- Email: [email protected]
New structural properties of supply chains with price-only contracts
Accepted Manuscript New structural properties of supply chains with price-only contracts George J. Kyparisis, Christos Koulamas PII: DOI: Reference:
S0167-6377(16)30125-0 http://dx.doi.org/10.1016/j.orl.2016.10.007 OPERES 6161
To appear in:
Operations Research Letters
Received date: 18 May 2016 Revised date: 11 October 2016 Accepted date: 12 October 2016 Please cite this article as: G.J. Kyparisis, C. Koulamas, New structural properties of supply chains with price-only contracts, Operations Research Letters (2016), http://dx.doi.org/10.1016/j.orl.2016.10.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
NEW STRUCTURAL PROPERTIES OF SUPPLY CHAINS WITH PRICE-ONLY CONTRACTS George J. Kyparisis∗ and Christos Koulamas Department of Information Systems and Business Analytics Florida International University, Miami, Florida 33199, USA ∗
corresponding author, Email: [email protected] Revised: October 2016 Abstract
We show that the manufacturer’s problem in two-stage push supply chains with price-only contracts can be reformulated as a newsvendor problem with a modified demand distribution. This reformulation provides additional managerial insights, facilitates the solution of a problem with competing retailers under more general assumptions and reduces a number of three-stage problems to equivalent two-stage problems. Key Words. newsvendor problem, wholesale price contract, competing retailers
1
Introduction
We consider two-stage push supply chains with a single retailer, a single manufacturer and a price-only contract. In the basic fixed retail price model, studied by Lariviere and Porteus [4], to be called LP in the sequel, the manufacturer acts as a Stackelberg leader and determines the wholesale price. Subsequently, the retailer, who faces an uncertain demand, determines the order quantity in the newsvendor framework. Afterwards, the random demand is realized. We also analyze an extension of this model with multiple competing retailers studied in [7] and show that the problem is solvable under more general assumptions on the random demand distribution. For the model with one retailer, the retail price is p > 0, the manufacturer’s unit cost is c > 0, c < p, the wholesale price is w, c ≤ w ≤ p, and the retailer order quantity is q. We 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
assume zero salvage price and zero shortage cost. All supply chain participants are assumed to be risk-neutral. For a nondecreasing function s(x), we define the elasticity of s(x) with respect to x as Es(x) = ′
Es (x) =
′
xs (x) s(x)
and the elasticity of the slope of s(x) with respect to x as
′′
xs (x) . s′ (x)
The random demand for the end product, x, has cdf F (x), F¯ (x) = 1 − F (x), and pdf f (x) with support on [0, ∞) so that f (x) = 0 for x < 0 and f (x) > 0 for x ≥ 0. We assume that f (x) is twice differentiable for x > 0 and that limx→∞ f (x)x = 0. The failure rate is defined as h(x) = f (x)/F¯ (x) and the generalized failure rate as g(x) = xh(x). F is IFR ′
′
(increasing failure rate) if h (x) ≥ 0 and F is DFR (decreasing failure rate) if h (x) ≤ 0 ([1]); ′
F is IGFR (increasing generalized failure rate) if g (x) ≥ 0 (LP) and, similarly, F is DGFR ′
(decreasing generalized failure rate) if g (x) ≤ 0. The partial expected value of x with upper limit z is defined as G(z) =
Rz 0
xf (x)dx.
In the next section, we study a supply chain with a single retailer and a single manufacturer and its extension to a supply chain with competing retailers.
2
A Two-Stage Supply Chain with a Fixed Retail Price and One Retailer
We assume in this section that the pdf f (x) has support on [A, ∞), A ≥ 0. For a fixed w, the retailer’s newsvendor profit maximization problem is max Πr (q) = pS(q) − wq, q≥A
(1)
where S(q) = E[min(q, x)] denotes the expected sales given q; the maximization in (1) is restricted to q ≥ A because the random demand x is at least A. The unique optimal order quantity is given by q(w) = F¯ −1 ( wp ) (observe that since w ≤ p, q(w) = F¯ −1 ( wp ) ≥ F¯ −1 (1) =
A). Since w = pF¯ (q(w)), the optimal retailer profit at q(w) is Πr (q(w)) = pG(q(w)). Given
q(w), the manufacturer’s profit maximization problem is maxw Πm (w) = (w − c)q(w) and it can be reformulated in terms of q as max Πm (q) = [pF¯ (q) − c]q. q≥A
2
(2)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
LP proved that if F is IGFR, then Πm (q) is unimodal and the problem maxq≥A Πm (q) has a unique optimal solution q ∗ . In contrast to LP, we do not require that F is IGFR. In order to simplify the exposition somewhat, we assume that g(x) ≤ 1 for all x ≥ A. We show next that the unique optimal solution q ∗ of the manufacturer’s problem (2) is also the solution of a modified single-stage
newsvendor problem stated as max Πc (q) = pSM (q) − cq, q≥A
where SM (q) = EFM [min(q, x)] =
FM (x) =
Rq 0
(3)
F¯M (x)dx. FM is a modified distribution defined as if x ≤ 0,
0, g(A+ ) Ax ,
if x ∈ (0, A],
(4)
1 − F¯ (x)[1 − g(x)], if x > A,
where g(A+ ) = limx→A+ g(x) (the modified distribution FM was originally proposed in [3]). Our main result in this section is that problem (2) has a unique optimal solution (proved by LP for IGFR distributions) (and also that problem (3) has a unique solution) under the following less restrictive assumption: (A1) Ef (x) + 2 =
′
xf (x) f (x)
+ 2 ≥ 0 for x > A.
The assumption (A1) was stated as condition (C2) in [9] and as condition (D4)(i) (with a strict inequality) in [8]. Ziya et al. [9] showed that (A1) implies the concavity of the manufacturer’s profit function Πm (q). This assumption will also be used to facilitate the solution of a two-stage supply chain with competing retailers later in this section. Theorem 1 (Solvability of the Manufacturer’s Problem). If (A1) holds and g(x) ≤ 1 for all x ≥ A, then the modified distribution FM given by (4) is well defined. Moreover, the profit functions in (2) and (3) are unimodal on [A, ∞) and the
unique optimal solutions q ∗ of problems (2) and (3) exist and are identical. If g(A+ ) ≥ 1 − pc , then q ∗ = A and if g(A+ ) < 1 − pc , then q ∗ > A, g(q ∗ ) < 1 and q ∗ satisfies the condition pF¯ (q ∗ )[1 − g(q ∗)] = c. 3
(5)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Proof of Theorem 1. FM (x) is continuous on (−∞, ∞) because FM (A− ) = FM (A) = FM (A+ ) = g(A+ ),
limx→∞ F¯ (x) = 0 and limx→∞ F¯ (x)g(x) = limx→∞ f (x)x = 0. FM (x) is nondecreasing
on (−∞, ∞) if and only if F¯ (x)[1 − g(x)] = F¯ (x) − f (x)x is nonincreasing for x > A. This is in turn equivalent to (A1). Thus, since FM (x) is continuous and nondecreasing on (−∞, ∞), the modified distribution FM is well defined. The pdf for the modified distribution FM is given as
fM (x) = g(A+ ) A1 , Since
Ru l
if x ≤ 0,
0, ′ F¯ (x) {g (x)x x
if x ∈ (0, A), + g(x)[1 − g(x)]}, if x > A.
F¯ (x)[1 − g(x)]dx = uF¯ (u) − lF¯ (l), for q > A we have that Z
q
A
F¯M (x)dx =
Thus, SM (q) =
Z
0
q
Z
q
A
F¯ (x)[1 − g(x)]dx = q F¯ (q) − AF¯ (A).
F¯m (x)dx =
2
q , q − g(A+ ) 2A
if q ∈ (0, A],
q F¯ (q) − g(A+ ) A2 , if q > A.
Since (A1) implies that F¯ (x)[1−g(x)] is nonincreasing for x > A, dΠm (q)/dq = dΠc (q)/dq = pF¯ (q)[1 − g(q)] − c is nonincreasing for q > A. Thus, d2 Πm (q)/dq 2 = d2 Πc (q)/dq 2 ≤ 0 for q > A. Consequently, Πm (q) and Πc (q) are concave on (A, ∞) and thus unimodal on [A, ∞).
Therefore, if g(A+ ) ≥ 1 − pc , then dΠm (A+ )/dq = dΠc (A+ )/dq = pF¯ (A)[1 − g(A+ )] −
c = p[1 − g(A+ )] − c ≤ 0. Thus, since dΠm (q)/dq and dΠc (q)/dq are nonincreasing for
q > A, dΠm (q)/dq ≤ 0 and dΠc (q)/dq ≤ 0 for q > A which implies that the unique optimal
solution of both problems is given by q ∗ = A. On the other hand, if g(A+ ) < 1 −
c p
< 1,
then, since dΠm (A+ )/dq = dΠc (A+ )/dq = p[1 − g(A+ )] − c > 0 and limx→∞ dΠm (x)/dq = limx→∞ dΠc (x)/dq = −c < 0, the unique optimal solution q ∗ is such that q ∗ > A, g(q ∗) < 1
and it satisfies the first order optimality condition dΠm (q ∗ )/dq = dΠc (q ∗ )/dq = 0 which can be written as (5). This completes the proof.
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
′
′
xf (x) f (x)
If F is IGFR then g (x) ≥ 0 for x > A or, equivalently, that
+ g(x) + 1 ≥ 0 for
x > A. Thus, since we assumed that g(x) ≤ 1 for x ≥ A, (A1) holds so that the IGFR assumption on F implies (A1). Xu and Bisi [8] state that a non-IGFR distribution given in Example 2 in Ziya et al. [9] satisfies (A1) which shows that (A1) is satisfied by some nonIGFR distributions. Paul [6] and Xu and Bisi [8] obtained additional conditions, satisfied by some non-IGFR distributions, under which Πm (q) is unimodal. The following example presents a DGFR distribution that satisfies (A1) and for which problem (3) is solvable. Example 1. Let f (x) =
a x(a+ln x)2
> 0, a ≥ 2 with support [1, ∞). Thus, g(x) =
1 a+ln x
is
decreasing for x ∈ [1, ∞). Since a ≥ 2, (A1) holds for x ∈ [1, ∞). We can use the modified distribution FM to relate condition (5) to an analogous condition for the single-stage newsvendor problem maxq Π0 (q) = pS(q) − cq. The well-known fractile
solution of this problem can be written as F¯ (q ∗ ) = pc . Since S(q) represents the expected
′ sales and F¯ (q) = S (q) represents the marginal expected sales, the fractile formula states
that the marginal expected sales are equal to the profit margin ratio pc . Similarly, since F¯M (x) = F¯ (x)[1 − g(x)] for x > A, condition (5) can be written as F¯M (q ∗ ) = pc . Therefore
′ SM (q) represents the expected sales for the problem (3), F¯M (q) = SM (q) represents the
corresponding marginal expected sales and condition (5) states that the marginal expected sales for problem (3) are equal to the profit margin ratio pc . The following result extends Theorem 4 in LP (who assumed that F is IGFR which implies (A1)) on the division of supply chain profits by specifying when the manufacturer captures less than half of the total supply chain profit.
Proposition 1 (Division of Supply Chain Profits). If (A1) holds and g(A+ ) < 1 − pc , then the profit ratio is
Π∗m Π∗r
= EG(q ∗ ). Thus, if F is
IFR or G(x) is convex for x ≥ 0 such that g(x) ≤ 1, then Π∗m ≥ Π∗r ; if G(x) is concave for
x ≥ 0 such that g(x) ≤ 1, then Π∗m ≤ Π∗r .
5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Proof of Proposition 1. It follows from Theorem 1 that, if g(A+ ) < 1 − pc , then the equilibrium order quantity
q ∗ is such that q ∗ > A ≥ 0, g(q ∗) < 1 and it satisfies the first order condition (5). Thus,
Π∗r = pG(q ∗ ) , Π∗m = [pF¯ (q ∗ ) − c]q ∗ = pF¯ (q ∗ )g(q ∗)q ∗ and
Π∗m pF¯ (q ∗ )g(q ∗ )q ∗ [f (q ∗ )q ∗ ]q ∗ G (q ∗ )q ∗ = = = = EG(q ∗ ). Π∗r pG(q ∗ ) G(q ∗ ) G(q ∗ ) ′
′
If G(x) is convex (concave) for x ≥ 0 such that g(x) ≤ 1, then G(0)−G(x) ≥ (≤) G (x)(0−x), ′
which can be written as G (x)x ≥ (≤) G(x), or, equivalently as EG(x) ≥ (≤) 1, and thus,
since q ∗ > 0 and g(q ∗ ) < 1,
Π∗m Π∗r
≥ (≤) 1. This completes the proof.
LP state in their Theorem 4 that Π∗m ≥ Π∗r if F (x) is convex and that Π∗m ≤ Π∗r is possible if F (x) is concave. The assumption that F (x) is convex implies that F is IFR which in turn implies that G(x) is convex for x ≥ 0 such that g(x) ≤ 1, which is assumed in Proposition 1. Conversely, the concavity of G(x) for x ≥ 0 such that g(x) ≤ 1 is needed to prove that ′
Π∗m ≤ Π∗r . If G(x) is convex for x ≥ 0, then G (x) and g(x) are nondecreasing for x ≥ 0 and ′
thus F is IGFR. Also, if F is DGFR, then g(x) and G (x) are nonincreasing for x ≥ 0 and thus G(x) is concave for x ≥ 0. In summary, our Proposition 1 indicates that the division of supply chain profits depends on the curvature of G(x).
2.1
A Two-Stage Supply Chain with a Fixed Retail Price and Competing Retailers
In this subsection, we consider a two-stage supply chain with a single manufacturer and multiple competing retailers analyzed in [7]. We show that the introduction of a modified demand distribution provides additional managerial insights for the problem and facilitates its solution under more general assumptions. The two-stage model in [7] is based on the single-stage competitive newsvendor model in [5] with multiple competing retailers. We assume in this subsection that the pdf f (x) has support on [0, ∞), that is A = 0 but we do not require that F is IGFR in contrast to Perakis and Roels [7]. As before, we also assume that g(x) ≤ 1 for all x ≥ 0.
6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Following Lippman and McCardle [5], Perakis and Roels [7] assume that customers move as a herd and visit the n retailers sequentially until the total demand is met, where all sequences (permutations) of the n retailers are equally likely. This implies that the retailers are symmetric and thus the manufacturer offers the same wholesale price w to all retailers and, in turn, each retailer orders the same optimal quantity q(w). The retailer’s problem is identical for all retailers and it can be stated as max Πr (q) = p q≥0
n X
1 ( )E[min(q, (x − (k − 1)q(w))+ ] − wq, k=1 n
where q(w) is the optimal solution of (6) which uniquely solves the equation p
(6) Pn
1 ¯ k=1 ( n )F (kq)
=
w ([5]). Given q(w), the manufacturer’s profit maximization problem is maxw Πm (w) = n(w − c)q(w) and it can be reformulated in terms of q as max Πm (q) = p q≥0
n X
k=1
F¯ (kq)q − ncq.
(7)
We will show next that the unique optimal solution q ∗ of the manufacturer’s problem (7) is also the unique solution of a newsvendor problem stated as max Πc (q) = pSM (q) − cq, q≥0
where SM (q) =
Rq 0
F¯M (x)dx and FM is a modified distribution defined as FM (x) =
where H(x) = ( n1 )
(8)
Pn
¯
k=1 F (kx)[1
if x ≤ 0,
0,
1 − H(x), if x > 0,
(9)
− g(kx)].
Theorem 2 (Solvability of the Manufacturer’s Problem). If g(0+ ) = 0, g(x) ≤ 1 for all x ≥ 0 and (A1) holds, then the modified distribution FM given by (9) is well defined. The profit functions in (7) and (8) are unimodal on [0, ∞) and the unique optimal solu-
tions q ∗ of problems (7) and (8) exist and are identical. Moreover, q ∗ > 0 and q ∗ satisfies the condition p
n X
k=1
F¯ (kq)[1 − g(kq)] = nc. 7
(10)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Proof of Theorem 2. Suppose that the assumptions of Theorem 2 hold. Since g(0+ ) = limx→0+ g(x) = 0, H(0+ ) = 1 and FM (0+ ) = 0 and FM (x) is continuous on (−∞, ∞). Since (A1) is equivalent
to F¯ (x)[1 − g(x)] nonincreasing for x > 0, this implies that H(x) is also nonincreasing for x > 0 so that FM (x) is nondecreasing for all x and the modified distribution FM is well
defined. Thus, SM (q) =
Rq
pSM (q) − cq = p( n1 )
0
R P F¯m (x)dx = 0q H(x)dx = ( n1 ) nk=1 F¯ (kq)q for q > 0 and Πc (q) =
Pn
¯
k=1 F (kq)q
− cq = ( n1 )Pm (q) for q > 0. Therefore, the problems
maxq≥0 Πm (q) and maxq≥0 Πc (q) are equivalent. Since dΠm (q)/dq = p
Pn
k=1
F¯ (kq)[1 − g(kq)] − nc, dΠm (0+ )/dq = np − nc > 0 and
dΠm (∞)/dq = −nc < 0. Also, since (A1) implies that F¯ (x)[1 − g(x)] is nonincreasing for
x > 0, dΠm (q)/dq is nonincreasing for q > 0 and Πm (q) is concave for q > 0. Therefore, the profit function Πm (q) is unimodal on [0, ∞) and there exists a unique optimal solution q ∗ of
(7) such that dΠm (q ∗ )/dq = 0 which is equivalent to (10).
Finally, since the problems maxq≥0 Πm (q) and maxq≥0 Πc (q) are equivalent for x > 0, q ∗ > 0 is also a unique optimal solution of of (8). This completes the proof. Theorem 2 shows that the unique equilibrium exists under assumption (A1). This result generalizes a similar result in [7] stated under the IGFR assumption. Since F¯M (x) = H(x) = ( n1 )
Pn
¯
k=1 F (kx)[1−g(kx)]
for x > 0, condition (10) can be written
as F¯M (q ∗ ) = pc . Therefore, as in the single retailer case, SM (q) represents the expected sales ′ for problem (8), F¯M (q) = SM (q) represents the corresponding marginal expected sales and
condition (10) states that the marginal expected sales for problem (8) are equal to the profit margin ratio pc . The next result partially extends Proposition 1 to the competing retailers case. Proposition 2 (Division of Supply Chain Profits). If g(0+ ) = 0, g(x) ≤ 1 for all x ≥ 0 and (A1) holds, then the profit ratio is
EGn (q ∗ ) +
G(nq ∗ ) Gn (q ∗ )
− 1, where Gn (x) = S(nx) − 8
Pn
k=1
F¯ (kx)x.
Π∗m nΠ∗r
=
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Thus, if Gn (x) is convex for x ≥ 0 such that g(x) ≤ 1, then Π∗m > nΠ∗r ; if Gn (x) is
concave for x ≥ 0 such that g(x) ≤ 1 and EGn (q ∗ ) +
G(nq ∗ ) Gn (q ∗ )
≤ 2, then Π∗m ≤ nΠ∗r .
Proof of Proposition 2. It follows from Theorem 2 that the equilibrium order quantity q ∗ satisfies the condition (10). Furthermore, one can show that nΠ∗r = pGn (q ∗ ) = p[S(nq ∗ ) − Π∗m = p
Pn
k=1
F¯ (kq ∗ )g(kq ∗)q ∗ . Thus,
Π∗m = nΠ∗r
Pn
¯
k=1
¯
k=1 F (kq
)g(kq ∗ )q ∗ . Gn (q ∗ )
k=1 F (kq
∗
P ′ Since (Gn ) (q ∗ ) = nF¯ (nq ∗ ) − nk=1 F¯ (kq ∗ )[1 − g(kq ∗ )], n X
Pn
′ F¯ (kq ∗ )g(kq ∗ )q ∗ = (Gn ) (q ∗ )q ∗ − nq ∗ F¯ (nq ∗ ) +
∗
)q ∗ ] and
(11)
n X
F¯ (kq ∗ )q ∗ .
(12)
k=1
Moreover, since G(nq ∗ ) = S(nq ∗ ) − nq ∗ F¯ (nq ∗ ), nq ∗ F¯ (nq ∗ ) −
n X
k=1
F¯ (kq ∗ )q ∗ = Gn (q ∗ ) − G(nq ∗ ).
(13)
Equations (11)-(13) imply that ′
(Gn ) (q ∗ )q ∗ Gn (q ∗ ) − G(nq ∗ ) G(nq ∗ ) Π∗m ∗ = − = EG (q ) + − 1. n nΠ∗r Gn (q ∗ ) Gn (q ∗ ) Gn (q ∗ ) If Gn (x) is convex (concave) for x ≥ 0 such that g(x) ≤ 1, then, in view of the proof of Proposition 1, EGn (x) ≥ (≤) 1. Thus, if Gn (x) is convex for x ≥ 0 such that g(x) ≤ 1, then,
since G(nq ∗ ) > Gn (q ∗ ), Π∗m > nΠ∗r . If Gn (x) is concave for x ≥ 0 such that g(x) ≤ 1 and
EGn (q ∗ ) +
3
G(nq ∗ ) Gn (q ∗ )
≤ 2, then Π∗m ≤ nΠ∗r .
Conclusions
The reformulation of the manufacturer’s problem can be also implemented to the assembly system comprising a downstream retailer who faces the newsvendor problem, a midstream assembler and n upstream simultaneous suppliers. By utilizing a modified distribution, this system can be reduced to an equivalent two-stage assembly system with a downstream assembler and simultaneous upstream suppliers (analyzed by Gerchak and Wang [2]), with the modified distribution FM in the place of F . 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Since Gerchak and Wang [2] require that F is IGFR in order for the unique equilibrium to exist for their assembly system, we also require that FM is IGFR in the three-stage system. Kyparisis and Koulamas [3] showed that if F is strictly IGFR and either g(x) is convex ′
for x > 0 or Eg (x) ≥ −1 for x > 0, then FM is IGFR on (0, C). One can verify that the condition that g(x) is convex for x > 0 is satisfied by most IFR distributions while the ′
condition that Eg (x) ≥ −1 for x > 0 is additionally satisfied by some IGFR distributions. In general, this type of reduction is feasible whenever a multi-stage system with a wholesale-price contract has a newsvendor problem in the last stage and a manufacturer’s problem in the next to last stage. It will be of interest to investigate whether the concept of the modified demand distribution applies to alternative contracts such as buyback contracts and to alternative supply chain configurations such as pull supply chains. Acknowledgement. We would like to thank the Area Editor and the two referees for their insightful comments which helped us improve earlier versions of this paper.
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
References [1] Barlow, R. E., F. Proschan. 1996. Mathematical Theory of Reliability. G. H. Golub, ed. SIAM Classics in Applied Mathematics. Society for Industrial and Applied Mathematics. Philadelphia. [2] Gerchak, Y., Y. Wang. 2004. Revenue-sharing vs. wholesale-price contracts in assembly systems with random demand. Production and Operations Management 13(1) 23-33. [3] Kyparisis, G. J., C. Koulamas. 2016. Assembly Systems with Sequential Supplier Decisions and Uncertain Demand. Production and Operations Management forthcoming. [4] Lariviere, M. A., E. L. Porteus. 2001. Selling to the newsvendor: An analysis of price-only contracts. Manufacturing & Service Operations Management 3 293-305. [5] Lippman, S. A., K. F. McCardle. 1997. The competitive newsboy. Operations Research 45 54-65. [6] Paul, A. 2006. On the unimodality of the manufacturer’s objective function in the newsvendor model. Operations Research Letters 34 46-48. [7] Perakis, G., G. Roels. 2007. The price of anarchy in supply chains: Quantifying the efficiency of price-only contracts. Management Science 53(8) 1249-1268. [8] Xu, Y., A. Bisi. 2012. Wholesale-price contracts with postponed and fixed retail prices. Operations Research Letters 40 250-257. [9] Ziya, S., H. Ayhan, R. D. Foley. 2004. Relationships among three assumptions in revenue management. Operations Research 52 804-809.
11
S0167-6377(16)30125-0 http://dx.doi.org/10.1016/j.orl.2016.10.007 OPERES 6161
To appear in:
Operations Research Letters
Received date: 18 May 2016 Revised date: 11 October 2016 Accepted date: 12 October 2016 Please cite this article as: G.J. Kyparisis, C. Koulamas, New structural properties of supply chains with price-only contracts, Operations Research Letters (2016), http://dx.doi.org/10.1016/j.orl.2016.10.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
NEW STRUCTURAL PROPERTIES OF SUPPLY CHAINS WITH PRICE-ONLY CONTRACTS George J. Kyparisis∗ and Christos Koulamas Department of Information Systems and Business Analytics Florida International University, Miami, Florida 33199, USA ∗
corresponding author, Email: [email protected] Revised: October 2016 Abstract
We show that the manufacturer’s problem in two-stage push supply chains with price-only contracts can be reformulated as a newsvendor problem with a modified demand distribution. This reformulation provides additional managerial insights, facilitates the solution of a problem with competing retailers under more general assumptions and reduces a number of three-stage problems to equivalent two-stage problems. Key Words. newsvendor problem, wholesale price contract, competing retailers
1
Introduction
We consider two-stage push supply chains with a single retailer, a single manufacturer and a price-only contract. In the basic fixed retail price model, studied by Lariviere and Porteus [4], to be called LP in the sequel, the manufacturer acts as a Stackelberg leader and determines the wholesale price. Subsequently, the retailer, who faces an uncertain demand, determines the order quantity in the newsvendor framework. Afterwards, the random demand is realized. We also analyze an extension of this model with multiple competing retailers studied in [7] and show that the problem is solvable under more general assumptions on the random demand distribution. For the model with one retailer, the retail price is p > 0, the manufacturer’s unit cost is c > 0, c < p, the wholesale price is w, c ≤ w ≤ p, and the retailer order quantity is q. We 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
assume zero salvage price and zero shortage cost. All supply chain participants are assumed to be risk-neutral. For a nondecreasing function s(x), we define the elasticity of s(x) with respect to x as Es(x) = ′
Es (x) =
′
xs (x) s(x)
and the elasticity of the slope of s(x) with respect to x as
′′
xs (x) . s′ (x)
The random demand for the end product, x, has cdf F (x), F¯ (x) = 1 − F (x), and pdf f (x) with support on [0, ∞) so that f (x) = 0 for x < 0 and f (x) > 0 for x ≥ 0. We assume that f (x) is twice differentiable for x > 0 and that limx→∞ f (x)x = 0. The failure rate is defined as h(x) = f (x)/F¯ (x) and the generalized failure rate as g(x) = xh(x). F is IFR ′
′
(increasing failure rate) if h (x) ≥ 0 and F is DFR (decreasing failure rate) if h (x) ≤ 0 ([1]); ′
F is IGFR (increasing generalized failure rate) if g (x) ≥ 0 (LP) and, similarly, F is DGFR ′
(decreasing generalized failure rate) if g (x) ≤ 0. The partial expected value of x with upper limit z is defined as G(z) =
Rz 0
xf (x)dx.
In the next section, we study a supply chain with a single retailer and a single manufacturer and its extension to a supply chain with competing retailers.
2
A Two-Stage Supply Chain with a Fixed Retail Price and One Retailer
We assume in this section that the pdf f (x) has support on [A, ∞), A ≥ 0. For a fixed w, the retailer’s newsvendor profit maximization problem is max Πr (q) = pS(q) − wq, q≥A
(1)
where S(q) = E[min(q, x)] denotes the expected sales given q; the maximization in (1) is restricted to q ≥ A because the random demand x is at least A. The unique optimal order quantity is given by q(w) = F¯ −1 ( wp ) (observe that since w ≤ p, q(w) = F¯ −1 ( wp ) ≥ F¯ −1 (1) =
A). Since w = pF¯ (q(w)), the optimal retailer profit at q(w) is Πr (q(w)) = pG(q(w)). Given
q(w), the manufacturer’s profit maximization problem is maxw Πm (w) = (w − c)q(w) and it can be reformulated in terms of q as max Πm (q) = [pF¯ (q) − c]q. q≥A
2
(2)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
LP proved that if F is IGFR, then Πm (q) is unimodal and the problem maxq≥A Πm (q) has a unique optimal solution q ∗ . In contrast to LP, we do not require that F is IGFR. In order to simplify the exposition somewhat, we assume that g(x) ≤ 1 for all x ≥ A. We show next that the unique optimal solution q ∗ of the manufacturer’s problem (2) is also the solution of a modified single-stage
newsvendor problem stated as max Πc (q) = pSM (q) − cq, q≥A
where SM (q) = EFM [min(q, x)] =
FM (x) =
Rq 0
(3)
F¯M (x)dx. FM is a modified distribution defined as if x ≤ 0,
0, g(A+ ) Ax ,
if x ∈ (0, A],
(4)
1 − F¯ (x)[1 − g(x)], if x > A,
where g(A+ ) = limx→A+ g(x) (the modified distribution FM was originally proposed in [3]). Our main result in this section is that problem (2) has a unique optimal solution (proved by LP for IGFR distributions) (and also that problem (3) has a unique solution) under the following less restrictive assumption: (A1) Ef (x) + 2 =
′
xf (x) f (x)
+ 2 ≥ 0 for x > A.
The assumption (A1) was stated as condition (C2) in [9] and as condition (D4)(i) (with a strict inequality) in [8]. Ziya et al. [9] showed that (A1) implies the concavity of the manufacturer’s profit function Πm (q). This assumption will also be used to facilitate the solution of a two-stage supply chain with competing retailers later in this section. Theorem 1 (Solvability of the Manufacturer’s Problem). If (A1) holds and g(x) ≤ 1 for all x ≥ A, then the modified distribution FM given by (4) is well defined. Moreover, the profit functions in (2) and (3) are unimodal on [A, ∞) and the
unique optimal solutions q ∗ of problems (2) and (3) exist and are identical. If g(A+ ) ≥ 1 − pc , then q ∗ = A and if g(A+ ) < 1 − pc , then q ∗ > A, g(q ∗ ) < 1 and q ∗ satisfies the condition pF¯ (q ∗ )[1 − g(q ∗)] = c. 3
(5)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Proof of Theorem 1. FM (x) is continuous on (−∞, ∞) because FM (A− ) = FM (A) = FM (A+ ) = g(A+ ),
limx→∞ F¯ (x) = 0 and limx→∞ F¯ (x)g(x) = limx→∞ f (x)x = 0. FM (x) is nondecreasing
on (−∞, ∞) if and only if F¯ (x)[1 − g(x)] = F¯ (x) − f (x)x is nonincreasing for x > A. This is in turn equivalent to (A1). Thus, since FM (x) is continuous and nondecreasing on (−∞, ∞), the modified distribution FM is well defined. The pdf for the modified distribution FM is given as
fM (x) = g(A+ ) A1 , Since
Ru l
if x ≤ 0,
0, ′ F¯ (x) {g (x)x x
if x ∈ (0, A), + g(x)[1 − g(x)]}, if x > A.
F¯ (x)[1 − g(x)]dx = uF¯ (u) − lF¯ (l), for q > A we have that Z
q
A
F¯M (x)dx =
Thus, SM (q) =
Z
0
q
Z
q
A
F¯ (x)[1 − g(x)]dx = q F¯ (q) − AF¯ (A).
F¯m (x)dx =
2
q , q − g(A+ ) 2A
if q ∈ (0, A],
q F¯ (q) − g(A+ ) A2 , if q > A.
Since (A1) implies that F¯ (x)[1−g(x)] is nonincreasing for x > A, dΠm (q)/dq = dΠc (q)/dq = pF¯ (q)[1 − g(q)] − c is nonincreasing for q > A. Thus, d2 Πm (q)/dq 2 = d2 Πc (q)/dq 2 ≤ 0 for q > A. Consequently, Πm (q) and Πc (q) are concave on (A, ∞) and thus unimodal on [A, ∞).
Therefore, if g(A+ ) ≥ 1 − pc , then dΠm (A+ )/dq = dΠc (A+ )/dq = pF¯ (A)[1 − g(A+ )] −
c = p[1 − g(A+ )] − c ≤ 0. Thus, since dΠm (q)/dq and dΠc (q)/dq are nonincreasing for
q > A, dΠm (q)/dq ≤ 0 and dΠc (q)/dq ≤ 0 for q > A which implies that the unique optimal
solution of both problems is given by q ∗ = A. On the other hand, if g(A+ ) < 1 −
c p
< 1,
then, since dΠm (A+ )/dq = dΠc (A+ )/dq = p[1 − g(A+ )] − c > 0 and limx→∞ dΠm (x)/dq = limx→∞ dΠc (x)/dq = −c < 0, the unique optimal solution q ∗ is such that q ∗ > A, g(q ∗) < 1
and it satisfies the first order optimality condition dΠm (q ∗ )/dq = dΠc (q ∗ )/dq = 0 which can be written as (5). This completes the proof.
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
′
′
xf (x) f (x)
If F is IGFR then g (x) ≥ 0 for x > A or, equivalently, that
+ g(x) + 1 ≥ 0 for
x > A. Thus, since we assumed that g(x) ≤ 1 for x ≥ A, (A1) holds so that the IGFR assumption on F implies (A1). Xu and Bisi [8] state that a non-IGFR distribution given in Example 2 in Ziya et al. [9] satisfies (A1) which shows that (A1) is satisfied by some nonIGFR distributions. Paul [6] and Xu and Bisi [8] obtained additional conditions, satisfied by some non-IGFR distributions, under which Πm (q) is unimodal. The following example presents a DGFR distribution that satisfies (A1) and for which problem (3) is solvable. Example 1. Let f (x) =
a x(a+ln x)2
> 0, a ≥ 2 with support [1, ∞). Thus, g(x) =
1 a+ln x
is
decreasing for x ∈ [1, ∞). Since a ≥ 2, (A1) holds for x ∈ [1, ∞). We can use the modified distribution FM to relate condition (5) to an analogous condition for the single-stage newsvendor problem maxq Π0 (q) = pS(q) − cq. The well-known fractile
solution of this problem can be written as F¯ (q ∗ ) = pc . Since S(q) represents the expected
′ sales and F¯ (q) = S (q) represents the marginal expected sales, the fractile formula states
that the marginal expected sales are equal to the profit margin ratio pc . Similarly, since F¯M (x) = F¯ (x)[1 − g(x)] for x > A, condition (5) can be written as F¯M (q ∗ ) = pc . Therefore
′ SM (q) represents the expected sales for the problem (3), F¯M (q) = SM (q) represents the
corresponding marginal expected sales and condition (5) states that the marginal expected sales for problem (3) are equal to the profit margin ratio pc . The following result extends Theorem 4 in LP (who assumed that F is IGFR which implies (A1)) on the division of supply chain profits by specifying when the manufacturer captures less than half of the total supply chain profit.
Proposition 1 (Division of Supply Chain Profits). If (A1) holds and g(A+ ) < 1 − pc , then the profit ratio is
Π∗m Π∗r
= EG(q ∗ ). Thus, if F is
IFR or G(x) is convex for x ≥ 0 such that g(x) ≤ 1, then Π∗m ≥ Π∗r ; if G(x) is concave for
x ≥ 0 such that g(x) ≤ 1, then Π∗m ≤ Π∗r .
5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Proof of Proposition 1. It follows from Theorem 1 that, if g(A+ ) < 1 − pc , then the equilibrium order quantity
q ∗ is such that q ∗ > A ≥ 0, g(q ∗) < 1 and it satisfies the first order condition (5). Thus,
Π∗r = pG(q ∗ ) , Π∗m = [pF¯ (q ∗ ) − c]q ∗ = pF¯ (q ∗ )g(q ∗)q ∗ and
Π∗m pF¯ (q ∗ )g(q ∗ )q ∗ [f (q ∗ )q ∗ ]q ∗ G (q ∗ )q ∗ = = = = EG(q ∗ ). Π∗r pG(q ∗ ) G(q ∗ ) G(q ∗ ) ′
′
If G(x) is convex (concave) for x ≥ 0 such that g(x) ≤ 1, then G(0)−G(x) ≥ (≤) G (x)(0−x), ′
which can be written as G (x)x ≥ (≤) G(x), or, equivalently as EG(x) ≥ (≤) 1, and thus,
since q ∗ > 0 and g(q ∗ ) < 1,
Π∗m Π∗r
≥ (≤) 1. This completes the proof.
LP state in their Theorem 4 that Π∗m ≥ Π∗r if F (x) is convex and that Π∗m ≤ Π∗r is possible if F (x) is concave. The assumption that F (x) is convex implies that F is IFR which in turn implies that G(x) is convex for x ≥ 0 such that g(x) ≤ 1, which is assumed in Proposition 1. Conversely, the concavity of G(x) for x ≥ 0 such that g(x) ≤ 1 is needed to prove that ′
Π∗m ≤ Π∗r . If G(x) is convex for x ≥ 0, then G (x) and g(x) are nondecreasing for x ≥ 0 and ′
thus F is IGFR. Also, if F is DGFR, then g(x) and G (x) are nonincreasing for x ≥ 0 and thus G(x) is concave for x ≥ 0. In summary, our Proposition 1 indicates that the division of supply chain profits depends on the curvature of G(x).
2.1
A Two-Stage Supply Chain with a Fixed Retail Price and Competing Retailers
In this subsection, we consider a two-stage supply chain with a single manufacturer and multiple competing retailers analyzed in [7]. We show that the introduction of a modified demand distribution provides additional managerial insights for the problem and facilitates its solution under more general assumptions. The two-stage model in [7] is based on the single-stage competitive newsvendor model in [5] with multiple competing retailers. We assume in this subsection that the pdf f (x) has support on [0, ∞), that is A = 0 but we do not require that F is IGFR in contrast to Perakis and Roels [7]. As before, we also assume that g(x) ≤ 1 for all x ≥ 0.
6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Following Lippman and McCardle [5], Perakis and Roels [7] assume that customers move as a herd and visit the n retailers sequentially until the total demand is met, where all sequences (permutations) of the n retailers are equally likely. This implies that the retailers are symmetric and thus the manufacturer offers the same wholesale price w to all retailers and, in turn, each retailer orders the same optimal quantity q(w). The retailer’s problem is identical for all retailers and it can be stated as max Πr (q) = p q≥0
n X
1 ( )E[min(q, (x − (k − 1)q(w))+ ] − wq, k=1 n
where q(w) is the optimal solution of (6) which uniquely solves the equation p
(6) Pn
1 ¯ k=1 ( n )F (kq)
=
w ([5]). Given q(w), the manufacturer’s profit maximization problem is maxw Πm (w) = n(w − c)q(w) and it can be reformulated in terms of q as max Πm (q) = p q≥0
n X
k=1
F¯ (kq)q − ncq.
(7)
We will show next that the unique optimal solution q ∗ of the manufacturer’s problem (7) is also the unique solution of a newsvendor problem stated as max Πc (q) = pSM (q) − cq, q≥0
where SM (q) =
Rq 0
F¯M (x)dx and FM is a modified distribution defined as FM (x) =
where H(x) = ( n1 )
(8)
Pn
¯
k=1 F (kx)[1
if x ≤ 0,
0,
1 − H(x), if x > 0,
(9)
− g(kx)].
Theorem 2 (Solvability of the Manufacturer’s Problem). If g(0+ ) = 0, g(x) ≤ 1 for all x ≥ 0 and (A1) holds, then the modified distribution FM given by (9) is well defined. The profit functions in (7) and (8) are unimodal on [0, ∞) and the unique optimal solu-
tions q ∗ of problems (7) and (8) exist and are identical. Moreover, q ∗ > 0 and q ∗ satisfies the condition p
n X
k=1
F¯ (kq)[1 − g(kq)] = nc. 7
(10)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Proof of Theorem 2. Suppose that the assumptions of Theorem 2 hold. Since g(0+ ) = limx→0+ g(x) = 0, H(0+ ) = 1 and FM (0+ ) = 0 and FM (x) is continuous on (−∞, ∞). Since (A1) is equivalent
to F¯ (x)[1 − g(x)] nonincreasing for x > 0, this implies that H(x) is also nonincreasing for x > 0 so that FM (x) is nondecreasing for all x and the modified distribution FM is well
defined. Thus, SM (q) =
Rq
pSM (q) − cq = p( n1 )
0
R P F¯m (x)dx = 0q H(x)dx = ( n1 ) nk=1 F¯ (kq)q for q > 0 and Πc (q) =
Pn
¯
k=1 F (kq)q
− cq = ( n1 )Pm (q) for q > 0. Therefore, the problems
maxq≥0 Πm (q) and maxq≥0 Πc (q) are equivalent. Since dΠm (q)/dq = p
Pn
k=1
F¯ (kq)[1 − g(kq)] − nc, dΠm (0+ )/dq = np − nc > 0 and
dΠm (∞)/dq = −nc < 0. Also, since (A1) implies that F¯ (x)[1 − g(x)] is nonincreasing for
x > 0, dΠm (q)/dq is nonincreasing for q > 0 and Πm (q) is concave for q > 0. Therefore, the profit function Πm (q) is unimodal on [0, ∞) and there exists a unique optimal solution q ∗ of
(7) such that dΠm (q ∗ )/dq = 0 which is equivalent to (10).
Finally, since the problems maxq≥0 Πm (q) and maxq≥0 Πc (q) are equivalent for x > 0, q ∗ > 0 is also a unique optimal solution of of (8). This completes the proof. Theorem 2 shows that the unique equilibrium exists under assumption (A1). This result generalizes a similar result in [7] stated under the IGFR assumption. Since F¯M (x) = H(x) = ( n1 )
Pn
¯
k=1 F (kx)[1−g(kx)]
for x > 0, condition (10) can be written
as F¯M (q ∗ ) = pc . Therefore, as in the single retailer case, SM (q) represents the expected sales ′ for problem (8), F¯M (q) = SM (q) represents the corresponding marginal expected sales and
condition (10) states that the marginal expected sales for problem (8) are equal to the profit margin ratio pc . The next result partially extends Proposition 1 to the competing retailers case. Proposition 2 (Division of Supply Chain Profits). If g(0+ ) = 0, g(x) ≤ 1 for all x ≥ 0 and (A1) holds, then the profit ratio is
EGn (q ∗ ) +
G(nq ∗ ) Gn (q ∗ )
− 1, where Gn (x) = S(nx) − 8
Pn
k=1
F¯ (kx)x.
Π∗m nΠ∗r
=
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Thus, if Gn (x) is convex for x ≥ 0 such that g(x) ≤ 1, then Π∗m > nΠ∗r ; if Gn (x) is
concave for x ≥ 0 such that g(x) ≤ 1 and EGn (q ∗ ) +
G(nq ∗ ) Gn (q ∗ )
≤ 2, then Π∗m ≤ nΠ∗r .
Proof of Proposition 2. It follows from Theorem 2 that the equilibrium order quantity q ∗ satisfies the condition (10). Furthermore, one can show that nΠ∗r = pGn (q ∗ ) = p[S(nq ∗ ) − Π∗m = p
Pn
k=1
F¯ (kq ∗ )g(kq ∗)q ∗ . Thus,
Π∗m = nΠ∗r
Pn
¯
k=1
¯
k=1 F (kq
)g(kq ∗ )q ∗ . Gn (q ∗ )
k=1 F (kq
∗
P ′ Since (Gn ) (q ∗ ) = nF¯ (nq ∗ ) − nk=1 F¯ (kq ∗ )[1 − g(kq ∗ )], n X
Pn
′ F¯ (kq ∗ )g(kq ∗ )q ∗ = (Gn ) (q ∗ )q ∗ − nq ∗ F¯ (nq ∗ ) +
∗
)q ∗ ] and
(11)
n X
F¯ (kq ∗ )q ∗ .
(12)
k=1
Moreover, since G(nq ∗ ) = S(nq ∗ ) − nq ∗ F¯ (nq ∗ ), nq ∗ F¯ (nq ∗ ) −
n X
k=1
F¯ (kq ∗ )q ∗ = Gn (q ∗ ) − G(nq ∗ ).
(13)
Equations (11)-(13) imply that ′
(Gn ) (q ∗ )q ∗ Gn (q ∗ ) − G(nq ∗ ) G(nq ∗ ) Π∗m ∗ = − = EG (q ) + − 1. n nΠ∗r Gn (q ∗ ) Gn (q ∗ ) Gn (q ∗ ) If Gn (x) is convex (concave) for x ≥ 0 such that g(x) ≤ 1, then, in view of the proof of Proposition 1, EGn (x) ≥ (≤) 1. Thus, if Gn (x) is convex for x ≥ 0 such that g(x) ≤ 1, then,
since G(nq ∗ ) > Gn (q ∗ ), Π∗m > nΠ∗r . If Gn (x) is concave for x ≥ 0 such that g(x) ≤ 1 and
EGn (q ∗ ) +
3
G(nq ∗ ) Gn (q ∗ )
≤ 2, then Π∗m ≤ nΠ∗r .
Conclusions
The reformulation of the manufacturer’s problem can be also implemented to the assembly system comprising a downstream retailer who faces the newsvendor problem, a midstream assembler and n upstream simultaneous suppliers. By utilizing a modified distribution, this system can be reduced to an equivalent two-stage assembly system with a downstream assembler and simultaneous upstream suppliers (analyzed by Gerchak and Wang [2]), with the modified distribution FM in the place of F . 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Since Gerchak and Wang [2] require that F is IGFR in order for the unique equilibrium to exist for their assembly system, we also require that FM is IGFR in the three-stage system. Kyparisis and Koulamas [3] showed that if F is strictly IGFR and either g(x) is convex ′
for x > 0 or Eg (x) ≥ −1 for x > 0, then FM is IGFR on (0, C). One can verify that the condition that g(x) is convex for x > 0 is satisfied by most IFR distributions while the ′
condition that Eg (x) ≥ −1 for x > 0 is additionally satisfied by some IGFR distributions. In general, this type of reduction is feasible whenever a multi-stage system with a wholesale-price contract has a newsvendor problem in the last stage and a manufacturer’s problem in the next to last stage. It will be of interest to investigate whether the concept of the modified demand distribution applies to alternative contracts such as buyback contracts and to alternative supply chain configurations such as pull supply chains. Acknowledgement. We would like to thank the Area Editor and the two referees for their insightful comments which helped us improve earlier versions of this paper.
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
References [1] Barlow, R. E., F. Proschan. 1996. Mathematical Theory of Reliability. G. H. Golub, ed. SIAM Classics in Applied Mathematics. Society for Industrial and Applied Mathematics. Philadelphia. [2] Gerchak, Y., Y. Wang. 2004. Revenue-sharing vs. wholesale-price contracts in assembly systems with random demand. Production and Operations Management 13(1) 23-33. [3] Kyparisis, G. J., C. Koulamas. 2016. Assembly Systems with Sequential Supplier Decisions and Uncertain Demand. Production and Operations Management forthcoming. [4] Lariviere, M. A., E. L. Porteus. 2001. Selling to the newsvendor: An analysis of price-only contracts. Manufacturing & Service Operations Management 3 293-305. [5] Lippman, S. A., K. F. McCardle. 1997. The competitive newsboy. Operations Research 45 54-65. [6] Paul, A. 2006. On the unimodality of the manufacturer’s objective function in the newsvendor model. Operations Research Letters 34 46-48. [7] Perakis, G., G. Roels. 2007. The price of anarchy in supply chains: Quantifying the efficiency of price-only contracts. Management Science 53(8) 1249-1268. [8] Xu, Y., A. Bisi. 2012. Wholesale-price contracts with postponed and fixed retail prices. Operations Research Letters 40 250-257. [9] Ziya, S., H. Ayhan, R. D. Foley. 2004. Relationships among three assumptions in revenue management. Operations Research 52 804-809.
11