increase their efficiency. On the other hand, a quantity discount
policy encourages retailers to order more, but the effect of retailer's efficiency on the whole
supply chain is still uncertain. Refer to Tsay, Nahmias and Agrawal (1999) and Anupindi and
Bassok (1999) for discussion of relevant supply contracts.
4
Porteus (1985) analyzes the effect of investment in reducing setup time on the EOQ model.
He uses two types of investment cost functions, logarithmic and power. Leschke (1996) verifies
the realism of those two investment functions via empirical research.
3.
Problem Setting in L&P
The following notation was used in L&P and will be used here.
ξ: random consumer demand for the product, with cumulative distribution Φ (ξ) with Φ (ξ) ≡
1− Φ (ξ), density function φ(ξ) and a finite mean µ.
c : manufacturer's constant marginal production cost per unit,
r : retail price per unit (given exogenously),
w : wholesale price per unit offered by a manufacturer to a retailer,
y : stocking level or order quantity determined by a retailer.
We use the superscripts I, M and R to denote the integrated system, the manufacturer and
the retailer respectively. Without loss of generality, we assume the salvage value to be zero.
Then the expected profit of an integrated firm stocking y units is:
y
Π I(y) = (r − c)y − r ∫ Φ ( ξ) dξ = − cy + r ∫ Φ (ξ)dξ
0
y
0
The problem is concave in the stocking level, and the optimal solution is given by:
yI = Φ
−1
(c / r ) .
For the decentralized system, one assumes a wholesale-price-only contract where the
manufacturer offers the good at a per unit wholesale price w, and the retailer then decides how
5
much to order. The problem is:
Max Π M ( w ) = ( w − c ) y( w )
w
y
s.t. y ( w ) = arg{max Π R ( y ) = − wy + r ∫ Φ ( ξ) dξ} .
y
0
(3-1)
The optimal solution, y*, satisfies:
w ( y*) = r Φ ( y*) , and
 y * φ( y*)  c
Φ ( y*)1 −
= .
Φ ( y*) 

 r
(3-2)
By introducing the generalized failure rate,
g(ξ) = ξφ(ξ) / Φ (ξ)
(3-3)
of the demand distribution, L&P defined the IGFR (Increasing generalized failure rate)
condition, g′
(ξ) > 0. Under IGFR condition, which holds for most demand distributions such as
normal, exponential, uniform, gamma, logistic and Weibull, the system has a unique solution for
w and y. They also provide an economic interpretation of IGFR. Since g(ξ) is the reciprocal of
price elasticity of the retailer's order quantity, IGFR means that the price elasticity decreases in
order quantity, which is similar to typical price-quantity curves in economics. L&P also showed
that the result could be easily generalized to situations with positive salvage value or positive
stock-out penalty.
4.
Extension of L&P
We extend the model of L&P as follows.
The manufacturer produces a quantity y of the good at total cost TC(y); We assume that
6
TC(0) = 0. The marginal cost, MC(y), equals TC′
(y) and we assume that MC(y) > 0, MC′
(y) < 0,
and MC′
′
(y) > 0, which means that MC(y) is decreasing convex as there are scale economies in
production. The main new ingredient is that the retailer incurs operating costs, denoted by h, per
unit ordered in each period, whether that unit ends up being sold or not. We can think of the
quantity-based operating cost as consisting of transportation cost, store area, number of sales
clerks required, etc. If downstream firm performs some production, assembly or finishing
operations, h will reflect these activities' costs as well. Later we shall allow these operating costs
to be non-linear, but for now they are assumed to be proportional to the number of units
ordered/finished ("constant").
We restrict this supply chain to wholesale price only contract as did L&P. The manufacturer
offers the good at constant wholesale price w, which is between its average unit production cost,
TC(y)/y, and the retailer's net sale price, r − h, and the retailer selects the order quantity y and is
responsible for excess or shortage of stock. The manufacturer wishes to choose a wholesale price
that maximizes her profit under the condition of reserving a profit, Π 0R , for the retailer, without
which it will not buy from this manufacturer (referred to as individual rationality (IR) in the
incentives literature; e.g., Kreps 1990, Salanié 1997). The manufacturer has a similar
requirement, say Π 0M .
Then the problem becomes:
Max Π M ( w ) = w ⋅y ( w ) − TC ( y ( w ))
w
y
s.t. y ( w ) = arg{max Π R ( y ) = − ( w + h) y + r ∫0 Φ (ξ)dξ} .
y
Π R ( y ) ≥ Π 0R
(IRR),
Π M ( w ) ≥ Π 0M
(IRM),
y ≥ 0 and w ≥ 0
7
(4-1)
For a given w, the retailer can obtain its optimal order quantity by the first order condition
corresponding to (4-1). The optimality condition for y and the corresponding retailer's profit are:
r Φ ( y) = w( y) + h ,
(4-2)
and
y
Π R ( y ) = r ∫ ξφ(ξ)dξ .
(4-3)
0
Π R(y) is increasing in y and thus y should be large enough so that the optimal retailer's profit
would be at least Π 0R . Since
∞
∫0 ξφ(ξ)dξ = µ , there is no feasible solution if
Π 0R ≥ rµ. Otherwise,
R
there must be a unique y min
satisfying
R
Π R ( y min
) = Π 0R .
(4-4)
Without considering IRR, the optimality condition for y is as follows, where MR is the
marginal increase in order quantity, in other words, the marginal revenue for r =1:
MR(y) = [MC(y) + h] / r
(4-5)
MR(y) ≡ Φ ( y ) − yφ(y) = Φ ( y ) ( 1 − g(y)).
(4-6)
where
Since φ(y) ≥ 0, MR(y) ≤ Φ (y) for all y > 0. Under the IGFR condition, MR(0) = 1 and
MR(y) is decreasing in y for 0 ≤ y ≤ y0 where y0φ(y0) / Φ (y0) = 1. Hence MR(y) is a distribution
with support on (0, y0), and it is stochastically dominated by Φ (y). Let m be the mode of the
demand distribution. The following Proposition characterizes the function MR(y) and/or the
location of y0 for some types of distributions.
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Proposition 1:
a. If the demand distribution is unimodal with Φ (m) ≤0.5 and IGFR, then y0 < m.
b. For a normal distribution the shape of MR(y) is affected only by its CV (coefficient of
variance).
c. For an exponential distribution, y0 equals to the mean and MR(y) is convex for 0 ≤y ≤y0.
d. For a uniform distribution on [a, b], y0 exists if 2a < b.
For proofs of the propositions, please refer to Appendix.
From Proposition 1, the MR(y) corresponding to an IGFR distribution function can be linear,
convex (e.g., for exponential distribution), concave (as for most unimodal distributions) or
concave/convex (for positive-skewed distributions). As to the nature of the optimal solution for y
various cases are depicted in Figure 1 (which is similar to the analysis in Karlin's 1958, p. 120
case II). In case I, where MC(0) + h < r, there is always unique solution to (4-5). When MC(0)
+ h > r, the decentralized system could have multiple solutions (case II) , or no feasible solution
(cases III and IV). For case III, however, solution(s) do exist in the integrated chain - implying
that the retailer has no rationale to outsource the goods and should use in-house facilities to
produce them.
Note that as h decreases, the RHS (right hand side) of (4-5) declines and the RHS curve
changes from case IV to case I (or II) in Figure 1. This provides an important insight about the
rationale of changing the integrated system to one of outsourcing. If the operating costs are
relatively high (as was the case in the past in some industries), outsourcing is not viable and it
was indeed rare. We will subsequently assume that the decentralized system has a solution, like
in cases I and II.
9
If MC(0) > r− h, we assume that there are one or two solutions1, say yA and yB where yA < yB
and that the optimal solution will always be yB, where Π M(yB) > 0 as we assumed TC(y)/y < w <
r− h. Otherwise, there will be a unique solution to (4-5) as shown in case I, where we set yA = 0.
Figure 1 : Cases of optimal solutions for equation (4-5)
Since Π M(y) is monotone increasing on [yA, yB] and monotone decreasing on [yB, y0], Π M(y)
M
M
M
and y max
, and thus the feasible region of y is [ y min
,
and Π 0M intersect twice, say at y min
M
M
M
R
R
y max
]. Assume that y min
< y min
< y max
and let ymin = y min
.
We can now convert problem (4-1) into the following:
Max Π M ( y ) = w ⋅y − TC ( y )
y
s.t. w = r Φ ( y ) − h
M
y min ≤ y ≤ y max
(4-7)
(IRR and IRM),
y
where r ∫ min ξφ(ξ)dξ = Π 0R and
0
M
y max
is the larger solution of (r Φ ( y ) − h) y − TC ( y ) = Π 0M .
From now on, yNIR refers to the solution of (4-5), that is, the soution "Without IR
constraints". The actual solution of the system (4-7), y*, is thus max(yNIR , ymin) so that IRR will
be satisfied. We now summarize the effect of parameters on the system (i.e., the comparative
statics).
1
For positive-skewed distributions, MR(y) is first concave and then convex in [0, y0] and thus there
maybe more than two intersections between MR(y) and MC(y). Such case occurs in Weibull distribution
with CV > 0.363 and in gamma distribution with CV > 0.511. If mode is positive (i.e., m > 0), the
convexity must be less than that of an exponential distribution with the same mean, say µ, and thus the
upper limit of convexity is 3/µ2. Since the convexity of MR(y) is near 0 for a relatively high µ, the
possibility of more than two intersections is very rare in practice even for the positive-skewed
distribution. Thus, we do not consider the case where there are more than two intersections between
MR(y) and MC(y) hereafter.
10
Proposition 2:
a. The optimal order quantity increases in retail price if yNIR > ymin, and remains constant
if yNIR ≤ymin.
b. The optimal wholesale price increases in retail price for constant manufacturing cost and/or
if yNIR ≤ ymin .
c. Retailer's profit increases in retail price.
Note that for concave manufacturing costs, since MC′
(y) < 0, dw*/dr depends on g ′
( y ) and
thus can be positive or negative. The degree of concavity in total manufacturing cost affects the
change of optimal quantity with respect to the change in retail price, though not the optimal
quantity itself. Since MC(y) = TC′
(y), for y* = yNIR> ymin we obtain dy*/dr from (4-5) as
follows2:
dy*/dr = − MR(y*) / [r MR′(y*) − TC″ (y*)].
If there are two different types of total production cost, say TC1(y) and TC2(y), satisfying
TC1′
(y*) = TC2′
(y*) and TC1′
′
(y*) < TC2′
′
(y*) for given r, it follows that dy*/dr for TC1(y) is
larger than for TC2(y) for yNIR > ymin .
5.
Effect of Linear Operating Cost (h)
As we observed in Section 4, under the IGFR condition on the demand distribution there
will be a unique optimal quantity yNIR. Due to the IRR constraint, the optimal solution must be
greater than or equal to ymin from (4-7).
2
Also refer to (A-1) in Appendix.
11
Proposition 3:
a. As retailer's efficiency improves, the optimal quantity will increase for yNIR > ymin.
b. If the wholesale price increases as retailer's efficiency improves, the demand distribution
satisfies the IGFR condition.
The converse of part (b) does not hold, as evident from the following example. Consider a
distribution with φ(ξ) = 3000 / (100 + 2ξ)2.5 for ξ > 0, where the average is 100, y0 = 100 and
g′
(ξ) = 75/(50 + ξ)2 > 0 , satisfying the IGFR condition. Assume r = 50, Π 0R =300, Π 0M =350
and TC(y) = 5y⋅[1+e(− y/200)]. Then ymin = 29.57 and we obtain the graphical result for y* and w*
with respect to h shown in Figure 2. We see that dw*/dh is positive in some range of h. As h
increases, the optimal quantity changes from y* to ymin, and finally after h > 3.76 to no feasible
solution because IRM is violated.
Figure 2 : The effect of retailer operating cost on optimal wholesale price and quantity
Proposition 4:
a. As retailer's efficiency improves, its profit will increase for yNIR> ymin.
b. As retailer's efficiency improves, the manufacturer's profit will increase for all y*.
Proposition 4(a) is intuitive since we can expect the retailer's profit to increase as its
operating cost decreases. However, retailer's profit remains same for yNIR ≤ ymin due to the fact
that dymin / dh = 0. We conclude that the profits of both manufacturer and retailer decrease with
the operating cost for yNIR > ymin. If y* ≤ ymin, the motivation for the retailer to reduce the
operating cost no longer exists unless it is reduced sufficiently to reach a breakeven point which
causes yNIR to reach ymin.
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6.
Non-linear Operating Cost
Suppose now that retailer operating costs are not necessarily linear; denote them by h(y).
Consider θh(y), where lower θ corresponds to improved operating efficiency for all y. We
assume that h(y) is twice differentiable, increasing and h(0) = 0. The optimal wholesale price for
the manufacturer is
w(y) = r Φ ( y ) − θh′
(y)
(6-1)
and the optimal quantity satisfies
MR(y) = [MC(y) + θ⋅MH(y)] / r
(6-2)
where MH(y) = d[y h′
(y)]/dy = h′
(y) + y h″(y).
By using (6-1), retailer's profit is now:
Π R(y) = − wy − θh( y ) + r ∫ Φ (ξ)dξ = r ∫ ξφ(ξ)dξ − θ{h( y ) − yh ′
( y )}.
0
0
y
y
(6-3)
We assume that there exists yNIR satisfying (6-2) and that Π R(y) of (6-3) is increasing in y
for a given θ as in the case with linear operating cost. Since in the special case h(y)=log(y) for y
≥ 1, one has MH(y) = 0, it follows that MH(y) < 0 if the operating cost is more concave than
logarithmic (MCL hereafter). If MH(y) > 0, then the operating cost is convex or less concave
than logarithmic (LCL hereafter)3.
Proposition 5:
a. ymin decreases in θ for convex operating cost, and increases for concave one.
3
If f(x) and g(x) are monotone increasing concave functions, f(x) is defined to be more concave than g(x),
if and only if there exists a monotone increasing concave function h(x) satisfying h(g(x)) = f(x).
Suppose h(y) = f[(log(y)] for y > 0 in our scenario, then MH(y) = h′
(y) + yh″(y) = f ″[(log(y)]/y. If
MH(y) < 0, then f″(x) needs to be negative and thus f(x) is increasing concave, since h′
(y)= f ′
[(log(y)]/y
> 0. Since in Figure 4 (IV), h(y) = 1− y− 0.1 and f(y) = h(ey) = 1− e− 0.1y are monotone increasing concave
functions, one can obtain f ″(y) = − 0.1e− 0.1y and MH(y) = f ″[(log(y)]/y =− 0.1y− 1.1 < 0 for y > 0 and thus
h(y) is MCL.
13
b. yNIR decreases in θ for convex or LCL operating cost, and increases for MCL one.
c. The optimal wholesale price decreases in θ for MCL operating costs when yNIR > ymin and
for concave one when yNIR < ymin.
Consider the following example with uniformly distributed demand between 0 and 200. If
r = 10 and TC(y) = 2y + 200[1 − e(− y/100)], and assuming that there are no operating costs, we
obtain the optimal quantity y*θ=0 = 70.08 and wholesale price w*θ=0 = 6.50. We now consider
IRR with Π 0R = 80 and five types of h(y) functions, four nonlinear and one linear. To ensure that
operating costs are comparable among the five cases, each function was adjusted to have the
same operating cost at y*θ=0. These functions are shown in Figure 3, where the average unit
operating cost at y*θ=0, was set to 1.
Figure 3: Non-linear h(y)
I:
y
II: 1− e
−
1
y
80
III: log(y)
IV: 1− y−
0.1
Table 1 summarizes the relevant results for optimal quantity and wholesale price for each
h(y). Since h(y)= 1 − y− 0.1 in case IV is MCL, MH(y) is negative and the resulting optimal
quantity decreases as retailer's efficiency increases. The optimal wholesale price increases in
retailer's efficiency, that is, as θ decreases.
Table 1: Summary of optimal quantity and wholesale price for several types of operating costs
Figure 4: Effect of retailer's efficiency on the optimal quantity and the wholesale price for nonlinear operating cost, where h(y) are
I:
y
II: 1− e
−
1
y
80
III: log(y)
14
IV: 1− y−
0.1
V: linear.
The optimal quantity and wholesale price are piecewise-linear (PWL, hereafter) in θ, as we
note in Figure 4. The optimal quantity, y*, ranges between yθ=1 and yθ=0, where MR(yθ=1) =
[MH(yθ=1)+ MC(yθ=1)]/ r and MR(yθ=0) = MC(yθ=0) / r. Note that in case IV where h(y) is MCL,
we have yθ=1 > yθ=0. If MH(yθ=1) is small relative to r, then (yθ=0 − yθ=1) is small and MR(y) is
PWL in [yθ=1, yθ=0]. The effect of y on MH(y) is also small and thus the optimal quantity and
wholesale price are PWL in θ.
7.
Return policy
Since the optimal quantity in a decentralized chain is less than in an integrated chain, the
manufacturer wishes to increase the quantity to eliminate a "double marginalization" and
compensate the retailer's loss when Π R(y*) < Π 0R . The incentive schemes may be a quantity
discount, a subsidy of partner's operating cost or a return policy.
Suppose that, as assumed by Pasternack (1985), the contract allows the retailer to return
unsold units at a buyback price of b up to a fraction A of the initial lot. Thus the retailer's
expected profit, assuming linear operating costs, is:
Π R(y | b, A)
= − ( w + h) y +
y (1− A)
∫0
∞
y
( rξ + bAy )φ( ξ)dξ +
∫y (1− A) ( rξ + b( y − ξ))φ(ξ)dξ + ∫y ryφ(ξ)dξ
y
Φ ( ξ)dξ
y
= − ( w + h − bA) y + r ∫ Φ ( ξ) dξ − b ∫
0
y (1− A)
(7-1)
and the manufacturer's expected profit is:
y
Π M(y | b, A) = (w − bA)y − TC(y) + b ∫y (1−
A)
Φ (ξ)dξ .
For an integrated chain the profit is independent of b and A, and thus still:
15
(7-2)
y
Π I(y) = − TC ( y ) − hy + r ∫ Φ (ξ)dξ ,
0
(7-3)
and the optimal quantity, yI, satisfies:
Φ ( y I ) = [MC(yI) + h] / r
(7-4)
yI
and let Π I* ≡ Π I(yI) = − TC(yI) + yI ⋅MC(yI) + r ∫ ξφ(ξ)dξ which is assumed to be positive.
0
In the decentralized system, the optimal quantity satisfies:
dΠ R ( y* | b, A)
= − (w + h − bA) + r Φ ( y*) − b[
Φ ( y*) − (1 − A) Φ ( y * (1 − A))]= 0.
dy *
(7-5)
However, even in the absence of operating costs, (7-5) depends on the demand distribution
and does not guarantee a unique solution. Hence we set A to 1, and also y* to yI in order to
coordinate the system4. Then, we have
w + h − b = (r − b)
MC ( y I ) + h
,
r
(7-6)
and for any w, the value of b which coordinates the channel is
(
)(
)
b = w − MC ( y I ) 1 − [ MC ( y I ) + h] / r .
(7-7)
Note that yI depends on both the distribution and the production cost function, but is unique
for any given data. If the production cost is linear, b can be computed directly from the system
parameters independent of the demand distribution.
Then the profits of both parties and optimality condition from (7-1), (7-2) and (7-6) are:
y
Π R(y | b, A=1) = − ( w + h − b ) y + ( r + b )∫ Φ ( ξ)dξ
0
= (1 − b / r) Π I(y) + (1 − b / r) [TC(y) − y⋅MC(yI)],
4
(7-8)
This means an unrestricted return quantity, in order to make the solution independent of the demand
distribution (as assumed by Pasternack (1985) in his examples, in conjunction with a normal distribution).
16
y
Π M(y | b, A=1) = ( w − c − b) y + b ∫ Φ (ξ)dξ
0
= (b / r)⋅Π I(y) − (1 − b / r) [TC(y) − y⋅MC(yI)]
(7-9)
For (7-8) and (7-9), both parties select yI as their optimal value, but manufacturer's profit is
less than that in the linear production cost, because b ≤ r and TC(yI) > yI⋅TC′(yI) = yI⋅MC(yI) due
to the concavity of TC(y). We define the "buyback ratio" as the ratio of buyback price and
wholesale price, that is, b/w. Note that achieving coordination with a low buyback ratio is
desirable for the manufacturer, because a low buyback ratio reduces the uncertainty in
manufacturer's profit.
Proposition 6:
With an unrestricted return quantity policy coordinating the chain:
a. The buyback ratio decreases as retailer's efficiency increases for a linear production cost.
b. An increase of buyback price raises the wholesale price and reduces the retailer's profit.
However, the marginal decrease in retailer's profit with respect to increasing buyback price
will be reduced as retailer's efficiency increases.
Note that for non-linear production cost, as retailer's efficiency increases, the direction of the
movement of buyback ratio is not clear5. However, Proposition 6(b) holds even in this general
case.
5
See (A-2) in Appendix. This means that as retailer's efficiency increases, the manufacturer may need to
increase the buyback price.
17
8.
Investment in Reducing Operating Costs
Consistent with the motivating philosophy of this work, we now allow the retailer to invest
in reducing its operating costs, assumed constant per unit. Let θ be the factor of operating
efficiency in [0,1] which changes the retailer’s operating cost to θh, and M(θ) be the investment
required for achieving the value θ, which has M′
(θ) < 0, M″(θ) < 0 and M(1) = 0.
8-1. Integrated Supply Chain
The expected profit of an integrated firm stocking y units, assuming linear production cost6,
is
Π I(y, θ) = (r – c – θh)y – r ∫0yΦ ( ξ )dξ – M(θ) = –(c + θh)y + r ∫0yΦ ( ξ )dξ – M(θ)
and the optimal conditions are
∂Π I / ∂y = – c – θh + r Φ ( y ) = 0,
(8-1)
∂Π I / ∂θ = – hy – M′(θ), and
(8-2)
0 ≤θ ≤1.
We will now discuss how to find θ from (8-1) and (8-2). Let yI(θ) be the y satisfying (8-1).
We use the popular demand distributions uniform, exponential, and unimodal. The unimodal
distributions include normal, logistics, Pareto, Weibull and gamma. For a unimodal distribution,
Φ ″(y) < 0 for y less than mode, and Φ ″(y) ≥ 0 otherwise.
Proposition 7:
d2yI/dθ 2 has the same sign as Φ ″(y) for every distribution.
6
Since it is extremely hard to analytically (or graphically) solve the problem with more than two nonlinear functions, we cannot handle non-linear manufacturing cost any more.
18
Proposition 7 implies that when the distribution is unimodal the optimal y is concave in θ for
y less than mode and convex otherwise. For the exponential distribution since Φ ″(y) > 0 for all
y, the optimal y is decreasing convex in θ. The optimal y is linearly decreasing in θ for a uniform
distribution.
Proposition 8: If the demand distribution is unimodal and the investment function M(θ) is
decreasing convex, there exist a finite number of solution to yI(θ) = − M′
(θ)/ h in 0 ≤θ ≤1.
Figure 5: Optimal order quantity as a function of the efficiency factor, θ
In order to find the solution for θ from (8-2), we assume the type of investment function,
M(θ), to be either logarithmic or power function, as did Porteus (1985). With the logarithmic
investment function, the firm needs to invest the same amount, say K, for reducing the current
operating cost by a fixed portion (1 − β). Then M(θ) = (K/lnβ)⋅lnθ where K > 0 and
0 < β< 1. The power function uses the simple form M(θ) = K(1− θ)α where K > 0 and α > 1.
Both of functions satisfy all the requirements: M(1) = 0, M′
(θ) < 0 and M″(θ) > 0. Note that
− M′(0) = ∞ for both investment functions.
Convex − M′
(θ) includes the logarithmic and power function with α > 2. In case of two
intersections7, say θ1 and θ2 (θ1 < θ2), the solution can be obtained by comparing Π I(θ1) and
Π I(θ =1). If there is only one intersection, there must be a unique optimal solution for θ in
0 < θ < 1 because yθ=0 < − M′
(0)/h = ∞ and yθ=1 > − M′
(1)/h. Concave − M′
(θ) includes the power
7
In theory, there may exist scenarios which have more than two intersections, but since such cases are,
based on our computational experiments, rare, we assume that there are at most two intersections.
19
function with 1<α<2. In case of no intersection, the solution for θ will be 1, because yθ=0 <
− M′
(0)/h = ∞ . By comparing the value at intersections and at end points, the solution can be
obtained. A graphical method for obtaining the optimal solution of θ is shown in Figure 6.
Figure 6: How to determine the optimal investment level (θ)
8-2. Decentralized Supply Chain
The expected profit of the integrated chain needs to be modified for a decentralized supply
chain with the wholesale price w, as follows:
Π R(y, θ) = (r – w − θh)y − r ∫0yΦ ( ξ )dξ − M(θ) = − (w + θh)y + r ∫0yΦ ( ξ )dξ − M(θ)
and
Π M(w) = (w – c)y.
Similarly to the integrated chain, the optimality conditions for the retailer become
∂Π R/∂y = − (w + θh) + r Φ (y) = 0, and
(8-3)
∂Π R/∂θ = − hy − M′
(θ) = 0.
(8-4)
If we substitute w of (8-3) to Π M(w), then we can obtain the same structure as our original
without-investment problem with constant unit production cost, as follows:
∂Π M/∂y = – c – θh + r MR(y) = 0.
(8-5)
Equations (8-4) and (8-5) show the same structure as the previous problem in the integrated
chain, except for using MR(y) rather than Φ ( y ) . The value of y(θ) of the decentralized chain is
less than that of integrated chain, as discussed in section 4. From Figure 6 (b), we can observe
that a decentralized system has higher optimal level of θ than an integrated chain.
20
With this level of θ, the manufacturer determines the optimal y(θ) and then wholesale price
by (8-3). This wholesale price makes the retailer select the optimal y and θ by (8-3) and (8-4),
which were originally computed by the manufacturer. Since the incentive to reduce operating
cost with investment is lower in the decentralized supply chain than in the integrated chain, the
manufacturer may motivate such investment using other methods, such as investment sharing,
revenue sharing, etc.
9.
Conclusion
In this paper, we analyzed the effect of production/operating cost incurred by the
downstream firm. As that firm becomes more efficient, the optimal quantity increases, the
wholesale price usually (not always) increases, and the profits of both parties increase. The
optimal quantity for a nonlinear retailer operating cost increases in improved efficiency when the
operating cost is more concave than logarithmic.
We extended L&P's model by adding operating costs, non-linear production costs and an
individual rationality constraint. The contract is still wholesale-price-based. Leadership (i.e,
identity of first mover) was attributed to the manufacturer, but the setting is equivalent to one
where the retailer first "lists" (w, y) pairs he is willing to contract on, and the manufacturer then
selects one of them. Thus, in particular, the implications of retail efficiency are independent of
who "leads". But things are likely to be quite different if a contract is based on revenue sharing
rather than wholesale price (e.g., Gerchak and Wang 2000, Cachon and Lariviere 2000).
We also explore the implications of retailer efficiency in a chain coordinated by a return
policy. In the future one might also explore how retail-price-sensitive demand with market
clearing affects downstream efficiency and is affected by it.
21
The way we modeled retailer efficiency - low operating costs - is just one dimension of the
potential advantage of independent retailers. Another dimension is better demand forecasts,
achieved by specialization, proximity to customers and better incentives. Yet another important
dimension is judicious choice of various efforts to increase demand through varying operating
hours, local advertising, promotions and other means. The simultaneous choice of sales effort
and order quantities in a newsvendor environment was modeled by Gerchak and Parlar (1987).
Cachon and Lariviere (2000, Section 4.2) review some relevant marketing literature and discuss
a stylized model. Gilbert and Cvsa (2000) discuss cost-reducing or demand-stimulating
investment by the downstream firm, which bears similarity to our notion of reducing operating
costs. But their model is one of market-clearing rather than newsvendor, and a centralized chain
is not considered. In the future, we intend to combine demand stimulating effort (investment)
selection with an L&P-type framework.
While our work may be nominally viewed as one of generalizing and economically
enriching the operations of the retail function of a supply chain and examining its implications
for coordination, out key motivation for doing that was much broader. If one wishes to reconcile
the advantages of decentralization, which make it so widespread, with the coordination
challenges it brings about, one has to, we believe, start by endowing the model(s) with
parameters which can be positively affected by decentralization. Ours is the first step towards
that, and it focuses on the chain's downstream operations.
Appendix
Proof of Proposition 1:
a. Since Φ (m) ≤0.5, Φ (m) ≤ Φ (m) and
MR(m) = Φ (m) − mφ(m) ≤ ∫0 [
φ(ξ)− φ(m )]dξ < 0.
m
22
b. For the normal distribution N(µ, σ2), we have φ(ξ), Φ (ξ), MR(ξ) and y0. In the same way, there are
φ0(ξ), Φ 0(ξ), MR0(ξ) and y00 for another normal distribution, i.e., N(0, (σ/µ)2). We know
Φ (y) =
∞
∫
y
=
1
2πσ
∞
∫
y /µ
e
−
(ξ− µ)2
2 σ2
dξ =
∞
∫
y /µ
1
1
2πσ
( x − 1)2
−
2
e 2(σ/ µ) dx =
2π(σ/ µ )
e
−
(µx − µ )2
2σ2
(µdx )
∞
∫φ0 (x )dx = Φ 0(y/µ).
y /µ
So,
1
φ0(ξ/µ)
µ
MR(y) = Φ (y) − yφ(y) = Φ 0(y/µ) − [y/µ]⋅φ0(y/µ) = MR0(y/µ)
φ(ξ) =
Since MR(y0) = MR0(y0/µ) = MR0(y00) = 0, y0 = µy00.
Also, from MR′(y) = (1/µ)MR0′
(y/µ) and MR′
′(y) = (1/µ2)MR0′
′(y/µ),
MR′(y0) = (1/µ)MR0′
(y00/µ) and MR′
′(y0) = (1/µ2)MR0′
′(y00/µ).
Hence, the shape of the new marginal revenue distribution is dependent on CV and not on µ.
c.
Φ (y) = exp(− y/µ)
From MR(y0) = Φ (y0) − y0φ(y0) = (1− y0/µ)exp(− y0/µ) = 0, y0 = µ.
Since MR′
′(y) = (3µ − y) / µ3⋅exp(− y/µ) > 0 for 0 ≤y ≤ y0 < 3µ, MR(y) is convex.
d.
Φ (y) = (b − y) / (b − a) for a ≤y ≤b.
From MR(y) = (b − 2y)/(b − a) and MR(y0) = 0, y0 = b/2.
Since MR(y) is defined in [a, b/2], there is no feasible y0 if b/2 < a.
Q.E.D.
Proof of Proposition 2:
a. From (4-5),
dyNR/dr = − MR(yNIR) / [r MR′(yNIR) − MC′(yNIR)].
(A-1)
Since d2ΠM(y)/dy2 = r MR′(y) − MC′(y) < 0 at yNIR, yNIR will increase in r for yNIR > ymin. But, if yNIR ≤
ymin, the optimal order quantity should be ymin to satisfy the IRR constraint, and thus remains constant.
b. From (4-2), dw*/dr = Φ (y*) − rφ(y*)⋅dy*/dr. For yNIR ≤ ymin, dw*/dr = Φ (y*) >0 from dy*/dr = 0.
For yNIR > ymin, dw*/dr = Φ (y*) +rφ(y*)⋅MR(y*) / [r MR′(y*) − MC′(y*)]
= − Φ (y*) [r Φ ( y*) ⋅g′
(y*)+ MC′(y*)]/ [r MR′(y*) − MC′(y*)].
(Q Φ ⋅MR′+ φ⋅MR = Φ [− Φ g′− φ⋅(1− g) ] + φ⋅Φ (1− g) = − Φ 2g′
)
If MC′(y*) = 0, then clearly dw*/dr > 0.
23
c. From (4-3), ∂ΠR(y*)/∂r =
y*
∫0
ξφ(ξ)dξ + r y*φ(y*)⋅dy*/dr > 0 for all y* > 0.
(Q dy*/dr is positive for yNIR > ymin and zero otherwise, and
y*
∫0
ξφ(ξ)dξ > 0 for all y* > 0.)
Q.E.D.
Proof of Proposition 3:
a. From (4-5), dy*/dh = [r MR′(y*) − MC′(y*)]− 1 < 0 for y* = yNIR > ymin. (See the proof of Proposition
2(a).) For yNIR < ymin, y* remains ymin and thus dy*/dh = 0.
b. From (4-2), dw*/dh = − 1 − rφ(y*)⋅dy*/dh. For yNIR < ymin, dw*/dh is negative since dy*/dh = 0.
For yNIR > ymin, y* = yNIR and
dw*/dh = − 1 − rφ(y*) / [r MR′(y*) − MC′(y*)]
= [r Φ (y*) g′
(y*) − rφ(y*) g(y*) + MC′(y*)] / [r MR′(y*) − MC′(y*)].
If dw*/dh < 0, then since it is necessary that g′
(y*) > [φ(y*) g(y*) − MC′(y*) / r ]/ Φ (y*) > 0, the
demand distribution must satisfy IGFR.
Q.E.D.
Proof of Proposition 4:
a. From (4-3) and Proposition 2 (a):
y*
∂ΠR(y*)/∂h = r ∫ ξφ(ξ) dξ ⋅dy*/dh, which is positive for yNIR > ymin and zero otherwise.
0
b. From (4-1) and (4-2), the manufacturer's profit is ∂ΠM(y*)/∂h = (dΠM(y*)/dy*)⋅(dy*/dh) − y*.
Since dΠM(y*)/dy* is zero for y* = yNIR > ymin and dy*/dh is zero for y* = ymin > yNIR. Thus for all cases,
∂ΠM(y*)/∂h = − y* < 0.
Q.E.D.
Proof of Proposition 5:
a. From (6-3), r ∫ min ξφ(ξ) dξ − θ{h( y min ) − y min h ′
( y min )}= Π 0R
0
and thus
y
′
dymin/dθ = θ[
h( y min ) − y min h′
( y min )] [y min (rφ( y min ) + θh′
( y min ))].
From the assumption of dΠR(y)/dy = y [r φ(y) + θ h″(y)] > 0 for a given θ, the denominator is positive.
If h(y) is increasing concave in y, h(y) =
y
y
( ξ)dξ > ∫h ′
( y )dξ = yh ′
( y ) and thus dy
∫0 h ′
0
min/dθ
> 0. On
the other hand, h(y) < yh′(y) and thus dymin/dθ < 0 for a convex h(y).
b. Here we just prove for y* = yNIR > ymin. Equation (6-2) came from dΠ M(y) / dy = r⋅MR(y) − [MC(y) +
θ⋅MH(y)] = 0. At the optimal solution y*, d2ΠM(y) / dy2 < 0 to maximize ΠM(y).
Since dy*/ dθ = MH(y*) / [d2ΠM(y*) / dy*2], dy* / dθ < 0 for positive MH(y), that is, for convex or
LCL h(y), and dy* / dθ > 0 for negative MH(y), that is, for MCL h(y).
24
c. From (6-1), dw*/ dθ = − h′
(y*) + dy*/ dθ ⋅[ − θ h″(y*) − rφ(y*)]
= − h′
(y*) − dy*/ dθ ⋅[ rΦ ( y*) − MC(y*) − θ h′
(y*)]/ y*.
(Q θh″(y*) = θ [MH(y*) − h′
(y*)]/ y* = [ rΦ ( y*) − MC(y*) − θ h′
(y*)] / y* − rφ(y*).)
Since rΦ ( y*) − MC(y*) − θ h′
(y*) is the first derivative of the profit in the integrated chain at y* and
y* < yI, it is positive. Hence, dw*/ dθ < 0 for dy*/ dθ > 0, but the direction of dw*/ dθ is not clear for
dy*/ dθ < 0. From (a) and (b), dy*/ dθ is positive for yNIR > ymin with MCM h(y), and for yNIR < ymin
with concave h(y).
Q.E.D.
Proof of Proposition 6:
a. Here we use the notation of y, instead of yI. From (7-4), dy/dh = [− r φ(y) − MC′(y)]− 1 = [d2ΠI(y)/dy2]− 1
< 0 at yI. From (7-7), b/w = r/w⋅[w − MC (y)]/(r − MC (y) − h)]. Here, w is not determined from the
system as in the previous cases, but by the manufacturer's policy. Therefore, we consider w as an
exogenous parameter.
∂(b/w)/∂h = r/w⋅[− dy/dh⋅(r − w − h)⋅MC′(y) + w − MC (y)]⋅[r − MC (y) − h]− 2
(A-2)
Here, since − dy/dh⋅(r − w − h)⋅MC′(y) < 0 and w − MC (y) > 0 for the concave production cost, the
direction of ∂(b/w)/∂h is not clear. (Q r > w − h, dy/dh < 0, MC′(y) < 0 and w > TC (y) / y > MC (y)).
However, if the production cost is linear, MC′(y) = 0 and thus ∂(b/w)/∂h > 0.
b. From (7-7), w = b⋅r⋅[r − MC (y) − h)] + MC (y) and thus ∂w /∂b = r⋅[r − MC (y) − h)] > 0.
From (7-8) and (7-9), ΠR* = (1− b / r)⋅[ΠI* + TC(yI) − yI⋅MC(yI)] and ΠM* = ΠI* − ΠR*. One can
obtain that ∂ΠR*/ ∂b = − [ΠI* + TC(yI) − yI⋅MC(yI)] / r < 0 (Q TC(yI) − yI⋅MC(yI) for concave TC(y)).
Since dΠI*/ dyI = 0, ∂ΠI*/ ∂h = − yI < 0 from (7-3). From the fact that MC′(yI) < 0 and dyI/dh < 0
from (a), ∂[TC(yI) − yI⋅MC(yI)]/ ∂h = dyI/dh ⋅[− yI⋅MC′(yI)] < 0.
Therefore ∂[∂ΠR*/ ∂b ]/ ∂h = − [ ∂ΠI*/ ∂h + ∂[TC(yI) − yI⋅MC(yI)]/ ∂h ] / r > 0.
Q.E.D.
Proof of Proposition 7:
Let y = yI(θ), y′= dyI(θ)/dθ and y″ = d2yI(θ)/dθ2, then y′= − h / rφ(y) < 0 from (8-1). So,
y″ = d[− h / rφ(y) ]/dθ = hφ′
(y)⋅y′/ rφ(y)2 = [− h / rφ(y)]⋅y′
⋅[− φ′
(y) / φ(y)] = − y′2⋅φ′
(y) / φ(y)
Since φ′
(y) = − Φ ″(y) and φ(y) > 0 for all y, the sign of y″ is same as that of Φ ″(y).
25
Q.E.D.
Proof of Proposition 8:
We can find yI(θ) from (8-1). Let the optimal order quantity at the end points of θ be yθ=1 and yθ=0 for
θ =1 and 0, respectively. Then yθ=1= Φ − 1 ((c + h) / r ) and yθ=0 = Φ − 1 (c / r ). Since Φ (y) is decreasing,
yθ=1 < yθ=0 for h > 0. For a unimodal distribution, the graph of yI(θ) can have three different shapes with
respect to the relation among mode (say m), yθ=1 and yθ=0, as shown in Figure 5.
The number of intersections between yI(θ) and − M′
(θ) depends on the shape of − M′
(θ). Since there is no
assumption about the third derivative of M(θ), M′
′
′
(θ) can be positive or negative. There exist at most 3
intersection points in 0 ≤θ ≤1 for convex or concave − M′
(θ).
Q.E.D.
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26
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Table 1: Summary of optimal quantity and wholesale price for several types of operating costs
h(y)
θ
I:
y*
y
w*
II: 1 − e
−
1
y
80
III: log(y)
IV: 1 − y−
0.1
V: Linear
y*
w*
y*
w*
y*
w*
y*
w*
0.0
70.08 6.50
70.08
6.50
70.08
6.50
70.08
6.50
70.08
6.50
0.2
69.52 6.42
69.90
6.38
70.08
6.45
70.12
6.46
67.85
6.41
0.4
68.96 6.35
69.72
6.26
70.08
6.40
70.16
6.42
65.62
6.32
0.6
68.39 6.28
69.53
6.15
70.08
6.36
70.20
6.38
63.39
6.23
0.8
67.82 6.20
69.33
6.03
70.11
6.31
70.88
6.31
61.15
6.14
1.0
67.65 6.11
69.12
5.91
73.30
6.11
74.21
6.11
58.90
6.05
1.2
69.85 5.91
68.90
5.79
76.42
5.92
77.45
5.92
56.65
5.97
1.4
72.04 5.71
68.67
5.68
79.48
5.74
80.60
5.74
56.57
5.77
28
1
(Case IV)
(Case III)
(Case II)
(Case I)
y
MR(y)
Φ ( y)
Figure 1 : Cases of optimal solutions for equation (4-5)
y*
y0
w*
Figure 2: The effect of retailer operating cost on optimal wholesale price
and quantity
29
II
h(y)
I
III
IV
y
Figure 3: Non-linear h(y)
I:
y
II: 1− e
−
1
y
80
III: log(y)
IV: 1− y−
0.1
IV
IV
III
III
I
I
V
II
II
V
Figure 4: Effect of retailer's efficiency on the optimal quantity and the wholesale price for
non-linear operating cost , where h(y) are
I:
y
II: 1− e
−
1
y
80
III: log(y)
30
IV: 1− y−
0.1
V: linear.
yI(θ)
yI(θ)
yθ=0
yθ=0
yI(θ)
m
yθ=0
m
yθ=1
yθ=1
yθ=1
m
1 θ
1 θ
(a) y1 > m
1 θ
(b) y1 < m < y0
(c) m > y0
Figure 5: Optimal order quantity as a function of the efficiency factor, θ
− M′
(θ) / h
1
Φ ( y)
y = yIθ=0
c+ h
r
yI(θ)
when θ = 1
y = yθ=0
MR(y)
c
r
y = yIθ=1
when θ = 0
yIθ=1
yθ=1
yIθ=0
y = yθ=1
y
0
θI*
θ*
yθ=0
(b) Obtain the optimal θ
(a) Obtain yθ=0 and yθ=1
Figure 6: How to determine the optimal investment level (θ)
31
1
θ