J
UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
IIil'VIDv"TORY PROBLEMS COl\TCERNIIiG CCMPOUl'ID PRODUC'rS
by
Ismail Nabih Shimi
June 1963
This research was supported by the Office of Naval
Research under contract No. Nonr-855(09) for research
in probability and statistics at the University of
North Carolina, Chapel Hill, N. C. Reproduction in
whole or in part is permitted for any purpose of the
United States Government.
Institute of Statistics
Mimeo Series No. 370
.,
.
ACKNOWlEDGEMENTS
,.
It is
my
pleasure to acknowledge
my
indebtedness to the
Chairman of my Advisory Committee, Professor Walter L. Smith,
for suggesting the problem considered in this dissertation,
for his constructive criticisms, and for his encouragement throughout my work with him.
I am extremely grateful to the Government of !Che United Arab
Republic and theOtf1ce.ofNaval Reaearc.h for fimancial assistance
during my studies.
I am also most grateful to Mrs. Doris Gardner for her patience,
and excellent typing.
To Martha Jordan is due
my
sincerest thanks
for never failing to lend a helping hand when it is most needed.
ii
TABLE OF CONTENTS
CHAPTER
PAGE
- - .. - .... - - - -
ii
INTRODUCTION - .. - - .. - - - - - .. ..
iv
ACKNOW'LEDGEME~'TS
SECTION I:
STATIC MODEL .. .. .. .. .. .. - ..
Introduction .. - - - .. .. .. ..
I
II
III
IV
.. - - - ..
1
- - - - .. - .. .. - - - - -
5
ORDERING COSTS WITH IlRED_TAPE" - - .. .. - - - - -
20
LINEAR ORDERING COSTS
CONVEX INCREASING ORDERING COSTS .. - ....
RANDOM SUPPLY
SECTION II:
V
VI
~
1
. ..
.. - - .. - .. .. - .. .. .. - -
DYNAMIC MODEL .. - .. - .. - .. - .... -
48
56
69
Introduction .. - .. - .. .. .. - - - - .. .. - -
69
DYNAMIC MODEL:
CASE OF LINEAR ORDERING COSTS- -
81
DTI1AMIC MODEL WITH LAG .. .. .. - .. - .. .. .. - - .. -
10;
BIBLIOGRAPHY - - .. - .. .. .. - .. .. .. - - - .. .. _..
116
iii
I NrHODUC'rION
Scientific methods are now widely employed by managements to
make policy decisions.
These methods are consequences of the new
scientific disciplines of operations research and management science.
They have led to the increasing use of mathematics, statistics, queueing theory, renewal theory, information theory and linear programing
in tbe business world.
One of the problems encountered in business management is the
inventory problem.
This ,is the general problem of deciding what
should be the quantities of goods to be stocked in anticipation of
future demands.
This inventory problem is a very particular case of
the general problem of decision making when we are facing uncertainties.
The policymaker decides on the stock levels according to a certain
criterion;
usually he tries to maximize his profit, or, equivalently,
to minimize his loss.
This loss, in general, is a random variable
which depends on a certain condition intrinsic to tbe system and on
certain random variables whicb influence that system.
Some of these
variables are under the control of the decisionmaker;
others are not
under his control and have in general a known probability distribution.
In all theories so far developed, at most one of the non-
controlled variables is assumed random and this is usually the rate of
demand for the commodities.
i'"
(1)
Losses are due to many factors:
The price the firm pays for obtaining goods, ready to
meet the demand, either by bUying or manufacturing them.
(2)
Inability to meet the demand, (e.g. store loses sales,
a factory runs out of
sp~re
parts, or a hospital runs
v
out of blood for transfusion, or soldiers run out of
ammunition);
this is called the penalty cost.
The stocking of non~demanded items;
(3)
this is called
the storage cost.
From all this it is evident that the optimal policy is aimed
at reaching some kind of a balance between overstocking and understocking.
The policy-maker does this by fixing the controlled vari-
ables so as to minimize the expected loss resulting from the randomness of the demands, about which>all he knows is its probability
distribution.
The inventory problem was recognizeq long before its first
mathematical discussion given by Arrow, Harris,
paper published in 1951.
~~rschak
in a
They showed that certain policies, actually
used in industry, are optimal under certain simplifying assumptions
on the costs involved (e.g. linearity assumptions) and they treated
both the static model and the dynamic model.
The static model is the
one when we are only concerned with satisfying demand for one period.
A production or :order1.pg
decision is made:' at "the J:>e.ginning· of the
period, and no further orders or deliveries can occur later on.
The
dynamic model is the one when we consider more than one period, and
we can add to our stocks at the start of each period.
In both cases
assuming the distribution of demand to be completely known.
This
paper was followed by one of Dvoretsky, Kiefer, and Wolfowitz, in
1952, where they gave the first discussion of existence and uniqueness theorems, and treated, in addition, the case of unknown distribution of demand.
In 1955 Bellman, Glicksberg, and Gass were the
vi
first to use the functional equation approach to determine the structure of the optimal policy, assuming linearity of of the different
costs.
This type of work was later extensively pursued and general-
ized to the cases when the different costs no longer have simple
linearity properties (see for example Arrow, Karlin and Scarf'
["2._71 ).
In nearly all of the previous work in this field the
authors were only concerned with one item to be demanded;
in a very
few cases they considered the possibility that there is also a substitute item such that if the firm runs out of the primary item
then demand can be satisfied by substitute items instead of the one
actually demanded.
The problem we are going to consider in this dissertation
involves the case when the demand is for two or more items, simultaneously, and if the firm runs out of ene of the items it will not be
able to satisfy any of the
d~mands.
The costs that we are going to consider in our work
storage costs;
penalty costs;
chase of the items.
manufacturing prices costs on pur-
We are going also to consider the selling price
r for each unit of the items demanded;
be included in the penalty costs.
models.
are~
in some cases this price can
We are going to study two basic
In section one we will study the static (one-stage) model,
and in section two the dynamic (multi-stage) model.
The reasons for studying the static model are:
(1)
Some situations call for just one period of production
of certain items, like production of particular items
1.
Numbers in square brackets refer to bibliography.
vii
for military use, or cases where we anticipate fluctuation in specifications of the tiems, or when the items
are perishable.
(2)
The insight gained into the relationship between the
variables involved in the static model proves helpful in
the later study of the more difficult dynamic modeL
(,)
If we can determine the value of the inventories left
over, called salvage gain (or cost), .which would represent its value for future use, and consider this cost
factor in our model, then solving the static model, can
give us a similar mathematical form of the solution to
the one we would get by solving the dynamic proolem.
The dynamic model is the one when we consider .more than one
period and we
c~~
add to our stocks at the start of each
have two cases here.
period~
We
One is when we assume that the time lag between
the ordering and. delivery of the commodities is negligible, in other
words delivery of commodities to the firm is assumed to be instantaneous.
The other case is when we assume that the delivery lag is
appreciable.
We are going to consider both cases in our studYT and
explain the difference more carefully when we come to the actual consideration later on.
We are going to make the following simplifying assumptions
throughout our work:
(1)
We assume that the kno'Wnprobability distribution of
demand is generated by a positive continuous probability
density function ¢(~) for
S ~ O.
I,..
ix
(4)
When the expected loss function has a unique minimum,
then small changes in the parameters of the model imply
slight variations in tpe solution for the unique
minimum, a desirable property for any usable policy.
II
SECTION I
STATIC MODEL
Introduction
The static or one-period inventory model is one in which we are
only interested in producing or buying items to meet demand over a
completely defined time period.
The manager of the firm can order
extra stocks only once at the beginning of the period; and after the
delivery of the ordered stocks nothing will be added to it during
the period.
This means that once the ordered stocks are delivered,
the amount of stocks to be used in supplying the demand in the
period is completely determined.
The level of stocks at any time
during the period depends on the stock levels which have been reached
by ordering or producing the extra stocks, and the demand up to this
time.
The goal toward which the initial decision is taken is to
minimize the costs (or maximize profits).
The demand in this period
is not known precisely, but is uncertain, and only the probability distribution of the total demand on the stocks in the period is assumed
known.
Given this knowledge of the probability distribution of demand,
the total expected cost that Will be incurred can be computed for each
possible initial stock level.
The optimal ordering policy is the policy
by which we select the initial stock level that has the lowest (minimum)
expected cost.
2
The problem that we are going to study here and for which 'Ive try
to find some kind of optimal policy is as follows:
Suppose a firm is
going to meet demands placed on two different products, during a known
period of time.
The demand is not for each product by itself but it is
going to be on the two products simultaneously.
This means that the firm
can no longer satisfy demands if it runs out of either of the products.
We assume that the demand is knO'l-Tn only to the extent that its probability distribution is known, and for simplicity let the known probability distribution be generated by a continuous density function.
The
firm is going to suffer losses due to the prices of the products, due to
storing and handling of the products, and a penalty cost if it fails to
meet demands by running out of any or both of the products.
We are going to assume at first, for simplicity, that a unit of
demand is only satisfied by equal unit amounts of each of the two products.
We shall discuss later possible extensions of this simple scheme.
Storage Costs:
The storage costs are assumed to be functions of the stocks on hand
at the end of the period of production, i.e. the storage costs for one
of the raw materials, (A) say, is given by the following
=
hl(Yl - g)
t
a positive value if Yl > g
o
where
Yl
is the initial stock level of
the period of production.
if Yl
(A), and
~
s
,
g
is the demand in
For the other raw material"
= I. oa
(B) say"
positive value
1
where Y2
is the initial stock level of
if Y2 >
S
if
s
Y2:S
(B)
Ordering Costs
For
(A) we have a cost function Cl(Zl)" where
ordered of
Yl" i.e.
(A)" to increase the stock level of
zl
= Yl
-
zl
is the amount
(A) from xl (say) to
xl~
For
(B) we similarly have C2 (Z2)" where z2 = Y2 - x2 and x2
is the amount on hand of (B) before ordering" and Y2 the corresponding
amount after delivery.
Penalty Cost
This is written
pee - Yl'S - Y2 )
pes - Yl )
if
Y1 < Y2
p(g - Y2 )
if
Yl > Y2
=
where
>
peE - Yi )
0
if Y. <
1.
~
-
is
(i
0
if Yi >S •
= 1)
2)
1
4
If
(Yl - xl)
is the amount ordered from A;(Y2 - x2 ) is the
amount ordered from B. .J and £ is the amount of demand in the period"
then the total loss in the period" denoted by L
(y "y )" is equal
l 2
XlX2
to
+
+
where
sold.
r
h~Y2- min (g"Yx27
p(s -
Y1 "s
-
Y2 )
is the selling price for each unit of the COIDpOunt product
CHAPTER I
LIN.E.AR ORDERING COSTS
Consider the case in which the ordering costs are linear; i.e.
·where
c >0
l
where
c > 0
2
and
If S is the amount demanded in the period, then the total loss
will be
+
h2 ~Y2-m1n(EIYx27
~ r min
where
r
(s,
+
P(E-Yl,S-Y2)
Yl ' Y2 )
is the sale price of the item demanded, and is assumed to
be constant.
Let the known probability distribution of dennnd be generated by
a continuous density function
h (.), h2(.), p(.),
l
second derivatives.
¢(E),
and assume that the functions
are continuous and possess piecewise continuous
From (1.1), the expected total loss is
6
f hlL-Yl-min(g,Ya27¢(g)d~ +f h~Y2-min(g,y~7
00
+
00
o
0
00
+
S
p(S-Yl ,S-Y2 )¢(S)dS
min(yl ,y2 )
min(g'Yl'Y2) ¢(g) dg ,
r (
o
thit is
where
Q:l
(1.3)
G(Yl ,y2 ) = cIYl
+c2Y2 +)
hIEyl -min(e,y)7¢(s )ds
o
00
00
+\ h~Y2-min(g,y~7¢(g)d, +J
p(g-y1 ,g-y2)¢(g)dg
min(y ,y )
l 2
o
co
-
r
J
min(s,Yl ,Y2 )¢(s)ds
o
NOvT the values of
xl and x2 are given. The optimal ordering
policy is defined to be a rule for choosing non-negative order sizes
such that E L
(yl,y,.,) is minimized.
Xl.1X2
c:
x
7
Hence we can determine the optimal ordering rule by determining the
values of Y
l
and Y
2
than xl and
x
values of
and Y tha.t minimize
2
Y
l
2
that minimize
respectively.
E L
xl ,x2 (yl ,y2 ), and are greater
',It is clear from (1.2) that the
G(Y ,y ) also minimize
l 2
E LXl'X2(Yl'Y2)' with respect to Yl and Y2 •
We are going to determine the conditions that
h (.), h (.) and
l
2
p(.)
must satisfy in order that minimum of G(Yl'Y ) may exist and be
2
unique and give us a simple optimal ordering rule. We shall also deter-
mine the values of
Y and, Y that give this minimum of G(y ,y ).
2
l
l 2
We are actually going to prove that the minimum of G(Y ,y ) is
l 2
attained in our case for some val\1e of Y
l
If we assume thatY
l
(1.4)
< Y2
~nd
Y
2
such that Y
l
= Y2 •
thetlequatio~ (1.;) Will have the form
Yl
Yl
1 (Y1-~ )¢(~)df f ~(Y2-~)¢(~)df
G(Y1'Y2)~OlY1+O~2+
.
h1
+
a
0
f
00
00
+ h 2 (y2-Y1 )
Yl
¢( g )dg +
..
, ...
"
1 P(~-Y1)¢(~)dl
Yl
However, if we put
o2 G(Yl ,y2 )
o Yl d Y2
from (1.4)
G"..
.; then
Yi'2
8
Yl
(1.5)
G~l = c 1 +J hi (Y1-e)¢(~)d~
t»
¢(~)a.g
2
- h (Y2-Y1)J
Yl
o
0:>
fp'(~-Y1) + :£7 ¢(~)a.g
-)
•
Yl
(1.6)
(1.i)
(1.8)
j
co
+
P"(~-Y1)¢(~)a.g
2
+ fhi(0)+h (y2-y1)+p' (0)+:£7¢(y1 ),
Yl
h;(Y2-~)¢(~)a.g
vIe
can
be sure that
co
+ h;;(Y2-Y1 ) (
G(y ,y ) will have a unique minimum,
l 2
:'if it is a strictly convex function of
it is defined.
The
)Yl
¢(~)d~.
Y
l
and Y
2
in the region where
conditions that will ensure the convexity of
9
However
I hi(Yl-g)¢<e)~ +~ p"(g-Yl)¢(g)~+fhi(O)+h2(Y2-Yl)
Yl
00
o
"1
+ p,(0)+r7¢(y)]
-
~
f
+
~
.
h"(y2-g Hi(g )dgf h;;(Y2-y1 )
f
o
i
r P"(g-Yl)¢(g)~
~
¢(g)~ +
Yl
:1
hi(Y1-')¢(' )d'
0
~
+
2
+ fhJ.(O)+h (Y2-Y1 )+P'(O)+.r7 ¢(y1)J •
Yl
It
is "clear
that if
h
l
{.!,
h (.)
2
and p(.)
are convex increasing
functions, then from (l.t), (1.8) and this last equation we find,
Gil
YlYl
i.e.
> 0, Gil
Y2Y2
> 0, and Gil
YlYl
Gil
_ (Gil
)2 > 0
Y2Y2
Yl Y2
G(yl ,y ) will be strictly convex in its domain
2
Y
l
,
< Y2 , •
In exactly the same way, and under the same assumptions on hI ( • ) I h ( ; )
2
and p(.), we can prove that
G(Yl'Y2 ) is convex in the domain
Y2 < Yl -
G' (0, y )<ofor every Y > 0 •.
2
2
Y1
If this condition is not sat:is.:f'ied the· commodity A should not be' ataolted,
At this point we shall assume t}1.at
.
in general; no demand would be satisfied, a.nd' we would incur the lOss.
due to the penalty cost and storage cost of'
Y2' units of commodity
B.
10
Since
h
2(.)
is a convex increasing function, we can see fram
(1.6)
. However" as we have seen above"
for all
Y > 0.
2
Thus if
then
for all
Similarly we can prove that
Yl > Y2
and.· Yl'~ 0,
Y2 >
°
a stlfficient condition, in the domain
under which G(Yl"O) is not a relative minimum
is also
Furthermore we.causee :from (104) that
Similarlj1 in the domain
Y
1
> Y2 '
11
G(y ,y ) -> co
l 2
if Y _>co
1
Hence if h (.), h (.), p(.) are convex increasing functions, and
l
2
then G(y ,y ) will have a unique minimum, if any, in both the two
l 2
Yl < Y2' and Y2 < Yl •
Now in the present case h (.) is a convex increasing function,
2
thus h ' (.) > O. Hence from equation (1.6), we can see that G' > o.
2
Y2
This means that G(y ,y ) is an increasing function of Y2 for
l 2
Y2 > Yl • Therefore the minimum, if eXisting, must be at same point
in the region Y2 :5 Y1" But in exactly the saine way, if we take
domains
Y2 < Yl ' we will find that G(y l ,y2 ) is an increasing function of
Y for Y > Y • Hence the minimum must be at same point in the
l
l
2
region Yl :5 Y2 ' Thus the minimum will be at same point in Yl ' Y2
plane where Y = Y •
l
2
Actually, if we take h (.) and h (.) to be linear functions (which
l
2
is the case in most of the practical cases, as far as we are aware) then
it is apparent immediately fram (1.6) that G! > 0; and for the similar
Y2
case when Y < Yl ' G' > 0; we then reach the same conclusions that the
2
Yl
minimum is attained at some point where Y = Y2 "
l
This result can be generalized (along exactly similar lines), to
the case in wrdch the demand is on equal amounts of any fixed number of
different products.
However~
let us next generalize our results to the case when the
items are demanded in different proportions.
Suppose one unit of the
12
end product is composed of
If we define
and h (X)
2
Yl = yl/a
= h2 (/3
x)
a
, and
units of A and /3
units of B.
Then
Y2 = Y 2 //3, and put hl (x) = h l (a x),
then we get
We can now proceed exactly in the same way as for the case when
a
= /3 = 1.
The unique minimum of E LX 'X (Yl' Y )
1 2
2
straight line
Yl =Y2 '
is at some point on the
that is to say that the unique minimum of
E L 'X (Yl ,Y2 ) is at some point on the line
xl 2
search for the minimum of
Yl =
~ Y2
0
Thus we
where
G(y) = (" 01+
~02)Y +~
t
y
ii1 (y-g )+ii2(y-g )
1
(Xl
¢(g)ag +1 p(g-y)¢(g)ag
o
y
Suppose the value of y which minimizes
Y*; then the values that minimize
is
y1
= a Y*,
and y2
= f3
G(y)(and E L
xl ,x2(y»
E L
xl ,x2 (Yl'Y2) are
y*.
By using the fact that equality of Yl and Y2
problem we go back to (1.1), and put Yl
= Y2 =y,
in our original
(say), and we find
that
(1.10)
= -. clxl
E Lx ,x (y)
1 2
- c2x2 + G(y)
where
y
(1.11)
G(y)
= (01+ 02)y + S ~h1(y-g)
r
+ h2(y-g)-r • g_7¢(g) dg
o
(Xl
+
~p(g-y)
- r • yJ ¢(g) ag
•
y
The value of y which minimizes
G(y) also minimizes E Lx x (y).
l' 2
Evidently
y
(1.12) G'(y)
= 01+02
~hi(y-g) + h~(Y-!17¢(g)ag
+)
o
-1~
(Xl
Y'
p' (g -y) +
B
¢( g) ag
14
y
(1.13)
G"(y)
= ) rb~
(y-g) + i12(y-g ,-7¢(g)dg + ~
~
p"( s -y)¢(g )ds
y
o
...
h (.), h (.'>, and p(.) are convex increasing functions, we can
l
2
see that G(.) must be convex; in fact G"(Y) > 0, from (1.13). Thus
Since
G'(Y) has at most one zero.
Let us now assume GI(O) < 0, for otherwise there would be no need
to stock the commodities.
We can see from (1.12) that the condition
that will insure G'(O) < 0 , is
Our earlier observation that G -->
equation
(ii)
~
as
Y _..>to
can be seen from
(1.11), i.e.
G(y)
---> ~
as
y
--->
~
•
Thus, if G(.) is convex, and if conditions (i) and (ii) are
will have a unique solution Y*, (say).
satisfied, then G' (y)
Y*
(1.14)
G'(Y*) = 01+02 +)
-r
Y*
rbi(y*-g) + b2(y*-g17¢(g) dg
o
LP'(g-y*)+r_7 ¢(g) dg = 0 •
Hence
15
Therefore, if we have
(I)
Y* - xl of
We order:
and y* ... x
A
of
2
B •
(II)
We do not order anything fromB, and we go back to equation
that will ~te G; (Y ,x )
l 2
1
(1.5) to find the value of Y
l
=0
•
(III)
VIe do not order anything from
A, and we go back to the equation
Y2
00
2
h (y2-s)¢(f)ds - hi(x -y )
l 2
o
co
LP'(S-Y2) +
r_7
¢(S) ds
Y2
which is true in the domain
that will make
GI (Yl ,x2 )
Y
l
amount
IYl Ji
='.4
1: - xl
y!
=
2
0,
and if
; - and if
amount
~
IY2-_.2
.4
J
- x
2
"
=
and
0
Y2; then we find the value of
is the value of
xl
~
Similarly in (c), if ~
G' (x ,y )
Y l 2
~
Y2
= o.
G' (x ,y )
Y2 l 2
In case (b), if
xl
y! '
x
>
~
x
< ~l 1
_
such that
we order from A the
then we do not order anything.
is the value of
and if
if
xl
Y
l
2
2
<~ ,
.
Y2
such that
we order from
B the
, then we do not order anything.
16
(IV)
....
We could go back to equation (1.5) and find the value of Y that
l
will make G' {y ,X ) = 0 i.e. the minimum of G(y ,x ) for this par~
l 2
Yl l 2
ticular value of
insure that
x •
2
G(Y ,x )
l 2
But if
'Vle
< G(Xl ,x2 )
want to get a value of Y
l
that would
without solving equation (1.5) for
the minimum as indicated above, we ord.erin this case from (A) up to
Y*, and do not order any of the
is implied by
Y1
= Y*
and Y2
(B)'s.
= x2
The fact that
G(Y*,x )
2
may be seen as follows.
< G(xl ,x2 )
From
. equation (1.14)
Y*
.00
01+02 +) L!lJ.(Y*-S) + h2(y*-s.27
¢(s)~-)· LP'(S-Y*)+.!7¢(s)~ = o.
o
Y*
Therefore
Y*
00
(1.15) °1 +) a hI(Y*-S)¢(S)dS-)y* LP'(S-Y*)+.!7
¢(S)~
Y*
= -
°2 -
~o h2(Y*-s)¢(s)ds
Also, from equation (1.5), if we have
y*
=
Yl=Y*
y =x
2 2
01 +)
Y = Y*
l
and Y = x 2
2
00
hI(y*-s)¢(s)~-h2("2-Y*) ~ ¢(S)~
Y*
o.
00
- ) y*Lp'(s-Y*)+rJ ¢
(S)~
1"7
Thus we get f~om (1.15)
y*
=-
G'
Yl
02 -
yl=y*
~ :X1
~ h2(Y*-~)¢(~)d£
o
0:>
- h2(X2 -Y*») .
Y*
¢(~)dl
and therefore
G(Y ,y ) is convex in Y, and G' (o,y ) < 0,
2
l 2
1
Yl
for all Y then G(Y*,Xg ) < G(Xl ,x ). So if xl < y* but x2 > y* ,
2
2
we order y* - xl of the (A) 's, but do not order anything from the
Since, as we have seen,
..
(B)' s.
(e)
In a similar way we can show here that the optimum policy is to
order y* - x
of the (B)'s, and not order anything from the (A)'s.
2
Suppose we were to introduce positive salvage gains vl(Yl-S) and
V2 (Y2-S), if
y.J .<
-
5 < Yl and 5 < Y2 respectively; and Vl(Yi-S) =
5, i = 1,2. Then
° if
18
Let the known probability distribution of demand be generated by
a continuous density function
•
¢(s),
and assume as before that the
functions
hI(.)" h2 (.), p(.) , vI( • ), and v2( • ) are continuous and
all possess piecewise continuous second derivatives. From equation
(1.16),the expected total loss is
where
00
(1.18) G(Y1'Y2)=Y1c1+Y2C2 +
~ \h1~Y1-min(g,Y~7-V1~Y1-min(g,Y~7J ¢(g)dg
o
00
+} [hJ""Y2-min(g'Y117-VaLY2-min(g'Y117j ¢(g)dg
o
r
+
p( g-Yl' g-Y2)¢(g)dg
min(Yl'Y2)
Let us assume that
-rr
.
0
Y < Y • Then
2
l
Yl
(1.19) G(Y1'Y2)=C1Y1+C2Y2+}o
L
.
~~(Y1-g)-V1(Y1-g17¢(g)dg
Yl
+
min (g, Y1'Y2)¢(g)dg
Lh2(y2-g)-v2(Y2-H7¢(g)dg
Now, as before, we evaluate
We £ind that
o GoG
oY ,
1
aY2
~ G and.Q G
~,
oYroY,.,
2
I
~
Y
2
19
2
G
G(Y ,y ) is a convex function if:
I 2
h (.) - v (.), p(.)
2
2
h (.) -v (.),
I
l
are convex increasing functions. In this case
o G(yl ,y2'
0
> 0 forY1 < Y2. SimiY2
0 G{y ,y )
l 2 > O. Then we deduce,
larly~ for Y
> Y2, we will find
()
I
YI
as before, that the minimum is at some point, where Y = Y = y.
I
2
it will also be true that
From this point on the argument fo110'\'18 identically the same argument
as in the first model with no salvage gains •
.
ClW'TER II
ORDERING COSTS vlITH "RED_TAPE"
In this chapter, in addition to the costs of the items, which
are assumed to be directly proportional to the amounts ordered, we
will have an administrative fixed cost associated with any amount
ordered from either of the products A or B.
This type of cost
may also represent "set_up" costs in lIlS.nufacturing processes.
So
we will have
and
where
__ \ k
Ki(Z)
and k i > 0, (i
[
= 1,2)
,
if z > 0
1
,
if z = 0
,
oi
•
Our assumptions about h (.) 1 h (.) 1 and p(.) are the same as
l
2
before, giVing us the conditions under which G(Yl'Y2) is a convex funcYl ~ Y2 ' and Y2 ~ Yl • Also G(yl ,y2 )
is strictly increasing with Y2 in the region Y2 > Yl ' and strictly
tion in any of the two regions
increasing with Yl in the region Y1 > Y2 • So G(Yl'Y2 ) has a unique minimum at the point (y* 1 Y*).
21
Let us imagine three planes parallel to the
(Yl'Y2)- plane,
and at heights
k + G(Y*,Y*), k + G(y*,y*), and k + k 2 + G(Y*,Y*)
2
l
l
above the Y Jy2-plane.
I
vIe
are going to prove some lemmas which show that each of the
three planes
in the
~efined
above determines uniquely two distinct regions
(Yl ,Y2 )-Plane.
These regions 'Will be of importance in
determining our optimal policies. First we are going to prove that
each of these three planes will cut the surface
continuous plane curve.
three curves on the
z
= G(Yl ,y2 )
in a
We will denote the projections of these
(Yl Y2 )-Plane by . (k ), (k2 ), and (kl +k 2 )
l
respectively.
Lemma A:
The plane parallel to the (Y Y2)-Plane, and at a height
l
say, where z > G(Y*,Y*), 'Will cut the surface z = G(yl ,y2 ) in
a continuous curve.
z,
Proof: We have seen before that the function G(Y 'Y2) is monotonically
l
decreasing along the line Y = Y as the coordinates increase to the
l
2
point (Y*,y*) .• and is then monotonically increasing, and tends to
infinity, when the coordinates increase beyond this point.
Thus there
eXists a point Y.l = Y2 = Yt (z ), say, > Y*, such that
(2al)
G(yt(~), yt(z»
=
-z
If
-z <
f~7
G(O,O)
then there eXists a point
G(YU(Z), y"('Z»
Yl
= Y2 = Ylie)
z
= 'Z
, say, <
"3*, such that
22
If condition (a) is not satisfied, then, 3ince G(O, Y2)
is
continuous in Y2' and strictly increasing to infinity with Y2'
N
then there exists a point Yl = 0, Y2= Y2' say, such that,
G(O, Y2) =
For similar reasons there will exist a point
z.
z.
Y2 = 0, such that G(Y1'O) =
Therefore if condition (a) is not
satisfied, we will take yll(Z) = O.
From (2.1) and (2.2), and the fact that G(y,y)
for y <
Y*,
and increasing for y >
all y"(z) < y < yt(z}.
Y*,
is decreasing
we have G(y,y) < z, for
But G(Y1'Y2} is continuous in Y1 and Y2'
and strictly increasing to infinity as
Y2 tends to infinity in the
region Y1 > Y2' and also increasing to infinity as
finity in the region Yl > Y2'
Hence for each Y1
Y1 tends to insuch that
y"(Z} ~ Y1 ~ yt(Z) there exists a unique Y2 ~ Y1' for which G(Yl'Y2}
= z.
Also for each Y2
such that y"(Z) ~ Y2 ~ yt(z} there exists
a unique Y1 ~ Y2' for which G(Y1'Y2) = z
Let us now consider the points
·
(Y1 'Y ) in the region Y2 > Y '
1
2
for which
We wish to prove that such points form a continuous curve.
the Implicit Function Theorem ~Goursat
"Let x
= x o,
Y
= Y0
l:§7 _7: -
be a set of values which satisfy the equation
F(x,y} = 0, and let us suppose that the functions
continuous in the neighborhood of the point
not vanish for
v1e will use
x = x o and y = y 0
F and oF/oy, are
(x ,Y ). If :;Floy does
o 0
then there exists one and only one
23
continuous function of the independent variable
the equation, and which assumes the value y
value x
o
o
x, which satisfies
when x assumes the
II
Let i
be some prescribed positive constant; write
Then the points vTe are considering satisfy the equation
Clearly G{Yl'Y2) is continuous in Yl and Y2' and oG/oY2 is continuous and always greater than zero in the region Y2 > Y1. Thus the
function F(y ,y ) satisfies the conditions of the implicit function
l 2
theorem at any point in the region Y ~ Y • Suppose the point yl=yr,
2
l
and Y2 = Y satisfies (2.3) where Y > yr. Define the neighbourhood
2
2
< a, and ly 2-y2' < b, where a and b are chosen as
large as possible, but so thatR lies entirely in the region Y > Yl •
2
R by: IY1-Yil
Then there exists a unique continuous function
dent variableY
l
l
~( •)
of the indepen-
such that
Hence if we start at any point M(yi, yp satisfying (2.3), and find
the corresponding neighborhood R, then Y2 = W(Y ) is uniquely defined
l
and continuous in this neighbourhood R. This proves that the points
=
in the region Y2 ~ Yl , and satisfying F(y ,y ) = 0 i.e. G(yl ,y2 ) i,
l 2
form a continuous curve. Similarly we can prove that the points in the
region Yl~ Y2' and satiSfYingG(yl ,y2 ) = i, form a continuous curve.
24
(yl(Z)" yl(Z»
(y"(z) "y"(z»
Putting
k
l
+ k
2
z
equal in turn to k
l
+ G(Y*,'Y*), k 2 + G(yf,Y*); and
+ G(Y*,Y*), we deduce from Lemma A that the three planes already
mentioned will out the Burf~ce z= 6(Yl'Y~) 1n three C~l;);tinuous plane
curves, with projections
(k ), (k ), and (k + k ) respectively.
2
l
2
l
For the curve (k ), the point corresponding to the point
l
(yl(Z), yl(Z» will be denoted by (yl(k ), yl(k ». For the curve
l
l
(k ) it will be denoted by (yl(k ), yl(k ». For the curve (k + k )
2
2
2
l
2
it will be denoted by (y I (k +k ), y I (k +k ) ) • Since the function
l 2
l 2
G(Y,y) is increasing with y, for
y >
y*,
we will have, if we assume
~
k
l
< k 2 , say, that Y* < yl(kl ) < y'(k2 ) < y'(k l + k 2 ). Condition (a)
implies for these three curves the following inequalities
k l + G(Y*,Y*)
< G(O,O),
for the curve
(k ),
l
k 2 + G(Y*,Y*)
< G(O,O),
for the curve
(k ) ,
2
<
for the curve
(k +k )·
l 2
k l +k2 + G(Y*,Y*)
For the curve
G(O,O),
(k ), the point corresppnding to the point (y"(z),
l
y"('Z», will be denoted by
it will be denoted by
(Y"(k ), y"(k ».
l
l
(y" (k ), y" (k ) ) •
2
2
For the curve
For the curve
(k
l
(k )
2
+ k )
2
25
it will be denoted by
G(y"y)
(y"(k +k )" Y"(k +k
l 2
l 2
is decreasing with y
k
Since the :f'unction
°< Y < Y*" we will have, assuming
for
with no loss of generality that
».
l
< k 2 " y"(k l +k 2 ) < y"(k2 ) <
y"(k l ) < Y* •
Let us consider the segment of the curve
region
Y2
~
Yl.
which is in the
It is important to notice. that no part of the curve
(kl ) lies in the region Y2 ~ Y
l
> Y'(kl ). Also, when k l + G(y*"y*)
< G(O"O)" that is to say when Y"(k l )
Y ~ Y " Y
2
l
l
from the region
(k )
l
the curve near the point
~ 0" the curve (k ) is excluded
l
< Y"(k1 ). For example" let us consider
(Y'(kl)"Y'(k
l
»"
and assume that the curve
turned back at this point and caltinued in the region
This would mean that we can find two points
Y ~ Y
2
l
> Y'(kl ).
and
q, <Yl;Y2) ,
P E (y1"Y2)
=
Y2 > Yl " Yl < Y'(k l )" and Y2 > Yl > Y'(k l ), with
these t,oJ'O points on the curve (k ). From this we can see that the line
l
say" such that
between the two points on the sruface
tions are the two points
from the y y -plane.
l 2
P
z
= G(yl "y2 ),
and whose projec-
and Q" will be at a height
If we take any point
k
l
+ G(Y*,Y*)
R, say, on the line P Q"
then we will always find a point which is the projection of a point on
the surface
= G(Yl'Y2)'
z
at a height k
l
+ G(Y*,Y*); that is to say
a point on the curve
(k ), with the same yl-coordinate as the point
l
R, but with smaller y -coordinate than the point R. Let us call
2
this point
S.
is equal to k
At the point S, the value of the function
l
+ G(y*"y*).
Since
G(y "y )
l 2
G(y ,y ) is increasing with Y
2
l 2
for fixed Y " then at the point R the value of the function
l
G(y ,y ) will be greater than the value at the point S, that is to
l 2
say, the value at
R will be greater than k
l
+ G(Y*,Y*).
This means
26
that the line between the two points on the surface, whose projections are
and Q, is under the surface at same of its points.
P
This contradicts the fact that the surface is convex.
proved that the curve
(k )
1
will not return to the region Y2
> yf(k1 ), after it reaches the line Y1
(yl(k ), yf(k
1
1
».
= Y2
applies to the curve
Y2
for a fixed
(k ) near the point
1
same is true for the segment of the curve
(k
at the point
It is obvious that the curve will not return to
strictly increasing with
and
> Y1
> Y1 , Y1 < yf(k1 ), since the function G(Y1'Y2 ) is
the region Y
2
Y1 ~ Y2·
Thus we have
Y1'
A similar argument
(y"(k ), Y"(k
1
1
».
The
(k ) 'Hhich is in the region
1
Similarly we can prove that for the other two curves
1
(k ),
2
+ k ).
2
Assuming with no loss of generality that k
1
<k , we have seen
2
that
Y* < Y'(k ) < yf(k ) < yf(k + k ). If k + k + G(Y*,Y*)
2
1
2
1
1
2
< G(O,O), then k + G(Y*,Y*) < G(O,O), and also k + G(Y*,Y*)
2
1
yU(k ), y"(k ), and y"(k +k ) all eXist, and,
2
1
1 2
as we have seen before, Y* > y"(k ) > y"(k ) > y"(k +k ). But if
1
2
1 2
< G(O,O); therefore
k1 + k
2
+ G(Y*,Y*) > G(O,O), and k
we assumed k
curve
1
< k ,
2
k
1
the two points
such that,
Y1
Y1-axis and ~he
(0, Y2 (k +k
1 2
»,
(k ) and
1
(k )
2
the two other curves
+ G(Y*,Y*) < G(O,O) then, since
+ G(Y*,Y*) < G(O,O).
(k1 + k 2 ) cuts the
(Y1(k +k ),O) and
1 2
2
= Y2 = Y"(k1 ),
Y* > Y"(k ) > Y"(k ).
1
2
So in this case the
Y2-axis at the two points
say, respectively.
will cut the line
In addition,
Y1
= Y2
at
and Y =Y2=yl/(k ), respectively,
1
2
Also" since G(O,y ) is monotoni2
cally increasing with Y ' and G(y ,0) is monotonically increasing
2
1
21
with Yl' if k l + G(Y*,Y*) > G(O,O) we will have k 2 + G(Y*,Y*)
> G(O,O), and k l + k 2 + G(Y*,Y*) > G(O,O) (remember k l < k 2 ).
Therefore, in this case, the points, (rl(kl),O), (0, Y2(k »;·
l
(Yl(k2 ),0), (0'Y2(k2
with
»;
and
(~1(kl+k2)'0), (0, ~2(kl+k2)' all eXist,
k
~l (k l ) < Yl (k2 ) < Y
l (k l +k2 ), and Y2{k l ) < Y2(k2 ) < 'Yi l+k 2) •
.Lemma B:
The curves
(k ), (k ), and (k +k ), do not intersect at
2
l
l 2
any point. Each curve will divide the Y y -plane in two parts. If
l 2
we call the part of the y y -plane in which the point (Y*,Y*) is
l 2
located: inside the curve, and the other part of the Y1Y2-plane:
outside the curve, then any straight line between any two points on
anyone of the three curves and in the same region Y1
~
Y2 or y 2
~
Y1
will be inside the curve •
Proof:
The proof of the non-intersection is clear from the fact that
the function G(Y ,y ) is a single valued function. The proof of the
l 2
second part follows from the fact that z = G(y ,y2 ) is a convex surface
l
Yl ~ y2 0r Y2
Any point on the surface z
in any of the regions
~
Yl •
Lemma C:
G(y ,y ) whose projection is a
l 2
point inside a curve (k) is at a height from the (Yl ,Y2 )-Plane which
is less than the height of the points on the surface whose projections
=
are on that curve.
Proof:
Let us consider the point
Y > Y • Join the points
l
2
we have two cases:
Q~ (Y ,Y ) inside
(k), and suppose
2
(Y*,Y*) and Q. by a straight line. Then
l
28
k + G(Y*,Y*) < G(O,O).
In this case the extension of this line will cut the curve
(k)
at some point P, say.
(ii) k + G(Y*,Y*) > G(O,O).
In this case, as we have seen in Lemma A, means that the curve
(k) cuts the Y2-axis at
(0, Y2(k»
where Y2(k) > 0.
this case, the extension of the line between
either cut the curve
(k)
(Y*,Y*)
Therefore, in
and Q will
at some poi~t, or will cut the Y2-axis at
f'J
(0, Y ), where Y2 < Y (k). Since G(0'Y2) is increasing
2
2
with Y2, then G(0'Y2) < G(0,sr2(k» = k + G(Y*,Y*). Thus in any case
the extension of the line (Y*,Y*) Q reaches a point P which is the
some point
the surface.
From this we can see that
G(yl ,y2 ) < k + G(Y*,Y*), i.e.
Q:=(Yl'Y2 ) is such that
Q is the projection of a point on the
surface which is at a height less than the height of the points on
the surface whose projections are the curve
(k).
For, if this were
28a
2.
(lA?J
L/
/(fJ.)
I
y~
I
FIGURE 2:
If
G(O,O) > G(Y*,Y*) + k
G{O,O) < G(Y*,Y*)
+
1
k2
and
29
Lemma ~:-
Any point outside any of the curves
(k ), (k ), or
2
l
(k +k ) is the projection of a point on the surface z = G(y ,y )
l 2
l 2
which is at a height greater than the height of the points on the
surface Whose projections are on that curve.
From these lemmas we can see now that the curves
(k), (k ) and
2
(k ) and (k ) is inside
2
2
(kl + k 2 ), are such that (k ) is inside
1
(k1 + k ). See figures (1) and (2) •
2
Lemma E: If a point Q?(y ,y ) is in the region of the (Yl Y2 )l 2
plane where Y2 > Yl ' and this point is outside the curve (k), then
there is no point P
or on the curve
(y1,y:P where Y > Y2' such that P
2
is inside
(k).
Proof: If the point P with the above properties does exist, then we
will have a contradiction.
For: (a)
the function G(y ,y ) is inl 2
creasing with Y2 in the region Y > Y ; (b) a point outside the
2
l
curve (k) is the projection of a point on the surface at a higher
level than points on or inside the curve
(k).
Thus, if P exists
then by increasing Y we have decreased, G(y ,y ), Which is non2
l 2
sense since we are in the region Y > Y ' Hence the lemma is proved.
l
2
For similar reasons, if Q == (y ,y ), where Y > Y2' is outside
l
l 2
the curve (k) (say), then there is no point
(Yi, Y2 ) where Yi > Yl '
such that P is inside or on the curve
(k).
Suppose Xl and x
are the initial stocks on hand, before
2
ordering, of A and B respectively; call this the "initial point",
If we did not order anything from A or B, we would get the expected
30
loss equal to
If we raise the stock level of A only to Yl (say}, then
If we raise the stock level of Bonly to y2 (say), then
.
If we raise the stock levels of both A and B to y1 and y2
respectively, then
(x 'x ) such that xl < x •
2
l 2
From the fact that G(Yl'Y2) is increasing as Y increases, when
2
Y2 > Y ' it can be shown that we could not get to a better point
l
(i.e. minimizing G) by increasing the B stock only_ In the case
Suppose we have the.initial point
xl < x2 ' if we have the point
(xl 'Y2) where Y2 > x2 , G(xl ,x2 )
< G(xl 'Y2)' and so G(xl ,x2 ) < G(Xl ,y2 ) + k 2 - The right hand side
of this last inequality is the loss plus ~clxl + c2x2-7 which we
would expect to incur if we were to increase the stock of B from
x2 to Y ' and the left hand side is the expected loss plus
2
c Xl + c2x2-7 if we do not increase the stock _ Hence we have
l
proved that by increasing the stock of B from x to any level
2
I
y2 > x2 we have increased our loss.
However, in the case when
31
Xl < x ' we might be able to get to a better point than (Xl ,x2 ) by
2
either increasing the A stock only, or by increasing both the A
stock and the
B stock.
We shall see in a moment the different
situations in which either one of these two alternatives is optimal
Also in the case Xl > x2 '
is the initial point, we cculd see that we are not
or it is better not to order anything.
when
(x 'X )
I 2
going· to get to a better point by increasing the stock of A only,
since G(Yl'Y2)
is increasing when YI is increased in the region
Y > Y • Thus we have a situation similar to when Xl < x2' and we
2
I
have to increase either the stock of B only, or both the stocks of
A and B.
There are
four different situations that may arise as a result
of our initial stock levels
(X 'X2 ). The optimal policy that we
I
should follow to minimize the expected loss in each case we label
I - IV and proceed to consider them successively with various sUbca.ses.
As we have seen before, we are not going to minimize the loss by
increasing the stock of B only.
the point
(i)
Now for the different positions of
(Xl 'X ) under the condition Xl < x2 ' we have:
2
the poin·!.; (X 'X ) is outside the curve (k + k ).
2
l 2
l
This means that G(Xl ,x2 ) > G(Yl'Y2) for any point (YI'Y2)
inside or ope the curve (k + k ). If we increase our stock of both
2
l
A and B to y* and y* we get the value G(y*,y*) + k +k • This
l 2
means we get a value that is equal to that of points on the curve
32
(k +k ), which means that we could improve our situation, i.e.
l 2
decrease
our loss, by increasing the stock of both A and B to
the level Y* and y*.
We can see that this is the best we can do
by ordering from both A and B.
For 1fwe order any other amounts
to get the stocks to the levels say Yi and Y2' other than Y* and
Y* which give the unique minimum of G(Yl'Y2)' then G(Yi,Y2 )
> G(Y*,Y*), and so G(Yi,Y2 ) + k l +k2 > G(y*,Y*)+k l +k 2 • This shows
that the loss in this case will be greater than the expected loss
incurred had we increased our stocks to the levels
Y* and y*
respectively.
We will now try to see if, by increasing the stock of A only,
we can incur a loss which is less than the one incurred by increasing
both A and B stocks to the levels
Y*
and y*.
In this connection
we have to recognize four different situations:
1)
If by increasing the stock of A from Xl but leaving the
stock of B at
2)
x
we could not get inside any of the
2
curves (k l +k2 ), (k2 ), or (k l ).
If by increasing the stock of A from Xl but leaving
the stock of B at x
3)
we could get inside the curve
2
(k +k ) only, but not the other two curves.
l 2
If by increasing the stock of A from Xl but leaving the
stock of B at
x
2
we could get inside the curves
(k +k )
l 2
and
(k ), but not inside (k ).
2
l
4) If by increasing the stock of A from Xl but leaving the
stock of B at
x
(k +k ), (k ) and
l 2
2
2
we could get inside the three curves
(k ).
l
;;
,e
We will need the following lemma.
For a fixed value of Y2 = x2 " say, where x2 ·< Y*".the
point in the region Yl ~ Y2 for which G(y "y ) is minimum is
l 2
Lemma F: -
Yl
= Y2 = x2 •
Proof:
If we imagine a plane parallel to the
(Y ,Y )-Plane" which
l 2
passes through the point on the surface whose projection is the point
(Y ,x ) where Y < x " is outside the curve
l 2
l
2
which is the projection of the intersection of this plane with the
yl =y2=x2 , then any point
surface.
The reason for this is that this curve is at most a tangent
to the line parallel to the
y -axis" and passes through the point
2
(from LeJDllla E) and will never cross this line at any point.
•
Yl =y2=x
2
The minimum is not going to be in the region Y < Yl since G(y "y )
2
l 2
is increasing in this region with Y • The result then f~cws from
l
LeJDllla D.
For Case (1).
We can see in this case that by ordering any
amount of A only we are not going to decrease the expected loss
and improve the situation over the one when we ordered from both A
and B amounts
Y* - Xl"
and Y* - x2" respectively, since we are
always going to be at a point outside the curve (k +k ), i.e. the
l 2
value of G at such points is greater than the value for points on
the curve
•
(k +k ) which we could actually achieve by ordering
l 2
Y*-x and Y*-x
of both A and B respectively. Therefore in
l
2
this case the best policy is to order Y*-x of A and y*-x of B.
l
2
I.
;4
For Case (2).
Since we could only get to points inside the
(kl +k ) but outside (k ), and since by lemma F the point
2
2
that would minimize G for this particular value of Y2=x is
2
Yl=Y2=x2 then by lemma D, G(x~'X2) > G(Y*,Y*) + k 2 , and so
curve
G(X ,x ) + k > G(Y*,Y*) + k + k • The left hand side of this
2 2
l
l
2
last inequality is the expected loss if we order from A only the
amount x -x ' which is the best point if we do not order from B;
2 l
the right hand side is the expected loss if we order Y*-X from
l
A and Y*-x from B. Therefore increasing the stock of A only.
2
to x will not improve the situation (in the sense of decreasing
2
the loss) over what happens when we order from both A and B to
reach the stock levels
Y* and Y*
the best policy is to order
For Case (3).
Y*-Xl
respectively.
Thus in this case
of A and Y*-x
2
0f B.
Since we could get to- points inside the curve
but outside
(k )
2
(k ), and by Lemma F the point that would minimize G
l
for this particular value of y2=x is Y1=y2=x , then by Lemma C,
2
2
G(x ,x ) < G(Y*,Y*) + k , and so G(x ,x ) + k < G(Y*,Y*) + k +k •
2 2
l
2
2 2
l 2
Therefore increasing the stock of A to x will improve the situa2
tion over the one when we order from both A and B to reach the
stock levels
Y*
and y*
respectively.
So in this case the best
policy is to order only from A the amount x -x ' and do not order
2 l
any from B.
3'
For Case (4).
..
Since we could get to points inside the curve
~l)­
and by lemma F the point that would minimize G for this particular
value of Y2=x2 is Yl=Y2=x2" then by Lemma C" G(x2"x2 ) < G(Y*"Y*)+k.
and so G(x2 "X2 )+k l < G(y*"y* )+kl+k l • But we assumed k < k 2j hence
l
G(x2"x2 ) + kl < G(y*"y*) + kl +k 2 " Therefore increasing the stock of
to x2 but leaving the stock of B at x will improve the situa2
tion over the one when we order from both A and B to reach the
A
stock levels y* and Y* reepectively.
Hence in this case the best
policy is to order only from A the amount x2-xl " and not order
anything from B.
(ii) The point
(x "x2 ) is inside the curve (k l +k 2 ) but outside
l
(k2 )·
This means that G(xl ,x2 ) > G(Yl'Y2) for any point (Yl ,Y2 )
inside or on the curve (k~), and G(X ,x2 )< G(y ,y ) for any point
l
l 2
(Y ,Y ) outside or on the curve (k + k ) •. We know that if we increase
l
2
l 2
the stocks of both A and B to y* the expected loss will be
G(y*"y*) + k + k , which is exactly the value of G(Y ,y ) for
l 2
l
2
points (Yl'Y2 ) on the curve (k l +k 2 ). Hence G(xl ,x2 ) < G(y*,y*) +
k + k , and from this we can see that we could not decrease the ex2
l
pected loss by increasing the stock levels of both A and B to Y*
and y*.
We will now see if we can decrease the expected loss, by ordering
from A only, to a value that is less than the expected loss at
·v
(x ,x ). There are three different situations:
l 2
;6
(1)
By increasing the stock of A from xl but leaving the
stock level of Bat x
we could not get inside any of the curves
2
(k ) or (k ) - By Lemma F the point that would minimize G for
2
l
this particular value of Y2=x2 is Yl=Y2=x2, which is outf!ide the
(k ) - Thus, from Lemma D, G(x ,x ) > G(Y*,Y*) + k , and so
2
2
2 2
G(X2,x ) + k > G(Y*,Y*) + k +k • But we have seen that
l 2
2
l
G(xl ,x2 ) < G(Y*,Y*) + k +k , and so G(x ,x ) < G(x2,x ) + k - From
l
2
l 2
l 2
this it follows that we could not decrease the expected loss by incurve
creasing the stock level of A to x - Thus in this case the best
2
policy is not to increase the stocks of A or B, but to stay with
the stocks on hand without ordering.
(2)
By increasing the stock of A only we can get inside the
(k )
2
(k ) _ By Lemma F the best point
l
is Yl=Y2=x2 which is inside (k2 ). Thus, by Lemma 0, G(x2 ,x2 ) <
G(Y*,Y*) + k 2 and so G(x ,X2 )+k < G(y*,Y*)+k +k2 - However, we
2
l
l
know that G(xl ,x2 ) «( G(Y*,Y*) + k +k - Thus we could not decide
l 2
directly as in the last case Whether it is better not to order anycurve
only but not inside
thing or to order from A only, and we have to compare G(x ,X )+k
2 2 l
with G(x ,x ), to decide. Thus if G(x ,x ) < G(x ,X )+k , then the
l 2
l 2
2 2 l
best policy is nat to order anything and operate with the stocks on
But ifG(x ,x ) > G(x ,X )+k , then the best policy is to
2 2 l
1 2
order only
. from . A to raise its stock to x 2 •
hand_
(;)
(k l ).
By increasing the stock of A only we can get inside the curve
By Lemma F the best point is
Yl=Y2=x2 , which is inside (k ).
l
Thus by Lemma 0, we have G(x ,x2)< G(y*,Y*)+k , and SO,G(x ,x ) + k
2
2 2
l
l
< G(Y*,Y*) + kl+kl < G(y*,Y*)+k l +k2 - But G(xl ,x2 )< G(y*,Y*)+k l +k2 •
37
II
Thus in this case also we have to compare G(x ,X )+k With
1 2 l
G(X ,x ) to decide what to do. Hence if G(x ,x ) < G(x ,x ) + k ,
l
l 2
2 2
l 2
then the best policy is not to order anything and operate with the
But if G(x ,x ) > G(x ,x )+ k
then the best
l
2 2
l 2
policy is to order only fram A to raise its stock to x •
2
stocks on hand.
(iii)
(x 'x ) is inside the curve
l 2
(k ).
l
The point
curve
(k ), but outside the
2
From Lemma. D, G(xl ,x2 ) > G(Yl'Y2) where (Yl'Y2) is aQ'
point, inside or on the curve (k ). In particular G(x ,x ) >
l 2
l
G(Y*,Y*) + k l • But from Lemma C G(xl ,x2 ) < G(yl "y2 ) where (Yl "Y2)
is any point outside or on the curve
(k ). In particular
2
• It is clear that by ordering from both
G(xl ,x ) < G(y*"y*) + k
2
2
A and B, the loss is not going to be less than the loss incurred
if we do not order anything.
We will now see if we can decrease the
expected loss, by ordering fram A only, to a value that is less than
the expected loss at
(1)
(x 'x ). There are two different situations:
l 2
By increasing the stock of A only we can not get inside
(k l ) • By Lemma F the best point is
yl=y2=x which is outside
2
Hence fram Lemma D G(x ,x ) > G(Y*,Y*) + k therefore
2 2
l
(k l ) •
(2.4)
But G(x ,x ) > G(Y*,Y*) + k and so we oove to compare G(x ,X2 )+k l 2
l
l
2
with G(x ,x ) to decide, as before. Yet there is a special case in
l 2
which the decision is direct, this is when k - k < k • For then
2
l
l
from (2.4) we will have G(x ,x ) + k > G(Y*,Y*) + k , and we know
2 2
2
1
that G(X ,X ) < G(Y*,Y*) + k ; thereforeG(x ,x ) < G(x ,X )+k •
2
l 2
l 2
2 2 l
Hence the best policy is not to order anYthing ani operate with the
stocks on hand,
(2)
curve
xl and x2 •
By increasing the stock of
(k l
?'
A only we can get inside the
By Lemma F the best point is
yl=y2=x2 which is in-
(k ). Hence from Lemma C, we have G(x ,x ) < G(Y*,Y*) + k ;
2 2
l
l
therefore G(x2,x2 ) + k l < G(Y*,Y*) +kl+k < G(Y*,Y*) + k + k • But
l
2
l
G(x ,x ) < G(Y*,Y*) + k + k r so that we have to compare
l
l 2
2
G(x2,x ) + k with G(x ,x ) to decide as before. If G(x ,x ) <
l
l 2
2
l 2
G(x ,x ) + k then the best policy is not to order anything and
2 2
l
operate with the stocks on hand. But if G(x ,x } > G(x ,x ) + k
l 2
l
2 2
then the best policy is to order only fram A to raise its stock
side
to the level x '
2
It can be seen very easily that the best policy is not
to order anythipg at all and operate with stocks on hand xl of A
and x
Yl
2
of B.
Since in this case when Xl > x2' G(Yl'Y2) is increasing as
increases, Thus, as we have seen before, we are not going to
minimize the expected loss by increasing the stock of A only.
we increase the stocks of both A and B to Y*
If
then the expected
39
Here, from Lemma D, we have G(X ,x ) > G(y*,y*) + k + k
so that
l 2
l
2
the expected loss incurred when we increase the stocks of A and B
to Y* and Y*
is less than the expected loss incurred if we do not
increase the stocks.
As in(~) this is the best we can do by ordering
Like ease ~(I), (i17,
from both A and B.
by increasing the
stoc~
we will try to see if,
of B only, we can incur a loss which is less
than the one incuwred by increasing both the A and B stocks to
the levels
Y* and y*.
We have four different situations here also,
and by similar reasoning we will have the following:
(1)
By increasing the stock of B from x ' but leaving
2
the stock of A at xl' we could not get inside any of the curves.
In this case the best policy is to order Y*-x
l
of A and Y*-x
2
of
B.
(2)
By increasing the stocks of B from x ' but leaVing
2
the stock of A at xl~ we could get inside the curve (k +k2 ) only
l
but not (k ) or (k ). In this case the best policy is to order
2
l
Y*-Xl
of A and Y*-x2 of B.
(3) If by increasing the stock of B from x2 but leaving
the stock of A at
inside
(k ).
l
Xl' we could get inside
(kl +k ) and (k 2 ) but not
2
From Lemma F, the point that would minimize G for this
particular value of Yl
= Xl
is Y2=Yl=xl , which is outside
(k l ).
Hence
40
G(Xl~Xl)
from Lemma DI
G(Y*IY*) + k
and. Y*-x
2
l
+ k~.
> G(Y*,Y*) + k l ; therefore I
So the best policy is to order
>
G(xl,xl) + k 2
y*-x
of B. (Notice the difference 'between this and
of A
l
1 (I),
(i),
('17. )
(4) By increasing the stock of B onlYI we can get inside
all three curves.
From Lemma F the point that would minimize
this particular value of yl=xl
is
Y2=Yl=~
I
G for
which is inside
Then the best policy is to order from B an amount
(kl ).
x -x ' but do not
l 2
order from A. '
(ii)
The point
(XlI X ) is inside
2
(k +k ) but outside
l 2
(k )·
2
We can prove as in case
ill
(iij7 that we could not decrease the ex-
pected loss by increasing the stock levels of both A and B.
There
are three different situations:
(1)
inside (k ).
2
A or
B only we could not get
The best policy in this case is not to order from either
B.
(2)
side
By increasing the stock of
By increasing the stock of B only we could get in-
(k2 ) I but not inside
(k;t).
The best policy in this case is not
to order frcm either A or B.
(,)
s ide
(k ) •
l
By increasing the stocks of B only we could get in-
From Lemma F J the point that would minimize G for this
particular value of Yl=x
fore ,
but
G(X1IX1 )
G(x ,x )
l 2
<
l
< G(Y*JY*)
is
y =y =x l which is inside (k ).
2 l l
l
There-
and G(xl,X2) + k 2 < G(Y*IY*)+k +k ;
l
l 2
G(Y*,y*) + k +k • Therefore in this case we have to
l 2
+ k
41
compare G(Xl , xl ) + k2, With G(x1, x ). If G(xl,X2) < G(xl, X1 )+k2
2
then the best policy is not to order anythiss and operate With the
1
But if G(X ,x ) > G(Xl, x ) + k2 then the best
l
l 2
policy is to order only from B to raise its stock to xl.
stocks on hand.
(iii) The point (Xl' X ) is inside the curve
2
outside the curve (kll.
-
(k2 ), but
We can prove as in case rII. (iii)7 that we could not decrease the ex-
-
pected loss by increasing the stock levels of both A and B. There
are two cases:
(1) By increasing the stock of B only but not A, we
could not get inside
(k )2 In this case the best policy iS I not
l
to order anything from either A or B.
(2) By increasing the stock of B only we could get
inside
(k ). In this case the best policy is not to order anything
l
from either A or B.
(iv) The point
(X ,x ) is inside the curve
l 2
the best policy is not to order anything from A or Bo
(kl ). Then
There is a simple method to find out if by increasing the stock
of A from Xl to x2 we can get inside the curve (kl +k2 ) or
(k2 ) or (kl ), when Xl < x < y*. Let the points of intersection of
2
the curves (k l +k2 ), (k2 ) and (kl ) With the line YI=Y2 be yll(kl +k )1
2
ytf(k 2 ) and ylt(k) res:pectively, such that y"(k +k ) < yll(k ) < yUCk )
1 1 2
2
I
(Lennna (A». If any of the curves does not intersect the line YI=Y2
at such :points Lfigure
(217
then we take its yll(.)
= O.
If
42
X < y't(k . + k ), then ,-re will not be able to get inside any of
2
l
2
the curves. If y"(k +k ) < x < y"(k ), then we will be able
l 2
2
2
to get inside the surves
curves.
If Y"(k )
2
side the curve
<
(k +k ) only, but not the other two
l 2
x < y"(k ), then we will be able to get inl
2
(k ), but not inside
2
(k ).
l
we will be able to get inside the curve
~f
(k1).
y"(k )
l
< x2 ' then
Similarly if
x2
< Xl
< Y*, then by increasing the stock B from x 2 to Xl' we. will not
be able to get inside any of the curves, if
Xl
< y" (k l +k 2 ). We will
be able to get inside the curve (k +k ) only if Y"(k +k ) < Xl <
l 2
l 2
Y"(k ). He will be able. to get inside the curve (k ) but not (k ),
2
2
l
if Y"(k ) < Xl < y"(k ).
l
2
Y"(k )
l
< Xl.
and Figures
(III)
'Th~
stock of B.
Proof:
(k ), if
l
All these results are deduced from the Lemmas
A to F,
(1) and (2).
Xl < x 2 and X
2
Lemma G:
We will be able to get inside
> Y* •
optimal policy in this case calls for not raising the
4~
Suppose we raise the stocks of A to Y1 1 and of B to Y 1 then:
2
(1) Suppose Yl > x2' and Yl < Y - Since:lin this region
2
G(yl ,y2 ) is increasing with Y2' G(yl,yl ) < G(y ,y ). But G(Y,y)
l 2
is increasing with Y for Y > Y*, therefore G(x ,x ) < G(yl,y ),
2 2
1
and so G(~21X2) < G(yl ,y2 )· Thus G(x2,x2 ), + k l < G(yl ,y2 ) + k l +k2 •
(2) Suppose Yl > x2 ' and Yl > Y2 • Since, in this region
G(Yl'Y2 ) is increasing with Yl ' G(Y2 IY2) < G(yl ,y2 ). But G(y,y) is
increasing with Y for y > Y*, therefore G(x2,x2 ) < G(Y2'Y2). Thus
G(x2,x2 ) < G(yl ,y2 ), so that ?(x2,x2 ) + k l < G(Y l ,y2 ) + k l + k 2 •
(3) Suppose Yl < x2 • Since G(yl ,y2 ) is increasing ~th Y2'
G(yl ,x2 ) < G(yl ,y2 )· Thus G(yl ,x2 ) + k l < G(Yl 'Y2) + k l + k 2 •
Similarly if x2 < ~, and xl > Y*, then the best policy is not
to raise the stock of A, whether we raise the stock level of B or
not.
From all this we can see that we can only decrease the value of
the expected loss below the value of the expected loss at the point
(xl ,x2 ), by increasing the stock level of ~, if xl < x 2' or by increasing the stock level of B, if xl > x •
2
Now back to the case xl < x 2' where x > y*. G(Yl ,y2 ) is a
2
convex function in each of the variables, when the other one is fixed~
For y2,;::x2 ' let the value of Yl that minimizes G(Yl ,x2 ) be yJ,;::yt,
say. For the different positions of the point (x 'X ) 'tve have:.
l 2
(i) The point (X 'X ) is outside the curve (k +k2 ).
l
l 2
(1) If (Y*,x2 ) is outside the curve (kl+~2)' and Xl < yt,
then we have to comJ;lare G(y!,x2 ) + k l with G(xl ,x2 ) • If
G(x ,x ) < G(yt,X )+k , then the best policy is not to order anything.
l 2
2 1
44
~ut if
G(Xx.:a.) ~_ G~X2) ...+ k l , then the best policy is to order
only from A the amount :l.:-~L
(2) If
(y!, X2 ) is inside the curve (kl +k2 ) and
xl < yt, then we have to compare G(X ,x2 ) 'With G(yt,x2 ) + k and
l
l
decide as in (1).
(,)
If (yt,x ) is inside the curve (k ) but outside (k l ),
2
2
and Xl < yt, then from LeIl31lla C we have G(Y1.' x2 ) < G(Y*,Y*) +k2 •
Therefore G(yt, X2 ) + k l < G(Y*,Y*) + k l + k • Hence the best policy
2
is to order from A only, an amount Y1. - xl.
(4) If (Y1.'X2 ) is inside (kl ), and xl < y!, then the
best policy is to order only from A the amount
If in any of these four cases,
Y! - xl'
xl > y!, then since the function
G(Yl'x2 ) is convex in Y., we can see that the best policy is not to
.1:
order anything at all.
(y!,x2 ) is outside (k2 ), and xl < yt, then
the best policy is not to order anything~
(1)
If
(2)
If
(yt,
x ) is inside
(k ) but outside (k ), and
2
l
xl < yt, then we have to compare . G(yt,X )+k with G(x ,x ) and
2 l
l 2
decide as in case LIII, (i) (l)}.
2
If (yt,x2 ) is inside (kl ) and xl < yt, then we
.
.
have to compare G(yt, X2 )+k with G(x ,x ) and decide as in (2)
l 2
l
above.
(,)
If in any of these three cases
xl >
yt~
then the best policy is
not to order anything ...
(iii)
(xl"Xg ) is inside the curve
The point
(k ) but outside
2
(y!"x2 ) is outside (kl )" and xl < yt" then
.
the best policy is decided after we compare as before G(X "x ) with
l 2
(1)
If
G(yt"x2 ) + k l •
(2)
If
(yt,x2 ) is inside (k l )" and xl < yt" then we have
to compare G(yt"X2 )+k l with G(xl "x2 )" and decide as before.
If in these two cases
xl>
yt,
then the best policy is not to order
anything.
(iv)
The point
policy is not to order
(IV)
(XI"X2 ) is inside the curve (kl ).
The best
anythi~~.
x2 < xl" and xl >
:-r:
From Lemma G the best policy is not to raise the stock of A,
whether we raise the stock level of B or not.
VIe mayor may not be
(X 'X ) by inl 2
For y1=x fixed" let the value
l
able to decrease the expected loss below its value at
creasing the stock level of B only.
of Y2 that minimizes G(Xl "Y2) be Y2=~.
(i) The point (X "X ) is outside the curve (k +k )
l 2
l 2
(1) If (xl"~) is outside the curve (k l +k 2 ) and x2 < ~,
then we have to compare G(xl"~) + k 2 with G(X ,x2 ). If G(X ,x2 )
l
l
< G(Xl"~) + k 2" then the best policy is not to order anything. But if
.
_G(Xl ,x2 ) > G(Xl"~) + k 2, then the best policy is to order only from B
~-.
the amount
~
- x2 •
..,
46
(2)
If· (X.t'~) is inside the curve
(k1+k2 ) but
outside (k 2 ) and x2 <~, then we have to compare G(xl'~) + k 2
with G(x ,x ) and decide as in (1).
1 2
(:;) If (xIJ~) is inside the curve (k2 ) but outside (lt1 ),
and x2 <~, then we have to compare G(xl'~) + k2 with G(x1,x2 )
and decide as in (1).
(4)
If
(xl'~) is inside the curve (k1 ), and x2
<~ ,
then G(xl'~) < G(Y*,Y*) + k l fro~ Lemma (c). Therefore
G(xl'~) + kg < G(Y*,Y*) + k + k • But G(x ,x2 ) > G(y*,Y*)+k + k2
l
2
1
l
by Lemma (D). Hence the best policy is to order from B the amount
~ - x2 •
-(ii) Th=-~_o~t (xl 'X2 ) is inside the curve (k l +k 2) but outside
the curve (k ) •
2
(1) If (Xl'~) is outside the curve (k2 ), and x2 < ~,
.then the best policy is not to order
(2)
If
anYthil?-~.
(Xl"~) is inside the curve (k ) but outside the
2
then the best policy is not to order anything.
(k ), and x <~,
l
2
(:;) If (xl'~) is inside the curve (kl ), and x2 < ~ ,
then we have to compare G(xl'~) + k 2 with G(X ,x2 ) and decide as in
l
(i) (1).
curve
(x 'X2 ) is inside (k ) but outside (k ).
2
l
l
(~,,~) is outside (k ), and x <~, then the
2
l
(iii) The point
(1)
If
best policy is not to order anythign.
(2)
If
(xl"~) is inside (k )" and x2 <~" the '. <,
best policy is not to order anything.
l
(iv)
The point
(Xl 'X2 )
is inside the curve
(k ).
l
The best
policy is not to order anything.
In any of the three cases (1)-(111), if x2
policy is not to order anything.
~ ~ , then the best
CHAPTER III
CONVEX INCREASING ORDERING COSTS
In this chapter we shall deter.mine the nature of the optimal
policy if the ordering costs
Cl(Zl) and C2 (Z2) are convex increasing functions, keeping the assumptions on h (.), h (.), and
2
l
p(.) unchanged from the previous chapter. If xl units of A,
and x
units of B are the initial stocks, and zl=yl-xl units
2
of A, and z2=y -x units of B are ordered, then the expected
2 2
loss is
~
where
G(y1,y2)
~
=)0 h~Yl-min(g'Ya17¢(,)dI +)0 h~Y2-min("Yl17¢(,)a,
f.
~
+
p( g·Y1"-Y2)¢(g
)~-
~
r (
min! g,y1,y25 ¢(, )a, •
min(y ,y )
)0
1 2
Given the initial stocks (x ,x ), the optimal ordering policy is del 2
. fined, as before, to be a rule for choosing non-negative order sizes
Zl=Yl-Xl and z2=Y2-x2 ; such that E LXl'X2(Yl'Y2) is minimized.
Hence we can determine the optimal policy by deter.mining the levels
Yl and Y of the stocks that will minimize E L
xl ,x2 (yl ,y2 ), SUbject
2
to being not less than xl and x respectively.
2
Let us assume that Y1 < Y2 " Then
h
G(Yl'Y2)
h
00
=~ hl(Yl-S)¢(S)~ +1 ~(Y2-S)¢(S)~ +h2(Y2-Yl)( ¢(s)~
o
)Y1
0
00
+ (
p(s-Y1)¢(s)ds
)Y1
From this we get
00
-(
tt
~pl(S-Y1) + E7 ¢(g)ds
JY1
and
=)
Y1
00
h2(Y2-s)¢(s)ds
+
h2(Y2-Y)1
. 0
Y
1
¢(S)~
·
and
oE
Lx x (Y1 IY2 )/o Yn = C (Yn- xn) + G' (Y1,Yn) •
11 2
c;.
c;
c;
Y2
c;
2
But I since C (.)1 h (.)
2
2
2
C (y2-x2 ) >
°
are convex increasing functionsl then
and G;2(Y1 IY2 ) >
OJ
therefore 0 E LX1Ix2(Y1IY2)/OY2 >
01
50
for all
Y " y
l
2
where
< y 2 .Hence the minimum" if existing" must
Y
l
be at some point in the region
way" if we take
< Yl "
Yo
c:;
increasing function of
~
Y '
l
Y
l
> Y2 "
for
Y
l
Y
l
< Y2 '
E L
(Y "Y ) is an
x l "x2 l 2
Hence the minimum must be
Thus the minimum will be at
(y "y )-plane where
l 2
In the case when
But in an exactly similar
we will find that
at some point in the region
some point in the
Y
2
Y = Y '
2
l
Yl =:l2=Y" say" we may write
Y
E L ,x (Y) =
x1 2
Cl(Y-Xl)+C2(Y-X2)+)o~~(Y-~)+h2(Y-~)-r • ~7¢(~)ag
L~p(~-y)-r
CIO
+
•
x7 ¢(~)
ag •
If we put
Y
G(y)
=L ~~(Y-~)+h2(y-~)-r'
then, since
hl(O)
= h2 (0)
CIO
~7¢(~)d~+1 ~p(~-y)-r • X7¢(~)d~
= p(O) =0 ,
Y
~ G(Y)=C'(Y)=)o ~hi(Y-~)+h2(Y-~17¢(~)ag
and
G"(y) =
Since
I
t
Y
~hl(Y-~ )+h~(y-n7¢(~)ag
h (.), h (.) and p(.)
l
2
CIO
-Jy
~p'(~-y)+~7¢(~)d<
Lp"(~-y)¢(~)ag
CIO
+
are convex increasing, G"{Y) > 0; Le.
G(y) is strictly increasing with
y.
Now
,
51
(J E Lx
x (y)
l' 2
oy
Let us assume that
This simply means that when we have initially zero stocks then we
should order positive amounts to reduce
~ur
expected loss.
Studying the graphs of' either Ci(y-xl ) and -£C:'/y-x2 )+Gr(y17,
or C (y-x2 ) and - £Ci(Y-'tl ) + Gi(y)}, we will be able to determine
2
(301).
the characteristics of the zeros of
Gr(O)
From
(3.2)
we can see that
< O. Since Cl (.), C2 (.) are convex increasing, then
Ci(y-x ) + C (y-x ) is increasing with y, and _Gr(y) is decreasing
2
l
with y, so that oLE L . (y) 7/a y vanishes at most once. Let
xl ,x2 us denote by y(x ,x ) the unique zero of Ci(y-x )+C (y-x ) + Gr(y),
1
2
1 2
whenever it exists. We must have y(x ,x )::: max (x ,x ). Thus we
1 2
l 2
have
2
2
By partial differentiation, we obtain the following two equations,
+ C~ £ y(xl ,x2)-x2-7
+ G''fy(xl ,x2 L7
·
o y(xl 'x2 )
d xl
o y(xl 'x2 )
0 xl
=
0
52
and
= o.
From
(,.3)
we get
•
From
(3.4)
we get
Since Cl (.)" 02(.)" and p(.)
and (3.6) we discover that
0<
o y(xl "x2 )
o Xl
<
are strictly convex" then from (3.5)
1
,
and
(3.8)
o <
The inequalities
o y(xl "x2 )
o x2
(,.~) and
<
1.
(3.8) prove that for fixed x2" y(xl "x2 )
is strictly increasing with Xl; and for fixed Xl" y(x "x ) is
l 2
strictly increasing with x • The actual amount ordered from A is
2
53
- 1
<
0
,
and
by (3.8).
This proves that although the level to which we should raise our
stocks of A for fixed x2 is an increasing function of Xl' yet
the actual amount ordered fron A is a decreasing function of xl.
Similarly for the stocks of B.
From (:5.2)
-G'(O)
> Ci(O)
and also
-G'(O) > C (0).
2
Next, to define the optimal policy, we are going to study a set
of graphs of _~Gf(y) + C'(y-x )_7 and Ci(y-xl ); and a second set of
2
graphs of -fG'(y) + Cl(y-xl L7 and C (y-x2 ). We will superimpose
2
these two sets to simplify comparisons which appear later.
We define
the point x* to be such that if x > x* then the ordinate of
2
= x2 is smaller than
ordinate.ffi f (y )+C 1.(y-x 17, at
l
_~Gt(y) + C2(y-x~7
xl > X*, then the
C (0).
2
at
y
Cl(O); and also if
is smaller than
l
Then, fram this definition of x*, it follows that x* is
determined by the equation
y=x
j-4~~~_~-tl_-=__ -r---------"""--/
IV~
~-
..fJ
t-
//1
/,(l)
/~'
/~
!
/
i.e.
From (:3.9)" .(;'.10) and (';.11)" we obtain,
However
_G'(.)
is decreasing, so
Similarly we can prove that" for
If we define the point
which the curve
,..J
xl
Ci(y-'Xl )
Xl
< x*
,
to be the non-negative value of
metts the c~e
the point y~ we can see, since Ci(.)
that
rV
~l
at all.
is unique.
Also, for any
We can then determine
~
xl =0.
equation
for
_fGI(y) + C2(y-x2t7 at
is strictly increasing,
tI'J
Xl < xl ' the curves do not meet
rJ
xl from the equation
with the understanding that if the root of
define
xl
In a similar way we define
,
(';.14) is negative then we
......
X ' by means of the
2
55
(1)
The optimal
policy~
If xl
x2
or
From the graphs" we can see that
is greater than
x*" then nO.positive ordering
should be done.
and x < :x.* ,we examine the graphs of Ci(y-x )
2
l
and -FGI(y) + C;)y-x2t7.
(2)
If xl < x*
If }rl ~ xl ~Xl(X2) } then we order to raise both stocks
F2.17
of A and B up to Y(X "X ) ~ max (X "X ).
l 2
l 2
F2.~7 If
x*
> xl > xl(x2 )"then we examine the graphs of
2
)_7.
2
C (y-x ), and -FGI(y) + C (y-x
2
2
(2.2.1) If X'2< x < X (X ) then we order to raise the
2
2 l
stocks up to Yl(X "X ) ~ max(x ,x2 ).
l 2
l
(2.2.2)
If x < ~2 " then we do not order anything.
2
Notice that in this case i.e, when xl > Xl(x )" x
could not
2
2
be greater than X (x ). The reason is that, since xl(X ) > x "
2 l
2
2
and X2 (X1 )
> xl" then X2 (X1 ) > ~ > xl(X2 ) > x 2 •
12.2.7 If xl < ';C1
' then we examine the graphs of C (y-x )
2
and -FGt(y) +. c~(y-x2t7 •
(2.;.1)
2
If ;
<
~2 ~ x2(xl )"
then 've order to raise the
stocks up to y(X;L"X ) •
2
(2.;.2)
If x
2
> X2(~1) , then we do not order anything.
.
N
Notice that in this case" i.e. when xl < xl" x
less than
then
rv
,,-v
x 2' the reason here is that" since
AI
x2 < xl < xl < x2
•
rv
2
could not be
tv
x2 < xl" and xl < x2
CHAPTER IV
RAl'IDOM SUPPLY
In our studies so far we have assumed that, when we order any
amount of A or B, we Will receive exactly the amounts we order
before the demand on the commodities start.
It happens in some
cases of production, especially if the materials
A and Bare
some kind of agricultural products, or if A and B are some kind
of rare or complex and delicate dammodities, that this assumption is
inappropriate.
In many cases the decision-maker can only "order" or
"not order" a material, and if he "orders" a material he "'ill receive
a random amount of that material governed by some probability distribution.
If he decides to add stocks of A, say, then he will receive
a random amount z 1"
If he decides to add stocks of B he will
receive a random amount
variables
zl and z2
z2.
are
We are going to assume that the random
in~ependent
probability distribution functions
and have completely known
Fl(zl)' and F2 (z2)' respectively"
We can see that the decision maker has only four alternative decisions:
. ~ (i) Not to' order anything"
(ii)
Wo emderonly from A.
(iii)
To order only from B.
(iv)
To order from both A and B.
51
The expected losses:
We are going to find the total expected losses
when the decision-maker decides on any of the above four decisions.
our earlier work.
(i) Expected loss with, nO ordering:
We have three different cases:
(1)
If xl is the initial stock of A and x2 is the
initial stoCk of B, such -that xl < x2 ' then
i
~
(4.1) E L 'X (x1,X2 )
X1 2
=
~l
h1 (":L- S)¢(S) +
j
o
h2(X2-s )¢(s )ds
0
+ h2(X2-"l) j
00
00
¢(s)as
+}
xl
p(S-X1)¢(s)dS.
xl
~
=(
)0
hl(xl-s)¢(s)ds + hl (X1 X2 )
o
¢(r;)dE
2
x2
+1
r)X
00
00
h2(x2"s)¢(s)dS
+(
)X2
p(S-'2)¢(s)dS.
L~(x-~
x
(4.3 J E LXI' "2("l,X2 J
=
J¢(
~ Jd~
L (x-~
x
+
h2
58
~
J¢( Jdf
L
co
+
p(~-xJ ¢(~ Jdf
(ii) Expected loss when we order only from A but not from B.
In this case we will add a random amount
zl to the stock of A for
the period of productionj
zl has the known density dF (zl). The
1
expected total 10s6 is, where, as before, h (Y), h (Y), and p(y)
1
2
all equal zero for y ~ 0 ,
co
E L 'X (ZI+"l,x2 J
X1 2
5 Fc
=
co
1(ZIJ +
zl=o
S hI(zl+x1-min ~ ~ ,x2l )¢(~)df
E=o
co
+5
h2(x2-mintE,zl+X1! )¢(s)dE
s=O
f P(~-(ZI+XIJ, ~-X2J¢(~)df_7dFl(ZIJ
+
s=o
•
Therefore
co
S
=
co
C1 (ZI)dF1(ZI J
Zl=O
co
.j
Zl=O
co
f
+5
h1 (Zl+ X1- X2)d(s)ds dF1 (Zl)
Zl=O S=X2
59
1 I.
QI)
QI)
h2(X2-Zl-Xl)¢(~)d~
+
elFl (zl)
zl=oS=zl+x1
1 f.
00
co
+
P(~-Zl-Xl' ~-"2)¢(~)df
elFl (Zl)'
Zl=O s=min(zl+xl ,x2 )
We must distinguish between two different cases:
(a)
If xl ~ x 2' then
(4.4) will reduce to
(4.5) E L ,x (zl+xl ,x2 )
xl 2
QI)
=i~
Cl (zl)elFl (zl)
zl=0
1 hl(Zl+"J.-X2)¢(~)df
00
+1
Zl=O S=X2
J i
00
+
(b)
If xl < x 2' then
(4.4) becomes
X2
f.
(zl+"J.-~)
•
h2(X2-~)¢(·)d'elFl(Zl)
elFl(Zl)+L
Zl=O s=O
d' elFl(Zl) •
Zl=O S=X2
r
hl
zl=o s=o
00
00
P(~-X2)¢{')
+f~
x
60
co
=1
(4.6) E L ,x (zl+\'''2)
x1 2
co
+
Zl=O
X
f.
J
C1(Zl)dF1 (Zl) +
2
h1{Zl+X1-,)¢(,)dg dF1 (Zl)
Zl=X2 -X1 s=O
j
.
co
f.
Zl=X2-X1
co
h1 (Zl+ X 1 X 2)¢(,)d, dF1(Zl)
s=x2
~l+xl
J
x -x
+) 2 1
i
zl=o
+
J~
zl=x2 -X1
h2(X2-g)¢(g) ds
dF1(Zl)
s=o
co
P(,-Zl-X 1)¢(') dg dF1(Zl)
S=Zl+x1
r
S=X2
p(,-X2 )¢(,) dg dF1 (Zl) •
61
The limits on the integrals in
fact that the functions
(4.5) and (4.6) are determined by the
hl(Y), h (y), and p(y)
2
are zero for non-
positive argument.
(iii)
(4.5) and (4.6) can be found, when
Similar expressions as
we order only from B and not from A, with appropriate changes.
(iv) Expected loss when we order from both A and B.
In this
case we add a random amount zl to the stock of A and a random
amount
z2 to the stock B.
Then the expected total loss is
co
co
.J J
~Cl(Zl)+C2(Z2)
E LX1'X2(Zl+Xl'Z2+X2) =
zl=o z2=0
co
+(
hl (Zl+Xl-min
Js=o
i S,z2+X2} )¢(e )ds
co
+f
h2(z2+x2..min \ e,zl+x1
s=o
1)¢(s )de
co
+
J
P(g-(Zl+X1 )'S-(Z2+X2»)¢(g)d17aF1 aF2
s=o
Therefore
62
(4.7) E ' , x (zl+"1,z2+"2)
2
ii
Q:l
+
Q:l
i
=
f f
c:o
f
Q:l
C1 (zl)aF1 (zl)
zl=o
f.
zl=o z2=o
Q:l
Q:l
C2(z2)aF2(z2)
z2=o
.
~(z2+x2-(zl+xl) )¢(g)<\<
aF}zl)aF2(z2)
s=zl+x1
c:o
P(g-(zl+xl),g-(z2+"2))¢(g)dg.
z2=o s=m1n zl+x1 ,z2+ x21
l
6:;
Now, using the fact that
h1(y), h (Y), and p(y) are zero for zero
2
or negative arguments, equation (4.;) becomes
<Xl
(4.8) E L ,x (zl+x1 ,z2+x2)
x1
2
<Xl
=J
C1(zl)dF1
zl=o
+1..
C2 (z2) dF2
2=0
z2+ X 2
~o
) E h1 (Zl+"J.-E )¢(E)ag dF2 dF1
<Xl
<Xl
.r L h2(Z2+X2-(Zl+X1)~
z2=zl+x1 -x2
¢(f)dE] dF2(z2)dF1 (zl)
s=zl+x1
64
~l+:lt -X
~
) 1
x.
1
2
l"(S-VX 2)¢(S)dI: dF2(Z2) dF1(Zl)'
a;:z2+ x 2
z2=o
x
00
J
(4.9) E Lx]!x (zl+x1,z2+ x2) =
2
~
~
c1 (zl)dF1(zl)
zl=o
f
~
~
00
+X)J
h1(Zl+ x1-(z 2
z2=o zl=z2+ x2- xl
+J
C2 (z2)dF2 (z2)
z2=o
¢(s)dl:J dF1(zl)dF2 (z2)
g=z2+x 2
The optimal policy:
h (.)" h (.), and p(.)
2
l
are convex increasing functions, we are going to define a procedure
that Will give
initial stocks
xl
U6
xl
Assuming as before that
the minimum expected
of A and x
> x2 • Then" since
2
1066
of B.
~(.), h (.)" and
for any values of the
Let us suppose that
p(.) are non-negative and
2
strictly increasing and C (.) also is non..negative, we can see that
1
66
Similarly, when xl > X2
m
(4.11)J
f
m
1
m
~(Z~+X~."2)¢(E)dgdF~(z~):::f h~(":i."2)¢(g)ag
Zl=O e=X2
S=X2
ThUS, from equations
(4.5) and (4.2), using (4.10) and (4.11), we
can see that
By similar logic, if xl = x2 = x, say, and we compare (4.5) and ~'4.:3.)
with xl = x2 = x , we can prove that E Lx ,x (x,x) < E Lx,x
• (Zl+X'X).
l 2
From the last two inequalities, we can see that if xl ~ x2 ' then
it would be a bad decision to order only from A, since we would actually incur a smaller loss by not ordering anything at all.
Similarly
if x2 > Xl' we Will not order from B only. This means that if
Xl > x we Will have to choose between only three alternative decisions:
2
(i)
Not to order anythiMj
(ii) To order from B only
j
(iii) To order from both A and B.
If we try to find regions in which one of these three decisions is
optimal, we have to make very strong restrictions on the functions
hI( .), h (.), and p(. ), over and above the fact that they are con2
vex increasing, and this would no doubt decrease the practicality of
our analysis, ~compare this with the one-dimensional case treated
in Anow ~~7_7.
If x =x =x' say, then lie have only two possible
I 2
alternative decisions:
(i)
(ii)
Not to order anything;
To order from both.
In this case we can prove that the difference between
(4.8)
and
(4.;)
E Lx,x (zl+x, z2+x) - E Lx,x (x,x), is such that
d
-dx rE L
x
>0 •
x,x (zl+x, z2+ ) - E Lx,x (x,x)-7-
-
Therefore E Lx,x (Zl+x, Z2+X) - E Lx,x (x,x) changes sign at most once,
say at x*. From this fact we can see that, if xl =x2=x' say, t~
for X < x*, it is best if we order from both A and B, while for
x > x*, then it is best we do not order anything.
We can find xl!'
solving the equation
d
dx
IE
L
x
x,x(zl+x, z2+ ) - E Lx,x (x,x)- 7 = 0 for
X•
We can treat the case X > Xl similarly, and we will have only
2
three possible alternative decisions to make:
(i)
Not to order anything;
(ii)
To order from A only;
(iii)
To order from both A and B.
by
68
The procedure we are going to follow in determining the best policy
in either of the two cases
xl > x ' or
2
x > xl' is by comparing, in
2
xl > x ' say, E Lx ,x (xl,xnc; ) given by (4.2), with
2
l 2
E L
(Xl,Zn+X2)
given
by
expression
similar to (4.6) and vTith
xl ,x2
c;
.
E L
given by (4.8). Then we adopt the procedure
xl ,x2 (Zl+xl,zn+X2)
c;
.
corresponding to the smallest of these three possible losses. The
the case of
task of computing the values of
as it looks.
thes~
three losses may not be as bad
Certainly not muchharder than solving for
= x2 = X.
x*
in the
> Xl' then our procedure
2
is to compare the values of E L
(x "X ) given by (4.1) and
. xl ,x2 1 2
case when ,Xl
E Lx
Similarly" when x
X (Zl+xl "x2 ) given by (4.6) and E L
(zl+x.. "Z2+X2) given
~,x2
~
l' 2
by (4.9).
We then adopt the procedure corresponding to the smallest
of these three values of the expected total losses.
SECTION II
DYNAMIC MODEL
Introduction.
We have examined in the previous chapters the static or onestage inventory model.
We were only concerned with production in one
period 1 when there is only one chance to add to the stocks 1 at the
'beginning of this period", In this section we a.re going to consicler
the dynamic inventory mode1 1 or what is sometimes called the multista.ge inventory model.
We will be concerned with production in more
than on period1 finite or infinite in number.
characterized as follows:
The situation is
There exists specific points in time which
we shall call epochs 1 and which might be determined by the process
itself.
The time span between two consecutive epochs, is one period
in which we anticipate demands.
At each one of these epochs we have
to make a decision 1 whether to order further stock and how much of it
to order 1 or not to order such stock and satisfy ensuing demand from
the stock on hand.
The demand is not known exactly1 but its probe.-
bility distribution is assumed known.
Stock left over in one period
can be used in the following periods.
Let us make the following three assumptions:
(1)
There is no lag in delivery of orders.
Thus if we order
further stock at a given epoch then this stock is supposed to arrive
instantaneously and therefore be available to satisfy demand in the
next period.
(2)
The distribution of demand in a period is assumed to be
known and to have a continuous probability density function ¢($)1 for
70
for
~ ~
o.
(3)
The probability distribution of demand is the same from
period to period, and successive periods are statistically independent.
~Regarding
the first ·a.ssumption of no lag,there .are· cases' in .which
the delivery lag is very small compared to the period of production, or
to the period over which demands on the product is being received.
In such cases we can neglect the effect of lag.
In other cases how-
ever, we cannot omit the effect of the time lag without forcing on
the model serious oversimplifications.
However, by neglecting the
time lage in delivery we simplify the mathematical procedures, and
our results do provide, in an approximate way, some insight into the
properties of even those processes for which the lag is appreciable.
We are going to consider the effect of lag, in greater detail, later
in this work.
The assumption that the distribution of demand has a continuous density ¢(~) is introduced to simplify the exposition.
All the
results can be shown to hold for the case in which the demand is discrete and all functions are defined on integer values.
These are the
same assumptions made in the study of the one-dimentional case given
by Arrow, Harris and Marscbak
["1_7,
see also Arrow ["2, Cahpter
9_7
and Bellman ["3_7.
We want to determine the optimal ordering policy of the
dynamic model at each stage.
This optimal policy will minimize some
average function of the overall cost of the process, which is determined by the various costs and the uncertain demand.
Some reasonably
realistic restrictions on the cost functions are going to be made so
71
as to make the optimal policies as simple in form as possible. This
is of utmost importance in practical applications.
Furthermore it
turns out that such mild restrictions as we shall make render the
mathematics of the problem analyt:i,cally manageable.
In our proofs we
are going to use Bel1.ma.n' s principle of optimality, which is:
"An
optimal policy has the property that, whatever the initial state and
initial decision are, the remaipingdecisions must constitute an
optimal policy with regard to tbe state resulting from the first
decision. II Bellman
£3,
p. 83_7 ~
We are going to use the functional equation approach to determine the structure of the optimal policy.
Bellman, Glicksberg and Gross
the structure of
~he
£4_7,
This approach is due to
where it is uaGd in detellltliJlling
optimal policy when we are concerned only with
one product, and this method is used also in Arrow 12, Chapter
9_7.
By means of the symmetries and recursive properties of the dynamic
model, an analytic expression of tbe recursive situation will be found.
The functional equation:
If Xl is the initial stock of A, and x2 the initial stock of
B, and we ordered the amount y1 - Xl of A and y2 - x2 of B in the
initial period, then the expected total loss for the initial period is
72
hl (Yl-m1n( S,Y2
»
¢(~)d~ I'D ~(Y2-min(~'Yl»¢(~)d~
+
+
f
s=o
.a
.P(~-Yl'~-Y2)¢(~)d~
s=min(.Yl 'Y2)
,
as we have proved in Chapter I.
Suppose we are interested in an infinite (unbounded) member
of periods.
Let f( x ,x ) = the minimum discounted expected loss. wbich
l 2
will be incurred during an infinite time period, starting with initial
stocks xl of A, and x
infinite time period.
2
of B, using an optimal policy throughout this
If we order at tbe first period the amounts
Yl - xl of A and Y2 - x2 of B, and then follow an optimal policy from
the second period on, then the expected discounted loss from the
second period on, discounted to tbe present value is as follows:
Yl
(2)
+
J f(Yi~'Y2-~)¢(~)dSl
£=0
,
73
e
co
(3)
" [ f(YI-Y2'O)
f
¢(g)dg
+t
f(Yl- g'Y2- g)¢( S)dS]
,
s=o
s=Y2
where a is the discount factor, which is introduced to prevent infinite costs from entering, and for which there exist economic justifications.
When we order stocks and then store them, we are laying
out money that we could have otherwise invested giving us a return
in the future.
We have assumed that if demand exceeds supply of any
of A or B in one period, then the stock level of this product is zero
at the start of the next period.
Suppose we are only interested in a finite number of periods
of production, n, say.
Let us define f (Xl'X ) as the minimum disn
2
counted expected loss for the n periods starting with an initial
supply Xl of A and x of B, and suppose we are using an optimal policy
2
throughout. If we order at the first period y1 - Xl of A and y2 - x
2
of B, and then follow the optimal policy from the second period on,
then the expected discounted loss from the second period on discounted
to the present value as follows:
J
Yl
00
(4)
" [fn-I(O'Y2- YI)
¢(s)ds +
S=Yl
s=o
J
Y2
CIl)
(5)
" [fn_ I (YI -Y2'O)
J
g=Y2
¢(s)ds +
f
g=o
fn-I(YI -S,Y2-s)¢(s)df]
fn-I(YI -S,Y2-s)¢(s)df]
·I.et us now consider the infinite :period case.
Sup:pose that the optimal :policy calls for Y2 > Yl ; then the optimal
policy must be such that f(Xl ,x ) satisfies the functional equation.
2
(6)
f
Yl
¢(.ldg +
f(Y1 "Y2-. l ¢(,ldg]
J
¢
.
where, in this case,
Yl
(Y ,Y ) = Cl(Yl-Xl ) + C2(y2-x2) + \
xl ,x2 l 2
)
EL
Yl
+
f
hl(Yl-s)¢(s)ds
s=o
ro
":!(Y2-· l ¢(')d' + ":!(Y2-Y1lJ
s=o
Yl
Yl
p(.-y1l¢(')d' - {
.¢(.ldg
o
Similarly, for the n period of production case, we have
¢n>d~
75
(8)
and we give similar expressions if the optimal policy calls for Y1'? Y2'
To simp1if'y subsequent work let us denote the expression between
brackets in (6) by T(Yl'Y2; X1"x2 ,f).
Then equation (6) becomes
and equation (8) becomes
(10)
A function f(·) which satisfies (6), for example, is called a solution
of
(6).
We then prove:
Theorem:
There exists a unique bounded solution to (6), and this
solution·· is continuous.
If we define the sequence
f
n
as
follows:
,
n
= 0,
1, 2, ••• ,
76
then f(xl "x2 )
=
lim
f (Xl"X2 ) exists for 0 ~xl" 0 ~ x2 "
n->oo n
and is a solution of
•
Proof:
Let fo(~"X2) be any non-negative continuous function in the
two variables defined over (0 ~ xl and 0 ~ x2) • For each n ? 1" let
Y2n = Y2n(Xl"~) be values for which T(y1 "y2 ; xl "x2 '
f n) attains the minimum. Since f l (X ,x2 ) is continuous" by our assumpl
tions" we see inductively that each element of the sequence
Yln
I
f ("J.'''2) J
f
.
= Yln(Xl"~)"
n
is continuous.
It is easy to see that
.
J
therefore" for large enough Yl' Y2"
Given xl"x2 " and a large /). >T(Xl'~; Xl"~"fn)" there exists Yl "such
that
for all Y > Yl
,
and there exists Y2 " such that
for all Y > Y2
Therefore we can easily see that the minimum of T(Yl "Y2; xl ,x2 ,fn )
is attained within some finite region of the yl y2-plane" where
Y2? Y1 •
Now" eVidently"
77
and
fn =
T(Y1 ,n- 1'~2 ,n- 1; x1 ,x
. 2 ,fn- 1) -< T(YJ.n'~2 n; x1,x2 ,fn- 1)
Therefore
,
where
and
Hence
,
where
YJ.n
V1
=" f
Ifn(Yln-~'Y~)- fn-l(Y
ln -
g, Y2n - g) /¢(g)iIt +
o
If
<Jl)
"/fn (O,y2n-yln) - fn-l(O,y2n-yln)
¢(g)ag
YJ.n
and
Y1 ,n-1
V2 =
aJ . /fn(Y1,n-l- g'Y2,":~:- g) - f n_1(Y1,n-i g,y2,n:. l~g) .\i( g) dl
o
+
"'fn (O'Y2;~iYl,n-l) -
fn,a(O'Y2,';:i Yl,n-ij
If
00
¢(g)dg
Y1,n-1
Therefore
co
[
00
Thus the series
»,
Z (f"n+l(Xl 'X2 ) - f n(Xl ,x2
n=o
any finite rectangle in the positive quadrant.
converges uniformly in
So f n (xl,xr:c ) converges
to f(Xl'~) for all Xl? 0, x2 ~ o. Since f n (xl ,x2 ) is continuous,
then f(X ,x2) is also continuous.
l
To prove uniqueness, let F(xl ,x2) be another solution for
Also
,
and so
,
79
where
and
Therefore
Max /Fn +l (Xl'x2 ) o~l~
fn+l(~,X2) / S o~<oo/Fn (Xl ,X2 ) - f n (Xl'X2 )
_.~
O~~
05:x2S00
I ,
and so
Thus we have shown that F(x ,x2 ) - f(x ,x2 ) is identically zero.
l
l
To show that the sequence ff (x ,x )
n l 2
let us assume that /fo (xl ,x2 )
I < M,
J
is uniformaly bounded,
for all xl'x2
?
O.
Then we have
that /fl (xl ,x2 ) / S T(Xl'~; x l ,X2 ,fo )
+ aM
-< ELxl ,x2 (xl,xl"\)
c;
co
<00
~ J [hl(.)+~(.)J
o
Therefore
¢(.)
d.j
o
p( .)¢(.)
d'
+ aM
80
r
00
sup
/f("J.'''e) I ~
G( E) ¢( E) dE + aM
o
h1(s) + h (s} + p(s).
2
inductive argument~ that
where G(s)
From this we see, by means of an
=
~ ELxl'x (x.. ,x2 )
2
00
+ a( (
J.
G( s)¢( e) dg + aM)
)
o
J
00
~
f
00
G(E)¢(E)dE +"
o
GW¢(E)dE + rPM
0
Hence
00
Ifn (X1,"e) I ~
(1 + " + ,,2 + '"
+ <>,,-1)
S
G( E)¢( E) d / +
o
00
~ ~--/S G(E)¢Wds + rl'M
o
r
00
and so if
o
G(E)¢(E)dE
< 00
then /fn (X1 ,"e)
I < 00.
rl'M
CHAPrER V
THE DYNAMI:C MODEL:
CASE OF Ul'JEAR ORDERING COWS.
Let us consider the case when Cl(Yl-Xl ) = cl.(Yl-Xl ) and
ciY2-X2) = c .(y -x ) from equations (6) and (l), we will then have,
2 2
2
(5.1)
=
min [ -clxl - c 2x 2 + G(Yl'Y2) ]
,
Yl~l
Y~
where
Yl
(5.2)
G( y l'Y2 ) = elYl +
e~2 +J
[hl(Yi E) + h2 (Y2- E) -
r.~
o
¢WdE
+joo[
P(E-Yl ) + h2(Y2-Y ) ... rylJ ¢(E)dE
l
Yl
Yl
¢( E) dE +
~
f(Yl- E,Y2- E)¢( E) dE
o
and we have assumed that Y2
Lemma 1:
f(X
l
Y '
l
+ 8x ,x ) - f(Xl'X ) ~ -c 8x
whenever
Proof:
~
l
eXl
2
2
1
l
and 8x are small positive numbers.
2
From equation (5. 1) we have
82
and hence
Thus
A similar argument will show that
~his
proves the lemma.
Theorem 1:
Proof:
In the region Yl~ Y2' G(yl ,y2) is strictly increasing
with Y2 •
From (5.2) we see that
Y1
0(Yl,y2+llY2)
= e1Y1
+
ei Y2+6Y2)
+J [~(Yl-~) ~(Y2+llY2-~)
+
o
-
r;~ J¢(~)d~
co
+1
[P(~-Yl)
+ h2(Y2+6Y2-Yl)
Y1
(l)
¢(~)d~ + Ctf(O'Y2+6Y2-Yl)f ¢(~)d~
- r Y1 ]
O
Y1
Y1
J f(Yl-~'Y2+llY2-~) ¢(t)d~
,
+ '"
o
and therefore
.Y1
= e26Y2
G(y1 ,y2 +lly2 ) - 0(y ,y2 )
1
+
f
[h2 (y2 +6;y2- t )
o
-
f
~(Y2-t)J ¢(~)dt +
co
[~(Y2+llY2-Yl)
Y1
- h2(Y~:rYl)J ¢(g)dg + d' [f(O,y2+8y2-y1)
00
-
f(O'Y2-Yl)~ ¢(~)d~ + a)i
Y1
Y1
0
[f(Y1- g'Y2+8Y2- g)-f(y1- g"Y2- g)J ¢( g) dg .
84
Now from lemma 1, we have
G( Yl'Y2<ay2) - G( yl'Y2)
f [~(
Y1
~ c28y2 +
Y2+8y2- g)-Ii:!( y2- g)J ¢( g) dg
o
fOO [~(Y2<aY2-Yl)-~(Y2-Yl)]¢(g)ag
+
Y1
Y1
co
-Ki[-C28Y2Jj
¢(g)dg +OJ
Y1
0
[-c28y2 J
Thus
Y1
G(yl ,y2 <ay2 ) - G(Yl ,y2 )
~J [h2(Y2+8y2-g)-~(Y2-g)J ¢(g)ag
o
co
+/
[~(Y2<aY2-Yl}-h2(Y2-Yl)J¢(g)dg
Y1
Therefore, since h (o) is an increasing function, and a
2
see that
< 1, we can
85
Similarly we can see that ify2
~
yl ' the corresponding
G(Yl'Y2) will be increasing with Yl in the region Y2 ~ Ylo We can
therefore conclude that the minimum of G(Y ,y2) is at some point where
l
Yl = Y2 ' Therefore, for sufficiently small xl' x2 (we will explain
later what we mean by sufficiently small), it follows from equation
(5.1) that
,
where
J .[hl(Y·~)
Y
(504)
G(Y)
= (c1+c2l:r +
+ h:a(y-t) • rot] ¢(t)dt
o
00
J
+
[p(~.y).
roy]
¢(~)d~ + a f(O,O)~
y
CD
Y
1
Y
+a
f(y·~,y·t) ¢(~)d!
o
Let us assume that for very small y, G(Y) is decreasing as y
increases.
We will later find conditions that will ensure the satis-
faction of this assumption.
It is undoubtedly a reasonable assump-
tion, for if it were not satisfied it would be more profitable not to
stock the commodities and to suffer the penalties incurred when we do
not satisfy the demands.
that
Then let y* be the largest value of y such
86
= min G(y)
G(Y*)
...
y,?o
Since G(y) tends to infinity as y tends to infinity, as we can see
from
(5.4),
and since G(y) is decreasing as y increases (When y is
very small), then we can see that 0 < y* <
00.
If Xl < y*, X < y*,
2
then we have from (5.3)
Therefore, if Xl <
'1*,
X'
2
< y*, then
,and
Thus we can see immediately from
(5.4)
that
d~~y)
exists for all y < y*,
and is given by
G'(Y)
= dG~) = 01
Y
+ °2
+J
[hi(Y-') + h2(Y-') +
"1 fi(y-.,y·.J
o
co
+
f
2(Y-"Y-') J ] ¢(.)a.<
•
J
[p'('.Yl + rJ ¢(')d'
y
Define a new function M(y) by the following
(5.6)
M(y)
•
•
37
..
It follows that for y < y*, G' (y) -is identical with M( y) • We would
also like to prove that Gi(y} , exists at y
= y*
and for this we will
need the Theorem 2 proved below.
A condition that ensures that G(y) is initially decreasing,
i.e. G'(O) <0, is
E
This is clear from
[p'(s)] >
(5.6)
c
l
+ c2 - r
by putting y ~ O.
(I)
Condition (I) is satisfied
if the marginal revenue r is larger than the sum of the marginal ordering costs c l and c2 ' and at the same time if the penalty cost p(.)
is an increasing fun~tion. If we do not consider the revenue factor
in our model, then the Condition (I) is satisfied if tbe marginal
expected penalty exceeds the sum of marginal ordering costs.
NOtice
that Condition (I) is the same one we introduced in our study of the
one-stage model.
Theorem 2:
Proof:
lim
y->
Qh')4G(Y*)
~
exists, and is given by M(y*)
y.::.y*-
We have that
y
G(Y)
~ (OJ.+02)Y +1
[hJ.
(y-~) + ~(y-~)
~ ¢(~) d I
- ro ]
o
00
+1
y
[p(~-y)
00
- ryJ
¢(~)d~ +Qf(O,O~ ¢(~)d~
y
88
f f(y-~,y-~) ¢(~)d~
y
... a
o
Therefore" if we put f(z"z)= f(z) for simplicity, we will have for
1[hl(Y-~)'" ~(H)
Y
G(y) - G(Y*)
= (C1+c2)(y-y*)
...
-
r~O ¢(~)d~
o
[p(~-y) -
! [p(~-Y*)
1¢(~)~ f ¢(~)d~
ry ]
00
¢<sld~ -
y-'*
¢(~)d~ ... af(O)
00
00
-
Y
y*
J :f(y-~) ¢(~)d~ -1- f(y*-~) ¢(~)~
Y*
y
... a
o
Thus
0
- ry* ]
Y*
G(Y)-G(Y*)=(01+02)(Y-Y*)+!O ~h1(y-e)-~(Y*-e)+h2(y-e)-~(Y*-e17¢(g)ag
Y
+fy*~~(y-g)+h2(Y.e)-re_7 ¢(e)a<
co
+)y ~p(g-y)·p(e-Y*)-r(y-Y*17¢(e)ae
ff
Y
+
p(s-Y* )-rY*_7¢(s )d~-O: f(O)]
Y*'
¢(g )dg
Y*
Ie r(y-e)¢(g )a<+0
Y-Y*
+a
Y
Y
J
Y-Y*
r(y-g)¢(g)ag-a
Y-Y*
r(y*-g)¢(, )ag
0
Y*
.. 0:(
f(y*-g)¢(g)dS ,
)Y-Y*
i
Y*
= (01+0 2)(y-y*)+ [h1(y-g )-h1(y*-g )+h2(y-e )-h2(y*-g17¢( g)a<
r
+ Jly*~h1(y-g)+~(Y-e)_rg_7 ¢(g)a<
1~
co
+
p(g-y)-p(e-Y*)-r(y-Y*17¢(e)a<
Y
Y
+ ( f:p(S .. Y*)-r~7¢(s)~"
)y*
.
y
0:
f(O)l ¢(s)dg
Y*
90
y-Y*
Y*
aJ ,[f(y-g)-f(Y*-s17¢(s)ds + af.
+
ff(Y-s)-f(Y*-s17¢(g)dg
Y-Y*
o
J f(Y-~ )¢(~)elf
Y
+ '" Y*
•
From (5.5), if xl =x2 =x < Y*, then f(x) = -(c l +c 2 )x + G(Y*). We
can also see that if Y-Y* ~ g ~ Y*, then Y-S < Y*, and if
-
- -
Y* -< s < y < 2'Y*, theny-g < Y* .-Thus, for y-Y* < S < y*
.
f(y-g)-f(y*-g)
.
,
we have
= -(c l +c 2 )(y-s)+G(Y*)+(c l +c 2 )(y*-g)-G(Y*)
= -(c l +c 2 )(y-Y*) •
Also, for Y*
~
g~ y ,
From the first mean value theorem of integrals we find
Y*
G(y)-G(Y*) = (c i +c 2)(y"Y*) + { fhi (Y-~)-hi(Y*-~ )+h2(Y-~ )-h2(Y*-~17¢(~)elf
00
+
J(, fp(~"y)-p(~-y*)-r(y-Y*17
¢(~)elf
y
+ f p(y"-Y* )-rY"3¢(y")fy-Y*] - '" G(Y*) fy* ¢( ~ )d~
91
Y*
y
- a(c 1+c 2 )(y-y*)l¢(s)ds + a G(Y*)
y-Y*
where
f..
¢(s)ds
Y*
yf, y", yl", and yTV are all in (Y*,y), and we have used
the fact that
functions.
h (.), h (.),
2
1
Therefore
+
p(.), eJ(.),
rh1(y-y' )+h2(y-y' )..
-
and
ryf
-
r(.)
7¢(y')
+ LP(ylf_y*) - ~7 ¢(yll)
f.
Y*
- a{c 1+c 2 )
¢(g)dg
y-Y*
+ (c +c )(y_y.fl) ¢(yl1')
1 2
+
r:v
r- f(y-y)
- f(Y*-y
rv ) 7¢(yrv ).
-
are continuous
92
Thus i f y decreases to Y*1 then y', y", y' I I , ylV tend to Y*,
and since h (.), h (.), p(.), ¢(.) are continuous and
l
2
p(o) = hl(O) = h2 (O) = 0, and also since f(x ,x )
is continuous,
l 2
it follows that
Q)
-- ( Lp'{s-Y*) + E7 ¢(s)ds
)y*
-
Y*
- 0(°1+°2)
J(. ¢(.)dg
We have therefore proved that the derivative of G(y)
right eXists at Y* and, comparing with
hand derivative equals M(Y*).
(5.6),
from the
we see this right-
A similar, but slightly easier argument
will dispose of the derivative to the left,
Thus GI(Y*) exists and
equals M(Y* ) •
Suppose that G'(Y*)
y*
< O. Then G(y) < G(Y*) for y > Y*, and
would not be the largest value of y
for which G(Y*)
= min
G(y).
y~o
This contradicts the definit10n of y*.
Gi(Y*) > O. Therefore G'(Y*)
can be used to find Y*
functions.
For x < Y*
= O.
Similarly we cannot have
The fact that G'(Y*) = M(Y*) = 0
numerically, since M(y)
involves only known
Therefore, for
Theorem 3: -
x
< Y*
For any y
> M(Y) •
D G(y) = 11m inf -G(y+h) - G(y)
+
J::i ... ~
h
-------
Proof:-
We see that
1
Y+oY
G(Y+5Y) - G(y)=(c l +c 2 )5Y +
fh (Y+Oy-S )+h (Y+5Y-S )-r17¢(g )ds
2
l
o
y
. .5. Lhl (y-g )+h2(y-S )-rs_7¢(s )ds
o
.
co
+
r
L-P(s-Y-5y)-r(Y+5y17 ¢(g)ds
~+5Y
co
- 5. .rp(~-y)
-
'77 ¢(~) dO
y
co
+
Ct
f(O,O)
f
J
¢(s )ds
y+oy
-1
co
¢(s )ds_7
Y
y+6y
J: f(Y+6Y-~. Y+6Y-~)¢(~)dO
+a
-J
Y
f(y-S,Y-S) ¢(s)dg •
o
Since h (.) and h (.)
l
2
are increasing and -positive, then
10
and'
Thus
Y
G(Y+5y)-G(y) ~ (C l +C 2)5Y +j(~hl(Y+8Y-S)-hl(Y-s)+h2(Y+5Y-S)
o
-h2(y-s17¢(s )ds
Y+oY
. f. r, ¢(,)o.;-1
Y
00
L r(y+oy)-p(,-Ny17¢(,)o.;
.y+8Y
-f[":p(s-y)-:rx7¢(s )ds ..
Y+8y
00
a f(O,O)
Y
.f
¢(S )ds
Y
Y
..
+aj[ Lf(Y+6Y-"Y+6y-,)-f(y-"y-,17 ¢(,)
+
0.;
y+5y
a
f(Y+oY-S,Y+5Y-s) ¢(S) ds •
f.
Y
By using the mean value theorem for integrals, and the fact that
Y
f
G(y+OY )-G(y) ~ (C l +c2 )5Y+ Lh (y+&y-~ )-h (Y-S)+h2 (Y+8Y-S )-h2(Y-~17¢(~ )a.g
l
l
o
co
oY~
-r Y'¢{y')
co
p(t-Y-Oy)¢(g)dg-j'r{Y+5y)¢{g)dg
Y+OY
y
S r(y+oy)¢(g)dg -1 LP{g-y)-rx7 ¢(g)dg
Y+Oyco
+
y y
f
y+oy
- '" G{Y*)
Y
¢(g)dg +
Y
"'1
ff{Y+5Y-g)-f{y-g)7¢{g)dg
0
1
Y+oY
+ '"
f-{C 1+C 2 ){Y+5Y-g) + G{Y*)7 ¢(g) dg •
•
Y
Therefore
Y
G(y+5y )-G(y)
~ (C l +C 2 )5y+ J-Lhl (y+5y-S )-hl (y-S )+h2(y+5y-S )-h2(y-~17¢(s )dg
o
co
-ry'¢{Y')8Y +j[ fp{g-y-8y)-p{g-y)-r817 ¢{g)dg
y
y
+ r. (y+oy)¢(y")5y+a
J L f(y+5y-~ )-f(y-g17¢(g )dg
o
where
yf, yll, yf f
f
are in the interval
(y 1 y+5y).
Therefore
G(y+oy )-G(y)
fY r hl (y+oy-g )-hl (y-S)
h2(-y+oy-S )-h2 (Y-S)
>
c
+c
+
.
0
+
0 y -7
o y
l 2
y
o
• ¢(g) ds
00
-j.
_ ryi ¢(y')
rp(s-y)-p(s-y-OY) +
-
0 y
r7 ~
-
y
• ¢( g )dg + r(y+oy) ¢(y")
y
- Ct
f
(C l +C 2 )¢(s)ds - a(c l +c 2 )(y+oy-yrt.)¢(y.rt) ..
o
Hence" since
since
y%" y". and yt"
all tend to y
oy ->
as
0 ,
and
¢(.) is continuous"
y
D+G(y)
~ c l +c 2 +
J fhl(y-s)
o
-1
2
+ h (y-sL7 ¢(S) ds
Lp·(s-y)+r.3 \I(s) d< - o(c 1+c 2 )
y
Thus
Theorem 4: -
\I(s) d< •
0
~ M(y)" for
D+G(y)
1
y
00
If M(Y)
all y.
has a unique ·zero at
ordering policy includes the rules: (a) If
Y=Y*, then the optim~l
xl ~
Y*1
and x2 ~
then order to raise the stocks of A and B up to the level
for both.
(b)
If
xl
= x2
Y*
Y*
> y*, do not order anything.
Proof:-
From assumption (I) Gt(O) < 0; also M(y)
y ~ y*.
Thus M(a)
= G'(Y)
for all
< 0, and M(Y*) = O. Since from. the assumptions
·
97
of the theorem M(y) has a unique zero" then M(Y) > 0 for y > Y* e
y > Y*.
Hence putting xl =x2=x'
in (5 •.3) we can see that the minimum is attained for y=x" if
By theorem.3"
D+G(Y) > 0
for
say>
x =X =x> y*. That is no ordering is called for if the initial stock
l 2
levels are equal, and exceed Y* •
-
Conditions which imply that M(y)
has a unique zero, are that
2
h (.), h (.)" and p(.) are convex" and hi(O)+h (O)+p'(O)+r > a(c l +c 2 ).
2
l
This is clear from differentiation of M(y) given by (5.6). Next
the case when either Xl < x and x > Y*; or Xl > x2 and
2
2
Xl > y*. We try to find the stock levels to which. we should raise
conside~
our stocks" in order to minimize the expected loss.
Notice that as
far as applications are concerned" the procedure mentioned above for
finding the minimizing stock levels" is of very little interest, since
for only a very few initial periods" if any" will Xl or X be greater
2
than y*.
Let us consider first the case when Xl < x
2
and x > y*.
2
We have seen before in theorem 1, that in the region Yl
is strictly increasing with Y2.
5 Y2'
G(Yl"Y2)
Therefore the minimum in (5.1) is
attained for some value Yl ~ Xl and Y2 = x 2 • Also" since G(Yl 'Y2)
is strictly
incre~sing
see that Y15 x "
2
Yl ~ Y2" then we can
Let us put equation (5.1) in the following form
with Yl' in the
re~ion
~
.
(5.10) f(x l ,x2 )= min f E L ,x}Yl'Y2)+ a f(0,y 2-yl
l
xl
(
»)
¢(s)dS
~~~
Y2 ~
~
x2
+a
~1
f(Y1-"Y2-')
¢(.)d< J
·0
where
i
Yl
+
Lh1 (y1-') + h2(Y2-' )-r •
.]¢(. )a.
o
~
+
J
LP('-Y1 )+h2(Y2-Y1 ) - r • Y1]¢(·)d< ·
Yl
Suppose that
Sl
is the largest :ralue of Y
l
that would make the
expression between brackets in (5.10) minimum for
Then, f6r
O~·
Xl ~ Sl' we have
Sl
Y
2
=X
2
> Y* •
< x2 and
~
(5.11)
f(x l ,x2 )
= E Lxl,X
2
(Sl'x2 ) + a f(O, x2-S 1 )
Sl
Sl
+).
f(Sl-"
x 2-.) ¢(.) af
o
Pub Xl=0
S ¢(s) ds
in (.5,11)" and we find
00
(5.12) f(O,x 2 )
= E LO,X
2
(Sl,x2 )+a f(O,x 2-S 1
)1
¢(g)dS +
Sl
Sl
+ oJ f(Sl-g,X2- g )¢(g)cjf
o
Put x1=Sl in (5.11), and we will have
00
(5.13) f(Sl'x2 )
=E tsl,~(~,x2)+a f(O,~-Sl)Jl
¢(g)cjf
1
f f(Sl-g,~-g)¢(g)
Sl
+
0<
cjf
·0
Then from (5.12) and (5.1;), we discover that
Hence
This result which is intuitively obvious, since if the initial stock
(B) is x2' we i~~diately order an amount
Sl of (A), at a cost of c1 per unit, i.e. With cost c 1S1 But by
of
(A) is zero, and of
o
lemma (I)
ThUS, frcm this result and (5.14),we can see that
100--
and hence,
where K(x )
2
is a function of x
2
only~
So far in our discussion we ha,ve avoided the assumption that
f(x ,X ) differentiable in each of its variables.
l 2
Such an assumption
of differentiability would be similar to the ones Bel1J:na.n
Arrow
f
~7 make throughout their theor4,es.
f'!!7
and
feel that this is a
vle
strong assumption, and might not be quite as general as they suppose.
\-le
can now see that following our previous procedures we could prove
their results for the one commodity case,without the assumption of
differentiability.
when
of/oX
l
exists
0 < xl < Sl' but at this point in our argument, however, we need
to assume that
of/oX2
where
We have been able to prove that
for
x
l
f(X ,x ) also possesses the partial derivative
l 2
x 2 > Y*, when Xl ~ x • Thus for
2
:5 x2
to find the minimum with respect to Yl'
pression between brackets in
wi th respect to
x2 > Y*, and 0
yl'
(5.10),
when Y2
= x2 '
:5
xl
:5 Sl'
of the ex-
we differentiate
we then have
J
<:Xl
oE
~1'X2 (Y1'X2 )!o Y1 + "[-K'(X2-Y1 )
¢(g )dg-K(x2- y1 )¢(y1 L7
Yl
[J \-°
Yl
+ "
1\
o
¢(g)dE + K(x2-y1 )¢(y1 L7
101
(Xl
=
a E L"i'X
2
(Y1,x2 )!a Y1 - a K'(X2- Y1 )
S.
¢(g) dg
Yl
Yl
- a c1
J
¢( g)ag ,
o
and on putting Y = 8 " the value at which the minimum is attained,
1
l
we will have
(Xl
- a Ki (x2 -8 1 )
y1 =Sl
S.
¢( g )dg
Sl
Sl
- a c1
J
¢(g) ag
=
O.
o
Therefore
From this" putting
x 2 - Sl
= "1",
aLl -
f
say" we have
(x2 - ,. L7
and we can at least in theory, solve this differential equation for
Kh).
Substituting this Value of K(,.)
in
(5.12), we find
102
K(~)
co
=
E La,X (S1'X2 )
2
f
+
aK(x2 -S1 )
Ji
¢(.J a.
51
51
+ a
L -°1(Sl-' J+K(x2-.17¢(.)
d<
o
and from this 'He can solve to find 5 1 - Notice that as far as
applications are concerned, the procedure mentioned above for
finding 51' is of very little interest, since for only very few
initial periods,
xl
or x
2
will be greater than y*.
If x15 x2, and x2 > Y*, th~n if Xl ~ 51 (given above)
we order only from A the amount 81 - Xl - 5imilarly if x2 5 Xl'
and Xl > y*, then if x2 :5 52 (defined similar to 51) we order only
Theorem 5:-
from B the amount 5 2-x2 -
CHAPTER VI
DYNAMIC MODEL HITH LAG
The time lag between the placing of an order and the delivery
is an important factor in the study of inventory problems.
time lag may be a constant or a random variable.
The
The effect of the
lag in delivery on the optimal policy is crucially dependent upon the
treatment of the excess demand.
We might assume that all excess de-
mands may be satisfied out of future inventories,; or we might assume
that all excess demands are either lost sales, or are being satisfied
by priority shipment (the last tv10 alternatives are essentially equivalent from the point of view of satisfying future demand).
of inventory control can be tackled in two different ways •
The problem
The first
way is by determining optimal ordering policies, (this is just what
we have been doing so far).
Generally, the problem of determining
optimal policies is exceedingly difficult.
The other way is to study
closely policies of "simple form", under var.:i. CUB conditions of cost,
supply, and demand (see Arrow-Harris-Marschak
Ch. 13, and Ch. 14).
117,
and Arrow f~7
The justifications for studying such "simple
Policies It are the follovling
(1)
Procedural simplicity
(2)
Many of these policies are being used in practice, so
that we have a large amount of accumulated data concerning their operation.
104
(3)
Simplicity of the evaluation of such policies, analytically and probabilistically.
In the present chapter we limit our study to a class of policie3
'.
of simple form that depend on two parameters, and seek to choose wi thin
this class the policy which minimizes the average long run costs.
By
specifying the class of simple policies, the study of the inventory
model turns out to be a stud¥ of certain associated stochastic processes.
We are going to consider a class of policies which are based
on the idea of maintaining a fixed level of stocks.
In our calcula-
tions concerning the stochastic processes which result from such
policies we shall assume the existence of "limiting" or "steady-state"
(xlO ' x20 ), (xll ' x2l ), (x12, x22 ), ••• , denote the
stocks of A and B, at the beginning of successive time periods. Let
conditions.
Let
H(xlm, X2m ) denote the expected costs conditional on
the
(m+l)-st period:
H(X , x
1m
2m
n
->
0:>
We shall suppose that
H{xlO 'X20 ) + H(xll,x2l ) + ••• + H{Xln,Xan )
=~
n+l
eXists, where P signifies the policy followed.
P in such a way as to minimize
policies P.
lm
) depends on the demand distribution,
the cost functions, and the given policy.
(6.1) 11m
(X ' X2m ) for
~
Our aim is to choose
wi th respect to the class of simple
It is reasonable to suppose that the sequence
(X
2n )
converges in distribution function to a limiting vector random variable
ln
' x
(zl,z2)' whose distribution is independent of the initial amount on
hand (xlO ' x ), and depends only on the policy followed.
20
can write
Thus we
105
(6.2)
We can see that
H(Zl,Z2).
(Zl,Z2) can be determined independently of
After determining
(Zl,Z2) ~ ~
can be explicitly calcu-
lated for a very general class of cost functions.
distribution of
The fact that the
(zl' z2) is independent of the cost functions is an
attractive feature of this type of policy.
We are going to study in this chapter a simple model similar to
the one first treated by T. M. 1'lhitin, and J. H. T. Youngs
then extended in Arrow
15.7 Chapter
14.
117,
H~ will assume the probability
distribution of demand for each period is known j'(g).
The policy
followed is to maintain the stock size of A at a fixed level
and of B at
8 , where we will choose
2
mize the expected costs.
as fo110101S:
and
8
1
and 8
2
8 ,
1
so as to mini-
The ordering policy is being characterized
vlhenever a demand comes in, an order is immediately placed"
on both of the commodities
A and B, equal to the demand.
Hhen a de-
mand or :P8.rt of it cannot be satisfied immediately we assume that it can
be satisfied later on, from deliveries coming in, but only after all
previous unsatisfied demands are fulfilled.
He will assume that
supplies arrive always at the start of each period.
Let us assume that the lag time in delivering ordered goods of A
is
!
periods, and for goods of B is
.E
periods.
This means that
goods of A ordered in the i-th period are delivered at the beginning
of the (i+a)-th period, and goods of B ordered in the i-th period
are delivered at the beginning of the (i+b)-th period.
no loss of generality that
b > a •
We assume with
10,6
Let
Y
be the amount of stocks available of A at the end
lm
of the m-th period, and Y
2m
the amount of stock available of
at the end of the m-th period.
mand on A in the
m~th
Also let
period, and
on B in the m-th period.
d
2ni
d
lm
B
be the amount of de-
be the amount of demand
Since, as before, we assume that demand
in any period is equal on both A andB, then
d
1m
= d2m = dm
j
say, for all m
= 1,2, •••
Therefore we can see that
m
Y2m = 8 2 - E d. +
i=l J.
A
Hence
m
In
where
zlm = E
d , and z2m = E
d
i
m-a+l
m-b+l i
We have the understanding that, if m < a, then the sum of the
d
from
1 to
sum of the
di 's
i
IS
Y
lm
or Y
2m
~-a
from
is taken to be zero; also, ifm < b
1
to rn-b
is taken to be zero.
thenthe
If any of
is negative this would simply mean that this negative
value is the amount of demand t:b.at needs to be satisfied, and a positive value of Y
1m
or Y
means the amount of stock actually on
2m
107
hand that requires handling and storage.
The demand in anyone period
is independent of demand in any other period.
[d \ is
i .J
a sequence of independently distributed positive random variables; thus
the sequence
a
tdi ~
The sequence
represents a renewal process.
is the sum of
Zlm
independent identically distributed random variables, therefore its
probability distribution is given by the a-fold convolution of
denoted by
.Ia) ( • ) •
f<.),
Similarly the probability distribution of z2m
b
is given by the b-fold convolution of j(.), denoted by
)(.).
f
We notice that this is independent of m, so
and z2m
we
will put
zlm
= zl
'
= z2·
We shall consider only three cost factors in this study, and
•
they are:
(2)
(1)
h (.) the storage and handling of stock from Ai
l
the storage and handling of stock from B; (3) The
h (.)
2
penalty cost p(.).
We shall write Sl and S2 for the levels
of A and B which
'\ole
wish to maintain under our simple policy.
Let us first suppose that
H(Sl,S2)
b
= a.
Then
= z2 = z,
zl
(say).
If
is the expected loss for a typical period, a function of
the stock levels
8
1
(6.5) H(Sl,S2) =
and S2' and if we assume Sl < S2' we will have,
S
S
1 hl(Sl-z) aj.a)(z) + 1 h2(S2-z)~(a)(z)
J
r
Jo
o
+
+
~(S2-Sl)
r
Sl
J
00
d la)(z)
Sl
P(z-Sl) d ;fa)(z)
108
Hence
2
= f 1 h (8 -.) <1
2
j<a~.)
+ h (8 -8 )
2
2
1
o
S2
2
=)
Sl > S2' we have
~(81 -.)a/a)(.)
S2
+(
Jo
o
00
+
a)(.).
Sl
Similarly, if we assume
(6.6) H1 (8 1 ,8 )
r cl
h (8 1 -8 2
1
»)
<1
f
h (8 -.)aj(a)(.)
2 2
~
a
1'(.-82 )<1
)(.) + )
S2
f a )(.)
S2
Hence
~
S2
o H(.Sl,S2)
oS.
_
-
1
Bu~,
;r( )
hi(Sl-z)d l·a(z) + hi(Sl-S2)
~
00
d
P
( )
a (z)
S2
0
assuming as before ~hat h (_)
2
see that
is s~ric~ly increasing, we can
Therefore the values of Sl and S2
that minimize H(Sl,S2)
be such that Sl
~
S2-
But similarly if we
~ke
Sl
must
> S2" we can
:prove that
by ~king h (.)
l
strictly increasing.
mum of H(Sl"S2) is at some value
Hence we deduce tha~ the mini-
Sl=S2=S, say" and "We will have
109
If we assume as before that
h (.), h (.), and p(.)
l
2
creasing, then R(.) is convex.
1
are convex in-
Thus, to find the value of S that
minimizes R(S), we differentiate
d H(S)
d S
= 0, and solve for S.
R(S) with respect to S,
put
We can extend this model to the case when the time lags of delivery of both A and Bare equal, but a random variable.
this case the probability distribution of
z
In
is given by
(6.8)
where
Ai
units.
is the probability that the time lag in delivery is
The analysis is exactly as before, with j(z)
i
in place of
ta)(z).
Let us consider now the case when b > a.
In this case we will
have
where
z; is a positive random variable with probability distribution
J(b-a)(z;).
is equal to
'Ilhe stock on hand of A at the end of the
m-th period
110
51 - zl
if 52 - Z2 ~ 0
51
if
51 -(8 2-Z3 '
,
z3 > 52
if 8 - z <
2
2-
,
;
o•
The first and second cases of this equation are obvious.
for the third case is that, if 52 - z2
stops at some point when
52 = Z2
= zl
5 0,
+
The reason
then the production
z3.~
zl=8 2-z
3
and the stock of A will be 51 - zl = 81 - (8 2 - z3). We also note
that the stock on hand of B at the end of the m-th period is
,
Therefore
if 81 - z1> 0,
~81 in this case, since b > a •
2
Now the expected cost due to handling of stock from A in the
Notice that we have chosen 8
m-th period is equal to
=
\
111
e
rr
hl ( Sl-S2+ Z 3 ) dfa)( zl ) d _~b~a)( z3 )
S25 zl+ z3
+
rr
•
d
f
a
)(zl) d
f b- a )(z3)
S2 $ zl+ z3
Therefore
S
Sl b (Sl-zl)j<b-a)(S2-zl)df a )(zl)
l
(6.9)
II =
zl=o
S
'.
) 1 f b- a )(S2- Zl)d fa)(zl)
z =0
1
~
+ bl(Sl)
t
j
d Ib-a)(Z3)
zo;r.=S2
~
S
.J bl(Sl-S2+Z3)J:l-ta)(S2-Z~7 ~b-a)(Z3)
e
2
+
z3=S2- S1
f2
£1 -/a)(S2-z3t7d Ib-a)(z3)
Z3=S2- S1
The expected cost due to handling of stock from B in the m-th
period is equal to
I 2 = E£h2 (S2- z 2)ISl- z l
SJ'.
zl _ 6
=
~ g7+ ELh2 (S2- z3-S1)ISl-z l <g7
b2(S2-Zl-Z3)afa) (Zl)d!b-a) (Z3)
1
112
=
I
Therefore
113
The expected cost due to penalty in them-th pertod is equal to
•
ff
=
,
a
l'(Zl-Sl>afb-a)(z,>a.I >(Zl)
Sl-Zl <
0
S2-z2>
0
if
d i(b - a)(Z3)d i(a)(Zl)
Sl-Zl<
0
S2-Z2 ~
0
fJ
Sl-Zl ~
S2-Z2 <
+
1'(Z2-S2>~b-a>(z,><i ja>(Zl)
0
0
f!
Sl-Zl ~
S2-Z2 <
<i lb-a)(z,> <i ta>(Zl)
0
0
114
d
¢<b-a)(Z )
Therefore
(6.11)
3
d
¢<a)(Z )
-
1
Hence the expected loss for a typical period is
Thus to find the values of Sland 8
,
we differentiate
that would minimize
2
H(Sl,S2)'
H(Sl,S2) partially with respect to Sl' and also
with respect to S2' and solve the following two equations in Sl and
S2
'
(6.1')
°
=
and
(6.14)
=
°
It is impossible to write out an explicit analytic for.mulae for
the values of Sl and S2
although
which yields the minimum of H(Sl,S2)'
(6.1') and (6.14) are implicit repres~ntations of the
solution, for the case when b> a.
.
The situation is different
and much easier for the case when a = b.
hl(S-z)
= hi(8-z),
h2 (S-z)
= h2
• (S-z), and p(z-S)
from (6:0 the unique solution So
(6.15)
j(a)(So)
For example, if
=p
ddH~S)
of
+
~
+h
1
2
=
•
=p
• (z-S), then
0, is given by
116
BIBLIOGRAPHY
•