..........
_....
•
,
~e
AN INVENTORY MODEL WITH SPECIAL SALE
by
Frank K.M. Hwang and B. B. Bhattacharyya
Institute of Statistics
Mimeograph Series No. 583
May
1968
iv
TABLE OF CONTENTS
Page
LIST OF TABLES .
v
1.
1
INTRODUCTION.
1.1
1.2
1.3
1.4
2.
2.6
2.7
4.2
4.3
4.4
4.5
...
e
6.
9
12
15
18
25
29
32
35
The General Setup
An Optimal Policy
Some Ordering Properties of Critical Numbers . •
.
....
.
Summary
Suggestions for Future Study.
LIST OF REFERENCES
35
38
51
66
Infinite Period Model When Demand Distributions
in Different Time Periods are Independent and
Identical . . . . . . . . . . . . . . . . . .
Critical Numbers of the Infinite Period Process
with Independent and Identical Demand Distributions.
Infinite Period Model When Demand Distributions
in Different Time Periods are Independent but
Varying . • • . . . . . . . . . . . . . . . •
Some Theorems on Critical Numbers of Varying Demand
Distributions Case.
. . . .
A Generalization. •
CONCLUSIONS.
5.1
5.2
'lit
Model • • . . . •
Mathematical Formulation. •
Optimal Special Sale Policy
Optimal Procurement Policy.
A Verbal Interpretation of the Dependence of
Optimal Policy on Cost Structure. . . • •
Some Further Results of Optimal Policy. .
The Case 0 ~ p < q and the Case q < 0 ~ p .
THE INFINITE-STAGE PROCESS .
4.1
5.
9
THE MULTISTAGE PROCESS . .
3.1
3.2
3.3
4.
1
4
6
7
THE SINGLE-STAGE PROCESS
2.1
2.2
2.3
2.4
2.5
3.
Description of an Inventory Model
..•.
Some Realistic Aspects of the Model .
Motivation and Methodology.
Review of Literature.
.......
....
....
66
83
99
106
122
125
125
126
128
v
TABLE OF CONTENTS (continued)
If
Page
7.
APPENDICES • . • • • •
7.1
7.2
Derivatives of the First and Second Derivatives
of the Function glN(Y'z) •.
Glossary of Terms • • . . • • . . . • • . . • . .
130
131
141
vi
LIST OF TABLES
Page
1.
Table of optimal policy corresponding to various cost
s true tures . . . . . . . . . . . . . . . . . .
30
CHAPTER 1:
1.1
INTRODUCTION
Description of an Inventory Model
We consider an inventory process dealing with the procuring,
storing, and selling of a single commodity.
The inventory process
is divided into "review periods" of equal length.
Stock level is
reviewed periodically and is known only at the beginning of a review
period.
We will refer to the set of points where the stock level is
known as "review points."
Procurement decisions are made at review
points, and delivery of goods is instantaneous.
By "instantaneous,"
we mean that delivery takes a negligible amount of time compared to
the length of a period.
There are two modes of selling within each period.
One is
called the "regular sale" and the other the "special sale."
The
regular sale is more or less a regular type of selling as usually
encountered in inventory literature.
The regular sale period covers
the whole review period except perhaps when the special sale is in
effect.
Sales are made in the regular sale period whenever the stock
is adequate to meet the demand.
The part of demand which exceeds
stock is considered lost and not carried over to the next period.
A
suitably defined shortage cost is charged in this case.
The special sale can be given at the beginning of a period after
the procurement and before the regular sale.
Goods are sold in the
special sale usually under a lower-than-regular price.
The underlying
assumption is that there exists demand which cannot be attracted to the
regular sale, but can be attracted to the special sale through lower
pricing.
Some of the reasons why a special sale may be desirable are:
2
a.
When large stock has been piled up and high holding cost is
expected;
b.
When goods are perishable and possible loss of goods in stock
is expected;
c.
When the special sale has promotional effect for the succeeding
regular sale;
d.
When the special sale itself is a profitable business.
To effectively relieve the situations stated in a and b, as well
as not to interfere with the regular sale too long, it is required that
the special sale be a quick sale - "quick" in the sense that the special
sale period is almost instantaneous compared to the length of the regular
sale period.
This requirement can be realized if the special sale price
is sufficiently low.
Besides the above timing requirement, we also require that the
special sale and the regular sale be conducted under one roof so that
goods allocated to the special sale but not sold there can be immediately
transferred back to the regular sale with zero cost.
This "under-one-
roof" requirement does not preclude the possibility to sell the commodity
to a second market at a different locationo
For instance, if the sale
to the second market is through a local dealer who can be contacted by
telephone, then we consider that the special sale is conducted where the
telephone is.
Conceivably, the inventory situation can be such that a special
sale is not desirable at all for some particular period; or if it is,
then the best strategy is to allocate only a part of the physical stock
to the special sale.
These questions are subject to management's decision.
3
By "management," we mean the party who is responsible for the optimization
•
of the baml cost of the inventory process.
The amount allocated to the
special sale will be referred to as the special stock.
During the special
sale period, sales are made as long as the special stock is adequate to
meet the demand.
The part of demand which exceeds the special stock is
considered lost but no shortage cost is charged.
The hypothesis is that
management has no obligation to satisfy demand in special sales, while
it does have the obligation in regular sales.
Since inventory decisions are made periodically, we will be
concerned only with the total demand in the special sale period and the
total demand in the regular sale period.
The former will hereafter be
referred to as the special-sale demand and the latter, the regular demand.
The special-sale demand and the regular demand of any period are random
variables.
As it is likely that the special sale will have some effect
on the succeeding regular demand, we make the assumption that the regular
demand depends on the amount of goods sold
~n
the special sale.
This
amount will be defined as zero when there is no special sale in the
period.
way.
The dependence on the special sale can, of course, go either
When the special sale has promotional effect, then the larger the
special sale, the larger will be the succeeding regular demand.
In
other cases, a large amount of special sale may hurt the succeeding
regular demand.
Demand in the i
th
time period is independent of demand
in any other time period.
The relevant inventory costs include the usual procuring cost,
holding cost, shortage cost, and sale revenue (the last item is inter-
..
preted as a negative cost).
All costs are assumed proportional to the
•
e
•
4
quantity involved.
Thus we can talk about cost on a unit goods basis.
The objective is to optimize the expected total cost where the expectation
is taken over all possible demand realizations.
Decision variables
subject to optimization are:
a.
The amount to be procured in each period (including zero as
a possibility).
b.
The amount to be allocated to the special stock (again
including zero as a possibility).
1.2
Some Realistic Aspects of the Model
Let us look at some of the real-life examples in which special
sale is offered alongside with regular sale.
a.
Inventory-Type Special Sale.
The motivation for this type
of special sale is mainly to relieve high inventory costs
or to dispose of goods which may perish soon.
The special
sale itself may lose money but is still desirable as the
lesser of two evils.
The inventory-type special sale can
be found in practice in all kinds of stores.
When special
sale is given to relieve inventory cost, certainly no
procurement will be made for that particular period.
That
positive procurement and special sale do not occur
together is a characteristic of the optimal policy for this
type of special sale.
b.
Supplementary-Type Special Sale.
The motivation for this
type of special sale is the same as that for the regular
sale - profit.
Basement sales in a department store, or
special sales which are offered quite regularly, usually are
.
5
e
mere attempts to catch as many customers and make as much
,
profit as possible and may not be necessitated by large
inventory situation.
Another example is the dairy store
which delivers dairy products to subscribing customers, but
also sells to customers who occasionally visit the store.
The two sales differ in service and could as well differ in
price.
But it is likely that the dairy store makes money on
both of them.
c.
The Promotional-Type Special Sale.
The motivation for this
special sale is either to introduce a new product or to keep
reminding customers of an old product.
The sale itself is
treated as some sort of advertising and thus is not required
to make immediate profit by itself.
It is still a profit-
oriented sale as distinguished from the inventory-type special
sale (which does not seek profit at all) in the sense that
the promotional special sale is believed to be responsible
for the profits made in the succeeding regular sale.
Many
soft-drink companies use the strategy of lowering the price
of a beverage once in a while to lure customers back.
In
some extreme cases, companies send out free samples to
potential customers, and thus incurring even a negative sale
revenue.
Now we comment on the particular set-up of the special sale in our
model.
review.
Let us arbitrarily say that a time period starts with a stockSince the question "how many goods should be procured" can be
answered only if we know how many goods are there in stock, it seems
logical that we adhere to the usual assumption that the procuring point
-e
6
follows the review point immediately.
There are now two possible cases:
either the special sale proceds the regular sale, or vice versa.
For
both cases, it is in general desirable to have a decision variable
controlling the amount of stock allocated to the first sale, since an
optimal allocation is not necessarily the whole stock.
However, it is
not possible to place a similar control on the second sale simply
because the stock level at the second sale is assumed unknown.
In
this study we will consider the case when special sale preceds regular
sale.
In those cases when starting inventory are high and an inventory-
type special sale is called for, then it is desirable to have special
sale preceding regular sale so that saving of holding cost and of loss
of perished goods could be achieved during the time period.
1.3
Motivation and Methodology
In mathematical inventory literature, the historical interest has
been overwhelmingly concerned with the procuring activity.
"Selling"
has rarely been considered as a decision variable except possibly in
Karlin and Carr's article on pricing policy [15].
In this study, we
treat selling as a decision variable from a different viewpoint.
We
still consider price as fixed in our model, but we allow two modes of
selling which differ not only in price but also in other aspects, and
work on the optimal combination of them.
The problem is further
complicated by the fact that the regular demand depends on the amount
of goods sold in the preceding special sale.
By the introduction of a selling-decision variable in addition to
the procuring-decision variable, we now have a two-dimensional
...
optimization problem.
Our approach is first to work on the special
7
case - the single-stage model, where the inventory process consists of
•
a single period.
Then we apply Bellman's "Principle of Optimality" and
construct recurrence equations for the multistage model.
This is the
classical "functional equation method" so popular in inventory
literature.
Optimal policies are found to be of simple forms
characterized by critical numbers.
Analytic solutions for critical
numbers are obtainable for the infinite-stage model when demand
distributions do not vary over time.
They are useful not only in
providing approximate solutions of critical numbers for the finitestage process, but also in revealing the structural features of optimal
policies.
When demand distributions do vary over time, we extend Karlin's
work from his one-dec is ion-variable case to our two-dec is ion-variable
case.
..
In this case we are able to obtain more general results due to
the special features of our particular model .
1.4
Review of Literature
Arrow, Harris, and Maschak [1] popularized the functional equation
method in dealing with mathematical inventory problems.
Their
attention, in their 1951 paper [1], was focused on a special-type policy.
Dvoretzky et al. [7] examined the results of Arrow et al., with full
generality.
Bellman [4], Bellman et al. [5], Karlin [13], and Iglehart
[11], [12] considered the existence and uniqueness of the solution to
the functional equation in an infinite-stage model.
For the one-decision-variable optimization inventory problem,
Bellman et al. [5] established conditions under which the optimal
policy will have the simple form of a series of critical numbers
'"
..
e
8
x ,x , ..• , called a base stock policy.
l 2
If the stock level at the
beginning of period n is below x , a procurement in the amount of the
n
difference is made; otherwise no procurement is made.
When demand distributions vary from period to period, it is in
general difficult to obtain analytic solutions for the critical numbers.
Karlin [14] showed that when demand distributions are increasing
stochastically over time, then the optimal policy of the first period
is identical to the first-period optimal policy of a problem in which
demand distributions are identical to the first period.
Veinott [18],
[19] extended Karlin's results to the case when demand distributions
are "stochastically increasing in translation."
i
th
period demand variable and
Namely, let D. be the
l.
~D. its distribution function, then
l.
where a.
>
l. -
~D
~ ~D
(E) holds for all E and i - 1,2,3, ..•
i
l.
i+l
0 is the minimum value D. can take.
the inequality
(E+a.)
l.
For the treatment of two-dimensional (two-decision-variables)
optimization inventory problems, Fukuda [8] considered the optimal
procuring and disposal policy in a multiechelon inventory system.
He
excluded explicitly the possibility that procuring and disposal may
occur in the same period.
Phelps [16] considered a similar problem
for the single-echelon system, except that in his model there are two
modes of procuring.
The one with a cheaper procuring cost has a limited
supply whose amount depends on previous policy.
Barankin [3], Daniel
[7], and Fukuda [9] considered regular and emergency procuring policies
where a premium is paid for shorter delivery time.
For general reference on inventory control theory, see [2], [10],
[17], and [20].
9
CHAPTER 2:
•
2.1
THE SINGLE-STAGE PROCESS
Model
In the single-stage process the decision maker wants to determine
his optimal procurement and sales policy when his time horizon is
limited to a single period only.
These policies will obviously depend
on the nature of the market demand and supply functions.
From the supply side, it is assumed that the firm can procure
instantaneously an unlimited quantity of the particular commodity at
a fixed price c per unit quantity.
Let x be the stock level at the
beginning of the period and let y be the inventory level after procurement.
Then Max{O,y-x} is the amount procured.
amount of goods allocated for special sale.
Let Max{O,y-z} be the
Then the stock level
available for regular sale is at least z.
In this study we shall assume that the special-sale demand D* is
a random variable with a known probability distribution function
at the special sale price r* per unit quantity.
D* is a known number a* where a*
>
the special sale, then obviously I
0.
~D*
The minimum value of
If "I" is the amount sold in
= Min{D*,y-z}.
The demand distribution at the regular sale will depend upon the
amount sold in the special sale.
If the two sales are competitive,
then regular sale is likely to be reduced; whereas if the special sale
is promotional in nature, then regular sale is likely to increase.
Let
the random variable D(I) denote the regular demand at price r per
unit quantity where the quantity sold on special sale is known to be I.
In this discussion, we shall assume D(I)
=D-
qI where the parameter
q is known and D is a random variable following known probability
.
e
•
10
distribution function cP
D
where ¢D is independent of 'r
*.
D
Note that
D(O) = D, thus D can be thought of as what the regular demand will be
if nothing is sold in the special sale.
demand distribution function is
=
~D
Given I - i, then the regular
., _i._e., P (DI/I=i)<e) =
-q1
r
-
~D
-q
iCe)
~D(e+qi).
The parameter q in our model will be negative if the special sale
is promotional in nature where as if it is competitive, then q will be
positive.
Demands in regular sale and special sale will obviously be
independent if q = O.
In this study, we shall consider only those
distributions which have continuous densities and the expectations of
the associated random variables are finite.
One of the basic assumptions about the random variable D(I) is
that it is nonnegative.
This introduces a certain restriction on the
maximum value "I" can take when q is positive.
...
of D be a
>
O.
Let the minimum value
Then D(I) will be a nonnegative random variable for
every realization of I if we assume that the upper bound of D* is a/q
where q
>
O.
(It is not necessary to put any such restriction on the
range of D* if q < 0.)
model when q
>
O.
Thus there is an obvious limitation on our
But in many such situations, "a" is a relatively
large number and "q" is a small positive fraction, so that the assumption
~D*(a/q)
= 1 is not likely to be a severe restriction.
In order to introduce more flexibility in the model, we shall
introduce a parameter p to take into account any kind of leakage within
the system.
special sale,
If there is no leakage, then the left-over amount after
~.£.,
y-I, will be available for regular sale.
But in
our model we shall assume that (l-p)(y-I) amount of goods will not be
...
available at the regular sale where p lies between 0 and 1.
The
11
proportion l-p takes into account any kind of leakage within the system
•
during the regular sale period.
Since the special sale is conducted
for a short duration at the beginning of the time period under study,
the loss due to leakage during this period will be assumed to be
negligible.
For a system without leakage, p, of course, will be one.
Besides the procurement cost, a second set of costs is associated
with the stock of inventories on hand.
Storage or holding costs may
be incurred by the actual maintenance of stocks or the rent of storage
space.
Each unit of commodity sold in the special sale is charged a
holding cost hI
~
O.
Each unit of goods not sold in the special sale
is charged a holding cost hI + h
Z
~.hl.
We have assumed that storage
or holding cost is proportional to the size of the stock of inventory.
However, other cases may be considered.
For example, if storage takes
place in a warehouse, the unit storage cost may jump from a low value
to a high value if the stock exceeds a certain quantity.
The holding cost arises when supply exceeds demand.
And when
demand exceeds supply (since it may be impossible or too costly to
guarantee that demand will be met under all circumstances), another
type of cost known as penalty cost or shortage cost arises.
The
failure to meet demand generates cost though in different ways under
different situations.
Often, it involves loss of good will on the part
of the customer, thereby giving rise to the ticklish problem of
determining the monetary value of goodwill.
assumed that the firm incurs a cost s
>
a
In this study we have
when demand exceeds supply
by unit amount and the total shortage cost is proportional to the
total excess demand.
sale.
Moreover shortage cost applies only to regular
--
12
Finally, there is the problems of disposal of goods left at the
end of the period.
It is assumed that they can be disposed of with
zero cost.
Within the framework outlined above, the firm wants to determine
the procurement level y and the amount to be allocated for special
sale y-z in such a way that the expected total cost is minimized.
Naturally for a policy to be admissible, y should be at least as
large as x and z should lie between 0 and y.
2.2
Mathematical Formulation
Let t (y,z,e,e*) denote the total cost of the single-period process
x
given a policy (y,z) and the initial stock level x when the special-sale
demand is e* and the regular demand is e.
Then
e* .::. y-z
.
t (y,z,e,e*)
x
=
c(y-x) + h 1Y - r*e* + h 2 (y-e*) - re
=
c(y-x) + h Y - r*e* + h (y-e*)
1
2
for
e .::. p(y-E:*),
e* .::. y-z
- rp(y-e*) + s(e-py+pe*)
for
E > p(y-E:*)
=
c(y-x) + h y - r*(y-z) + h z
1
2
- re
- rpz + S(E-pZ)
•
E*
>
y-z
E
.::.
pz,
E*
>
y-z
E
.::.
pz •
for
for
(2.2.1)
13
Let
L (y,z)
...
x
= E[~ x (y,z,s,s*)],
where the expectation is over the regular demand and the special demand
variables.
Then
J [00
+
rp (y-e*) + s (e-py+pe*) Jon (e+q e*) de
~n* (e*) de*
p(y-s*)
00
+J
- r*(y-z) + h z +
2
JPZ
- rs
$D(s+qy-qz)d~
a-q(y-z)
y-z
00
+J
[- rpz + s(E-pz)]$D(s+qy-qz)ds
1JJ
*
D
D*(s*)ds*
(2.2.2)
pz
where
1JJ
and sD denote the density function of the random variables D*
and D respectively.
We also assume that D* and D have finite mean
values so that L (y,z) is finite iff y is finite.
x
that L (y,z) is a continuous function of y and z.
x
It is also evident
14
Definition:
If a policy (y', z') is such that
L (y'
x
'
z') =
Min
Min
L (y,z),
x
x~<oo O::..z~
then (y', z') is an optimal policy.
Let
g(y,a)
00
..
- r*(y-z) + hZz + JlPZ
-rE 0n(E+qy-qz)dE
a-q(y-z)
00
+ Jl
[- rpz + O(E-pZ) lOn (E+qy-qZ) dE
pz
(2.2.3)
Then
L (y,z) = - cx + g(y,z)
x
(2.2.4)
In order to obtain the optimum value of y and z, we use the
technique of stepwise minimization.
Let z
y
be the value of z at
which g(y,z) attains its minimum when y is a finite known nonnegative
15
number.
Since g(y,z) is a continuous function of z and the range of
z is a closed interval, there exists a z
at which g(y,z) attains its
y
minimum.
Let W(y)
= g(y,z).
y
Now if W(y) attains its minimum at a finite
value y, then (y,z-) will be our optimum policy.
For if (y,z) be an
y
arbitrary feasible policy, then
g(y,z) > g(y,z )
-
2.3
y
~ W<Y) = g(y,z-).
y
= W(y)
Optimal Special Sale Policy
Let z
y
denote an optimal special sale policy when the procurement
level y is given.
when p
~
q
~
O.
Then the structure of z
The structure of z
y
y
when p
is given by Theorem 1.1
~
q
~
0 is not true will
be discussed later in the chapter.
Theorem 1.1:
.
If p
~
q
~
0 and
(a)
if r* + h Z + sq - rp - sp
(b)
if r* + h Z - rq
(c)
if r* + h
~
~
0, then Zy
0; then Zy
Z + sq - rp - sp
<
=0
y
0
<
r* + h
Z - rq;
then there exists a uniquely determined number Z, a
z
Proof:
y
<
Z
<
00, such that
Z
=0
for - < y <
=~
p-q
Z
Z
for - < y < p- -q
=
for 0
q
y
~
y
00
Z
< -
P
Since L (y,z) = -cx + g(y,z), it is sufficient to minimize
x
g(y,z) to obtain the minimum of L (y,z) for a given value of x and y.
x
Differentiating g(y,z) with respect to z, we obtain:
D g(y,z)
2
=
[r* + h
2
+ sq - rp - sp +
16
(p-q)(r+s)~(E+
fPZ
qy-qz)dE]
a-q(y-z)
• [1 - If'D*(y-z)]
= V(qy+pz-qz)[1
- If'D*(y-z)]
for y-z
<
a
-q
(2.3.1)
where
V(6)
r* + h
2
+ sq - rp - sp +
(r+s)(p-q)~D(6).
(2.3.2)
and
D g(y,z) = 0
2
for y-z
>
a
q
1 - If'
for y-z
>
a
q
(2.3.3)
since
--
(a)
D*
(y-z) = 0
Now
r* + h 2 + Sq - rp - sp + (r+s)(p-qHD(qy+Pz+qz)
>
r* + h
since p
~
2
+ sq - rp - sp
q and
~D ~
Moreover 1 - If'D*(y-z)
D g(y,z)
2
~
o.
~
o.
0 in (a).
g(y,z) is a monotone-nondecreasing continuous function of z for
Now if D g(y,z) is strictly positive then z = Min z
2
y o~z::y
Otherwise there may be a closed interval of optimal values of z.
= o.
Since in any case, 0 will be included in the closed interval, z
=0
a fixed y.
is an optimal policy. in (a).
y
17
(b)
r* + h
Z
+ sq - rp - sp +
-< r* + h Z - rq
Dzg(y,z)
~
(r+s)(p-q)~D(qy+Pz-qz)
~D(qy+pz-qz) <
as
1
0 in (b)
g(y,z) is a monotone-nonincreasing function of z for every finite
value of y.
The minimum of g(y,z) is attained either at z
= y or there may be
a closed interval with y as the right end point such that at any point
of the interval, g(y,z) attains minimum.
In any case z
y
=y
is an
optimum policy.
o
- rq,
Z
then because of the continuity of ~D' there exists a unique number Z
(c)
If r* + h
such that V(Z)
qy + (p-q)z
>
Z
+ sq - rp - sp <
=0
>
r* + h
and V(8)
<
0 for all 8
Z for every 0
~
z
for every admissible z.
Dzg(y,z)
<
0 if Y
>
Z
q;
Since
~
~D
<
Z
If y > - , then
q
Z.
y; and if y
<
Z
-, then qy + (p-q)z
Z
q
<
y < 00, and z
Z
Now when -
Y
<
p-
Dzg(y,z)
<
=y
and DZg(y,z)
~
0 if Y
<
Z
p.
Accordingly as in
y
=0
if
Z
when 0 < y < -.
-
P
Z
-:
y <
-q
0 when qy + (p-q) z < Z or z < ~ , and
DZg(y,z) > 0 when z ->~
p-q
Z
is a monotone-nondecreasing function,
case (a) and case (b), an optimum solution of z is given by z
-
<
P
- p-q
.
g(y,z) will attain minimum when Z --~
p_q •
In this case also, there may be a range of values of z for which
will be an optimal z.
g (y,z) will attain minimum, but ~
p-q
18
2.4
Optimal Procurement Policy
Substituting z
y
for z in g(y,z) and calling the new function W(y),
we will show that W(y) is convex in y.
For easier reference, we list the results of Dlg(y,z), D g(y,z),
2
Dllg(y,z), D g(y,z) and D g(y,z) here.
12
22
For their derivations,
consult Appendix 7.1.
=
og(y, z)
oy
y-
=
J
Z
[c + h
1
+ h
2
- rp - sp
a*
+
J
00
[c + hI - r* - sq
y-z
(2.4.1)
og(y,z)
oz
=
[r*+h +sq-rp-sp +
2
f
pz
(p-q) (r+s) cP (E:+qy-qz)dE: ]
D
a-q(y,z)
• [l-'fD*(y-z)]
(2.4.2)
19
= d2 g(y,Z)
dy2
•
pz
+ [r*+h +sq-rp-sp+
2
J
a-q(y-z)
+ (r+s)q
=
2
~D(qy+pz-qz)
. [1 -
~D*(Y-z)]
(2.4.3)
2
p g(y, z)
2
dZ
Pz
[r*+h 2+sq-rp-sp+
J
(p-q)(r+S)~D(E+qy-qz)dE]~D*(Y-Z)
a-q(y-z)
+ (r+s)(p-q)
2
~D(qy+pz-qz)
. [1 -
~D*(Y-z)]
(2.4.4)
~D*(Y-Z)].
(2.4.5)
=-
+ (r+S)(P-q)q¢D(qy+pz-qZ) . [1 -
Theorem 1.2:
o~ y
Proof:
•
<
W'(y) is a continuous monotone-nondecreasing function for
00.
Note that:
(2.4.6)
20
and
.
W"(y)
d~
dz
D
(
)
+
2D
(
)
-y
+
D
(
)
(-y)2
g
g
g
= 11 y,Zy
12 y,Zy dy
22 y,Zy
dy
(2.4.7)
dZ
whenever
df-
2-
d Z
and ---L
dy2
exist.
Proof will be partitioned into three cases.
a.
o . ::.
r* + h Z + sq - rp - sp
From Theorem 1.1, Z
y
2dz
d Z
-y
= 0, and --Z = O.
dy
dyZ
= O.
From (2.4.6),
(2.4.8)
From (2.4.7),
W"(y) = D
11
Substituting Z
since y-z
b.
r* + h
>
2
Z/q
- rq
= 0 in (2.4.3), then
a/q.
>
<
-
g(y,0).
~D*(Y-z)
(2.4.9)
= 1 and
~D*(Y-z)
=0
Hence
O.
From Theorem 1.1, z
y
=
y.
From
(2.4.6),
•
(2.4.10)
21
From (2.4.7),
•
=
(r+s)q
(r+s)p
c.
r* + h
2
2
2
~D(qy)
+
~D(qy) >
2(r+s)(p-q)q~D(qy)
+ (r+s)(p-q)
o.
+ sq - rp - sp
2
~D(qy)
(2.4.11)
< 0 <
r* + h
- rp
2
From Theorem 1.1
z
dz
~
dy
Z
o
y
for -
q
<
y
z
Z
-q
=~
p-q
-<Y<-
=y
z
o -< y -p
< -
p-
.
eX1sts
except on t h
e two ·
p01nts -Z an d -Z
p
For ~
q
dz
<
y, ~
dy
= 0,
d2z
and---Z
dy
o.
q
Thus from (2.4.6) and (2.4.7):
(2.4.12)
(2.4.13)
Substitute z
y-z = Z/q
>
a/q
~
=0
in (2.4.3).
a* and
~D*(Y-z)
The second term is now zero since
= O.
Hence (2.4.9) is nonnegative.
dz
For y
<
~ ~
p' dy
=1
2
d z
and ~
dy
22
= O.
From (2.4.6) and (2.4.7):
•
(2.4.14)
(2.4.15)
as shown in case b.
Finally for ~ < y < ~, note:
p- -q
o
Thus from (2.3.1) and (2.3.3), D g(y, ~)
p_q
2
= O.
Hence
W'(y) = D1g(y, ~
p_q )
w" (y)
•
-- D11g(y, ~
p-q) - -S....
p-q
(2.4.16)
12 g ( y, ~)
p-q
D
23
+ (r+s)l4>D(Z) [1 -
IjI
(~)]
D* p-q
-
(r+s)q24>D(Z)
[
1-
IjI
D*
(~)]
p-q
(py-Z) / (p-q)
=
(r+s)p24>D(PY-PE*+qE*)~D*(E*)dE* ~ 0
J
(2.4.17)
a*
From (2.4.12) and (2.4.16), W'(y) is continuous at
Z-nv
p-q
is continuous in both y and z, and
~
=0
at y
Z
= -.
q
--
proved.
W(y) is a convex function for 0
Theorem 1.3:
+ h
=>
When 0
2
q
~
p, then:
for all 0
~
y
Y
<
W'(y)
•
<
~
y
<
00.
00.
there exists a uniquely determined number
Y, 0
=>
~
Y=
<
<
00,
such that:
0 for all 0
Max(x,Y) •
~
y
W'(Y)
<
Y
= ~p = y at
Furthermore from (2.4.13),
(2.4.15), (2.4.17), W'(y) is monotone-nondecreasing.
Corollary:
D g(y,z)
1
From (2.4.14)
and (2.4.16), W'(y) is also continuous at ~ since ~
p
p-q
Thus W'(y) is a continuous function.
qZ since
=0
and
Theorem 1.2 is
*-
24
Proof:
(a)
When 0
~
r* + h + sq - rp - sp, from (2.4.8)
2
= D1g(y,0).
W'(y)
It W'(M)
M -+0
It d1g(M,0)
=
M-+oo
W'(O)
= D1g(0,0) = c
When r* + h
It W'(M)
M-+oo
2
= It
M+oo
=
- rq
~
=
c + hI + h 2
+ hI - r* - sq
0, from (2.4.10)
D1g(M,M) + It D g(M,M)
2
M-+oo
(c+h1-r*-sq) + (r*+h 2+sq-rp-sp)
c + hI + h 2 - rp - sp •
•
Z5
When r* + h
Z + sq - rp - sp
<
a
<
r* + h
Z - rp
It W'(M) = It Dlg(M,O)
M-HO
M-+oo
=
c + hI + h Z - rp - sp.
Thus (a) is proved.
-e
=>
a~
W'(y)
for all
a~
y <
00
=> W(y) is monotone-nondecreasing for
=> Y =
(c)
a~ y
<
00
Min y = z.
x:3Y<oo
Statement is obviously true by the monotone-nondecreasingnes$
of W' (y).
Z.5
A Verbal Interpretation of the Dependence of Optimal Policy on
Cost Structure
Given a policy (y,z), now consider the case when an additional
6 amount of goods is procured where 6 is an infinitisimal small
positive number, and we want to determine whether this 6 should be
allocated to the special sale or not.
First of all, when the special-sale demand is not greater than
y-z, then whether this 6 amount of goods is allocated to special
26
sale or not is of no consequence.
sale demand is greater than y-z.
So let us assume that the specialIn other words, this
~
amount of
goods will be sold in the special sale if it is made available.
Two cases now have to be considered.
is greater than z, then the cost of
~
When the regular demand
if allocated to the special
sale is:
(c+hl-r*-sq).~.
Because by selling an additional
~
(2.5.1)
amount of goods in special
sale, the regular demand is lowered down by the amount
saving of shortage cost in the amount of
On the other hand, the cost of
~
s·q~
q'~,
and a
is realized.
if withheld from the special
sale is
-e
(c+hl+h2-rp-sp)~
(2.5.2)
Due to the leakage factor, the amount
the regular sale.
sp~
A revenue of
r·p.~
p~
only is available to
and a saving of shortage cost
are thus realized.
Now if c + hI - r* - sq
r* + h
2
~
c + hI + h
2
- rp - rq, or equivalently,
+ sq - rp - sp ~ 0, it says that even if this ~ amount of
goods can be sold in the regular sale, it is still better to sell it
in the special sale.
This being the case, then all goods will be
allocated to the special sale.
This is the conclusion in Theorem
case a.
When the regular demand is not greater than z, then the cost of
~
if allocated to the special sale is:
(2.5.3)
27
This is because by selling an additional 6 amount of goods in
the special-sale, the regular demand is lowered by the amount q6, and
thus a revenue of rq6 is lost.
On the other hand, the cost of 6 if withheld from the special
sale is simply:
(2.5.4)
Now if c + hI + h 2 -~ c + h 1 - r* + rq, or equivalently
r* + h
2
- rq
~
0, it says that even if the 6 amount of goods will
not be sold in the regular sale, it is still preferable to withhold
it from the special sale.
This being the case, then no goods will
be allocated to special sale.
This is the conclusion in Theorem 1.1,
case b.
-e
Finally, when r* + h
says:
2
+ sq - rp - sp
~
°
<
r* + h
2
- rq, it
having an 6 amount of goods sold in the regular sale is
preferable to having it sold in the special sale; while having it
sold in the special sale is preferable to reserving it for the
regular sale but not being able to sell it there.
This cost
structure is thus seen to be a somewhat normal situation.
In this
normal caSe, there will exist some equilibrium points at which the
cost of allocating an 6 amount of goods to the special sale is
indifferent to the cost of reserving it for the regular sale.
And
since both costs are monotone nondecreasing in marginal cost, the
equilibrium points will be optimum points if they are admissible.
In Theorem 1.1, case c, the equilibrium point is solved and its
admissibility checked for all ranges of y.
We have discussed the problem of how to allocate a 6 amount of
goods procured for the inventory system.
Now we turn our attention
28
~
to the problem whether this
all.
Recall that
amount of goods should be procured at
(c+hl-r*-sq)~
is the cost of this
~
amount of goods
when it is sold in the special sale while it could be sold in regular
sale; and
(c+hl+h2-rp-rq)~ is
it is sold in the regular sale.
amount of goods when
If Min(c+h 1 -r*-sq ' c+h 1+h 2-rp-sp) ->
then procuring cannot be profitable.
is to procure nothing.
~
the cost of the
So the optimal procuring policy
If Min(c+hl-r*-sq, c+h +h -rp-sp) < 0, then
l 2
there exists a level Y such that procuring up to that level will be
profitable.
When this Y is less than the initial stock level x,
then again no procurement is made.
explained.
Theorem 1.3 is thus verbally
Let
f(x)
-e
Min
Min
xy<oo 02.zy
L (y,z)
(2.5.5)
x
The minimum exists since for all finite x, L (y,z) is finite
x
iff Y is finite.
Then
f(x)
=-
cx + W(x)
for Y
<
x
= - cx + W(Y)
for x
<
Y
(2.5.6)
f(x) is also a convex function.
Taking derivative:
f' (x) = - c
=-
+
W' (x)
for Y
c
From Theorem 1.3, it follows trivially:
Theorem 1.4:
When
°2. q 2. p,
then:
x
<
x
<
Y
(2.5.7)
°,
29
(a)
f'(x) is a continuous monotone nondecreasing function.
(Implying that f(x) is convex and f"(x) exists a.e.)
It
(b)
f'(M)
M-+oo
(c)
2.6
= h1 +
h
2
f'(x) = -c for all x
<
Y.
In particular f'(O)
= -c.
Some Further Results of Optimal Policy
Summarizing Theorem 1.1 and Theorem 1.3, we obtain the following
table.
The last case,
c+ hI - r* - sq
c+h +h -rp-sp)
l 2
~
~
i.~.,
~
c + hI + h 2
c + hI - r* + rq,
c + hI + h 2 - rp - sp, and Min(c+hl-r*-sq,
0, is more or less a normal cost structure.
We are
able to obtain more specific results in this case by further breaking
down the cost structure.
Substituting y + zip and z
y
= Z-qy/p-q = Y in W'(y) and recall-
= g1 (y,Zy)
for zip
ing from Section Z.3, S'(y)
=c
+ hI - r* - sq +
=c
+ hI - r* - sq - ~ (r*+hz+sq-rp-sp)
(since r* + h
Z
~
y
~
Z/q, we obtain:
(r+s)q~D(Z)
+ sq - rp - sp +
(r+s)(p-q)~D(Z)
= V(Z) = 0)
r*p+hzq-rpq
=c +
hI -
p-q
(2.6.1)
•
e
Table 1.
Table of optimal policy corresponding to various cost structure
c+hl+h Z
c+h -r*-sq
C+hl-r*-sq,)
1
Min
>
-
c+h -r*+rq
1
e'
2:. c+hl+hZ-rp-sp
(
c+h1+hZ-rp-sp
~0
y =
x
z =
yes
no
no
for all x
x
o
yes
no
yes
Y< x
x < Y
x
Y
o
o
no
yes
no
for all x
x
x
no
yes
yes
Y
x
x
Y
Y
x
o
x
Z-qx
p-q
yes
yes
no
< x
x < Y
Z
-<x
q
Z
-
Z
< x < -
p-
-p
x<~
p
yes
yes
yes
~
q
Z
-
<
Max(x, Y)
<
Max(x,Y)
p-
<
x
x
Max(x,Y)
o
ZZ
-I
Max(x,Y)
-q
Z
Max(x,Y) -p
< -I Max(x,Y)
q
Max(x,Y)
p-q
Max(x,Y)
w
o
31
and
w' (~)
Dlg(~,
=
J
0)
Z
q
=
[C + hi + h Z -
rp - sp
+
(r+S)P.D(~ -
pc*
+
qC*)] 0D* (c*)dc*
a*
(c+h 1-r*-sq)'10't'D* (E*)dE*
=
J~
q
[r*
+
hZ
+
sq - rp - sp
+
(r+s)P.D(~ Z -
pc*
+
qC*~OD*(C*)dC*
a*
(2.6.2)
+ (c+hl-r*-sq)
Since W'(y) is monotone-nondecreasing in y and Z/q
(2.6.2)
Furthermore, from W'(Y)
=
~
0, we will know the ordering of ~
Z
p
case in Table 1 can be stated as:
(a)
When a ~ q
~
p, and
If
z
a
+ (c+hl-r*-sq) >
a
zip, clearly
(2.6.1)
and - by knowing the signs of (2.6.1) and (2.6.2).
Theorem 1.5:
~
q
Hence the last
Y,
32
then
(b)
c + h
1
-
y
x,
z
=a
for Y
<
x
y
Y,
z
=a
for x
<
Y
If
r*p+h q-rpq
2
p-q
<
a
Z
<
J<1
[c
+ h 2 + sq -
+ (r+s)pOD
rp - sp
a*
(~
Z - po* - qo*)
J. D* (o*)do*.
+ (c+h -r*-sq)
1
then
y
= x,
y
x,
y = Y
(b)
z
Z
=a
for -
<
x
for Y
<
x<-q
for x
<
Y.
<
x
<
x
q
Z = Z-qx
p-q
Z = Z-qY
p-q
Z
If
a
< C
r*p+h q-rpq
2
+ hI p_q
then
2.7
The Case
a~
p
= x,
=a
-z = Z-qx
Z
for q
Z
for p
Y
= x,
z
=x
for Y
<
z
= Y,
z
=Y
for x
<
<
q and the Case q
y
= x,
Y
z
p-q
<
a~
Z
< -q
Z
x <
p
Y.
p
Results in the previous sections correspond to the case
However, analysis in the other two cases
a~
p
<
q and q
<
a~
a~
q
~
p is
p.
33
essentially identical, and results are similar except perhaps
differing on the boundary conditions of the V(8) function and the
W(y) function.
Let us see how the ordering of 0, q, and p affects
the results.
From (Z.3.3):
v (8)
=
r* + h
°
p
<
When
~
r* + h
+ sq - rp - sp + (r+s) (p-q)<Pn (8).
Z
q, then V(8) is monotone-nonincreasing in 8 and:
Z
- rq
~
V(8)
~
r* + h
Z
+ sq - rp - sp
Corresponding to Theorem 1.1, we now have: Theorem 1.6.
--
°2. p
<
q and
(a)
i f r* + h
(b)
i f r* + h
(c)
i f r* + h
Z
- rq
Z
+ sq - rp - sp
Z
~
- rq
0, then z
°
<
<
z
y
=
y
°
Y
~
0, then z
<
Z
for
<
such that
00,
Z
-<
p
y
= ~
Z
< y
for -q-
=
for
p-q
°
=y
y
r* + h Z + sq - rp - sp, then there
exists a real number Z, a
Proof:
If
°
:2..y
<
00
Z
< -
~p
<
Z
(Z.7.l)
q
Follows proof of Theorem 1.1 immediately.
When q
~
°
<
r* + h
p, then V(8) is still monotone-nondecreasing and
Z + sq - rp - sp -< V(8)
<
r* + h
Z
- rq
34
Hence we have:
Theorem 1.7:
When q
(a)
i f r* + h
(b)
i f r* + h
(c)
i f r* + h
2
~
~
p and
+ sq - rp - sp .::.. 0, then z
~
rq
2
2
0
0, then z y
z
y
Z
<
2
- rq, then there
such that
00,
Z
P
y
for y
~
for -Z < y
p-
p-q
Proof:
<
=0
=y
+ sq - (r+s)p < 0 < r* + h
exists a real number Z, a
Y
< -
<
(2.7.2)
00
Follows proof of Theorem 1.1 immediately.
Theorem 1.2, Theorem 1.3, and Theorem 1.4 apply also to the
case 0
~
q
<
p and q
~
0
<
p.
Proofs are identical except perhaps
the partitioning into different ranges of y should follow (2.7.1) and
(2.7.2), respectively.
Theorem 1.5 applies to the case q
~
with the part W(Z/q) omitted, and will apply to the case 0
~
if all inequalities stated in (a), (b) and (c) are reversed .
•
0
<
P
p < q
-
e
35
CHAPTER 3:
3.1
THE MULTISTAGE PROCESS
The General_ Setup
Without loss of generality, we assume that the multistage process
is composed of N periods.
In each period we have a single-stage procesS
as described in the last chapter, except that goods left at the end of
period i are carried over to period i + 1, for i + 1, 2, 3, ..• , N - 1.
Goods left at the end of period N are disposed of with zero cost.
The following is also assumed:
a.
Each period is of identical length, and the cost structure
is stationary over time.
-e
b.
The leakage factor "p" is stationary over time.
c.
Demands from period to period are independent but not necessarily
identical random variables.
sale demand
Typically the i
th
period special-
D~ has a probability distribution ~i' and the i th
period regular demand D., given that there is no special sale
~
in period i, has a probability distribution
* a., respectively.
minimums a.,
~
~
~.,
~
* Di have
D.,
~
(Within each period, the
dependence of the regular demand on the amount sold in the
special sale is the same as what we proposed for the singleperiod process.
d.
The parameter "q" is stationary over time.)
A discount factor a is charged on future cost.
Specifically,
i l
cost incurred in period i is discounted by the factor a to period one, i
=
1, 2, 3, ... , N.
Since our inventory system is such that at each stage the system is
characterized by the stock level, and the purpose of the process is to
minimize some function of the stock level.
And furthermore, the past
36
history of the system, given the present stock level, is of no importance
in determining future actions, Bellman's l1Principle of Optimalityll
applies.
It says "An optimal policy has the property that whatever the
initial decisions are, the remaining decisions must constitute an optimal
policy with regard to the state resulting from the first decision."
(P. 83 of [5].)
On this basis, we construct the following recurrence equation for
the N-stage model.
Definition:
fij(X ) is the expected total cost from period i to period j under
i
an optimal policy where x. is the starting stock level at period i.
~
--
Then we can write:
(3.1.1)
where H (x ) denote the cumulative distribution function of x given
yz 2
2
the first period policy (y, z).
Again let £* and £ denote the special demand outcome and the
regular demand outcome of the first period, then
for £
~
y-z and £
~
for £
~
y-z and £
> p(y-~*),
for £
>
y-z and
p(y-c*), we have X
z
p(y-s*) - £;
z=0
we have X
pz
we have X
pz - £;
for £ > y-z and £ > pz
we have X
o.
£: ~
z
z
37
Let:
+Joo
{-
rp(y-e:*)
+s[e:
- p(y-e:*)]
+afZN(O)}
p(y-e:*)
+
JOO
<"r*(y-z) + hZz
y-z
+Joo [-
rpz
+s(e:-pz) +afZN(O)<j>l[e: +q(y-z)]~1/Jl(e:*)de:*.
pz
Definition:
A policy for the N-stage process is a sequence of N pairs of
. d Max {O 'Yi-xi }
numb ers ( Yl,zl'YZ,zZ""'Yn,zn ) such t h at at t h e 1.th per1o,
is to be procured, and Max{O,y.-z.} is to be allocated to the special
1
sale.
policy.
1
th
The i th period component of a policy is called a i
period
--
38
3.2
An Optimal Policy
We have discussed in the single-state process the-various results
corresponding to different parametric structures.
Since our goal in
this chapter is mainly to see the link between the single-stage
process and the multistage process, we will restrict our attention
only to a particular parametric structure.
Theorem 3.1:
For the N-stage model, assuming the following:
o
< r
+ s - ac,
O..::.q..::.p,
--
r* + h 2 + sq - rp - sp
<
°..:. r* + h 2 -
c + hI - r* - sq
<
0
<
- rp - sp
<
0 is implied.)
(Note that c + hI + h
2
rp - (p-q)ac
(1- p)c + hI + h 2
then:
(a)
There exists a uniquely determined critical number ZlN'
a
l
..::. ZIN
<
such that a first-period optimal allocation
00,
policy given y is:
z
(b)
for
0
Y
=
ZlN- qy
p-q
=
Y
ZIN
q
<
Y <
ZIN
p - y
-- <
o<
y
00
ZIN
q
< --
<
ZIN
p
There exists a uniquely determined critical number YIN'
o -<
Y <
ln
00,
such that a first period optimal
policy is Y + Max(x,Y
lN
).
p~ocurement
39
(c)
fiN (x) is a continuous monotone-nondecreasing function,
(implying that:
fIN (x) is convex
f" (x) exist and f" (x) > 0 almost everywhere.)
IN
IN
N-l
(hl+h )
L:
Z j=O
fiN (x)
=
-c
for x ~ YIN.
(In particular fiN(O)
Proof:
= -c,
and f~(x) ~ -c for all x ~ 0.)
The parameteric structure superimposed on Theorem 3.1 should
be recognizable from our discussion in the single-stage process.
A
fraction term of c is sometimes added to correspond to the fact that
goods remaining in the stock at the end of period one are not disposed
of but carried over to period two.
Proof is by induction.
the single-stage process.
We have seen that Theorem 3.1 holds for
Let us make the inductive assumption that
Theorem 3.1 holds for n = Z,3, ... ,N-1.
We proceed to show that it
also holds for n = N.
Note that the inductive assumptions:
fi N-l (x) is continuous monotone-nondecreasing,
,
f
1,N-l (x)
fi'N-l (0)
exists and> 0
= -c
a.e.,
and fi,N-l (x)
>
-c for all x > 0 is equivalent to
the assumptions:
f" (x) is continuous monotone-nondecreasing,
ZN
fZN(x) exists and> 0
fiN(O)
= -c
a.e.,
and fiN (x) > -c for all x > O.
40
Now the proof follows closely our procedure in the single-stage
process.
Taking derivative of glN(y,z) with respect to z, we obtain:
~* + h 2 + sq -
rp
sp
PZ
+
f
(P'-q) [r + s + o:f
2N (pz-r) J
a--q (y--z)
1
.1 (, +qy+qZldE}
=
f*
+
[1 - '1 (y-z) J
+ h 2 + sq - rp - sp
f
qy+(p-q)z
(p-q)
a
[r + s + af 2N (qy-Pz- qz-E)]
l
[1 .- 'J! (y·z)]
1
=
V1N(qy+pz-qz)[1
o
'J! (y-z)
1
for y-z .-
a
q
1
a
for y- z '.
-q
(3.2.1)
where:
v
IN
(8)
r* + h
Z
+ sq - rp - sp
(3.2.2)
41
Since:
J
•
8
ViN(8)
=
a
(p-q) [r + s +
af~N(8-E:)]<Pl (E:)dE:
1
+
(p-q)(r+s-ac)~l
(8)
>
O.
V (8) is a continuous monotone-nondecreasing function
1N
It V (M)
1N
= r*
+ h
Z + sq - rp - sp
= r*
+ h
Z
M-+oo
+ sq - rp - sp
N-Z
~ (ap)j]
j=O
>
and V1N (8)
<
r* + h
Z
- rq
° for all 8
<
~
0.
ZlN.
ZlN
V (qy+pz-qz) ~ V (ZlN)
1N
qz ' 1N
1 - \{II (y,z) 2:. 0, glN(y,z) is thus nonnegative.
Now for y
> --
=
z
y
Min
02.-z'::y
z
=°
0.
Since
--
42
For
ZlN
p
2.. y 2..
ZlN
q'
then V1N(qy+pz--qz) = V1N(ZlN) = 0 at z =
ZlN- qy
p-q
z
y
=
o<
zlN- qy
p-q
Z -qy
ln
p-q
is a minimum point.
ZlN
for - -
_< Y
P
<
-
ZlN- qy
p-q
ZlN- qy
p-q
Furthermore it is admissible since
ZlN
y
< -- •
-
q
o
z
y
Max
for all
z = y
O.:.z2Y
Part (a) is proved.
(b)
Substitute Zy in glN(y,z) and define
(3.2.3)
Note that due to the inductive assumptions on fZN(x) and fZN(x),
the following DlglN(y,z), DZglN(y,z), DllglN(y,z), D1ZglN(y,z) and
DZZglN(y,z) are all well defined.
For derivations, consult Appendix
7.1.
1jJ
+
f~
y-z
~+
L
+f pz
1
(E:*)dE:*
hI - r* - sq
q[r + s + afiN(PZ-CllOI(C+qy-qZld:r °I(c*ldc'
z -q(y-z)
1
(3.Z.4)
DZ"LN(Y'z) -
~'+hz+Sq-rp-sp+
L
f
43
pz
a
1
.
-q(y-z)
[r+S+"f2N(PZ-E)].1(E+qy-qZ)d~
[l-'¥l (y-z)]
0.2.5)
+ rp(YE*)
Ja -qE*
1
[q
2
(r+s-ac)~l (qy+pz-qz)
pz
+
J
·l-q(y-z)
0.2.6)
(p-q)[r+s+af2N(pz-E)~1(E+qy-qZ)d}~l(Y-Z)
+ fPZ
a1-q(y-z)
+ fOO
[q(p-q)(r+s-ac)~l (qy+pz-qz)
y-z
f
pz
+
(p_q)2 af "2N(pz-E*H 1
a -q(y-z)
(qy+pz-qz)dd~l (E*)dE*.
1
0.2.7)
44
+ fPZ
a -q(y-z)
(p-q)[r+s+af2N(PZ-E)]¢1(E+qy-qZ)d1~1(y-z)
1
+
f
00
[q(p-q) (r+s-ac)¢l (qy+pz-qz)
y-z
(3.2.8)
From (3.2.3), we obtain:
dz
wiNey) = D1g 1N (y,Zy) + D2g1N (y,Zy) ~
(3.2.9)
and
dz
W~N(Y) = D11g 1N (y,Zy) + D12 g 1N (y,Zy) ~
(3.2.10)
2-
dz
whenever ~
z
2
d
and ---..:J...
dy
= 0
d z
and --.:i.. exist.
dy
ZlN
For -q-
<
dz
y, then Zy = 0, ~ = 0,
•
(3.2.11)
45
(3.2.12)
Substitute z = 0 in (3.2.6).
since y-z > Z1N/ q > a /q.
1
Then
~1
(y-z) = 1 and $1(Y-z) = 0
Hence
2
[p (r+s-ac)~1(pY-PE*+qE*)
For 0 2.. y
(3.2.13)
+
•
2
J:
[q (p-q) (r+s-ac).pl (PY)
46
2
[(p-q) (r+s-ac)¢l(PY)
J
PY
+
a
2
p af 2N (PY-£)¢1(PY)d£]$1(£*)d£*
~0
1
(3.2.14 )
by the inductive assumption f
2N
(x)
>
0
a.e.
ZlN- qy
. Note:
p-q
~-'--
=0
From (3.2.1), D g (y,z)
2 1N
.
(3.2.15)
= O.
Hence
(3.2.16)
--
47
J
p(Y-E*)
+
p2af2N(pY-PE*-E)~1 (E+qE*)]W 1
(E*)dE*
a1-qE*
00
+
J
(PY-Z
1N
) / (p-q)
2
pZIN-pqy
q af" (
- E*)
2N
p-q
2
q af" (
2N
•
pZIN-pqy
p-q
- E*)
-e
48
J
p{y-e:*)
+
p2afZN{py-Pe::*-e::)<P1 {e::+qe::*)]1/J1 (e::*)de::* ..:::. O.
(3.2.17)
a -qe::*
1
.
ZlN
From (3.2.11) and (3.2.16), WINey) is cont1nuous at ---,
ZlN- qy q
since D g {y,z) is continuous in both y and z, and
p_q
= 0 at
1 1N
Z
y=-.
q
From (3.2.13) and (3.2.16), WINey) is also continuous at
ZlN- qy
ZlN
ZlN .
= --- =
--- S1nce
p
p-q
p
zlN
-)
P
O.
Thus wINey) is a continuous function.
From (3.2.12), (3.2.14),
and (3.2.17) WIN{y) is monotone nondecreasing.
It
W1N{M)
=
It
M-+oo
M-+oo
=
It
M-+oo
•
Furthermore:
49
N-2
L (ap)j]
j=O
N-1
=
c + (h +h )
1
2
L
(ap)j > 0
(3.2.18)
j=O
and
=
f~
(c+h
o
=c
+ h
-r*-sq)~ (e*)de* +
(r*+h +sq-rp-sp)
1 1 2
1
+ h 2 - rp - sp
<
0
(3.2.19)
Thus there exists a unique Y1N' 0
WiN(Y 1N )
=0
and WiN(Y)
<
0 for all Y
<
<
Y1N
Y1N •
< ~,
such that
Since WiN(Y) is a
monotone nondecreasing function, Y
is an optimal procuring level
1N
if admissible.
(~)
Thus part (b) is proved.
By definition of f
1N (x),
for Y ..::. x
1N
for x
<
Y
1N
(3.2.20)
50
Taking derivative:
for YIN":' x
=-
c
for x
YIN
<
0.2.21)
Since WiN(x) x t Y ) WiN(Y 1N ) = 0, fiN(x) is continuous.
IN
is monotone nondecreasing since WiN (x) is.
It
From (3.2.21) and (3.2.18):
N-1
It
N-+oo
- c + c + (H +h )
1 2
~
j=O
N-l
(h +h )
l 2
E
(ap)j
j=O
Q.E.D.
Part (c) is thus proved.
Note that since N is arbitrary, Theorem 3.1 will obviously
still be true if results are stated in ZlN' YiN and fiN(x).
Lemma 3.2:
Under the assumptions of Theorem 3.1, then
r*p+h q-rpq
2
c + hI - - - --p-q
Proof:
Z'N
iN p
W,(_J._)
•
< 0 <=>
ZiN
-- < Y
p iN
for all i
= 1,2, ••• ,N.
51
=c
+ hI - r* - sq + -S- (r*+h +sq-rp-sp)]
p-q [VoN(ZoN)
~
~
2
= c
+
h
1
-
r*p+h q-rpq
2
p-q
----.:;:~--
Since WiNey) is monotone nondecreasing, Lemma 3.2 follows.
3.3
Some Ordering Properties of Critical Numbers
In this section, we shall derive some ordering properties of
= 1,2, ... , N, when demand distributions
the critical numbers YiN,ZiN' i
are identical from period to period.
distributions, and
Theorem 3.3:
Let
{~,~}
denote the common
{~,~}
the densities.
r*p+h q-rpq
2
Suppose c + hI p_q
<
0 and
~i
=
~,
~i
= ~.
Then under the assumptions of Theorem 3.1, ZiN and YiN are monotone
nondecreasing in i.
Proof~
We will first prove ZIN ~ Z2N' YIN ~ Y2N •
From Lemma 3.2,
ZiN
-- <
P
(3.3.1)
52
0
>
> -
Z2N -
E ~
Y2N for all
E ~
o.
But VlN (Z2N) ~ V2N (Z2N) = 0 = VlN(ZlN) implies Z2N ~ ZIN since
V (0) is monotone-nondecreasing in 0.
lN
The proof for Y part is similar.
~
W2N (Y 2N ).
Again we show WiN(Y2N)
However, since W1N(y), W (y) have various functional
2N
forms at different ranges of y, and we do not know in which range
Y~N is~
we have to consider all possibilities.
Z2N
From (3.3.1),
~ Y ; also we have proved Z2N
2N
-p-
Y can lie in any interval of the following sequence:
2N
~
ZIN.
Thus
53
Z
Z
2N < Mine IN
p p
First, 1e t
Z2N
q
-) <
Z
IN < y
q
2N
<
00.
Max
Z2N
ZlN
(p- , -q)
ZZN
Then q
<
Y2N'
ZlN
<
-< p
00.
From (3.2.11),
But:
Y
=
t:
2N
a*
[
J
J
p(Y
+
-e:*)
2N
a-qe:*
•
+ hI + h Z - rp - sp
per + s +
1
af3N(pY2N-Pe:*-e:)]~(e:+qe:*)de: ~(e:*)de:*
54
>
"r"
0
ZlN
Next let Uax(-p
-e
Z2N
~
---) <
q
-
Y <
2N -
ZlN
-- ,
q
then from (3.2.16) and
(3.2.11):
and
Bvt Dlg1N(y,z) i~ monotone-pondecreasing in z for z
Z
<
--qy
l~_q
sinee D1Z8IN(Y'z) is nonnegative.
either 0 or -V1N(qy+pz-qz)
~
(The first term in (3.2.7) is
ZIN- qy
-V 1N (ZIN) = 0 for Z <
p_q .)
lienee
That the last difference is nonnegative has been proved in the
Third, let
"
•
Z
IN
p
<
-
Y
2N
Hax
Z2N
< ---
q
by supposing
Z2N
= --q
55
From (3.2.16):
Again by nOting:
f
'N(P •
3
-e
f 'N(P •
2
whil~
the last equality is because from
Z2N- qY 2N
- E)
p-q
Z2N- qY 2N
p-q
E) = -c,
Z2N
Y :
p - 2N
~ <
Z2N
Z2N-qY2N
Z - q --< p'
2N
P
= Z
< Y
P • p-q
p-q
2N - 2N'
But:
> -c
56
"e
-e
Z2.N- qY 2N
p-q
a-q •
57
PY2N -(P-q)£*
+
f
a
p[r+s+
f
Z2N
p[r+s+af3N(Z2N-£)]~(£)d£~(£*)d£*
a
58
Finally for the other two cases:
Z2N
.......... < y
q
-
2N
ZIN
P
ZIN
P
< - - when l1ax (--
Z2N
we have - p
< y
-
ZN
<
ZIN
= -P
ZZN
and - -
Y
P ZN
Z
IN in both cases.
P
Hence:
Z
Z
>
WI ( 2N) _ WI ( IN)
.
(c
2N
P
IN
r*p+hzq-rpq
+ hl -
p-q
.. 0
by
P
~sing
Lemma 3.2.
) - (c
+ hI -
r*p+h q-rpq
2
)
p-q
<
ZIN
< --
P
59
Thus we have proved W (Y )
ZN 2N
possible ranges.
~
W (Y ) for Y lying in all
1N 2N
2N
But:
o
implies YIN
~
Y2N since WiN(y) is monotone-nondecreasing.
We have succeeded in proving YIN
N.
~
Y2N , ZlN
~
Z2N for arbitrary
Since YiN ~ Yi+l,N and ZiN ~ Zi+l,N are equivalent to
Yl N-i+l ~ Y2 N-i+l and Zl N-i+l ~ Z2 N-i+l for all i
,
,
,
'-
= 1,2, ••. ,N-l,
Theorem 3.4 is proved.
Theorem 3.5:
for all i
Suppose c + hI -
= 1,2, .•• ,N.
r*p+h q-rpq
2
p-q
>
0 and
~.
1
= ~,
~.
1
=~
Then YiN are monotone nondecreasing in N.
Proof:
~
Since now we do not have the inequality Zl
Z2' the essential
ingredient in proving Part b of Result 3.2.1, we have to resort to
a different approach.
Define
Py
rl (y)
=c
+ hI + h 2 - (r+s)p +
f
a
p(r+s-ac)~(£)dc;
then rl(y) is clearly continuous and monotone-nondecreasing.
Also
rl(M)
and
~
c(l-ap) + hI + h 2
>
0
(3.3.2)
60
So there exists a unique yQ, ~ < y~
pand
~(y)
<
0 for all y
<
<
00
such that ~(y~)
0,
Q
Y •
r*p+h q-rpq
z
2
>
0
implies
~N > Y2N from Lemma 3.2.
Now c + h l p_q
By Theorem 3.3, Zy
= Y2N.
2N
From (3.2.13):
By noting
we obtain:
PY
~
c + h
l
+ h
From
ingness of
Z
~(Y
- r+s)p +
Q
)
~(y),
2N
J
p(r+s-ac)~(€)d€
a
= 0 = W2N (Y2N)
we conclude y
~ ~(Y2N)
Q
~
Y2N.
and the monotone-nondecreas-
61
Now we will show Y
1N
Supposing Y
1N
Zy
1N
= Y1N •
=c
Y by using a contradiction argument.
ZN
ZlN
Y ' then f (Y -s) = -c. Also since Y < -p-'
ZN
ZN 1N
1N
<
~
So from (3.Z.13):
+ h
+ h Z - (r+s)p +
1
f
~rn
a
p(r+s-ac)~(£)d£
= Q(Y 1N )
we conclude Y
1N
But Y
1N
~
Y •
~
Q
Y
Q
and the way Y is selected,
Y is contradictory to Y < Y '
ZN
1N
ZN
~
Again since N is arbitrary and YiN
~
Y + ,N is equivalent to
i 1
Y1 ,N-i+1 ~ YZ,N-i+1' Theorem 3.4 is proved.
Theorem 3.5:
When
~i
=
~,
~i
=
~
for all i
= 1,Z,.,.,N, then under
the assumptions of Theorem 3.1, YiN are uniformly bounded for all i
and N, i
<
N.
Proof:
r*p+hzq-rpq
When c + h
1
-
of Theorem 3.4, Y
ZN
> 0, then we have seen in the proof
p-q
Q
Q
~ y when y is the number determined from
62
Q
Q(y )
=
0, Q(y) ~
° for all y
<
yQ.
Since N is arbitrary an~ YiN is
equivalent to Y2,N-i+2' we conclude YiN
when c + h -
r*p+h q-rpq
2
p-q
When c + hI -
>
~ yQ for all
i and N, i
° holds.
r*p+h 2q-rpq
p_q
ZIN
~ 0,
from Lemma 3,2,
-p- ~ YIN'
ZIN
First assume --q-- < YIN' then from (3.2.11) and (3.2.4):
=
JYlN
a*
t
c + hI + h 2 - rp - sp
Define:
[c + hI + h 2 - rp - sp
~ N,
63
p(y-e:*)
+
J
p(r+s-ac)~(e:+qe:*)de:]$(e:*)de:*
a-qe:*
(c + h -r*-sq)$(e:*)de:*
1
(3.3.4)
Then since:
+ V(qy)ljJ(y)
>
0
z.l.
for -
q
< Y.
-
QI(Y) is monotone-increasing.
It
~l (M) ~
M+oo
Furthermore:
c + hI + h Z - rp - sp +
p(r+s-a~)
> 0,
and
Z
QI(
~N)
Z
~N)
2. WiN (
<
WiN(Y IN )
= O.
QI ZIN
there exists a uniquely determined number Y ,--q
such that Q (yQI)
l
= 0
and Ql(Y)
>
<
Ql
Y
<~,
0 for all Y > yQl,
We find: WiN(YlN) ~ Ql(YlN)
QI
Also, Ql(Y ) = = WiN(Y 1N ) ~ Ql(Y lN )'
Compare (3.3.3) and (3.3.4).
•
°
since f2N(pYlN-ps*-s) > -c.
QI
y ~ YIN by the monotone nondecreasingness of Q, ~nd our
selection of yQl.
ZIN
Next assume --- < y
P
-
ZIN
q
We will show the
< ---
IN -
~niform
boundedness of YIN by showing that ZIN are uniformly bounded.
Define
QZ(8)
=
r* + h + sq - rp - sp +
Z
(p-q)(r+s-ac)~(8)
Then QZ(8) is monotone-nondecreasing in 8 and
Qz(O)
..
= r*
+ h Z + sq - rp - sp
~
O.
(3.3.6)
65
Q2
There exists a Z ,0
Q2(0)
>
0 for all 0
>
~
Q2
Z
~
M, such that Q2(Z
)
z
0
~nd
ZQ2.
Compare (3.2.2) with (3.3.6), we note VIN(ZIN)
fiN (x)
Q2
~ Q2(~lN) si~ce
~
-c for all x.
Q2
But Q2(Z ) = 0 = VIN(ZIN) ~ Q2(ZIN)·
Since Q is monotone-
2
nondecreasing and from the way we select zQ2, we conclude zQ2 > Z
- IN,
r*p+h q-rpq
2
Thus we have proved that when c + hI < 0,
p-q
YIN
~
Max (y
QI
, zQ2/q).
Again since N is arbitrary and YiN is
equivalent to Yl , N-i+l' we conclude YiN are uniformly bounded for
all i and N, i
•
<
N.
--
66
CHAPTER 4:
4.1
THE INFINITE-STAGE PROCESS
Infinite Period Model When Demand Distributions
Time Periods are Independent and Identical
Theorem 4.1:
i~
Different
Suppose
+
s - ac > 0
(a)
r
(b)
p ~ q ~ 0
(c)
r* + h Z + sq - rp - sp
(d)
c + hI - r*-sq
(e)
demand distributions
<
0
<
~
0
~
r* + h
Z
(l-ap)c + hI + h
{~i'~i}
- rq - (p-q)ac
Z
are identical for every time
period and the demand on special sale or
the i
th
regula~
sale in
time period is independent of the demand in the
special sale and regular sale in any other time period.
Then fiN (x) converges uniformly for every finite i to a unique
function fl(x) for every finite close interval 0
~
x
~
b.
Moreover,
fl(x) is a bounded continuous convex function in the interval
o
<
x
<
b for every finite b.
And fl(x) is the unique solution among
the class of continuous and uniformly bounded solution of the infinitestage functional equation.
Proof:
Under the assumptions stated, we know from Theorem 3.1 there
exists for every finite i and N (i
~
N) critical numbers 0
~
YiN
< ~.
From Theorem 3.6, YiN are uniformly bounded for every i and N.
Let (yiN,ziN) be the i
policy.
•
Then:
th
period component of the N-stage optimal
·e
67
= Lx. (y-N,Z-N)
1
1
f.1 1 •N(x·+l)dH- (X·+
1
y,N,Z'N
1 l)
1
1
1
(y.'
N- l'Z,1N-1) + a
-< Lx i
1
,
JOO
o
f'+ 1 N(x.+l)dH(X.+ )
y.1, N_1'Z'1, N-1 1 l
1 ,
1
Similarly:
..
. dH(x. )
y.1, N'Z'1, N 1+1
Hence:
If·N(x.)
1 I
1
1 - f.1, N- 1(x.)
..
dH(X.),
yiN-lziN-l 1+1
68
<
(4.1.1)
CI.
where the supreme is taken over all possible x + •
i l
Now x + ~
i l
Yi = Max(xi,Y iN )
and in general,
<:
That such a "B" exists is because of 0
~
xi
~
00
b, and the uniform
boundedness of YiN.
Let
u.
1.
(4.1.2)
=
Since (4.1.1) is true for all 0
~
xi
~
B, in
part1e~lar:
(4.1. 3)
In general, we will have:
69
Now LX(YNN~ZNN)' the single-stage minimum expected cost is
continuous in x and is finite for every 0 2 x
exists a number m
<
such that
00
I
Sup
~
B.
Therefore there
Lx(yNN,zNN) 12m, where m
O<x<B
depends on the demand distribution only.
(4.1.5)
00
00
Since N~i DiN 2 N~i a
N-i
m
m = I-a ' fiN (x) converges uniformly to
a function fi(x) for all 0
~
x 2 B.
fi(x) is a continuous function.
fi(x) is convex.
fi(x)
= fl(x).
Since the convergence is uniform,
Since fiN(x) is convex for each N,
Since i is arbitrary and fiN(x)
= fl,N-i+l(x),
Let us denote this common limiting function by fI(x).
That fI(x) is a solution of the infinite-stage functional equation:
= Lx (Y,z)
f(x)
+
aJOO f(x)dH-y,x-(x)
(4.1.6)
o
is due to the uniform convergence of fiN to fl.
(see p. 15 of [5]).
To prove the uniqueness of fI (x) as solution of (4.1.6):
Let Fi(x ) be any other solution such that FI(x ) is continuous
l
l
and uniformly bounded for all 0 2 x 2 B
1
=L
xl
<
00.
Then:
(y,~) + aJoo F-l (x2 )dH-y,z-(x 2 )
(4.1.7)
o
An optimal policy (y,~) exists since the right-hand side of
•
(4.1.7) is continuous and finite for finite
a solution:
y.
Since fI(x) is also
70
(4.1.8)
By the definition of (y,~) and
(y,z)
(4.1.9)
and
f-(x )dH= =(x )
1 2 y,z 2
(4.1.10)
- f-(x )ldH= =(x )
y,z 2'
1 2
- f-(x ) dH- -(x
1 2
y,z
2}~
(4.1.11)
The maximum exists since F1 and f1 are both continuous and
finite in the interval 0 ~ x ~ B where B = Max{x
2
1
u = Max IF1(x) - f1(x)
O<x<B
o ~ xl ~ B, we obtain:
•
I.
,y,y}.
Let
Then since (4.1.11) is true for all
u < au
(4.1.12)
71
For 0
<
a
<
1, u
= o.
Or equivalently, FI(x)
= fI(x)
for all x
lying in a finite interval.
Theorem 4.2:
(a)
Under the assumptions of Theorem 4.1:
There exists a uniquely determined number ZI' a
<
Z1
<
00,
Y1
<
00,
such that an optimal allocation policy given y is:
z
Z1
Y
= 0, -
q
< y <
Z--qy
Z1
ZI
< y <P - q
1
= ---"---
-
p-q
= y,
(b)
00
Z1
0 < y < -
-
P
There exists a uniquely determined number YI' 0
such that an optimal procurement policy is y
(c)
f~l (M) --~
M-+-oo
f~(x)
1
Let
•
= Max(x'Y1).
ft(x) is a continuous monotone-nondecreasing function,
•
. Proof:
<
= -c
(h +h ) /l-ap
1
2
for x ..:::.. Y1.
72
+foo
•
[-
rp(y-e:*) + s(e:-py+pe:*) + afI(O)]
p(y-e:*)
+J
f-
00
r*(y-z) + hZz +
f
y-z
pz
[- re: + afI(pz-e:)]
a-q(y-z)
•<j>(e:+qy-qz) de:
+
f:z [- rpz + s(o-pz) + afI(O)l~(O+qy+qZ)d~ .(o·)do·
(4.1.13)
Then:
fI(X)
=-
ex + Min
Min
gI(y,z)
(4.1.14)
x~<oo 0:5..z~
From (4.1.13):
•
•
+
f
p(y-e:*)
}
p[r + s +
a-qe:*
afl(py-pe:*-e:)]~(e:+qe:*)de: ~(e:*)de:*
+
+
Jy-~ (+L
m
f
73
hI - r* - sq
p~
q[r + s +
a-q(y-z)
.f1(PZ-£)].(£+qy-qZ)d~ o(£*)d£*
(4.1.15)
sp
+f
pz
(p-q)[r + s +
aft(pz~£)]
z-q(y-z)
.00£+qy-qZ)d£}[1 -
=
{:*
~(y-z)]
qy+(p-q)z
+ h Z + sq -
rp - sp +
f
(p-q)
a
o[r + s + .f (qY+PZ-qZ-£)]0(£)dj [1 1
=
VI (py+pz-qz)[1
-
~(y-z)]
~(y-z)]
a
for y-z < -
-q
a
for y-z < -
q
(4.1.16)
74
where
V-(0)
1
= r*
+ h
2
+ sq - rp - sp
(4.1.17)
From Theorem 4.1, f (x) is a convex function defined on any
1
finite close interval of x.
Therefore ft(x) exists almost everywhere,
&nd D g1(y,z), D g1(y,z) and V1 (0) are well defined continuous
1
2
functions.
Furthermore, ff(x) is monotone-nondecreasing and bounded
~
between -c and (h 1+h 2 )/(1-ap) since by having an additional
amount
of goods in the starting stock, the best could happen is that a
procuring cost of
•
c~
will be saved; and the worst could happen is that
no procurement will be made and this
~
amount of goods will not be
sold hereafter, and thus incurring a holding cost (h l +h 2 )
00
j~O
(ap)
j
~
=
(hl+h2)~/(1-ap).
Now v1(0) is monotone-nondecreasing since the !ntegrad
(~-q)[r
+ s + aff(0-£)] is monotone-nondecreasing and nonnegative.
Moreover:
V1(a) = r* + h 2 + sp - rp - sp
V-(M)
1
<
0
---+
M+w
Hence there exists a uniquely determined Z1 such that VI(ZI)
and VI(0)
<
0 for all 0
<
ZI'
=0
..
75
e
Considering (4.1.16) and using arguments similar to the singlestage process or the N-stage process, we obtain part (a).
•
Let
Again using arguments similar to the single-stage process or
the N-stage process, we have:
Zy
for q
y
<
<
y
< -
00
Z-
Zy-qy
--=-)
p-q
<
for
1
~
p
Z-
-
1
q
•
Zy
for 0 .::. y
<
p
and wt(y) is continuous.
To show the monotone nondecreasingness of wt(y) , we cannot take
Wr(y) since we can say nothing about the existence of ff(x) and
thus Dllgy(y,z), Dlzgy(y,z) and Dzzgy(y,z) are not well defined.
However, we can take the difference of wt(y) at any two points
Yl > yZ and show that wt(Yl) ~ Wi~Yz)'
> -l and let
First let y be any point -q
•
•
then:
~
be any positive number,
76
•
e
W.l(y+li) - W.l(y)
1
1
•
+
+
j
P (y+li-E:*)
p[r
+
a-qE:*
JOO
(c+hl-r*-sq)ljJ(E:*)dE:* _
y+li
•
J
{c + h
l
+ h
Z- rp
- sp
a*
p (y-E:*)
+
jY
p[r
+ s +
afi(pY-PO*-O)]~(O+qO*)d~ ~(o*)do*
a-qE:*
P (y+li-E:*)
.
•
+
J
a-qE:*
p[r
+
s +
afi(PY+P8-PO*-O)1~(O+qO*)d0Jt .(e*)do*
77
•
+
JI
p[r + • + afY(PY+PA-pe.+qe.-el],(elder $(e·lde·
a
y+t.
->
1
P (y+t.)- (p-qh:*
JI
V-(Z-)~(E*)dE*
1 1
Y
= O.
(4.1.18)
z-
z-
Next let y be any point lying in the interval [pI. ql)
Z'i
a positive number such that y + t. < --.
-q
W.1(y+l1) - W.1(y)
1
1
Z--qy
•
•
1
p-q )
Then:
and t.
=
•
78
Z--qy-qll
J
1
Y ll
p-q
+ -
a*
+ Jy+OO. _ ':'I-qy-qll
---=-_ _
L.I
{C + hI
- r* - sq
p-q
p.
+
p-q
J
ZI-qy-qll
a-q(y + II - -=p-_-q--)
•
1
p-q
•
--=---p-q
.~(E + qy+qll-q·
Z--qy
•
ZI-qy-qll
q[r + s + af-I' (p •
E)]
Z--qy-qll }
1
)dE ~(E*)dE*
p-q
79
ZI-qy {
c + h
- r* - sq
1
p-q
+
J
p'
ZI-qy
q[r + s + afi(p .
p-q
_Z
p~I~_:_y
- E:)]
a-q(y _ ZI-qy )
p-q
.~(E + qy - q.
Z--qy
}
1
)dE
p-q
~(E*)dE*
•
J
1
p(y+~)-(p-q)E*
+
per +
a
s
+ afi(pytpo-po*+qO*-OlJo(old
ZI-qy-ql:l
y+~--=--­
-J
p-q
1[c + hI ZI-qy
y-~-
p-q
r* - sq
J
~(E*)dE*
--
80
•
ZI-qy-qll
Y
:f
+
II -
-=---p-q
zr qy
y -
{:r*
+ hz +
sq -
rp -
sp
p-q
(Since p(y+ll)-(p-q)E*
~
p(y+ll)-(p-q)(y + II -
Z--qy-qll
1
)
p-q
= Z-l
and
e
faIr
+ s + ait(e-E)l,(E)dE is monotone nandecreasing in 0.)
ZI-qy-qll
Y + II -
-f
y -
=
•
--=---p-q
ZI-qy
p-q
°
(4.1.19)
Finally let y lie in the interval [0,
z-
number such that y + II
•
1
< --
-p
z-1)
p
Then:
and II be a positive
81
W.!(y+ll) - w.!(y)
1
1
•
00
•
=
f
>
0
P(y+ll)
{
0 c + hI - r* - sq +
q[r
+ s + aft(py+pO-£)]
(4.1.20)
From (4.1.18), (4.1.19), (4.1.20), and the continuity of wt(y)
Z1
at the points -p
nondecreasing.
•
fa
and
Z1
- - , we conclude that W-'(y) is monotone
q
1
Furthermore:
82
It
Wt(M)
It
=
M-+oo
M-+oo
and
=
(c+hl-r*-sq) + (r*+h 2+sq-rp-sp)
=c
+ hI + h 2 - rp - sp
<
o.
There exists a uniquely determined number YI' 0
that wt(YI) = 0 and wt(y)
<
0 for all y
<
Y1 .
•
YI
< ~,
such
Part (b) is proved.
Part c trivially follows from the fact
fI(x)
<
=-
cx + WI(x)
for Y
<
x
=-
cx + WI(Y)
for x
<
Y
83
sin~e ft(~) i~ monoton~~nondecre~sing and e~iets
Mpreover,
everywh~re~ fI(~)
il1terval of
4.2
now exists
a~mo~t
everywhere for any closed
~.
Critical Nu~bers of the Infi ite-Period Process with Inde endent
and Iqentical ~mand Oistributions
i
Since:
z-
-1.)
p
=c
+ h1 ~ r* - so~ + -Sp-q
=c +
h
1
...
[V-(Z-)
1 1 - r*+h 2+sq-rp-sp)]
r*p+hzq-rpq
p-q
(4.2.1)
i
We have:
Lemma 4.3:
i
.....,
o <=>
z-1
P
-
yl'
When
From
~
r*p+h 2Q-J1PCl
+ hI, .,. ..
and
(4.1.~7)
pl"'q ,,;
Tp~or,m
;.. O,tr:)Pl Lenun{i 4.3, ZI
~
z-1
P
84
~
YI'
4,2
=0
. ~1'"
1
41'"
+p+6P~r*~h2~sq'
(4.2.2)
< (p..,q) <;rt'"s:'qa»
I
;?y
To eolve
.
.
Xl'
l\otE.
that for. Yi
:>
(l
--
85
+
f
(c
ro
+ hI ..
r* -
sq) ljJ(e*)de:*
yI
" flI
a*
;=
0
(4.2.3)
•since t'y(pYI-pe:*-e:>
...... c;: from l'heorem 4.2.
2y
•
If the solution of Y1 frQm (4.2.3)
z."
clearly Y- < -!
l -q
"
Now
~s
not
~reater
than
q-'
then
-e
+
j
{c +
QO
86
t'~ ~q
h ...
1
...
pY--z1 1
p-q
z-.1 . . qY-1
p-q
q[F
pY--Z1 1
+s +
Z--qYfI(p •
1
1
p..,.q
p-q
pY--Z-
-e
=
+
j
j
~_ql
[c + h1 +
P~ -
rp - sp
+P(r+s-~c)~(pY--p£*-q£*)]
.
1
a*
QO
[c
+ h l - r* - sq +
q(r+s-qc)~(ZI)]~(£*)d£*
PYI-ZI
p-q
r*p+hzq-rpq
=c+h
=0
1
-
p-q
i
(4.2.4)
87
Since wt(y) is a monotone nondecreasing function, (4.2.3) and (4.2.4)
can be solved without gfeat
When c + h
l
-
difficul~y.
r*p+h2q~rvq
P~q
there is no easy way to solve
Z1
>
Z1
0, from Lemma 4.3,
p-
>
Y1.
Now
explicitly since the term fi(Zr- S ) in
VI(ZI) cannot be made equal to "_ C ."
However, once the stock level is
down to YI' then for all s4cce~~ing periods, Yl = Max(xi'YI) = YI' <
ZBut from Theorem 4.2, Y1 < ~ implies zi = Y = Yr. So the solving
i
of Z1 is only of academic in~erest.
YI can still be solved from:
W'!'(Y-)
--
1
1
=
+
{
r* + h Z + sq - rp - sp +
PY-
J
a 1
(p-q)
z-1
p
--
88
=
°
(4.2.5)
siQce ff(pYI-e)
= -c.
From (4.2.5), we solve;
1
-1 rp+sp-c-h l -h 2
YI = P ~ (p(r+s-ac)
)
Lemma 4.4:
(a)
(4.2.6)
If the following conditions hold
F , F , F , F are probability distributions with
3
I
2
4
densities f , f , f , f •
l
2
4
3
Li(e) = -t , Li(e) = t , L3(e)
l
2
(b)
Li(e:)
=
= -t 3 ,
L4(e) = t 4 , where
dLi(e:)
dE
and t l , t 2 , t 3 , t 4 are nonnegative
constants.
~
(c)
FI[Ll(e)]
F [L 3 (e)], F2 [L 2 (e)]
3
(d)
A + BF [L (b)]
3 3
~
0, F [L (a)]
2 2
constants; B is nonnegative, a
J
b
{A
+
BF1 [L 1 (E)]lf 2 (L 2 (E) ]dE .':.
Proof:
Using integration by parts:
..
a
=
~
F [L 4 (e)] for all E.
4
° where Z,
~
b.
B, a, bare
89
"Theorem 4.5:
Let Y , ZA be the critical numbers when demand
A
distributions are identically
~A
~A
and
at each period.
Let Y , ZB
B
be the critical numbers when demand distributions are identically
~B
and
~B·
Moreover let
~A(£) ~ ~B(£)
for all £, and
~A(£*) ~ ~B(£*)
for all £*, then under assumptions (a), (b), (c) and (d) of
Theorem 4.1, YA ~ Y •
B
Proof:
First consider the case c + hI
~
r*p+h q-rpq
2
p_q
>
O.
From (4.2.6):
90
Next consider the case c + h
l
-
rp+sp-r*-h -sq
2
(p-q) (r+s-ac)
r*p+h q-rpq
2
p-q
<
O.
From (4.2.2):
= ~B(ZB) ~ ~A(ZB)
(4.2.7)
ZB
From Lemma 4.3, -- < Y • We will consider the two possibilities
p B
ZB
ZB
Y < and -- < Y •
B
B - q
q
ZB
ZA
Assuming -- < Y , then from (4.2.7),
< Y •
B
B
q
q-
-e
Let a and a* be the minimum values of the distributions
~A
and
~A
respectively; and band b* be the minimum values of the distributions
~B
and
~B
respectively.
Then b
~
a, b*
>
a*.
By noting fl(x)
~
-c and
using Lemma 4.4:
1jJ
A
(c;*)dc;*
91
YB
~
J
[r* + h2 + sq - rp -
sp
a*
+
p(r+s-ac)~
(pY -p€:*+q€:*)]~ A(€:*)d€:* + (c+h 1-r*-sq)
A B
YB
~
J
[r* + h 2 + sq - rp - sp
a*
+
J
p(r+s-ac)~
(pY -p€:*+q€:*)]~ B(€:*)d€:* + (c+h 1-r*-sq)
B B
YB
=
[c + h + h2 - rp - sp
l
b*
+
J
00
Y
B
and
(c+h1
-r*-sq)~ B(e*)de*
92
YB
=
f
[c + h
1 + h 2 - rp - sp
b*
= -c.
since f'(pY -pE*-E)
B B
W' (Y ) =
A
B
ZB
ZB
Next assuming -p-~ YB ~ qZA
(4.2.7), --
Y .
B
p <
If YB ~
,
then W~(YB)
ZA
q-'
then Wl(Y B)
= D1gB(Y B,
= D1gA(Y B,
ZB-qYB
p_q ).
ZA-qYB
p_q ).
From
If
However, D1gA(y,z) is independent
of z for y - z
Y
B
ZA
a
> -- > -
4.4:
q
-
>
q-
>
a*.
a*.
So in either case
93
Z _ y [c + hI - r* - sq + q(r+s-acHA(ZA)]I/J(E*)dE*
A q B
p-q
'.
94
[r* + h 2 + sq - rp -
+
p(r+s-ac)~
A
Z -qY
B B
p-q
•
..
A
(Z A )
(pY B-pe*-q€*)]$A (E*)dE*
[r* + h 2 + sq - rp -
-.
sp-q(r+a-ac)~
sp-q(r+s-ac)~
.
A
(Z )
A
95
ZB- qYB
p-q
[c + hI - r* - sq +
q(r+s-ac)~B(ZB)]~B(€*)d€*
and
-e
·_qY
B B
p-q
Z
f
ZB"-qY B {
- •.-_._---p-q
+
hl
+
h 2 .- rp - sp
_+
h]
.-
r*- sq
.
.~
=
jYB-
[c + hI + h
2
B
(E + qY
B
- q •
- rp - sp
b*
[c + hI - r* - sq +
-.
q(r+s-ac)~B(ZB)]~B(E*)dE*
ZB-qYB
p-q
since
f~(pYB-PE*-E)
=
··c and fBI (p •
ZB-qYB
.-. E)
p-q
~---
=
-c.
(The latter
equality is due to:
p •
To summarize, we have proved that when c + hI WA(Y )
B
~ W~(YB)
regardless of the value of Y •
B
But
r*p+h q-rpq
2
p_q
~
0,
-•
97
implies Y
B
~
Theorem 4.5
Y since WA(y) is a monotone nondecreasing function.
A
is proved •
Theorem 4.6:
Under assumptions (a), (b), (c) and (d) of Theorem
4,1, then YIN
~
YI for all N
= 1,2,3,."
Where YIN is the first-period critical number of aN-stage
process with varying demand distributions over the time period,
Proof:
First assume c + hI ZlN
< -Hence
r*p+h q-rpq
2
p-q
> 0,
Then from Lemma 3,2,
P
-.
+
•
f*
+ hZ +
sq - rp - sp
j
+.
.PY-1
(p-q)
a
l
98
+ [r* + h Z + sq - rp - sp +
=
(p-q)(r+s-ac)~l(pYI)]
0
••
Next assume c + hI -
.
=
0
r*p+hzq-rpq
p_q
<
O.
99
•
(4.2.8)
To prove W1N(YI) - wt(YI)
~
0, we need to consider all possible
ranges of YI so that the difference can be written out explicitly,
However, the proof is exactly the same as the proof of the corresponding part for Theorem 3.
We will find that those terms ft(x) in
wt(YI) have arguments x less than YI and hence fI(x)
~
corresponding terms fIN(x) in W1N(YI) is
-c.
= -c
while those
This plus (4.2.8)
provides the tools needed for the proof.
4.3
Infinite Period Model Whe~ Demand Distributions in Different
Time Periods are Independent but Varying
Theorem 4.7:
•
Suppose
+
s - ac > 0
(a)
r
(b)
p ~ q ~ 0
(c)
r* + h
(d)
c + hI - r* - sq < 0 < (l-ap)c + hI + h
(e)
the i
2
th
+ sq - rp - sp
~
0
~
r*
+ h 2 - rq - (p-q)dc
Z
period demand distributions {9i'~i} are independent
but not necessarily identical to the demand distributions
of any other time period,
(f)
there exists two distribution functions
expectations such that 9.(e),
1
and
~.(e*) < ~(e*)
1
-
<
-
for all e*, i
{9,~}
with finite
9(e) for all e
= i,2,3, •.•.
Then fiN (x) converges uniformly for every finite i to a unique
function f.(x) for every finite close interval 0 < x < b.
1
Moreover,
100
fi(x) is a bounded continuous convex function in the interval
•
a
<
x
<
b for every finite b .
Proof:
Theorem 4.3 is a restatement of Theorem 4.1 except that now
demand distributions are allowed to vary over time.
The proof of
Theorem 4.1, aside from the portion dealing with the uniqueness of
the solution to the functional equation, will still be true here if
we can show YiN are still uniformly bounded for all i and N, i
<
N.
Let (Y,Z) be the critical numbers corresponding to the
infinite-stage process where demand distributions at each period
=
are identically
{~,~}.
i+l ... where
is the critical number when demand distributions
Y~
J.
are identically
(~.,~.)
J.
].
for all i = 1,2,3, ...
From Theorem 4.6, Y'J.
for all periods.
< Y~
N-
J.
for all N
From Theorem 4.5,
From Theorem 4.2, Y
<
00.
i,
Y~ <
J.-
Y
Thus we see that
YiN are uniformly bounded and Theorem 4.7 follows.
= It fiN(x) exists and is a bounded
From Theorem 4.7, f.(x)
N-+oo
1
continuous convex function over all finite close interval of x.
follows that
=
It
f~(x)
1
exists almost everywhere.
I
It
Therefore g.(y,z)
1
2
giN(y,z) exists and gi(y,z), gi(y,z) are well defined
N-+oo
continuous functions.
Also:
(4.3.1)
-e
101
are well defined and
= V.(qy+pz-qz)[l
-
~
~.(y-z)]
for y-z < a./q
-
1
=0
~
for y-z > ai/q
(4.3.2)
With the boundary conditions checked, then there exists a
uniquely determined number Z., a.
~
1
<
Zi
< ~,
From (4.3.2), we can construct the z
y
=0
such that V.(Z.)
~
~
function as we have done
many times now.
Let
W. (y) = g. (y,
~
1
zY)
(4.3.3)
We can show as before:
Zi
for -
<
y
<
Zi
for -
<
y
< -
q
Z.-qy
~
p-q )
P
-
r;1:)
Zi
- q
(4.3,4)
for
and that Wi(y) is continuous monotone-nondecreasing.
(See proof of
Theorem 4.2.)
So there exists a uniquely determined number Y , 0
i
such that
W~(Y.)
~
~
=0
and W.(y)
~
<
a for all y
<
Y .
i
<
Y
i
Now from:
<
r;1:),
102
fi(x)
=-
ex + Wi (x)
= -
ex + Wi(Y i )
for x
=-
c + W~(x)
for Y.] . <- x
<
(4.3.5)
Y
i
We obtain
fl(x)
= -c
Since
W~(x)
(4.3.6)
for x < Y
i
is continuous and
W~(Y)
= 0,
fi(x) is a continuous
function.
In short, we have shown that:
Theorem 4.8:
The results of Theorem 4.2 are still valid when the
assumptions are replaced by those of Theorem 4.7.
From Theorem 4.8, we obtain the functional form of Wi(y) for
all values of y.
Now an analogy of Theorem 4.6 for
demand distributions case could be proved by
4.6 step by step.
Le~a
4.9:
then Yi ~
varying
the proof of
Thus we have:
Yr"
Corollary:
r*p+h q-rpq
Proof:
follow~ng
Under assumptions (a), (b), (c), and
When c + hI -
~h~
2
p-q
< 0, Z. . < Z-i.
-
]. -
(~)
Qf Theprem 4.7,
-e
103
=0
Z1
Lemma 4.10:
~
Zi as Vi (8) is monotone-nondecreasing.
Under the assumptions of Theorem 4.7, then
+ hI -
c
r*p+h 2q-rpq
p-q
<
0
Zi
<=> -
p
< Y
i
Proof:
Z
W~ (-.!.)
~
=
p
c + h1 -
r*p+h q-rpq
2
p-q
(the derivation is omitted as it
has been done many times under similar situations), Wi(Y ) = 0 by
i
definition of Y , and the monotone-decreasingness of Wi{y) assure
i
Lemma 4.10.
Theorem 4.11:
{fi(x): i
Under the assumptions of Theorem 4.7,
= l,2,3, .•• }
th~n
is the unique solution of the infinite-stage
functional equation among the class of solution which is continuous,
uniformly bounded over x for all x lying in a finite interval, and
has uniformly bounded optimal procuring policy.
•
Proof:
Let {F. (x): i
~
= 1,2,3, ... }
be any other solution which belongs
to the class specified in the theorem.
Then we can write
104
(4.3.7)
where (y,~) is the i
th
period component of an optimal policy.
(The existence of which is guaranteed by the fact that the righthand side is continuous and finite for finite
y.)
Since {f. (x):
~
i = 1,2,3, ... } is also a solution of the infinite-stage functional
equation:
(4.3.8)
Now by the definition of an optimal policy:
(4.3.9)
(4.3.10)
From (4.3.7), (4.3.8), (4.3.9) and (4.3.10):
IF.1. (x.)
~
•
- f. (x.)
~
~
I
105
(4.3.11)
where the supreme is over all possible x + .
i l
Now x + ~ Max(y,y) and in general
i l
=
By our assumption Yk are uniformly bounded for all k.
From
Thus for all x.]. lying in a finite interval
x + is bounded by a number B <
i j
00
uniformly over j,
Let
Sup IF.]. (x) - f.]. (:It)
O<x<B
•
I
(4.3.12)
-
Then since (4.3.11) is true for aIr 0 < x.] . <
B; we obtain:
-
V.].
-<
aV'+
]. l
(4.3.13)
And in general:
(4.3.14)
=
Sup !Fi+n(X) - fi+n(x)I is uniformly bounded over
O<x<B
n since Fi+n(x) and-fi+n(x) are both uniformly bounded for all x
But Vi+n
•
lying in a finite interval.
Let n+oo in (4.3.14), we obtain Vi
Theorem 4.11 is proved.
= O.
Since i is arbitrary,
106
4.4
Some Theorems on Critical Numbers of Varying Demand Distributions
Case
When demand distributions are varying from period to period, it
is in general difficult to obtain analytic solutions for critical
numbers.
In this section we will show that if the demand distributions
· some s t oc h as t"lca 1 sense, then the l..th period cr •4 tical
are or d ere d 1.n
numbers will have the same solutions as those of a process where
demand distributions are identically
{~.,~.}.
1.
1.
For mathematical
convenience, we will consider the demand distribution sequence
{~1'~1'~2'~2'~3'~3"'" ~n'~n'~n'~n'~n'~n""}' Namely the demand
distributions are identical from the nth period on where n is an
arbitrarily large number.
For practical purpose, it does little
harm to approximate any demand distribution sequence by our specified
one while allowing n to be sufficiently large.
Theorem 4.12:
for i
Under the assumptions of Theorem 4.7, and furthermore
= 1,2, .•• ,n-l,
~ 1.. (£
P + a.)
1.
>
~ i+l (€)
for all
€
for all s*
and for i
•
~i
>
n
- ~i+l' ~i - ~i+l' then the following is true .
r'~p+h2q-rpq
(a)
When c + hI -
Z.
1.
z-;-,
Y.
1.
1.
- - - - - - < 0,
p-q
Y-;-, i
1
1,2,3, ..•
then
107
r*p+hzq-rpq
(b)
When c + hI -
Y.
~
Y~,
then
= 1,Z,3, ...
i
~
> 0,
p-q
Proof:
(a)
Since demand distributions are identical from nth period
on,
z. = Z~,
~
Z~, Y~.
by definition of
~
= Y-i
Yi
~
~
~
From Z.
~
= Z~
~
and (4.2.2)
z-
z
~ n-l (-.E.
p + a n- 1) -> ~ n (Z-)
n
(-.E. +
)
n-l p
a n- l
(4.4.1)
for all i > n
= ~ n- l(Z---l)'
n-
Hence
Z
(4.4.2)
-.E.+ a
>2-->Z
p
n-l - n-l - n-l
while the last inequality is by Corollary of Lemma 4.9.
No~
z
+
j
n-l
(p-q)[r + s + af'(Z
n n- l-E)]~n-.1(E)dE
an-I
r* + h
= V-I
n-
Z
(Z
+ sq - rp - sp +
n- 1)
(p-q)(r+s-ac)~
n- l(Zn- 1)
(4.4.3)
108
where f'(Z I-E)
n n-
= -c
since
Z
Z
n-1
E
<
Z
+ a n- 1)
p
- a
(-!!.
= -!!.
< Y
n-1
p - n
from (4.4.2) and Lemma 4.10.
= ~l(Z
nn- 1) implies Z---l
n- -< Zn- l'
And hence:
Z
n-1
(4.4.4)
= Zn_1
Using (4.4.4) then (4.4.2), (4.4.3), (4.4.4) can be proved when
n is replaced by n-1.
Repeat this argument, we obtain:
for all i = 1,2,3, •••
Y +a
For Y part, we will first show W' 1( n n+1)
np
i
(4.4.5)
>
W'(Y ).
n
n
From Lemma 4.10,
Z
n < Y
P -
(4.4.6)
n
From (4.4.2),
Z
-!!.>Z
p -
n-1
-a
n-1
(4.4.7)
109
Hence
Y +a 1
n n.,...
>
p
z
(~+
a n-.1)/P
p
Z
n-1
=-p
(4.4.8)
From (4.4.6) and (4.4.8), we have to deal with the following
four possible cases:
Z
-
n
q
< Y ,
n
z
~ < Y ,
q
n
z
z
~ < Y <~,
P n - q
z
z
~ < Y < ~
P - n-q'
Z 1
h- <
--_.
q
Z
Y +a
n n- 1
(4.4.9)
p
Y +a
Z
n-1
n n-1 < ----n-1
- < ......;;;;.---:;;;,...,;;;.
P
P
q
Z
--<
Yn+a n-.1
Z
n-1
Y +a
n n.,...l
n-1
q
(4.4.11)
p
Z
n-1
- - < ......;;;;.---.:;;;,...,;;;. < - P
P
P
First consider (4.4.9), then:
(4.4.10)
(4.4.12)
110
W'(Y)
n
n
=
DIg (Y ,0),
n
n
Now
-Ia*
Yn+an- 1
P
{c +
h1
+
h
z-
rp - sp
n-1
p[r + s + af'(Y
n n
(c+h
1
-r*-sq)~
n-1
+a1-pE*-E)]~
1(E+qE*)dE'~n-1(E*)dEj
nn-
(E*)dE*
Yn+an-1
~
I
[r* + h
2
+ sq - rp - sp +p(r+s-ac)
a*
n-1
.
.~
n-1
(Y +a
-PE*+qE*)]~
(E*)dE* + (c+h -r*-sq)
n n-1
n-1
1
111
(since f'(Y
n n+a n-1 -p£*-£)
y +a
=
f
n
>
~
-c)
n- 1
[r* + h 2 + sq - rp - sp + p(r+s-ac)
P a*
n-1
(obtained by the transformation u*
~f
=
p£*)
Y
n
[r* + h 2 + sq - rp - sp +p(r+s-ac)
pa*
n-1
• ep
.
Yn+an-1
>
f
pa*n- 1
•
pY -pu*+qu*
u* 1
(_--:n:.:...-+ a
)]1/J
(-)- du* + (c+h -r*-sq)
n-1
p
n-1
n-1 p p
1
[r* + h
2
+ sq - rp - sp +p(r+s-ac)
112
.~
n
(pY
n
-pu*+qu*)]~(u*)du*
n
+ (c+h -r*-sq)
1
(by Lemma 4.4)
~
f
Y
n [r* + h + sq - rp - sp +p(r+s-ac)
2
a*
n
.~
n
(pY
(since
n
-pu*+qu*)]~
pa~_l ~ a~
rt
(u*)du* + (c+h -r*-sq)
and 0
1
~
an _1 )
+ h 2 - rp - sp
+ fP(Yn-U*)
p[r + s + aftn- l(pY n
-pu*-£)]~n (£+qU*)d£} ~n (u*)du*
a -qu*
n
(c+h 1 -r*-sq)~ n (u*)du*
(where
= D1gn (Yn ,0)
f~+l (pYn-pu*-£)
(4.4.13)
113
W'
n-l
Y +a
(n n-l)
p
> W'(Y ) under (4.4.9).
n n
Next consider (4.4.10).
W'(Y)
n n
WI
n-1
(4.4.14)
Then:
= DIgn (Yn ,0)
Y +a 1
• n n2n - 1 - q
-=-:::.._---p"---)
p-q
Y +a
(n n-l)
p
Y +a
D9
(n n-l
0)
1 n-l
p
,
,
>
sinae D g (y,z) is monotone-nondecreasing in z for the range
1 i
2 -qy
i
z < .-;;;;--
-
p-q
So again from (4.4.13), we obtain:
Y +a
(n n-1) > WI(y ) under (4.4.10)
n-1
p
n n
(4.4.15)
WI
Next consider (4.4.12).
W'(Y) = DIg (Y,
n n
n n
Y +a
WI
(n n-l)
n-l
p
•
Then:
2 -qY
n
n)
p-q
Yn+an- 1
DIgn- 1 (
,
p
2
n-l
_ q'
Y +a
n
p
n-
1
.......;:;:......::;...--_......._~-)
p-q
114
Now:
Z
_ q.
n-l
-.=:;......;;;;-.
p-q
Y +a 1
n np
--J_~_)
p[r + s + af'(Y +a
n
J
pZ
+
a
n-1
I-PE*-E)]~ n- I(E+qE*)d:lJ ~n- I(E*)dE*
q[r + s +
Y +a
- q •
n
.~
..
n-
-qYn -qan-l
p-q
n-l
n
n-1
p-q
n-1
-Z
pZ
f'(
n
n-l
-qY -qa
n
p-q
n-l - E)
n-1
j
Z
Y +a .1- n- 1
(E + q • n fl )dE
p-q
~
n-1
(E*)dE*
115
y +a
n
~
+
f
-2
;=~ n-l
[c + hI + h Z - rp - sp + p(r+s-ac)
a*
n-l
f
[c + hI - r* - sq +q (r+s-ac) <1J n- 1 (2 n - 1 ) ]1jJn-l (E:*)dE:*
00
y +a
n
n-l
p-q
-2
n-l
(since
--
Yn+an-12n-l
p-q
f
f~(X) ~
-c)
[r* + h + sq - rp - sp - q(r+s-ac)
Z
a~_1
.<1J n- 1(2n- 1) + p(r+s-ac)<1J n- 1(Yn+a n-.1- P E:*+qE:*)]
-{a~
Yn+an-l -2 n-l
p-q
[r* + h
Z + sq - rp - sp - q(r+s-ac)<1J n-l (2n-l )
n-l
•
+ P(r+s-ac}<p
pY -'pu*+qu*
u* 1
1 (---::n~__~_ + a
)]1jJ
(-)- du*
np
n-l
n-l p p
116
(obtained by the transformation u*
Y +a
2
S:a~
n
m.. . l
-Z
= ps*).
n-l
p-q
[r* + h 2 + sq - rp - sp -
q(r+s-ac)~
n-l
+
p(r+s-ac)~
n
(pY
n
-pu*+qu*)]~
n
(u*)du*
(by Lemma 4.4)
J
pY -Z
_>
n
p-q
n
[r* + h 2 + sq - rp - sp -
q(r+s-ac)~n (2n )
a*
n
+
•
•
p(r+s-ac)~
n
(pY
n
-pu*+qu*)]~
+ [c + hI - r* - sq +
n
(u*)du*
q(r+s-ac)~
n
(Z )]
n
n- l(Z n- 1)
117
(since p(Zn- I-an- 1) -< Zn from (4.4.7), pa*n- 1 -< a*n
and ~n-l (Zn-l) = ~n-l(Zn-l) = ~n(Zn) = ~n(Zn)
from (4.4.4))
Z -qY
n
n
p-q
p[r + s +
+ hI - r* -
c
Z -qY
n
n
sq
{
p-q
.
+
Z -qY
f'
n
n
p-q
Z -qY
q[r + s + af' I(P •
Z -qY
a -q(Y
n
n
-
n-l
Z -qY
n
n - e:)
p-q
Z
~ < Y )
P -
+ qY - q.
n
Z
n
f~+I(pYn-Pu*-e:) = -c
(p
n
p-q
- e:)]
p-q
n
(where
n
n)
11;
.~ (E
f'
n-
-qY
}
n
n)dE
p-q
si.nce pY n - pu* -
-c since p .
Z -qY
n
n
p-q
-
E .::..
E <
-
Yn = Yn+ 1 and
pYn -< Y-n = Y + for
n 1
118
= D1 gn (Yn ,
Z -qY
n
n)
(4.4.16)
p-q
Y +a
WI (n n-1) > WI(y ) under (4.4.12)
n-1
p
- n n
Finally consider (4.4.11).
WI(y)
n n
Y +a
(Y,
1 n n
.
n
p-q
n)
Y +q
(n n-1
0)
9
1 n-1
p
,
=D
=Dg
-e
Then
Z -qY
=Dg
WI (n n-1)
n-1
p
(4.4.17)
1 n-1
Y +a
(n
n-1
p
Z -qY
n-1
0)
p-q
since D g (y,z) is independent of z when y - z
1 k
~ a~
and
Yn+a n- 1
p
From (4.4.16), therefore
Y +a
WI (n n-1) > WI(y ) under (4.4.11)
n-1
p
n n
From (4.4.14), (4.4.15), (4.4.17) and (4.4.18), we
for all values of Y :
n
Y +a
WI (n n-1) > WI (Y )
n-1
p
- n n
=
WIn- l(Yn- 1).
(4.4.18)
concl~de
that
/
119
Hence:
Y +a
n n-1 > Y
p
- n-1
(4.4.19)
or pY n- 1 - a n- 1 -< Yn
Note that in proving (4.4.13) and (4.4.16), the crucial facts
used are:
u*
>
-
e: > a
a*
-
n
n
and
Z -qY
f'
n+1
(p.
n
p-q
n - e:) = -c
e:
>
-
a
n
From (4.4.6) and (4.4.19)
•
f'(pY 1-pu*-e:) < f'(pY I-a 1) < f'(Y ) = -c
n
n- n
nn- n n
and
when u*
>
-
a*
and e: > a
n-1
- n-1
Thus we can prove inductively
120
u* -> a*i
£
£ >
-
for all i
> a
-
i
a.
1
= 1,2,3 ...
Utilizing (4.4.2m, we note that Wi(Y )
i
= Wi(Y i )'
and therefore
Wi(Yi) = Wi(Y i ) = Wi(Y i ) gives Yi = Yi for i = 1,2,3 .•.
(b)
From (4.4.1) and (4.2.6)
pY-
~ n- l(Y n+an- 1)
= ~ n- 1(~
+
p
=
~n-1(pYn-1)
Hence
>
Y + a
n
n-1 -
p~
Now from (4.4.21):
>
n-1 -
pY
n-1
or
a n- 1) -> ~ n (pY-)
n
-e
121
+
PY 1
a n-
J
q[r + s + af' (pY 1-d]Q> l(E)dE}
n
nn-
~
ll-
1 (Efc)de:*
n-1
J
qY
+ r* + h 2 + sq - rp - sp +
1
a n-
(p-q)
n-1
'[r + s + af'(pY
n
n- 1-e:)]~n- 1(e:)de:
= JOO
[c + hI - r* - sq
+q(r+s-ac)~n- l(PYn-.1)]$n- l(e:*)de:*
O
+ r* + h 2 + sq - rp - sp +
=
w-t--:n- 1
(Y
(p-q)(r+s-ac)~
n- l(qYn- 1)
n- 1)
But
y
n-l
=y
n-1
From (4.4.22) and an inductive argument, we obtain Y
i
i
= 1,2,3, .•.
Theorem 4.12 is proved.
(4.4.22)
= Yi
122
When
~l(E) ~ ~2(E)
for all E, the variable of
stochastically larger than the variable of
Theorem 4.12 is of the type
~i+l
E
~i(-
p + a.)
J. ->
~l'
~2
is said to be
The condition in
~.+l(E).
J.
The variable of
is stochastically larger than a linear transformation of the
variable of
~i'
~
Since a i
0 and 0 < p
~
1, the condition
~i(; + a ) ~ ~i+l(E) is more general than the condition ~i(E)
l
>
~i+l(E),
and contains the latter as a special case.
4.5
A Generalization
Supposing that demand at each period comes from n independent
customers (or n groups of customers), each has a demand sequence
with structure as suggested in Chapter 3 of D and D! (except now
i
we need a subscript j to index customer j), then what can we say of
the optimal policy with respect to the aggregated demand?
n
, D = E D. ., and Ii = Min(Df. Yt-z )
i
i
j=l J.J
i=l J.J
n
n
n
Then Di(I i ) = E Dij (I •• ) = E D.. + q E I ij = Di + ql i ·
. 1
J.J
j=l J.J
j=l
J=
n
•
Let Dr =
E
D~.
So the same pattern of dependence holds for the aggregated demand.
D is independent of D! since each D is independent of D!j'
ij
i
Apparently D. are also independent of each other.
J.
Hence results
in Chapter 2 and Chapter 3 apply to the aggregated demand.
Again, if for each j D.. (or
J.J
for all i, then D. (or
J.
D~)
J.
D~.)
J.J
are identical random variables
are identical for all i.
Results in
Section 4.1, Section 4.2 hold.
.
Finally, when D..
J.J
each j:
(D~.)
J.J
are not identical among i, but for
123
e:
{1J p
~ ..
+ a 1J
.. ) >
-
~.+l .(e:)
1
,J
for all e:,
i = 1,2,3, ...
for all e:*,
i
for all e:,
i = 1,2,3, ...
for all e:*,
i = 1,2,3, ...
= 1,2,3, ...
then we will show:
and consequently, results in Sections 4.3, 4.4 hold.
We will only prove the case when j
•
= 1,2
(and for
Q
only),
extension to the general case is immediate .
Lemma 4.13:
Let ~l = ~11*~12 and ~2
= ~2l*~22'
convolution, then:
for all e:,
and
for all e:,
implies
for all e:,
where "*" denotes
124
where a
1
= all +
a 12 is the lower range of the distribution function
..
Proof:
(obtained by the transformation v
= p(u-a12 »
(obtained by using Lemma 4.4)
•
125
CHAPTER 5:
5.1
CONCLUSION
Summary
In Chapter 1, we introduce the inventory model which has two
different modes of sale.
Then we discuss the realistic aspect of the
model and give some examples.
A discussion on motivation and method-
ology and a review of literature is included in this chapter.
In Chapter 2, the single-stage process is thoroughly studied and
optimal policies corresponding to each possible parametric structure
are either derived or indicated.
Some effort is accorded to the
physical interpretations of parametric structure.
The hope is to
give some insight to the inventory process which mathematical formulations and derivations fail to reveal.
Chapter 3 is an extension of the single-stage process to the
multistage process.
Though the extension works in general, we give
our attention to a particular parametric structure (which is more or
less a normal type) only.
The avoidance of discussion on the other
situations is mainly due to fear of redundancy.
Chapter 4 investigates the infinite-stage process.
First, an
existence and uniqueness result in given to validate further discussions.
Then we solve the critical numbers in the optimal policy when demand
distributions are identical from period to period.
distributions vary, we establish the result:
E
~.(~p
+ a.)
> ~.+l(E) for all
~-~
E
and
When demand
if for all i,
E*
~.(--) > ~.+l(E*)
~p-~
for all E*, where
~., ~. are the i th period regular demand distribution and special
~
~
demand distribution respectively, then the i
th
period critical numbers
126
are
~qqa1
to the critical numbers in a process where demand distributions
are identical to the i
5.2
th
period demand distributions.
Susgestions for Future Study
Since this work is a first attempt to investigate inventory models
with special sales, many simplifying assumptions are made to facilitate
th~
presentation ot the main feature in a distinct
~anner.
General-
izating some of the assumptions will be a natural second step.
Among
tnem, we mention the following:
a.
The classical generalization as followed by many other
inventory models, has several directions here.
We may
introdvce a delivery lag, usually a fixed number A, to
account for the time difference between placing a procurement order and its actual delivery.
iqtroduce a procurement setup cost.
Or we may
Another possible
generalization is to assume the existence of a transferment cost for goods transferred back to the regular
stock after special sale.
b.
~n
our model, special sale is assumed to precede regular
sale of the same period.
The model assuming the reverse,
namely, that regular sale precedes special sale, and a
comparison between this model and our model for optimality
will
~ertainly
be interesting to study.
The difficulty
pf the new model is, if we retain the assumption that the
regular demand depends on the amount sold in the previous
special sale, then the inventory process at each period,
will not be only characterized by the stock level, but
127
also the amount of goods sold in the special sale of
the preceding period.
c.
The particular dependence between the regular demand
and the special demand assumed in our model carries
along with it the restrictive condition a.*
1.
a.
< -1..
-
q
It
is desirable to introduce other kind of dependence
relationship between ordinary demand distribution and
special demand distribution where such restrictions
are pot necessary to impose .
•
128
CHAPTER 6.
LIST OF REFERENCES
1.
Arrow, K. J., T. Harris and J. Marschak. 1951.
policy. Econometrica 19(3):250-272.
Optimal inventory
2.
Arrow, K. J., S. Karlin and H. Scarf. 1958. Studies of the
Mathematical Theory of Inventory and Production. Stanford
University Press, Stanford, California.
3.
Barankin, E. M. 1961. A delivery-lag inventory model with an
emergency provision (The single-period case). Naval Res.
Logist. Quart. 8(3):285-311.
4.
Bellman, R. 1957. Dynamic Programming.
Press, Princeton, New Jersey.
5.
Bellman, R., I. Glicksburg and O. Gross. 1955. On the optimal
inventory equation. Management Sci. 2(1):83-104.
6.
Daniel, K. H. 1958. A delivery-lag inventory model with
emergency, Chapter 2. In H. Scarf, D. Gilford and M.
Shelly (ed.), Multistag;-Inventory Models and Techniques.
Stanford University Press, Stanford, California.
7.
Dvoretzky, A., J. Kiefer, and J. Wo1fowitz. 1952. The inventory
problem: I case of known distributions of demand.
Econometrica 20(2):187-222.
8.
Fukuda, Y. 1961. Optimal disposal policies.
Quart. 8(3):221-227.
9.
Fukuda, Y. 1964. Optimal policies for the inventory problem
with negotiable leadtime. Management Sci. 10(4):690-708.
Princeton University
Naval Res. Logist.
lQ.
Hadley, G. and T. M. Whitin. 1963. Analysis of Inventory
Systems. Prentice-Hall, Inc:, Englewood Cliffs, New Jersey.
11.
Iglehart, D. L. 1963. Dynamic programming and stationary
analysis of inventory problems, Chapter 1. In H. Scarf,
D. M. Gilford and M. W. Shelly (ed.), Multistage Inventory
Models and Techniques. Stanford University Press, Stanford
California.
12.
I~lehart,
~3.
Karlin, S. 1955. The structure of dynamic programming models.
Naval Res. Logist. Quart. 2(4):285-294.
D. L., and S. Karlin. 1962. Optimal policy for
dynamic inventory process with non-stationary stochastic
demands, Chapter 8. In K. J. Arrow, S. Karlin and H.
Scarf (ed.), Studies in Applied Probability and Management
Science. Stanford University Press, Stanford, California.
129
1960. Dynamic inventory policy with varying stochastic
Management Sci. 6(3):231-258.
14,
Karlin, S.
demands.
IS.
Karlin, Samuel and Charles R. Carr. 1962. Prices and optimal
inventory policy, Chapter 10. In Kenneth J. Arrow, Samuel
Karlin and Herbert Scarf (ed.), Studies in Applied Probability
and Management Science. Stanford University Press, Stanford,
California.
Phelps, E. 1962. Optimal decisions rules for the procurement,
repair or disposal of spare parts. RM-2920-PR, Rand
Corporation, Santa Monica, California.
16.
17.
Scarf, H. 1963. A survey of analytic techniques in inventory
theory, Chapter 7. In H. Scarf, D. M. Gilford and M. W.
Shelly (ed.), Multistage Inventory Models and Techniques.
Stanford University Press, Stanford, California.
18,
Veinott, A., Jr. 1963. Optimal stockage policies with nonstationary stochastic demands, Chapter 4. In H. Scarf,
D. M. Gilford and M. W. Shelly (ed.), Multistage Inventory
Models and Techniques. Stanford University Press, Stanford,
California.
19.
Veinott, A., Jr. 1965. Optimal policy in a dynamic single
product non-stationary inventory model with several demand
classes. Operations Res. 13(5):761-778.
20.
Veinott, A., Jr. 1966. The status of mathematical inventory
theory. Management Sci. 12(11):745-777.
130
,
APPENDICES
•
131
CHAPTER 7:
•
7,1
APPENDICES
Derivatives of the First and Second Derivatives of the Function
81N (Y,z)
r*e:* + h 2 (y-e:*)
+
JP
(y-e:*)
a -qe:*
1
+J""
[- rp(y-e:*) + s[e:-p(y-e:*)]
p(y-e:*)
•
+J"" {- r*(y-z) +
h 2z
y-z
Pz
+
J
a-q(y-z)
•
[- rs +
af2N(pz-e:)]~1 (e:+qy-qz)de:
132
+Joo [- rpz + s(e:-pz)
pz
•
(7.1.1)
The taking of partial derivatives will
~a8~ excep~
a straightforward
for the two terms:
J
pz
A·
b~
a1-<J(Y-~)
[- r~
+ af2N(pz-e:)]~1(e:+qy-qz)de:
(7.1.2)
and
•
B •
J
ClO
[-
rpz
,f-
s (e:-pz) + af 2N (O) H 1 (e:+qy-qz)de:
pz
~
J [00
rps + s(e:-qy+qz-pz) +
af2N(O)]~1(e:)de:
(7.1.3)
pz+qy-qz
aA az'
aA ay'
aB ay
aB f·~rst and t hen su b st i tute t hem
We wi 'l
~
compute ay'
ba~k
•
•
into D1g1N(y,z), P2 g1N(y,z)
133
•
(7.1.4)
•
fZ
~
J
.
[- rq +
a 1-q(y.-z)
a(p-q)f2N(pz-E)]~1(E+qy-qz)dE
+ (p-q)[- rpz +
•
eD
~B
l:' '"
Y
•
J
pz+qy-qz
,..
af2N(O)]~1 (pz+qy-qz)
(7.1.5)
•
*.f
134
(-rp-sp+sq)
0Cl
~1 (e:)de:
P<l:+qy-qz
- (p-q)[- rpz + af2N(O)]~1 (pz+qy-qz)
Now
~9nSidedng
(7.1. 7)
(7.1.2), (7.1.3), (7.1.4), (7.1.5), (7.1.6)
and (7.1. 7) I
• C
+ h1 +
-e
f
y-'l-t
h2 +
fP
a*
(y-e:*)
a -qe:*
1
1
•
+
flXl
(-rp-spH (e:+qe:*)de: - p[-rp(y-e:*) + af (O)]
1
2N
p(y-e:*)
+[
L
•
r"(y-z)
+
h z
2
+f
pz
a -q(y-z)
1
·~l(e:+qy-qz)de:
•
[-re: + af
2N
(pz-e:)]
135
+ q[-rpz +
+
af2N(O)]~1 (pz+qy-qz)
f~ -
Sq·l (c)do - q[ -rpz
pz+qy-qz
•
-
{- r*(y-z)
Z
Y
• c + h1
•
•
+
f
a*
1
{
h Z - rp - sp
+
.f1P(O) H
l
(pz+qy-qZ~
J
+
oo
f
{
- r* - sq +
136
fPZ
y-z
a -q(y-z)
1
•
.~ 1 (E+qy-qZ)dC} W1 (E*)dE*
(7.1.8)
[-rE + af
- fr*(y,z)
-e
•
f
J
f
00
+
r* + h
y-z
2
+ (p-q)[- rpz +
•
+
f
00
+
pz
(PY-P£*-E)]
[-
a -q(y-z)
1
af2N(O)]~1 (pz+qy-qz)
(-rp-sp+sq)~l (E+qy-qz)d£
- (p-q)[- rpz + af 2N (O)]
pz
•
2N
·~l (pz+qy-qz)
137
-<PI (E:+qy-qz)dE:
+
JOO
[-rpz + s(E:-pz) + af
2N
(0)]<Pl (E:+qy-qz) dE:} 1/1 (y-z)
1
pz
=
f..
JOO
L
y-z
+
J
+ hZ +
sq - rp - sp
P~
a -q(y-z)
1
(7.1.9)
•
Again it is straightforward to compute D g (y,z) from (6.1.8)
11 1N
and D g (y,z), D g (y,z) from (6.1.9) except possibly the term:
22 1N
12 1N
c = JPz
[r + s + af
a - q (y-z)
1
ZN (pz-E:)]<P 1 (E:+qy-qz)de:
(7.1.10)
Pz+qy-qZ
=
•
•
[r
J a
1
+
138
Now
•
•
(7.1.11)
•
(7.1.12)
139
TheJ;'l;
·f
Y
-:I1
..
e
1;1*
140
,
+
~. +
h 2 + sq • rp •
141
•
·J J
CIO
{
y-.
-{tit +
llz
a -q (y-.)
1
h + sq - rp - Sp·+JPZ
(p-q)[r + s + afZN(pz-E)]
2
al-q(y-z)
For the single-period process, N
fiN
ii
= 1.
By our convention
0 for ;f. >N.
For the infinite-stage process, then replace fiN by f i in all
above equations.
7.2
Glo$sarX of Terms
In this section, we will list some of the important terms and
functions frequently appearing in the text.
Again we will deal with
the N-stage functions only.
Xi • initial stock level at period i;
Yi - Xi • amount of goods procured at period i;
Y - Zi
t
= amount
period;
of goods allocated to the special sale of the i
th
142
~i
== i
th
period regular demand distribution when nothing is sold
in the special sale of the same period;
I
'l'
z
i
Yi
== i
th
period special-sale demand distribution;
= optimum
i th period
al~ocation
policy given Yi;
Y == optimum ~.th per iOO procurement policy;
i
z
i
Yi
= optimum i th period allocation policy;
==
i
Zi == i
p
th
th
period procurement policy critical number;
period allocation policy critical number;
= the leakage factor;
q == parameter reflecting the dependence of the regular demand
on the amount sold in the special sale;
c
hI
= procurement
= holding
cost;
cost applying to all goods available at the
beginning of the period;
h
2
==
holding cost applying only to goods not sold in the special
sale;
r
= regular
sale revenue;
r*
= special
sale revenue; and
s
==
shortage cost.
o
vINloIe) = r*
+ h2 +
sq - rp - sp
+
(7.2.1)
8
1N
(y,z) and its first and second derivatives has been listed
in the previous section.
8_
143
z
•
ZIN
for - q
=
Y °
=
ZIN- qy
p-q
<
y
<
00
ZIN
ZIN
for - - < y < - - q
p -
=Y
for
WIN(y)
°.:5..Y
ZIN
p
< --
= gIN(y,Zy)
(7.2.2)
(7.2.3)
Y < x
I N-
(7.2.4)
Y < x
I N-
x < Y
-c
(7.2.5)
also
(7.2.6)
where
L (y,x)
xl
- cx +
gn (y,z)
is the single period cost given xl and a policy (y,z).
(7.2.7)