NESCC 13-048 - Draft Report of the Welding Task Group

14 Pages·2013·0.26 MB·English

JOURNALOFLATEXCLASSFILES,VOL.*,NO.**,*********** 1
Optimal Inventory Decisions in the Multi-period
News-vendor Problem with Partial-Observed
Markovian Supply Capacities
Haifeng Wang, and Houmin Yan
Abstract—Thispaperconsidersamulti-periodnews-vendor I. INTRODUCTION
problem with partially observed supply capacity information
Inventorycontrolisoneofthemoststudiedtopicsforthe
which evolves as a Markovian Process. The supply capacity
supply chain management. Both demand and supply infor-
is fully observed by the buyer when the capacity is smaller
thanbuyer’sorderingquantity.Withadynamicprogramming mation is fundamental for any inventory control decisions.
formulation, we prove the existence of a unique optimal Most works in the supply chain management literature
ordering policy. In a two-state Markovian capacity case, we assumethatthesupplycapacityiseitherknowntothebuyer,
furtherdemonstratethatthebuyerordersmorethanrequired
or unlimited. Few of them consider the supply capacity
torevealsupplycapacity.Wealsoprovideanumericalexample
information uncertainty and its influence to the inventory
to demonstrate the characterization of the optimal policy.
policy. Yet the supply capacity is always unknown or only
Note to Practitioners—Supplier management is a critical
partially observable in practice. In such an environment,
task for inventory managers, in particular, when there ex-
mostofwell-knownresultsmaynothold.Itistheobjective
isting supply uncertainties. In such circumstances, inventory
managers have to make inventory replenishment decisions of this paper to explore the structural properties of an
based on their best estimation of supply conditions. Their inventory system with partially observed supply capacity
past experiences are the basis for estimation. Note that their and to study if there exists an optimal policy for the above
past experiences, to the best, only partially reveals the supply
problem.
condition.Thispaperdevelopsanewmodelinwhichitmakes
useofapartialobservedsupplycapacityinformation,whichis
derived in the order fulfillment process, to forecast the future
A. The Importance of Modeling the Partially Observed
supply capacity. There are two possible observations of the
Capacity
order fulfillment: (1) if the ordering quantity is greater than
the current period supply capacity, the buyer observes the The study of inventory systems with the partially ob-
valueofthecurrentperiodsupplycapacity;(2)otherwise,the served supply capacity is important in many real-life cases.
buyerknowsthatthecurrentperiodsupplycapacityisgreater
We shall introduce some of cases in a supply chain, where
thanitsorderingquantity.Basedonthesetwoobservations,the
capacities can be partially observed.
buyer updates the future supply capacity forecasting accord-
ingly.Comparedwithtraditionalmodels,wedemonstratethat Supplier’s capacity uncertainty: Unexpected machine
making use of the partial observed information significantly breakdowns, resulting unexpected maintenance, may lead
reduces costs of inventory systems. However, this inventory to a lower production capacity. Moreover, the uncertain
model assumes that all the leftover inventories are salvaged
repairingtimemayaffecttheavailabilityofcertainfacilities:
at the end of each period. This assumption is applicable to
the repair is not planned, even planned, then its repair-
perishableproducts.Forfutureresearch,extendingthismodel
to cases where leftover inventories to be carried to the next ing time is uncertain. These stochastic events lead to an
period is the future research direction. uncertain supply capacity. Further, the product quality can
also be uncertain. When the product quality is low or the
Index Terms—Partial information, Markovian process, in-
ventory planning and control rate of defects is high, the actual available products are
unknown. Since these uncertainties can’t be predicted, the
supply capacity is not observed directly.
ThisworkissupportedinpartbyRGCCompetitiveEarmarkedResearch
Grants,CUHK4239/03E,CUHK4167/04EandCUHK4177/06E. Buyers’ competition: A supplier serves multiple cus-
Haifeng Wang is with the Center for Intelligent Networked Systems, tomers. When the capacity is tight, the supplier reserves
Department of Automation, Tsinghua University, Beijing 100084, China
its capacity to individual customer. It is known in practice
(email:[email protected]).
Houmin Yan is with the Department of Systems Engineering and as the capacity allocation, or rationing. Orders from buyers
EngineeringManagement,ChineseUniversityofHongKong,Shatin,N.T., varyfromtimeandareunknowntotheotherbuyers.Hence
Hong Kong. He is also a non-resident member of Center for Intelligent
the reserved capacity for an individual buyer is uncertain,
Networked Systems, Department of Automation, Tsinghua University,
Beijing100084,China(email:[email protected]). evenunderconditionofknowingthesupplyallocationrules.
JOURNALOFLATEXCLASSFILES,VOL.*,NO.**,*********** 2
As a result, a buyer doesn’t know the capacity volume Hence, our finding suggests inventory managers pay a par-
reserved for him. ticular attention to the partially observed supply capacity
Multiple sources of supply: The buyer also orders from informationwhenthereexistsasupplycapacityuncertainty.
multiple suppliers. Each supplier has its own reserved
capacity for the buyer. Hence the buyer can only estimate C. Literature Review and Its Relation to Our Model
the capacity reserved for him, based on observed signals, There is an extensive literature in inventory decisions
such as the received ordering quantity. As a result, the with the given capacity constraint, such as the work of
available capacity is partially observed. Federgruen and Zipkin [5], Gallego and Scheller-Wolf [8],
Gallego and Toktay [9], and Gallien and Wein [10]. In ad-
dition, there are papers dealing with the uncertain capacity
B. Summary of Paper
constraint. Khang and Fujiwara [7] address a discrete time
This paper studies the optimal inventory replenishment inventory model where the maximum amount of supply,
policyatthebeginningofeachperiodtomeetthestochastic from which instantaneous replenishment orders can be
demand with the partially observed supply capacity infor- made, is a random variable. They also characterize the
mation. We allow the buyer to fulfill the out-of-capacity optimal ordering policies. Wang and Gerchak [13] extends
orders by paying an extra cost from other sources, such themodeltoincludethevariablecapacityandrandomyield
as a spot market. The demand is realized at the end of together in periodic inventory systems.
each period. The leftover inventory is salvaged while the The research on inventory decisions with the partially
unfulfilled demand incurs a penalty cost. observed information is quite recent. Bensoussan, Cakany-
We consider a multi-period problem where the capacity ildirim and Sethi [4] study a multi-period news-vendor
observationofthecurrentperiodwillinfluencethecapacity model with a partially observed demand. By assuming
distributionofthenextperiod,aswellasthevaluefunction that the leftover inventories are salvaged and unsatisfied
of the next period. The capacity allocated by the supplier demands are lost in each period, they decouple the peri-
maychangefromperiodtoperiod,evolvingasaMarkovian ods from the view point of the Beyesian demand update,
Process. The capacity for the current period is NOT fully and prove the existence of optimal ordering policy. Aviv
observedbythebuyeratthetimewhenthebuyerplacesits and Pazgal [1] introduce the partially observed Markovian
orders. When the order quantity is greater than the supply decision process. In their setting, to maximize its revenue,
capacity, the supplier provides its maximum capacity to the a buyer faces a dynamic pricing problem of selling a
buyer. Only at this time, the capacity is fully observed by given stock during a finite selling season. Bensoussan,
thebuyer.Thepartiallyobservedcapacityinformationlimits Cakanyildirim and Sethi [2] and [3] study the inventory
buyer’s capability in forecasting its needs as well as the system with partially observed inventory levels. Lu, Song
inventory optimization. and Zhu [11] and [12] study the news-vendor model with
In this paper, we prove the existence of an optimal censored demand data.
ordering policy for a general Markovian capacity process. To the best of our knowledge, the problem of inven-
With results for the general case, we specialize our study tory decisions with a partially observed Markovian ca-
to cases where the supply capacity has two possible states. pacity remains open. We model an inventory system with
We find out that revealing the supply capacity leads to a the partially observed Markov supply capacity. Similar to
cost reduction. Hence the buyer may purchase more than Bensoussan, Cakanyildirim and Sethi [4], the unnormal-
neededinordertofigureoutthesupplycapacity.Moreover, ized probability is used to linearized the highly nonlinear
ifthepossiblecapacityislow,thebuyer’sorderingpolicyis dynamic programming equation. We solve the inventory
equivalenttoamyopicinventoryorderingpolicy.Otherwise, managementproblemwiththepartialinformationofsupply
the buyer makes the tradeoff between the current and the capacity, while Bensoussan, Cakanyildirim and Sethi [4]
future period costs. solve the inventory management problem with the partial
We also study the performance of the partial-observation information of demand. The methodology in Bensoussan,
model and the non-update model. In the partial-observation Cakanyildirim and Sethi [4] is extended in our model to
model, the buyer dynamically updates the supply capacity prove the existence of optimal inventory policy. Moreover,
forecast according to the partial supply information based an extensive model with two possible supply capacities is
on the order fulfillment in each period. On the contrary, for studied and the buyer’s tradeoff between the current and
non-update model, this partially observed supply capacity future period cost is characterized.
information is ignored. We prove the existence of the
optimal inventory replenishment for the partial-observation D. Plan of Paper
model. And we use numerical experiments to demonstrate In the next section, we formulate the problem. In our
significant cost savings of this partial-observation model. model, the evolution of partially observed Markovian ca-
JOURNALOFLATEXCLASSFILES,VOL.*,NO.**,*********** 3
pacity is characterized, and the un-normalized probability B. Dynamic forecast updates for the supply capacity
is used to linearize the dynamic equation. In Section 3, we
In each period, the buyer observes a , which is the
t
provetheexistenceofanoptimalorderingpolicy.InSection
amount of ordering charged at the normal purchasing cost
4, we specialized the optimal inventory replenishment pol-
c. Then, two scenarios are as follows.
icy for the problem with two-state Markovian capacity. We
1) When the order quantity is greater than the supply
alsocarryoutnumericalexperiments.Wefindthatordering
capacity, i.e. q ≥ Q , the supplier can’t satisfy the
uptothesmallerpossiblecapacitycansavethefuturecost. t t
request. Then the supplier exhausts all of its capacity
Finally, we conclude the paper in Section 5.
to produce products, i.e. a = Q . For this scenario,
t t
the supply capacity reveals through the observed a .
t
The density probability function of the capacity for
II. PROBLEMFORMULATIONANDDYNAMIC
the next period can be obtained as follows:
PROGRAMMINGEQUATIONS
π (Q )=p(Q |a ); (1)
t+1 t+1 t+1 t
A. Sequence of Events and Cost Structure
2) On the other hand, the order quantity is less than the
The sequence of events is as follows: at the beginning of supply capacity, and the supplier satisfies the request
each period, the buyer informs the supplier for its intended completely, i.e. q ≤ Q and a = q . For this
t t t t
order quantity qt; thus the supplier checks the available scenario, the buyer knows that the capacity is greater
capacity reserved for this buyer and tries its best to satisfy thanitsorderquantity,i.e.Q ≥a ,butdoesn’tknow
t t
therequestedpurchase.Denotethereservedsupplycapacity theexactcapacity.Wedenotethiscaseasthepartially
as Qt. Note that {Qt} is a Markovian Process with a observed supply capacity. Then
transition probability p(Q | Q ), which is exogenously
t+1 t
decided as an input. The buyer gets 1) at = qt ∧ Qt πt+1(Qt+1) = Pr{Qt+1 |Qt ≥at}
at the normal unit purchasing cost c and 2) qt − at, if = Pr{Qt+1,Qt ≥at}
the unsatisfied order is greater than 0, at a higher unit Pr{Q ≥a }
t t
(cid:82)
purchasing cost c+e, where e≥0 indicates that the extra +∞
π (ξ)p(Q |ξ)dξ
ordercanbemoreexpensive.Notethattheextraordermay = at (cid:82)t t+1 , (2)
+∞
come from the same supplier by asking for an expensive, at πt(ξ)dξ
emergentorder,orcomefromsomeothersupplyresources,
where the second equality is due to Bayesian equa-
such as a spot market. At the end of this selling season,
tions, and the third equality is due to the condi-
all the unsatisfied demand is lost. Shortage cost at the unit
tionalprobabilityequations,whichisPr{Q ,Q ≥
cost of b, and salvaged value at the unit revenue of h are (cid:82)+∞ t+1 t
a }= Pr{Q |Q =ξ}π (ξ)dξ.
recorded. t at t+1 t t
Hence the supply capacity Q is partially observed.
The following is the list of notations used in this paper: t
This approach is similar to Bensoussan, Cakanyildirim and
Sethi [4] where the partially observed demand signal is
q : order quantity from the buyer;
t characterized.
Q : available capacity of the supplier;
t
α: discount factor;
Dt : demand of period t; C. Dynamic Programming Equations
p(Q |Q ): transition probability of a Markov
t+1 t With the sequence of events described above, it is possi-
process {Q };
t ble for us to derive one-period cost function as L(D ,q )+
Pr{A}: probability of the event A; t t
e(q −Q )+, where
c: normal unit purchasing cost; t t
(cid:189)
at : amount of order charged at normal cy−h(y−x), if x≤y
L(x,y) =
purchasing cost c; cy+b(x−y), if y ≤x,
(cid:189)
e: additional unit purchasing cost for the
cy−h(y−x), if x≤y
extra unit when q >Q ; = (3)
t t bx+(c−b)y, if y ≤x.
h: unit salvage value;
b: unit shortage cost; To eliminate trivial cases, it is reasonable to assume that
F(·): cumulative demand distribution function; 0≤h<c<b and 0<e. Then it is straightforward to see
π (·): probability density function of capacity that
t
(cid:189)
Q ;
t cy, if x≤y
V (·): value function. L(x,y) ≤ if y ≥0. (4)
t bx, if y ≤x
JOURNALOFLATEXCLASSFILES,VOL.*,NO.**,*********** 4
I
Recallthatπ (·)istheprobabilitydensityfunctionofthe otherwise =0. Define recursively that
t A
capacity Qt. Then the value function can be expressed as (cid:90) +∞
I
follows: ρ (x) := p(x|ξ)ρ (ξ)dξ
t+1 at=qt t
qt
Vt(πt(cid:90)(x)qt) ρ (x) := +πI(xat)<.qtp(x|at), t≥1;
= inf { E [L(D ,q )+e(q −Q ) 1 1
qt≥0 0 Dt t t t t
Also let
+αVt+1(p(x|Qt))]πt(Qt)dQt (cid:90)
(cid:90)
+∞ λ := ρ (x)dx.
+ E [L(D ,q )+ t t
Dt t t
qt (cid:82) Then
+∞π (ξ)p(x|ξ)dξ (cid:90)
αVt+1( qt (cid:82)q+tt∞πt(ξ)d(cid:90)ξ )]πt(Qt)dQt} λt+1 =Iat=qt qt+∞ρt(ξ)dξ+Iat<qt, t≥1;λ1 =1.
qt We can check recursively (it holds when t = 1; and by
= inf {E [L(D ,q )]+ e(q−Q )π (Q )dQ
qt≥0(cid:90) Dt t t 0 t t t t supposingthecorrectnessfort,checkthatitholdsfort+1)
qt that the following equation is true:
+α V (p(x|Q ))π (Q )dQ
t+1 t t t t
0 (cid:82)+∞ (cid:90) ρt(x) = πt(x)λt.
π (ξ)p(x|ξ)dξ +∞
+αV ( qt (cid:82) t ) π (Q )dQ }, Hence, we have
t+1 +∞ t t t
qt πt(ξ)dξ qt (5) πt(x) = ρtλ(x) = (cid:82) ρρt((xx))dx.
t t
where the first term on the right hand side is In what follows, we consider the infinite horizon case. To
the cost when the supply capacity Q is smaller simplify the notation, the sub-script is omitted.
t
than the buyer’s intended ordering quantity, and We define W(ρ) as
(cid:82)
+∞ (cid:90)
the sRecond term of + qt EDt[L(Dt,qt) + W(ρ) := ρ(x)dx·V((cid:82) ρ ). (6)
tαhVet+su1p(pql+ytR∞q+ctπ∞atp(πaξtc)(piξt()y·d|ξξQ)dξ)i]sπtl(aQrgte)rdtQhatn itshe tbhueyerc’ossitntewnhdeend It is obvious that W(·) can be obtaiρn(exd)dbxy V(·). By
t
ordering quantity. definition of W(ρ), we have
The buyer chooses a set of purchasing quantities {q } to
t W(ρ(·))
minimize its total cost. Ordering too much results in a high (cid:90) (cid:90)
purchasingcost;ontheotherhand,orderingtoolessresults = ρ(x)dx· inf { L(D,q)f(D)dD
q≥0
in a high penalty cost and failures to observe the supply (cid:90)
q ρ(Q)
capacity. + e(q−Q)(cid:82) dQ
ρ(x)dx
0(cid:90)
q ρ(Q)
+α V(p(·|Q))(cid:82) dQ
D. Unnormalized Probability ρ(x)dx
0
(cid:82) (cid:41)
It is difficulty to directly solve the dynamic equation +∞ρ(ξ)p(·|ξ)dξ (cid:90) +∞ ρ(Q)
(5) owning to complex fractional parts. Both denominator +αV( q (cid:82) ) (cid:82) dQ
+∞ρ(ξ)dξ q ρ(x)dx
and numerator include the decision variable qt. In this part, (cid:189)(cid:90) q (cid:90)
we use the similar technique of a variable substitution as
= inf L(D,q)f(D)dD ρ(x)dx
in Bensoussan, Cakanyildirim and Sethi [4] so that the q≥0
(cid:90)
equation (5) can be expressed in a simpler and solvable q
+ e(q−Q)ρ(Q)dQ
form.
0(cid:90)
From equations (1) and (2), we know q
+α W(p(·|Q))ρ(Q)dQ
(cid:82)
π (x) = I q+t∞(cid:82)πt(ξ)p(x|ξ)dξ 0 (cid:90) +∞ (cid:190)
t+1 at=qt +∞π (ξ)dξ +αW( ρ(ξ)p(·|ξ)dξ)
qt t q
I
+ p(x|a ), t≥1, := inf G(q). (7)
at<qt t q≥0
I
where the indicator function =1 when event A occurs, Note that Equation (7) is a Bellman equation in ρ.
A
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III. THEEXISTENCEOFANOPTIMALFEEDBACK period forecasted by p(·|·) is equal to the mean of current
SOLUTION period demand.
Inthissection,wefirstshowtheexistenceanduniqueness We need the following lemma. Its proof appears in
of the solution W for equation (7). Then we prove the Appendix.
existence of an optimal feedback control q∗. Lemma 3.1: If equation (7) has a solution W, the solu-
tion is in B.
A. Existence of a Unique Solution W Now define the map T(W) as,
In this subsection we start with a definition of a function (cid:90) (cid:90)
space B. Then we prove that, if there is a solution to T(W) := min{ L(D,q)f(D)dD ρ(x)dx
equation (7), it must exist in the above defined function q≥(cid:90)0
q
space. As a result, the existence and the uniqueness of the
+ e(q−Q)ρ(Q)dQ
solution follow. Note that similar definitions of function 0(cid:90)
spacesHandB areusedinBensoussan,Cakanyildirimand q
+α W(p(·|Q))ρ(Q)dQ+
Sethi [4].
(cid:90)0
Define function space H of function ρ: +∞
(cid:90) αW( ρ(ξ)p(·|ξ)dξ)}. (15)
∞
H:={ρ∈L1(R+): x|ρ(x)|dx<∞}, (8) q
0
Thenwecanobtainthefollowinglemma.SeeAppendixfor
where L1(R+) is the space of integrable functions whose
its proof.
domain is the set of nonnegative real numbers, and
Lemma 3.2: ||T(W)−T(W˜)|| ≤ αmax{1,c }||W −
B 0
H+ :={ρ∈H|ρ≥0}, (9) W˜|| .
B
Theorem 3.1: There exists a unique solution W of the
with the norm:
(cid:90) (cid:90) dynamic programming equation (7).
+∞ ∞
||ρ||= |ρ(x)|dx+ x|ρ(x)|dx. (10) Proof: From Assumption 3.1, we know that αc < 1,
0
0 0 such that αmax{1,c } < 1. Hence by Lemma 3.2, the
0
Also define the following space B of function φ: mappingT :B →B isacontractionmapping.Thenbythe
(cid:189) (cid:190)
Contraction Mapping Theorem [6], there exists a unique
|φ(ρ)|
B = φ(ρ):H+ →R|sup <∞ , (11) solution W such that T(W)=W. Hence we have proved
||ρ||
x>0
the desired results. (cid:164)
with the norm:
|φ(ρ)|
||φ|| = sup . (12)
B ||ρ||
ρ∈H+ B. The existence of Optimal Feedback Control
For the technical convenience, we need the following as-
By Equation (3), we can easily obtain L(x,y) ≥ (c−
sumptions:
h)y. Substitute it into Equation (7), we then can prove that
Assumption 3.1: Assume, for any ρ∈H+, (cid:82)
(cid:90) (cid:90) (cid:90) W(ρ) ≥ (c−h)q ρ(x)dx. With equations (52) and (56),
we know that
x p(x|ξ)ρ(ξ)dξdx≤c ξρ(ξ)dξ, with αc <1.
0 0
(cid:90)
(13)
(c−h)q ρ(x)dx≤W(ρ)
This assumption is necessary to complete proofs of the
(cid:181) (cid:182)
followinglemmasandthetheorem.Itissatisfiedbyspecific
bµ
probability transition function p(·|·), such that ≤ µD +αmax{1,c0}||W||B ||ρ||
(cid:90) (cid:90) (cid:90) (cid:90) (cid:181) Q (cid:182)
bµ bµ
α x p(x|ξ)ρ(ξ)dξdx≤αc0 ξρ(ξ)dξ < ξρ(ξ)dξ, ≤ D +αmax{1,c } D ||ρ||.
µ 0 µ (1−αmax{1,c })
Q Q 0
i.e.
(cid:82) R R
The discounted expected demand of next period Note that ρ(x)dx = R xρ(x)dx = xρ(x)dx, then
R R xπ(x)dx µQ
forecasted by p(·|·) R ||ρ|| = ρ(x)Rdx+ xρ(x)dx = 1+µ . Hence we obtain
ρ(x)dx ρ(x)dx Q
that
< The mean of current period demand. (14)
Since the discount factor is always strictly smaller than 1, bµ (1+µ )
0≤q ≤ D Q :=B. (16)
this assumption holds when the expected demand of next µ (c−h)(1−αmax{1,c })
Q 0
JOURNALOFLATEXCLASSFILES,VOL.*,NO.**,*********** 6
Now we define W (ρ) as values of Q and Q with Q < Q . Note that three or
B l h l h
more finite state problem can be analyzed by the similar
W (ρ)
B (cid:90) (cid:90) approach, with much more tedious forms and equations.
= min { L(D,q)f(D)dD ρ(x)dx We assume that the un-normalized capacities are Ql and
q∈[0,B] Q , with probability of θ , and θ , respectively. Hence the
(cid:90) h l h
q
distribution function can be written as
+ e(q−Q)ρ(Q)dQ
0(cid:90) ρ(x)=θ I +θ I . (21)
q l x=Ql h x=Qh
+α W (p(·|Q))ρ(Q)dQ
B
Suppose that the transition probability functions are
0 (cid:90) (cid:190)
+∞
I I
+αW ( ρ(ξ)p(·|ξ)dξ) . (17) p(x|Q )=β +(1−β ) ,
B l l x=Ql l x=Qh
q I I
p(x|Q )=(1−β ) +β , (22)
h h x=Ql h x=Qh
Note that W (ρ) can be shown to be a unique solution to
B
equation(17)sinceW(ρ)isshowntobeuniquebyLemma that is to say, the transition matrix is
(cid:181) (cid:182)
3.2.IfW (ρ)iscontinuousinρ,weobtainthecontinuityof
B β 1−β
G(q)(definedinequation(7))inq.Afterthat,theexistence B:= l l .
1−β β
h h
oftheminimizerq∗ isestablished.SeeAppendixforproofs
of the following lemma. ThenthedynamicprogrammingEquation(7)canbefurther
Lemma 3.3: For any ρ,ρ˜∈H+, we have reduced to
(cid:90)
q
|W (ρ)−W (ρ˜)| ≤ H ||ρ−ρ˜||, (18) W(ρ) = min{[cq−h (q−D)f(D)dD
B B B
q≥0(cid:90) 0
where H is a constant which is independent of ρ. +∞
B
+b (D−q)f(D)dD](θ +θ )
Theorem 3.2: There exists an optimal feedback control, l h
q
i.e, the optimal order quantity.
I
+ e(q−Q )θ
Proof: Recall the definition of G(q) in Equation (7), and Ql≤q<Qh l l
I
we can write + Qh≤q[e(q−Ql)θl+e(q−Qh)θh]
WB(ρ)=q∈m[0i,nB]G(q), (19) +αIQl<q≤QhθlW(p(·|Ql))
I
+α [W(p(·|Q ))θ +W(p(·|Q ))θ ]
where Qh<q l l h h
(cid:90) (cid:90) +αI W(θ p(·|Q )+θ p(·|Q ))
q≤Ql l l h h
G(q) = L(D,q)f(D)dD ρ(x)dx I
+α W(θ p(·|Q ))}. (23)
(cid:90) Ql<q≤Qh h h
q
+ e(q−Q)ρ(Q)dQ With the definition of W(ρ) in Equation (6), we obtain
0(cid:90)
q W(aρ(.))=aW(ρ(.)), if a is a constant. (24)
+α W (p(·|Q))ρ(Q)dQ
B
0 (cid:90) By collecting and simplifying terms, we have
+∞ (cid:90)
+αW ( ρ(ξ)p(·|ξ)dξ). (20) q
B W(ρ) = min{[cq−h (q−D)f(D)dD
q
q≥0(cid:90) 0
The first, second, and third terms of equation (20) are +∞
continuous in q. By Lemma 3.3, we know that W(ρ) is +b (D−q)f(D)dD](θl+θh)
continuous in ρ, which leads to the continuity of the fourth q
I
+ e(q−Q )θ
term of equation (20) in q. Hence G(q) is continuous in Ql≤q<Qh l l
q. Moreover, q belongs to a bounded and closed set [0,B]. +I [e(q−Q )θ +e(q−Q )θ ]
Qh≤q l l h h
Hence,byWeierstrass’Theorem[6],thereexistsaminimum I
+α [W(p(·|Q ))θ +W(p(·|Q ))θ ]
intheboundedandclosedsettominimizesuchacontinuous Ql<q l l h(cid:111) h
I
function. The minimum q∗ can be obtained. (cid:164) +α q≤QlW[θlp(·|Ql)+θhp(·|Qh)] . (25)
Note that the aim of observing current period supply
IV. OPTIMALPOLICYANALYSISOFTWO-STATE capacity is to predict supply capacity for future periods.
MARKOVCHAIN And with this prediction, it is possible for us to adjust our
Inordertofurtheranalyzetheoptimalpurchasingpolicy, purchasing quantity to minimize the cost function. In order
and reveal the supply capacity, we specialize our study to to characterize the purchasing policy which influences the
cases where the supply capacity can only take two possible costs for the current (myopic cost) and the future period
JOURNALOFLATEXCLASSFILES,VOL.*,NO.**,*********** 7
(cost-to-go from the next period), we divide the total cost Then P (q) = g (q)+g (q). The minimizer of function
c 1 2
function into two parts: P (q) and P (q). Define g (q) is
c f 1
(cid:90) b−c
q F−1( ).
P (q) := [cq−h (q−D)f(D)dD b−h
c
(cid:90) 0 g (q) is a piece-wise linear function with minimizers less
+∞ 2
+b (D−q)f(D)dD](θl+θh) than or equal to Ql.
q For the notation convenience, we further define
I
+ Ql≤q<Qhe(q−Ql)θl q¯ := F−1(b−c),
+I [e(q−Q )θ +e(q−Q )θ ], c b−h
Qh≤q l l h h
b−c−e θl
and q¯ := F−1( θl+θh),
e1 b−h
I
P (q) := α [W(p(·|Q ))θ +W(p(·|Q ))θ ] b−c−e
f Ql<q l l h h q¯ := F−1( ). (29)
+αI W [θ p(·|Q )+θ p(·|Q )]. e2 b−h
q≤Ql l l h h
Obviously, we have q¯ >q¯ >q¯ .
c e1 e2
Then the total cost can be expressed as follows:
1) Case 1: q¯ <Q : Since q¯ minimizes both g (q) and
c l c 1
W(ρ) = min{Pc(q)+Pf(q)}. (26) g2(q), the minimizer
q≥0
b−c
q∗ =q¯ =F−1( ). (30)
c b−h
A. The case of the Independent Capacity
2) Case 2: Q ≤q¯ <Q : In this case, since g (q) and
When the prevailing future capacity is independent to l c h 1
g (q)arenon-increasingforq ∈[0,Q ]andnon-decreasing
the current capacity, we denote this case as the case 2 l
for q ∈ [Q ,+∞), we can narrow down our search space
of the independent capacity. Mathematically, the transi- h
to [Q ,Q ]. That is
tion probabilities are chosen as β = 1 − β , which l h
l h (cid:90)
q
leads to p(·|Q ) = p(·|Q ). With Expression (24),
l h min {[cq−h (q−D)f(D)dD
W [θlp(·|Ql)+θhp(·|Ql)]=(θl+θh)W(p(·|Ql)). Hence (cid:90) q∈[Ql,Qh] 0
+∞
I
Pf(q) = α Ql<q[W(p(·|Ql))θl+W(p(·|Ql))θh] +b (D−q)f(D)dD](θl+θh)+e(q−Ql)θl}. (31)
+αI W [θ p(·|Q )+θ p(·|Q )] q
q≤Ql l l h l
Let us solve the first order condition, and we have
= α(θ +θ )W(p(·|Q )), (27)
l h l
b−c−e θl
which is independent of q. No matter how the purchasing F−1( θl+θh)=q¯ <q¯ <Q .
b−h e1 c h
policyis,thevaluefunctiondoesn’tchangeatall.Basedon
this, W(ρ) can be reduced as follows: Hence the optimality is obtained at the point q¯e1 or the
lower bound of Q , i.e.
l
W(ρ) = min{P (q)}+P . (28)
q≥0 c f b−c−e θl
q∗ = max{q¯ ,Q }=max{F−1( θl+θh),Q }.
Hence the myopic solution is optimal for the case of the e1 l b−h l
independent capacity. In other words, the buyer only needs (32)
tofocusonreducingthecurrentperiod’scost.Theresulting
3) Case 3: Q ≤ q¯: In this case, since g (q) and
myopic policy is also optimal in the view of multi-period. h c 1
g (q) are non-increasing for q ∈ [0,Q ], similarly, we
This result is intuitively right. In what follows, we figure 2 l
narrow down our search space to [Q ,+∞). Note that
out what exactly q∗ is, and how other factors influence the l
g (q)+g (q) is equal to equation (31) for Q ∈ [Q ,Q ].
optimal purchasing quantity. 1 2 l h
For q ∈[Q ,+∞), g (q)+g (q) is
Define h 1 (cid:90) 2
(cid:90) q
q
min {[cq−h (q−D)f(D)dD
g (q) := cq−h (q−D)f(D)dD
1 (cid:90) 0 q∈[Qh,+(cid:90)∞+)∞ 0
+∞
+b (D−q)f(D)dD](θ +θ )
+b (D−q)f(D)dD](θ +θ ); l h
l h
q
q
+e(q−Q )θ +e(q−Q )θ }, (33)
l l h h
its convexity in q is straitforward, and
which is convex in q and the solution of the first order
I
g (q) := e(q−Q )θ
2 Ql≤q<Qh l l condition is q¯e2 (defined in equation (29)). Hence the
I
+ [e(q−Q )θ +e(q−Q )θ . optimal purchasing in this case is as follows:
Qh≤q l l h h
JOURNALOFLATEXCLASSFILES,VOL.*,NO.**,*********** 8
Ql Qh Legend
priceisaswellgreaterthanthebiggerpossiblecapacityQ ,
h
scenario(1) q¯c we order up to the larger one between q¯e2 and Qh.
q¯e1 Moreover, when θ =0, i.e. the only possible capacity is
l
scenario(2) Qh, then it is easy to obtain from Theorem 4.1 that
(cid:189)
q¯, if q¯ ≤Q ;
scenario(3) q∗ = c c h
max{q¯ ,Q }, otherwise.
e2 h
Fig.1. IllustrationofpossiblescenariosinTheorem4.1and4.3 Further, if the additional unit purchasing cost e = +∞,
which indicates that it is impossible to make the additional
order, the optimal policy is
1) q¯ > Q : since q¯ > Q , we have q¯ >
e2 h e2 h e1 q∗ =min{q¯,Q }.
Q , which indicates that g (q) + g (q) is non- c h
h 1 2
increasing for q ∈ [Q ,Q ]. Then the optimal Theorem 4.1 characterizes a myopic policy and estab-
l h
purchasing quantity is order-up-to q¯ , i.e., q∗ = lishes a foundation for the subsequent discussions.
e2
q¯ .
e2
2) q¯ > Q and q¯ ≤ Q : it indicates that
e1 h e2 h B. General Case: Purchase More to Observe Supply Ca-
g (q)+g (q)isnon-decreasingforq ∈[Q ,+∞).
1 2 h pacity
q¯ > Q indicates that g (q) + g (q) is non-
e1 h 1 2 Inthissubsection,weconsiderthecaseswherethesupply
increasing for q ∈ [Q ,Q ]. Then the optimal
l h capacity of the next period is correlated to the capacity
purchasing quantity is order-up-to Q , i.e., q∗ =
h of the current period. If capacities are correlated, then the
Q .
h future cost function P has to be taken into consideration.
3) Otherwise q¯ ≤ Q : this is similar to Case 2 f
e1 h First, we need the following lemma.
andtheoptimalpurchasingquantityisorder-up-to
Lemma 4.1: The solution W(ρ) of equation (7) is con-
max{q¯ ,Q }, i.e., q∗ =max{q¯ ,Q }.
e1 l e1 l cave in ρ, i.e.
We conclude the optimal purchasing quantity as follows:
W(βρ+(1−β)ρ˜)≥βW(ρ)+(1−β)W(ρ˜).
(cid:189)
max{q¯ ,Q }, if q¯ >Q ; Proof: Note that the uniqueness of W(ρ) is proved in
q∗ = e2 h e1 h (34)
max{q¯ ,Q }, if q¯ ≤Q . Theorem 3.1. We use the mapping T defined in equation
e1 l e1 h
(15) and prove the lemma by value iteration according
To conclude this subsection, combining 3 cases together, to Wn+1 = TWn. Let W0(ρ) = 0, which is obviously
we have the following theorem: concave in ρ. Suppose Wn(ρ) is concave in ρ, then
Theorem 4.1: Theoptimalpurchasingquantityq∗ forthe
independent capacity case is as follows, Wn+(cid:189)1(cid:90)(βρ+(1−β)ρ˜) (cid:90)
scenarios Conditions Optimal purchasing q∗
= min L(D,q)f(D)dD (βρ(x)+(1−β)ρ˜(x))dx
(1) q¯c <Ql q¯c q≥(cid:90)0
Q ≥q¯ q
(2) andh q¯ >e1Q max{q¯e1,Ql} + e(q−Q)(βρ(Q)+(1−β)ρ˜(Q))dQ
c l 0(cid:90)
(3) Q <q¯ max{q¯ ,Q } q
h e1 e2 h
+α Wn(p(·|Q))(βρ(Q)+(1−β)ρ˜(Q))dQ
where q¯,q¯ ,q¯ are defined by equation (29) and Q ,Q
c e1 e2 l h 0 (cid:90) (cid:190)
are two possible supply capacities with Q > Q . See +∞
h l +αWn( (βρ(ξ)+(1−β)ρ˜(ξ))p(·|ξ)dξ)
Figure 1 for an illustration of the possible scenarios.
q
Comparing with possible supply capacities, when the ≥ min{βT (Wn)(ρ)+(1−β)T (Wn)(ρ˜)}
y y
demand is low (q¯ < Q and q¯ < Q ), we simply q≥0
c l e1 h
purchaseuptotheminimizerq¯ atthenormalpricewithout = βWn+1(ρ)+(1−β)Wn+1(ρ˜). (35)
c
considering the capacity constraint.
Since Wn(ρ) converges to the solution W(ρ), we have the
When the demand is moderate, i.e. the minimizer q¯ of
c desired result. (cid:164)
thenormalpriceisgreaterthanthesmallerpossiblecapacity
Then based on this fact, define
Q , and the minimizer q¯ of the high price is smaller than
l e1
the larger possible capacity Q , we order up to the larger K := αW(θ p(·|Q )+θ p(·|Q ))
h l l l h h
one between q¯ and Q .
e1 l −α[θ W(p(·|Q ))+θ W(p(·|Q ))]
l l h h
When the demand is forecasted to be high, i.e. the cost
> 0, (36)
minimizerq¯ ofthenormalpriceisgreaterthanthesmaller
c
possiblecapacityQ ,andthecostminimizerq¯ ofthehigh where the inequality is due to Lemma 4.1.
l e1
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Recall the definition of P , and it is straightforward to Remark 4.4: Whenthedemandis low(Scenario (1)),in
f
conclude the following theorem. order to save the cost-to-go from the next period, the buyer
Theorem 4.2: P (q ) = P (q )−K , for all q ≥ Q , purchases products only up to the smaller possible supply
f m f l l m l
and 0≤q <Q . capacity Q .
l l l
This theorem demonstrates that the buyer may try to pur- Remark 4.5: Whenthedemandislow(Scenario(1)),the
chase more to reduce the future cost at the amount of K . largerthegapisbetweenq¯ andQ ,thehigherthepossible
l c l
ThisisbecausepurchasingmorethanQ leadstothebuyer istoorderuptoq¯,whichisthesameasthemyopicpolicy.
l c
to figure out whether the current period capacity is Q . If it Remark 4.6: When the smaller possible supply capacity
l
isn’tQ ,theonlypossiblevalueofQisQ .Henceordering Q is very low (Scenario (2) and (3)), the optimal policy
l h l
up to Q can fully discover the exact value of the current is the same as the myopic one.
l
period’s capacity, which leads to a reduction of the cost-to-
go function.
C. Numerical Analysis and Results
Sincethecost-to-gofunctionremainsthesamewhenq ∈
[Q ,+∞), we know that the purchasing policy does not Itispossibletore-writeequation(25)inanexplicitform
l
change in scenarios (2) and (3) described in Theorem 4.1. in order to carry out numerical studies. By substituting
Equation (21) and using Equation (24), we have,
When scenario (1) occurs, we need to balance the tradeoff
betweenmyopicandcost-to-gofunction.Definethemyopic W(θI +θ I )=θ W(θlI +I )
cost reduction as l x=Ql h x=Qh h θ x=Ql x=Qh
Z h
q
K := g (Q )+g (Q )−g (q¯)+g (q¯) = min{[cq−h (q−D)f(D)dD
m 1 l 2 l 1 c 2 c q≥0Z 0
= g1(Ql)−g1(q¯c)>0. (37) +∞
+b (D−q)f(D)dD](θ +θ )
l h
The buyer can reduce this myopic cost by ordering up to q
I
thecostminimizerq¯c insteadofthesmallerpossiblesupply + Ql≤q<Qhe(q−Ql)θl
capacity Ql. +IQh≤q[e(q−Ql)θl+e(q−Qh)θh]
Remark 4.1: The larger the gap between Ql and q¯c is, +αI (θ(1−β)W( βl I +I )
the greater the myopic cost reduction Km is. Ql<q l l 1−βl x=Ql x=Qh
Remark 4.2: The greater the second derivative of the +θ β W(1−βhI +I ))
cost function g1(·), i.e. d2dgq12(q), the greater the myopic cost h h βh x=Ql x=Qh
saving Km is. +α(θl(1−„βl)+θhβh) «ff
Then the optimal ordering quantity is ·I W βlβl+θh(1−βh)I +I .
(cid:189) q≤Ql θ(1−β)+θ β x=Ql x=Qh
l l h h
q¯, if K >K ;
q∗ = c m l (38) (39)
Q , if K ≤K .
l m l
Further define
Theorem 4.3: The optimal purchasing quantity q∗ of the
multi-period news-vendor problem with partially observed θ = θl, Φ(x)=W(xI +I ), (40)
two-state Markovian capacity is as follows. θ x=Ql x=Qh
h
Scenarios Conditions Optimal purchasing q∗
(cid:189) and then we obtain
q¯, if K >K ;
(1) q¯ <Q c m l  Z
c l Q , if K ≤K . q
l m l Φ(θ) = min [cq−h (q−D)f(D)dD
(2) Qh ≥q¯e1 max{q¯ ,Q } q≥0Z 0
and q¯ >Q e1 l +∞
c l +b (D−q)f(D)dD](θ+1)
(3) Q <q¯ max{q¯ ,Q }
h e1 e2 h q
I
where q¯c,q¯e1,q¯e2 are defined in equation (29), Kl and Km + Ql≤q<Qhe(q−Ql)θ
are defined in equations (36) and (37), respectively. Ql +IQh≤q[e„(q−Ql)θ+e(q−Qh)] «
and Q are two possible supply capacities with Q > Q .
See Fighure 1 for the illustration of the above three phossiblle +αIQl<q θ(1−βl)Φ(1−βlβ )+βhΦ(1−ββh)
„l h «ff
scenarios.
Remark 4.3: When θl = 0, the only possible capacity +α(θ(1−βl)+βh)Iq≤QlΦ θθβ(1l+−(β1)−+ββh) .
l h
is Q . Then K = 0 and the policy is the same as the
h l (41)
one of the basic model, the case of the independent supply
capacity. The myopic policy is also optimal for the multi- Assumption 4.1: Assume a symmetric Markov chain of
period model. capacity: β =β =β.
l h
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I
Then we can rewrite (41) as: + [e(q−Q )·2+e(q−Q )]
(cid:189) (cid:90) Qh≤q (cid:181) l h (cid:182)
q 1
Φ(θ) = min [cq−h (q−D)f(D)dD +αIQl<q 2(1−β)Φn(2)+βΦn(2)
q≥0(cid:90) 0 (cid:181) (cid:182)(cid:190)
+∞ +α(2(1−β)+β)I Φn 5 .(44)
+b (D−q)f(D)dD](θ+1) q≤Ql 4
q
+I e(q−Q )θ And let θ = 1, then
+IQl≤q<[eQ(hq−Q )θl+e(q−Q )] 1 2 (cid:189) (cid:90) q
Qh≤q (cid:181) l h (cid:182) Φn+1( ) = min [cq−h (q−D)f(D)dD
+αIQl<q θ(1−β)Φ(1−ββ)+βΦ(1−ββ) 2 q≥0(cid:90) +∞ 0 3
(cid:181) (cid:182)(cid:190) +b (D−q)f(D)dD]·
I θβ+(1−β) 2
+α(θ(1−β)+β) Φ . q
q≤Ql θ(1−β)+β 1
I
+ e(q−Q )·
(42) Ql≤q<Qh l 2
1
Let β = 2,h = 1,c = 2,b = 4,e = 1,Q = 12,Q = +I [e(q−Q )· +e(q−Q )]
3 l h Qh≤q l 2 h
15,α = 0.95, demand D subjects to a normal distribution (cid:181) (cid:182)
1 1
with mean µ = 10 and variance σ = 1. Then we find +αI (1−β)Φn(2)+βΦn( )
Ql<q 2 2
that q¯c = 10.43 < Ql (Here we only do analysis of this (cid:181) (cid:182)(cid:190)
1 4
case because other cases’ policy is the same as the myopic +α( (1−β)+β)I Φn .
inventory decision according to Remark 4.6). 2 q≤Ql 5
We use value-iteration method to solve this equation. (45)
Start with Φ0(θ)=0 and
Also, let θ =1 and we have
Φn+1(θ)
 Z
q (cid:189) (cid:90)
= min [cq−h (q−D)f(D)dD q
q≥0Z +∞ 0 Φn+1(1) = mq≥in0(cid:90) [cq−h 0 (q−D)f(D)dD
+b (D−q)f(D)dD](θ+1) +∞
q +b (D−q)f(D)dD]·2
I
+ e(q−Q)θ q
Ql≤q<Qh l I
I + e(q−Q )
+ Qh≤q[e„(q−Ql)θ+e(q−Qh)] « IQl≤q<Qh l
+αI θ(1−β)Φn( β )+βΦn(1−β) + Qh≤q[e(cid:181)(q−Ql)+e(q−Qh)] (cid:182)
Ql<q 1−β β 1
„ «ff +αI (1−β)Φn(2)+βΦn( )
+α(θ(1−β)Z+β)Iq≤QlΦn θθβ(1+−(β1)−+ββ) , +α((Q1l<−qβ)+β)I Φn(1)(cid:111). 2 (46)
q q≤Ql
= min [cq−h (q−D)f(D)dD
q≥0Z +∞ 0 Note that Φ(45),Φ(45) and Φ(1) appear in equations
(44), (45), and (46), respectively. In order to carry out the
+b (D−q)f(D)dD](θ+1)
q iteration, we make the following approximations:
I I
+ Ql≤q<Qhe(q−Ql)θ+ Qh≤q[e(q−Ql)θ Φ(1)≈Φ(4)≈Φ(5).
+e(q−Qh„)] « 5 4
+αI θ(1−β)Φn(2)+βΦn(1) Then combining equations (44), (45), and (46) together,
Ql<q „ 2 «ff starting with Φ0(1) = 0,Φ0(2) = 0 and Φ0(1) = 0, we
2
+α(θ(1−β)+β)I Φn 2θ+1 . (43) solve the equation numerically and obtain
q≤Ql θ+2
1
Φ( )=416.8<Φ(1)=547.6<Φ(2)=756.8, (47)
2
Inordertocarryouttheiterationprocess,letθ =2,then
(cid:189) (cid:90) q where we find out that Φ(1) ≥ 2Φ(1)+ 1Φ(2) (demon-
3 2 3
Φn+1(2) = min [cq−h (q−D)f(D)dD strating concavity), and the optimal purchasing
q≥0(cid:90) 0
+∞ q∗| =12, q∗| =12, q∗| =10.43. (48)
θ=1 θ=1 θ=2
+b (D−q)f(D)dD]·3 2
q With the above calculation, the optimal purchasing is Q =
l
+IQl≤q<Qhe(q−Ql)·2 12orq¯c =10.43whenq¯c <Ql.Andalsothelargerθ = θθhl