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Inventory model for an inventory system with time-varying demand rate
ARTICLE IN PRESS Int. J. Production Economics 122 (2009) 269–275
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Inventory model for an inventory system with time-varying demand rate Michinori Sakaguchi Faculty of Economic Sciences, Hiroshima Shudo University, 1-1, Ozuka-Higasi-Ichoume, Asaminami-ku, Hiroshima-shi 731-3195, Japan
a r t i c l e in fo
abstract
Available online 21 June 2009
The standard inventory problems of the multi-period have been modeled under different situations. Specifically we have considered the demand subjects of a continuous distribution and a discrete distribution, and whether the demand of each period is unchanged or not. A method to get an economic order quantity in inventory systems with discrete and unchanged demand was presented in a previous paper, and this method has been generalized to an inventory model with varying continuous demand. However, it was not achieved due to there being many classified cases in the general situations. In this article the above method is discussed in the case discrete demand to determine whether it increases or decreases from period to period. A theoretical method is presented by using previous results and some examples are given which suggest how the concept can handle on inventory system. In order to make the decision, an algorithm is also presented under some conditions, and examples are shown by using the computer software program, Mathematica, which helps to explain the findings. In general cases, we view the optimal policy in the inventory problems in only a few periods. & 2009 Elsevier B.V. All rights reserved.
Keywords: Inventory model Probabilistic demand Dynamic programming and economic order quantity
1. Introduction Probabilistic inventory models of the multi-period have been studied in which some conditions are searched to help obtain an optimal policy, providing that the total cost function of a single period is known extensively (Sakaguchi and Kodama, 2002). In those models, demand of each period is assumed to be unchanged for simplicity. These facts were applied to the case varying demand. However, it remains difficult to get the precise economic order quantity. Moreover, it was not achieved because there were many cases in the general situation. Therefore, the study initially researched the inventory model in the restricted case where demand decreased over time (Sakaguchi and Kodama, 2005). A model with exponential demand is researched in Sakaguchi (2007a) since it is easier when demand is subjected to an exponential
distribution. In this paper we will continue to develop the method stated above where the case demand is discrete. The results in Sakaguchi and Kodama (to appear) inspire us to consider some examples in Sakaguchi (2007b) and subsequently generate a lot of demanding calculations. The model investigated in this article is as follows. Let h and p be the holding cost and the shortage costs per unit per period, respectively, and let c be the purchasing cost per unit. Let us denote by z the amount on hand in initial period after a regular order is received and assume that demand in a single period be a discrete random variable. The decision criterion of single period is the minimization of the expected cost which includes the purchasing, holding and shortage costs. That is, the expectation EfCðb; zÞg of the total cost Cðb; zÞ is EfCðb; zÞg ¼ cðpurchasing quantityÞ þ hEfholding quantityg
E-mail address: [email protected] 0925-5273/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2009.06.003
þ pEfshortage quantityg.
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Our objective is to obtain a value z at which EfCðb; zÞg is minimized. Then, when the initial inventory quantity is x, the amount of the replenishment quantity is 0 or zx. The basic facts about our model are stated in Sakaguchi (2007a) and we refer the general notion of the inventory management to Silver et al. (1998). Let N be the number of periods in the inventory model. In order to analyze the structure of our inventory models, many kinds of the functions are used in this article. The fundamental ones are Hi ðzÞ and wi ðzÞ ðz ¼ 1; 2; . . . ; NÞ which are defined by the equations
p
fi ðbÞ gðxÞ GðyÞ
a
k¼
shortage cost per unit per period with cop probabilistic density function of demand quantity b at period i ð1 i NÞ function which has a continuous derivative on [0, 1] with gð0Þ ¼ 0; gð1Þ ¼ 1 and g 0 ðxÞ40 ð0 x 1Þ Ry 0 gðxÞ dx, yð0 y 1Þ discount factor ð0oao1Þ
pc hþp
Suppose the following conditions hold:
EfCðbi ; zÞg ¼ cx þ Hi ðzÞ,
An order is instantaneously replaced without the fixed
DEfCðbi ; zÞg ¼ c p þ ðh þ pÞwi ðzÞ,
Each replenishment is made at the beginning of each
where bi is a random variable of demand quantity in the ith period. Two conditions: D1 and D2 on the functions Hi ðzÞ are assumed to get the fundamental Theorem 4 and it leads us to an algorithm of seeking an optimal policy. Two distributions, a uniform distribution and a Poisson distribution, are considered in this paper as making inventory examples. The statue of time-varying demand materializes by changing a parameter of distribution. We denote by DEC a decreasing demand from period to period and denote by INC an increasing one. Though we deal only with the case demand subjects to a discrete distribution, the model is built by using a continuous function gðxÞ that indicates occurrence of a demand quantity along time in one period. The functions win ðzÞ
ðn ¼ 1; 2; . . . ; N 1; i ¼ 1; 2; . . . ; N n þ 1Þ
are constructed inductively. Put
k¼
pc . hþp
In the case of DEC, a method, used to obtain the economic order quantity from period i through period i þ n 1 is achieved by solving an inequality win ðzÞ k þ
ac , hþp
where a is a discount factor. In the case of INC, it is too complicated and consequently we view only problems of a few periods. Finally, a lot of examples are shown that are computed by making use of the computer software Mathematica. 2. Inventory model The inventory models in this paper are the dynamic inventory ones in which demands are varying period by period.
2.1. Notation and assumptions Let us use notation as follows: t N c h
length of period number of periods purchasing cost per unit holding cost per unit per period
cost.
period. We often use a variable x which means the stock level before a regular order is taken at each period and a variable z presents the initial stock quantity after a regular order. Therefore a replenishment quantity is z x. Demand quantity b of ith period is subject to a discrete distribution with a probabilistic density function fi ðbÞ and they are independent of each other. Moreover if b is not an integer with b 0, then fi ðbÞ ¼ 0. Demand of ith period occurs according to the function gððT itÞ=tÞb at time Tðit T ði þ 1ÞtÞ. That is, the inventory level at time T is z gððT itÞ=tÞb.
2.2. Functions In order to get the optimal policy, the following functions will be in use: the sum of the expected total cost from period i to period i þ n 1 with the initial inventory level x at the time i before a regular order, provided an optimal policy is done at each opportunity Ii1 ðb; zÞ the average inventory quantity at period i when the initial stock after a regular order is z and the amount of demand in period i is b Ii2 ðb; zÞ the average shortage quantity at period i when the initial stock after a regular order is z and the amount of demand in period i is b C i ðb; zÞ the total cost of period i i
f n ðxÞ
The expectations of each function are given in Sakaguchi and Kodama (2005): 8 0 if z 0; > > >P z > > > ðz Gð1ÞbÞfi ðbÞ > > < b¼0 EfIi1 ðb; zÞg ¼ 1 P > > ðzg 1 ðz=bÞ > þ > > b¼zþ1 > > > : bGðg 1 ðz=bÞÞÞf ðbÞ if z 1: i 8 Gð1Þmi z > > > 1 < P f½Gð1Þ Gðg 1 ðz=bÞÞb EfIi2 ðb; zÞg ¼ b¼zþ1 > > > : z½1 g 1 ðz=bÞgfi ðbÞ
if z 0;
if z 1;
where mi is the mean of fi ðbÞ. As was stated earlier in the introduction our criterion in this model of a single period is to minimize the expectation of total cost.
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271
Now EfC i ðb; zÞg is written by EfC i ðb; zÞg ¼ cðz xÞ þ hEfIi1 ðb; zÞg þ pEfIi2 ðb; zÞg and the functions Hi ðzÞ ð1 i NÞ are defined by the equations EfC i ðb; zÞg ¼ cx þ Hi ðzÞ.
(1)
i
The function H ðzÞ is defined on the set of all integers Z. We denote by DHi ðzÞ the first difference Hi ðz þ 1Þ Hi ðzÞ of Hi ðzÞ and the second difference DHi ðz þ 1Þ DHi ðzÞ of Hi ðzÞ by D2 Hi ðzÞ. We abbreviate by Df ðz; bÞ the difference f ðz þ 1; bÞ f ðz; bÞ of a function f ðz; bÞ. Let introduce functions wi ðzÞ ð1 i NÞ as follows: wi ðzÞ ¼
z X
fi ðbÞ þ
b¼0
1 X
Dðzg 1 ðz=bÞ bGðg 1 ðz=bÞÞfi ðbÞ
b¼zþ1
(2) Then it follows Proposition 1. (Sakaguchi, 2007a). Let z be an integer. Then ( cp if zo0; i DH ðzÞ ¼ c p þ ðh þ pÞwi ðzÞ if z 0:
Fig. 1. Inventory level in Example 1.
the amount of demand in a single period. The situation is seen in Fig. 1. Since ! z X D fi ðbÞ bGðg 1 ðz=bÞÞ ¼ Dðz2 =ð2bÞÞ ¼ ð2z þ 1Þ=ð2bÞ. b¼0
It follows from (2) that wi ðzÞ ¼
z X
fi ðbÞ þ
b¼0
1 2z þ 1 X fi ðbÞ . 2 b¼zþ1 b
(5)
Demand with uniform distribution: Let fi ðbÞ be as in Section 3.1. Then it is shown by (5) that
3. Distribution of demand In this article inventory examples are considered in the case when demand of its distribution is a uniform distribution or a Poisson distribution.
wi ðzÞ ¼
8 > > <
Li 1 2z þ 1 P 1 zþ1þ Li þ 1 z b¼zþ1 b
! if z ¼ 0; 1; . . . ; Li 1;
> > :1
if z ¼ Li ; Li þ 1; . . . :
(6)
3.1. Uniform distribution Let L1 ; L2 ; . . . ; LN be a sequence of natural numbers and let fi ðbÞ be a probabilistic density function as follows: 8 < 1 if b ¼ 0; 1; . . . ; Li ; (3) fi ðbÞ ¼ Li þ 1 : 0 if otherwise:
Let l1 ; l2 ; . . . ; lN be a sequence of positive numbers. Let fi ðbÞ ð1 i NÞ be the probability density function of a Poisson distribution with mean li . That is,
fi ðbÞ ¼ e
lbi b!
if b ¼ 0; 1; . . . .
wi ðzÞ ¼
z X
eli
b¼0
(4)
4. Inventory example An example of our inventory model is given if a demand pattern function gðxÞ and probabilistic density functions fi ðbÞ ð1 i NÞ are presented. 4.1. Example 1 Let gðxÞ ¼ x. If so, it is simplest with g 1 ðxÞ ¼ x and GðyÞ ¼ 12y2 . The inventory level in a single period is given by the function hðTÞ ¼ z ðb=tÞT this is a line, where b is
lbi b!
þ
1 2z þ 1 X lb eli i . bb! 2 b¼zþ1
(7)
4.2. Example 2 Set gðxÞ ¼ Since
3.2. Poisson distribution
li
Demand with Poisson distribution: Let fi ðbÞ be as in Section 3.2. Then by (5)
pffiffiffi pffiffiffi x. Then g 1 ðxÞ ¼ x2 and GðyÞ ¼ 23y y.
Dðzg 1 ðz=bÞ bGðg 1 ðz=bÞÞÞ ¼ Dðz3 =ð3b2 ÞÞ ¼
3z2 þ 3z þ 1 3b
2
,
it follows from (2) that for an integer z ðz 0Þ z X
1 3z2 þ 3z þ 1 X fi ðbÞ . (8) 2 3 b¼0 b¼zþ1 b pffiffi pffiffiffi The inventory level at time T is z ðb= t Þ T which is a convex function (cf. Fig. 2). Demand with uniform distribution: Let demand of ith period be subject to uniform distribution. Then by (8)
wi ðzÞ ¼
wi ðzÞ ¼
8 > > <
fi ðbÞ þ
1 Li þ1
z þ 1 þ 3z
> > :
2
þ3zþ1 3
1 if z ¼ 0; 1; . . . ; Li 1
if z ¼ Li ; Liþ1 ; . . .
Li i P b¼zþ1
! 1 b2
(9)
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Fig. 2. Inventory level in Example 2.
Fig. 3. Inventory level in Example 3.
Demand with Poisson distribution: Let fi ðbÞ be as in Section 3.2. Then it follows from (8) that wi ðzÞ ¼
z X
eli
b¼0
lbi b!
þ
1 3z2 þ 3z þ 1 X lb eli 2i . 3 b b! b¼zþ1
(10)
4.3. Example 3 Let a function gðxÞ be gðxÞ ¼ x2 . Then g 1 ðxÞ ¼ GðyÞ ¼ y3 =3. Thus
pffiffiffi x and
and so by (2) z X b¼0
fi ðbÞ þ
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 2ððz þ 1Þ z þ 1 z zÞ X fi ðbÞ pffiffiffi 3 b b¼zþ1
5. Optimal policy The optimal policy in the dynamic inventory system of period N is obtained if we are able to decide an integer number x¯ 1N. Let us indicate this procedure. 5.1. Formulation of inventory system
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 Dðzg 1 ðz=bÞ bGðg 1 ðz=bÞÞÞ ¼ pffiffiffi ððz þ 1Þ z þ 1 z zÞ, 3 b
wi ðzÞ ¼
Then li lj if and only if wi ðzÞ wj ðzÞ for all integers z ðz 0Þ.
(11)
Using the theory of dynamic programming let us i introduce functions f n ðxÞ which are stated at Section 2.2. Suppose that n ¼ 1; 2; . . . ; N;
i ¼ 1; 2; . . . ; N n þ 1.
Let x be an integer. Set
for an integer z ðz 0Þ. The inventory level is wriiten by the function z ðb=t 2 ÞT 2 which is concave in Fig. 3. Demand with uniform distribution: Assume that demand of ith period subjects to a uniform distribution. Then by (11) 8 ! Li pffiffiffiffiffiffiffi pffiffi > > ððzþ1Þ zþ1z zÞ Pi p1ffiffi < 1 z þ 1 þ 3 b wi ðzÞ ¼ Li þ1 b¼zþ1 > > : 1 if z ¼ 0; 1; . . . ; Li 1 (12) if z ¼ Li ; Liþ1 ; . . . Demand with Poisson distribution: Let fi ðbÞ be a probabilistic density function of a Poisson distribution with the parameter li. Then by (11) pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 z X lb 2ððz þ 1Þ z þ 1 z zÞ X lb eli i þ eli pffiffiffii . wi ðzÞ ¼ b! 3 bb! b¼0 b¼zþ1 (13)
i f 1 ðxÞ
¼ min fcx þ Hi ðzÞg,
(14)
zx;x2Z
( i
f n ðxÞ ¼ min
zx;x2Z
cx þ Hi ðzÞ þ a
1 X
) iþ1
f n1 ðz bÞfi ðbÞ .
(15)
b¼0
In order to study the structure put the function F in1 ðzÞ as follows: F in1 ðzÞ ¼ Hi ðzÞ þ a
1 X
iþ1
f n1 ðz bÞfi ðbÞ,
(16)
b¼0 i
where f 0 ðxÞ ¼ 0. Then it is shown by (15) that i
f n ðxÞ ¼ min fcx þ F in1 ðzÞg zx;x2Z
(17)
and these equations allow us to use mathematical induction. 5.2. Fundamental analysis
The following propositions show fundamental fact with respect to the relations among functions wi ðzÞ ð1 i NÞ.
If we add the following conditions D1 and D2 on Hi ðzÞ, then the properties of the functions F in ðzÞ can be obtained.
Proposition 2. (Sakaguchi, 2007b). Assume that demand is subject to a uniform distribution in the example as above. Then Li Lj if and only if wi ðzÞ wj ðzÞ for all integers z ðz 0Þ.
D1
There is an integer R such that DHi ðRÞ ac for all i ð1 i NÞ.
D2
If z is an integer, then D2 Hi ðzÞ 0 for all i ð1 i NÞ.
Proposition 3. (Sakaguchi, 2007b). Suppose that demand is subject to a Poisson distribution in the example as above.
Theorem 4. (Sakaguchi, 2007a). Assume D1 and D2 hold. Let n ð1 n NÞ and i ð1 i N n þ 1Þ be integers. Then
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there is an integer x¯ in such that
Proposition 6. If l1 l2 lN , then the inventory system with Poisson distribution ffi ðbÞg at the three examples in Section 4 satisfies the condition DEC.
x¯ in ¼ minfz 2 ZjDF in1 ðzÞ 0g. Moreover let x be the initial inventory quantity at period i. Then the optimal policy at period i considering the expected total cost from period i through period i þ n 1 is as follows: If xo¯xin , then order x¯ in x and if x x¯ in , then do not order. One is able to see 0 x¯ in R. By Proposition 1 D1 and D2 are rewritten are:
All examples in Section 4 satisfy conditions D1 and D2. For each i ð1 i NÞ, it is able to find an integer x¯ i1 to solve the inequality DF i0 ðzÞ 0, that is, DHi ðzÞ 0. Precisely said (18)
This inequality is equivalent to an inequality wi ðzÞ k by Proposition 1. Therefore x¯ i1 ¼ minfz 2 Zjwi ðzÞ kg.
6.2. Algorithm An algorithm is given under the condition DEC. On account of this, let us introduce the functions win ðzÞ
ði ¼ 1; 2; . . . ; N; n ¼ 1; 2; . . . ; N i þ 1Þ
inductively as follows: For z x¯ iþ1 put n z X ac wiþ1 ðbÞ k fi ðz bÞ, win ðzÞ ¼ wi ðzÞ þ a n1 hþp iþ1
D1 There is an integer R such that wi ðRÞ k þ ac=ðh þ pÞ for all i ð1 i NÞ. D2 If z is an integer, then Dwi ðzÞ 0 for all ið1 i NÞ.
x¯ i1 ¼ minfz 2 ZjDHi ðzÞ 0g.
273
(19)
Let x be the initial inventory level before a regular order is placed. By virtue of Theorem 4 the decision of the inventory problem which is one of single period i is as follows: If xo¯xi1 , then order x¯ i1 x, and if x x¯ , then do not order. In the inventory problem of period N the solution is obtained by finding an integer x¯ 1N and it is carried out by considering the inequality DF 1N1 ðzÞ 0. 6. The case decreasing demand If we find an integer x¯ 1N , then the inventory problem can be solved. Theorem 4 exhibits to us the way how to get it. However, in general it is very difficult to seek x¯ 1N as there are a lot of cases when it has to be considered. 6.1. Condition DEC In Sakaguchi (2007b) an algorithm is presented where demand is decreasing. That is a generalization of the research in Sakaguchi and Kodama (2002) when demand distribution is static. The condition DEC is designated during this state. DEC: If z is an integer with 0 z R, then
DH1 ðzÞ DH2 ðzÞ DHN ðzÞ. It is also written by Proposition 1. DEC: If z is an integer with 0 z R, then w1 ðzÞ w2 ðzÞ wN ðzÞ. Propositions 2 and 3 yield the following propositions. Proposition 5. If L1 L2 LN , then the inventory system with uniform distribution ffi ðbÞg at the three examples in Section 4 satisfies the condition DEC.
b¼¯xn
(20) where wi0 ðzÞ ¼ wi ðzÞ þ ac=ðh þ pÞ. Now we have the following Proposition 7. Let i be an integer with 1 i N and n an integer with 1 n N i þ 1. If conditions D1, D2 and DEC hold, then ( c p ac þ ðh þ pÞwi ðzÞ if 0 zo¯xiþ1 n ; DF in ðzÞ ¼ iþ1 i c p ac þ ðh þ pÞwn ðzÞ if x¯ n z: The following algorithm follows from this proposition. Algorithm. Assume the conditions D1, D2 and DEC hold. Then Step 1: Find an integer R with w1 ðRÞ k þ ac=ðh þ pÞ. Step 2: Let n ¼ 1. that is minimal among an Step 3: Find an integer x¯ Nnþ1 n ðzÞ k þ ac=ðh þ pÞ. integer z with wNnþ1 n1 Step 4: If noN; then n :¼ n þ 1 and go to Step 3. If n ¼ N, then we stop. 6.3. Numerical illustration of DEC We illustrate the use of this algorithm on three types of examples in Section 4. Calculations are done by using a computer software Mathematica. Let N ¼ 3;
c ¼ 100;
p ¼ 200;
h ¼ 5;
a ¼ 0:975.
Then
k¼
pc 0:487805 hþp
and
kþ
ac 0:963415. hþp
First in order to get R solve the inequality 1
w ðzÞ k þ ac=ðh þ pÞ. Next solve the inequality w30 ðzÞ k þ ac=ðh þ pÞ;
w3 ðzÞ k
x¯ 31.
and find an integer Then the function w21 ðzÞ is captured and consider the inequality w21 ðzÞ k þ ac=ðh þ pÞ to obtain x¯ 22 . Finally we may find an integer x¯ 13 by w12 ðzÞ k þ ac=ðh þ pÞ.
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Hence:
Table 1 Uniform demand with DEC. Type
R
x¯ 31
x¯ 22
x¯ 13
Example 1 Example 2 Example 3
223 243 196
36 57 16
119 143 91
153 177 161
Find the minimal integer x¯ 21 with w2 ðzÞ k. If w1 ð¯x21 1Þ k þ ac=ðh þ pÞ, then x¯ 12 is the minimal
integer with w1 ðx¯ 12 Þ k þ ac=ðh þ pÞ. If w1 ð¯x21 1Þok þ ac=ðh þ pÞ, then x¯ 12 is the mimimal integer such that w11 ð¯x12 Þ k þ ac=ðh þ pÞ.
7.2. Numerical illustration of the 2-period model
Table 2 Poisson demand with DEC. Type
R
x¯ 31
x¯ 22
x¯ 13
Example 1 Example 2 Example 3
292 302 279
97 139 47
244 253 232
292 302 279
Keep some constants of 6.3 in this section. Set N ¼ 2;
c ¼ 100;
h ¼ 5;
a ¼ 0:975.
First we show procedures in the case DEC. Put L1 ¼ 300;
Uniform demand: Let L1 ¼ 300; L2 ¼ 250; L3 ¼ 200. It follows from Proposition 5 that a system of uniform distribution ffi ðbÞg satisfies the condition DEC. Therefore the solution x¯ 13 is obtained in Table 1 by using the algorithm. Poisson demand: Set l1 ¼ 300; l2 ¼ 250; l3 ¼ 200. Then this is the case in the state of DEC by Proposition 6 and there are solutions x¯ 13 in Table 2. It is amazed that x¯ 13 ¼ R.
p ¼ 200;
L2 ¼ 250
and
l1 ¼ 300;
l2 ¼ 250.
It is quite similar to Section 6.3. Uniform demand: Table 3 shows us the required number x¯ 12. Poisson demand: Table 4 gives us the number x¯ 12 in the problem with a Poisson distribution. In the case INC, two examples in Section 4.1 are shown by numerical analysis. Set e ¼ w1 ðx¯ 21 1Þ k ac=ðh þ pÞ. Uniform demand: Let us consider the inventory problem in Example 1 with a uniform demand. There are integers x¯ 12 in Table 5. Poisson demand: There are integers x¯ 12 in Table 6 that yield our decision in the case a Poisson demand.
7. Model of a few periods In Cheaitou et al. (2009) there are researches on the inventory system of 2-period. In this article the following observation is valid. When a parameter of demand distribution varies from period to period in our inventory model, it seems that there is a property of the functions wi ðzÞ ð1 i NÞ such that either an inequality wi ðzÞ wj ðzÞ holds for all integers z with z 0 or an inequality wi ðzÞ wj ðzÞ holds for all integers z with z 0. Because Propositions 5 and 6 suggest these facts.
7.1. 2-Period model
( ¼
c p ac þ ðh þ pÞw1 ðzÞ c p ac þ ðh þ
Let us consider three functions w1 ðzÞ, w2 ðzÞ and w3 ðzÞ that satisfy the following condition: If 0 z R, then wi ðzÞ wj ðzÞor wi ðzÞ wj ðzÞ for all i and j Then there are six cases of relations among w1 ðzÞ, w2 ðzÞ and w3 ðzÞ. One of them is the case DEC, and there are numerical analyses in Section 6.3. At present there is no way to deal with them together. Therefore it is too complicated and it needs
Table 3 Uniform demand with DEC.
Let N ¼ 2. Then there are two functions w1 ðzÞ and w2 ðzÞ. First assume that w1 ðzÞ w2 ðzÞ for all integers z ðz 0Þ. That is a condition DEC and it happens in the case a uniform demand with L1 L2 or in the case a Poisson demand with l1 l2 in Section 4. The algorithm in 6.2 works usefully to obtain x¯ 21 . On the contrary suppose that w1 ðzÞ w2 ðzÞ for all integers z ðz 0Þ. Denote by INC this condition. The algorithm as above does not work in this case. However Proposition 7 yields again a method to catch the number x¯ 21 . Because if N ¼ 2, then Proposition 7 hold without the condition DEC, that is
DF 11 ðzÞ
7.3. 3-Period model
pÞw11 ðzÞ
if 0 zo¯x21 ; if
x¯ 21
z:
(21)
Type
R
x¯ 21
x¯ 12
Example 1 Example 2 Example 3
223 243 196
54 85 24
117 146 111
Table 4 Poisson demand with DEC. Type
R
x¯ 21
x¯ 12
Example 1 Example 2 Example 3
292 302 279
121 174 59
292 302 279
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Table 5 Uniform demand with INC.
275
Table 7 x¯ i1 .
L1
L2
x¯ 21
e
x¯ 12
x¯ 11
x¯ 21
x¯ 31
250 100
300 300
223 223
0.030 0.037
186 75
54
36
45
Table 8 x¯ i2 .
Table 6 Poisson demand with INC.
l1
l2
x¯ 21
e
x¯ 12
250 100
300 300
146 146
0.379 0.037
244 100
considerable work. Our basis of getting the number x¯ 13 is shown the following inequalities in Sakaguchi (2007a): 8 > DH1 ðzÞ ac if zo¯x22 ; > < z 1 P DF 2 ðzÞ ¼ DH1 ðzÞ ac þ a DF 21 ðbÞf1 ðz bÞ if z x¯ 22 > > : b¼x¯ 2
x¯ 12
x¯ 22
119
93
Observe x¯ 22 x¯ 31 . It is able to gain a value of x¯ 13 by considering the inequality w12 ðzÞ k þ
ac . hþp
In fact we have x¯ 13 ¼ 133.
2
References
and
DF 21 ðzÞ
¼
8 > DH2 ðzÞ ac > <
if zo¯x31 ; z P
DH2 ðzÞ ac þ a DF 30 ðbÞf2 ðz bÞ > > : b¼x¯ 3
if z x¯ 31 :
1
7.4. Numerical illustration of 3-period model Let us examine x¯ 13 in the case Example 1 with uniform demand. Keep N ¼ 3; c ¼ 100; p ¼ 200; h ¼ 5 and a ¼ 0:975. Let L1 ¼ 300; L2 ¼ 200 and L3 ¼ 250: This is the case w1 ðzÞ w2 ðzÞ and w2 ðzÞ w3 ðzÞ. Then the numbers x¯ i1 ði ¼ 1; 2; 3Þ are obtained in Table 7 by solving the inequalities wi ðzÞ k ði ¼ 1; 2; 3Þ. Since L1 L2 , it is a problem of 2-period with DEC to get the numbers x¯ 12 . It is also a problem of 2-period with INC to obtain x¯ 22 . This is due to L2 L3 . As in Section 7.1 we have x¯ i2 ði ¼ 1; 2Þ in Table 8.
Cheaitou, A., Delft, C.V., Dallery, Y., Jemel, Z., 2009. Two-period production planning and inventory control. International Journal of Production Economics 118, 118–130. Sakaguchi, M., Kodama, M., 2002. On the dynamic probabilistic inventory problems with piecewise cost functions which may not be piecewise smooth. In: Bulletin of Informatics and Cybernetics, vol. 34. Research Association of Statistical Sciences, Kyushu University, pp. 75–90. Sakaguchi, M., Kodama, M., 2005. Inventory models with a varying demand distribution. In: Proceedings of the International Workshop on Recent Advances in Stochastic Operations Research, Canmore, August 25–26, pp. 237–244. Sakaguchi, M., 2007a. On the probabilistic inventory models with timevarying demand, system sciences for economics and informatics. Monograph series of Institute for Advanced Studies, vol. 39, Hiroshima Shudo University, Kyushu University Press, Japan. Sakaguchi, M., 2007b. Optimal ordering for an inventory system with decreasing demand. In: Proceeding of Stochastic Operations Research II, Well On, Nagoya, Japan, pp. 251–258. Sakaguchi, M., Kodama, M., Sensitivity analysis of an economic ordering quantity for dynamic inventory models with discrete demand. International Journal of Manufacturing Technology and Management, to appear. Silver, A.E., Pyke, F.D., Peterson, R., 1998. Inventory Management and Production Planning and Scheduling, third ed.. Wiley, New York.
Contents lists available at ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Inventory model for an inventory system with time-varying demand rate Michinori Sakaguchi Faculty of Economic Sciences, Hiroshima Shudo University, 1-1, Ozuka-Higasi-Ichoume, Asaminami-ku, Hiroshima-shi 731-3195, Japan
a r t i c l e in fo
abstract
Available online 21 June 2009
The standard inventory problems of the multi-period have been modeled under different situations. Specifically we have considered the demand subjects of a continuous distribution and a discrete distribution, and whether the demand of each period is unchanged or not. A method to get an economic order quantity in inventory systems with discrete and unchanged demand was presented in a previous paper, and this method has been generalized to an inventory model with varying continuous demand. However, it was not achieved due to there being many classified cases in the general situations. In this article the above method is discussed in the case discrete demand to determine whether it increases or decreases from period to period. A theoretical method is presented by using previous results and some examples are given which suggest how the concept can handle on inventory system. In order to make the decision, an algorithm is also presented under some conditions, and examples are shown by using the computer software program, Mathematica, which helps to explain the findings. In general cases, we view the optimal policy in the inventory problems in only a few periods. & 2009 Elsevier B.V. All rights reserved.
Keywords: Inventory model Probabilistic demand Dynamic programming and economic order quantity
1. Introduction Probabilistic inventory models of the multi-period have been studied in which some conditions are searched to help obtain an optimal policy, providing that the total cost function of a single period is known extensively (Sakaguchi and Kodama, 2002). In those models, demand of each period is assumed to be unchanged for simplicity. These facts were applied to the case varying demand. However, it remains difficult to get the precise economic order quantity. Moreover, it was not achieved because there were many cases in the general situation. Therefore, the study initially researched the inventory model in the restricted case where demand decreased over time (Sakaguchi and Kodama, 2005). A model with exponential demand is researched in Sakaguchi (2007a) since it is easier when demand is subjected to an exponential
distribution. In this paper we will continue to develop the method stated above where the case demand is discrete. The results in Sakaguchi and Kodama (to appear) inspire us to consider some examples in Sakaguchi (2007b) and subsequently generate a lot of demanding calculations. The model investigated in this article is as follows. Let h and p be the holding cost and the shortage costs per unit per period, respectively, and let c be the purchasing cost per unit. Let us denote by z the amount on hand in initial period after a regular order is received and assume that demand in a single period be a discrete random variable. The decision criterion of single period is the minimization of the expected cost which includes the purchasing, holding and shortage costs. That is, the expectation EfCðb; zÞg of the total cost Cðb; zÞ is EfCðb; zÞg ¼ cðpurchasing quantityÞ þ hEfholding quantityg
E-mail address: [email protected] 0925-5273/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2009.06.003
þ pEfshortage quantityg.
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Our objective is to obtain a value z at which EfCðb; zÞg is minimized. Then, when the initial inventory quantity is x, the amount of the replenishment quantity is 0 or zx. The basic facts about our model are stated in Sakaguchi (2007a) and we refer the general notion of the inventory management to Silver et al. (1998). Let N be the number of periods in the inventory model. In order to analyze the structure of our inventory models, many kinds of the functions are used in this article. The fundamental ones are Hi ðzÞ and wi ðzÞ ðz ¼ 1; 2; . . . ; NÞ which are defined by the equations
p
fi ðbÞ gðxÞ GðyÞ
a
k¼
shortage cost per unit per period with cop probabilistic density function of demand quantity b at period i ð1 i NÞ function which has a continuous derivative on [0, 1] with gð0Þ ¼ 0; gð1Þ ¼ 1 and g 0 ðxÞ40 ð0 x 1Þ Ry 0 gðxÞ dx, yð0 y 1Þ discount factor ð0oao1Þ
pc hþp
Suppose the following conditions hold:
EfCðbi ; zÞg ¼ cx þ Hi ðzÞ,
An order is instantaneously replaced without the fixed
DEfCðbi ; zÞg ¼ c p þ ðh þ pÞwi ðzÞ,
Each replenishment is made at the beginning of each
where bi is a random variable of demand quantity in the ith period. Two conditions: D1 and D2 on the functions Hi ðzÞ are assumed to get the fundamental Theorem 4 and it leads us to an algorithm of seeking an optimal policy. Two distributions, a uniform distribution and a Poisson distribution, are considered in this paper as making inventory examples. The statue of time-varying demand materializes by changing a parameter of distribution. We denote by DEC a decreasing demand from period to period and denote by INC an increasing one. Though we deal only with the case demand subjects to a discrete distribution, the model is built by using a continuous function gðxÞ that indicates occurrence of a demand quantity along time in one period. The functions win ðzÞ
ðn ¼ 1; 2; . . . ; N 1; i ¼ 1; 2; . . . ; N n þ 1Þ
are constructed inductively. Put
k¼
pc . hþp
In the case of DEC, a method, used to obtain the economic order quantity from period i through period i þ n 1 is achieved by solving an inequality win ðzÞ k þ
ac , hþp
where a is a discount factor. In the case of INC, it is too complicated and consequently we view only problems of a few periods. Finally, a lot of examples are shown that are computed by making use of the computer software Mathematica. 2. Inventory model The inventory models in this paper are the dynamic inventory ones in which demands are varying period by period.
2.1. Notation and assumptions Let us use notation as follows: t N c h
length of period number of periods purchasing cost per unit holding cost per unit per period
cost.
period. We often use a variable x which means the stock level before a regular order is taken at each period and a variable z presents the initial stock quantity after a regular order. Therefore a replenishment quantity is z x. Demand quantity b of ith period is subject to a discrete distribution with a probabilistic density function fi ðbÞ and they are independent of each other. Moreover if b is not an integer with b 0, then fi ðbÞ ¼ 0. Demand of ith period occurs according to the function gððT itÞ=tÞb at time Tðit T ði þ 1ÞtÞ. That is, the inventory level at time T is z gððT itÞ=tÞb.
2.2. Functions In order to get the optimal policy, the following functions will be in use: the sum of the expected total cost from period i to period i þ n 1 with the initial inventory level x at the time i before a regular order, provided an optimal policy is done at each opportunity Ii1 ðb; zÞ the average inventory quantity at period i when the initial stock after a regular order is z and the amount of demand in period i is b Ii2 ðb; zÞ the average shortage quantity at period i when the initial stock after a regular order is z and the amount of demand in period i is b C i ðb; zÞ the total cost of period i i
f n ðxÞ
The expectations of each function are given in Sakaguchi and Kodama (2005): 8 0 if z 0; > > >P z > > > ðz Gð1ÞbÞfi ðbÞ > > < b¼0 EfIi1 ðb; zÞg ¼ 1 P > > ðzg 1 ðz=bÞ > þ > > b¼zþ1 > > > : bGðg 1 ðz=bÞÞÞf ðbÞ if z 1: i 8 Gð1Þmi z > > > 1 < P f½Gð1Þ Gðg 1 ðz=bÞÞb EfIi2 ðb; zÞg ¼ b¼zþ1 > > > : z½1 g 1 ðz=bÞgfi ðbÞ
if z 0;
if z 1;
where mi is the mean of fi ðbÞ. As was stated earlier in the introduction our criterion in this model of a single period is to minimize the expectation of total cost.
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271
Now EfC i ðb; zÞg is written by EfC i ðb; zÞg ¼ cðz xÞ þ hEfIi1 ðb; zÞg þ pEfIi2 ðb; zÞg and the functions Hi ðzÞ ð1 i NÞ are defined by the equations EfC i ðb; zÞg ¼ cx þ Hi ðzÞ.
(1)
i
The function H ðzÞ is defined on the set of all integers Z. We denote by DHi ðzÞ the first difference Hi ðz þ 1Þ Hi ðzÞ of Hi ðzÞ and the second difference DHi ðz þ 1Þ DHi ðzÞ of Hi ðzÞ by D2 Hi ðzÞ. We abbreviate by Df ðz; bÞ the difference f ðz þ 1; bÞ f ðz; bÞ of a function f ðz; bÞ. Let introduce functions wi ðzÞ ð1 i NÞ as follows: wi ðzÞ ¼
z X
fi ðbÞ þ
b¼0
1 X
Dðzg 1 ðz=bÞ bGðg 1 ðz=bÞÞfi ðbÞ
b¼zþ1
(2) Then it follows Proposition 1. (Sakaguchi, 2007a). Let z be an integer. Then ( cp if zo0; i DH ðzÞ ¼ c p þ ðh þ pÞwi ðzÞ if z 0:
Fig. 1. Inventory level in Example 1.
the amount of demand in a single period. The situation is seen in Fig. 1. Since ! z X D fi ðbÞ bGðg 1 ðz=bÞÞ ¼ Dðz2 =ð2bÞÞ ¼ ð2z þ 1Þ=ð2bÞ. b¼0
It follows from (2) that wi ðzÞ ¼
z X
fi ðbÞ þ
b¼0
1 2z þ 1 X fi ðbÞ . 2 b¼zþ1 b
(5)
Demand with uniform distribution: Let fi ðbÞ be as in Section 3.1. Then it is shown by (5) that
3. Distribution of demand In this article inventory examples are considered in the case when demand of its distribution is a uniform distribution or a Poisson distribution.
wi ðzÞ ¼
8 > > <
Li 1 2z þ 1 P 1 zþ1þ Li þ 1 z b¼zþ1 b
! if z ¼ 0; 1; . . . ; Li 1;
> > :1
if z ¼ Li ; Li þ 1; . . . :
(6)
3.1. Uniform distribution Let L1 ; L2 ; . . . ; LN be a sequence of natural numbers and let fi ðbÞ be a probabilistic density function as follows: 8 < 1 if b ¼ 0; 1; . . . ; Li ; (3) fi ðbÞ ¼ Li þ 1 : 0 if otherwise:
Let l1 ; l2 ; . . . ; lN be a sequence of positive numbers. Let fi ðbÞ ð1 i NÞ be the probability density function of a Poisson distribution with mean li . That is,
fi ðbÞ ¼ e
lbi b!
if b ¼ 0; 1; . . . .
wi ðzÞ ¼
z X
eli
b¼0
(4)
4. Inventory example An example of our inventory model is given if a demand pattern function gðxÞ and probabilistic density functions fi ðbÞ ð1 i NÞ are presented. 4.1. Example 1 Let gðxÞ ¼ x. If so, it is simplest with g 1 ðxÞ ¼ x and GðyÞ ¼ 12y2 . The inventory level in a single period is given by the function hðTÞ ¼ z ðb=tÞT this is a line, where b is
lbi b!
þ
1 2z þ 1 X lb eli i . bb! 2 b¼zþ1
(7)
4.2. Example 2 Set gðxÞ ¼ Since
3.2. Poisson distribution
li
Demand with Poisson distribution: Let fi ðbÞ be as in Section 3.2. Then by (5)
pffiffiffi pffiffiffi x. Then g 1 ðxÞ ¼ x2 and GðyÞ ¼ 23y y.
Dðzg 1 ðz=bÞ bGðg 1 ðz=bÞÞÞ ¼ Dðz3 =ð3b2 ÞÞ ¼
3z2 þ 3z þ 1 3b
2
,
it follows from (2) that for an integer z ðz 0Þ z X
1 3z2 þ 3z þ 1 X fi ðbÞ . (8) 2 3 b¼0 b¼zþ1 b pffiffi pffiffiffi The inventory level at time T is z ðb= t Þ T which is a convex function (cf. Fig. 2). Demand with uniform distribution: Let demand of ith period be subject to uniform distribution. Then by (8)
wi ðzÞ ¼
wi ðzÞ ¼
8 > > <
fi ðbÞ þ
1 Li þ1
z þ 1 þ 3z
> > :
2
þ3zþ1 3
1 if z ¼ 0; 1; . . . ; Li 1
if z ¼ Li ; Liþ1 ; . . .
Li i P b¼zþ1
! 1 b2
(9)
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Fig. 2. Inventory level in Example 2.
Fig. 3. Inventory level in Example 3.
Demand with Poisson distribution: Let fi ðbÞ be as in Section 3.2. Then it follows from (8) that wi ðzÞ ¼
z X
eli
b¼0
lbi b!
þ
1 3z2 þ 3z þ 1 X lb eli 2i . 3 b b! b¼zþ1
(10)
4.3. Example 3 Let a function gðxÞ be gðxÞ ¼ x2 . Then g 1 ðxÞ ¼ GðyÞ ¼ y3 =3. Thus
pffiffiffi x and
and so by (2) z X b¼0
fi ðbÞ þ
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 2ððz þ 1Þ z þ 1 z zÞ X fi ðbÞ pffiffiffi 3 b b¼zþ1
5. Optimal policy The optimal policy in the dynamic inventory system of period N is obtained if we are able to decide an integer number x¯ 1N. Let us indicate this procedure. 5.1. Formulation of inventory system
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 Dðzg 1 ðz=bÞ bGðg 1 ðz=bÞÞÞ ¼ pffiffiffi ððz þ 1Þ z þ 1 z zÞ, 3 b
wi ðzÞ ¼
Then li lj if and only if wi ðzÞ wj ðzÞ for all integers z ðz 0Þ.
(11)
Using the theory of dynamic programming let us i introduce functions f n ðxÞ which are stated at Section 2.2. Suppose that n ¼ 1; 2; . . . ; N;
i ¼ 1; 2; . . . ; N n þ 1.
Let x be an integer. Set
for an integer z ðz 0Þ. The inventory level is wriiten by the function z ðb=t 2 ÞT 2 which is concave in Fig. 3. Demand with uniform distribution: Assume that demand of ith period subjects to a uniform distribution. Then by (11) 8 ! Li pffiffiffiffiffiffiffi pffiffi > > ððzþ1Þ zþ1z zÞ Pi p1ffiffi < 1 z þ 1 þ 3 b wi ðzÞ ¼ Li þ1 b¼zþ1 > > : 1 if z ¼ 0; 1; . . . ; Li 1 (12) if z ¼ Li ; Liþ1 ; . . . Demand with Poisson distribution: Let fi ðbÞ be a probabilistic density function of a Poisson distribution with the parameter li. Then by (11) pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 z X lb 2ððz þ 1Þ z þ 1 z zÞ X lb eli i þ eli pffiffiffii . wi ðzÞ ¼ b! 3 bb! b¼0 b¼zþ1 (13)
i f 1 ðxÞ
¼ min fcx þ Hi ðzÞg,
(14)
zx;x2Z
( i
f n ðxÞ ¼ min
zx;x2Z
cx þ Hi ðzÞ þ a
1 X
) iþ1
f n1 ðz bÞfi ðbÞ .
(15)
b¼0
In order to study the structure put the function F in1 ðzÞ as follows: F in1 ðzÞ ¼ Hi ðzÞ þ a
1 X
iþ1
f n1 ðz bÞfi ðbÞ,
(16)
b¼0 i
where f 0 ðxÞ ¼ 0. Then it is shown by (15) that i
f n ðxÞ ¼ min fcx þ F in1 ðzÞg zx;x2Z
(17)
and these equations allow us to use mathematical induction. 5.2. Fundamental analysis
The following propositions show fundamental fact with respect to the relations among functions wi ðzÞ ð1 i NÞ.
If we add the following conditions D1 and D2 on Hi ðzÞ, then the properties of the functions F in ðzÞ can be obtained.
Proposition 2. (Sakaguchi, 2007b). Assume that demand is subject to a uniform distribution in the example as above. Then Li Lj if and only if wi ðzÞ wj ðzÞ for all integers z ðz 0Þ.
D1
There is an integer R such that DHi ðRÞ ac for all i ð1 i NÞ.
D2
If z is an integer, then D2 Hi ðzÞ 0 for all i ð1 i NÞ.
Proposition 3. (Sakaguchi, 2007b). Suppose that demand is subject to a Poisson distribution in the example as above.
Theorem 4. (Sakaguchi, 2007a). Assume D1 and D2 hold. Let n ð1 n NÞ and i ð1 i N n þ 1Þ be integers. Then
ARTICLE IN PRESS M. Sakaguchi / Int. J. Production Economics 122 (2009) 269–275
there is an integer x¯ in such that
Proposition 6. If l1 l2 lN , then the inventory system with Poisson distribution ffi ðbÞg at the three examples in Section 4 satisfies the condition DEC.
x¯ in ¼ minfz 2 ZjDF in1 ðzÞ 0g. Moreover let x be the initial inventory quantity at period i. Then the optimal policy at period i considering the expected total cost from period i through period i þ n 1 is as follows: If xo¯xin , then order x¯ in x and if x x¯ in , then do not order. One is able to see 0 x¯ in R. By Proposition 1 D1 and D2 are rewritten are:
All examples in Section 4 satisfy conditions D1 and D2. For each i ð1 i NÞ, it is able to find an integer x¯ i1 to solve the inequality DF i0 ðzÞ 0, that is, DHi ðzÞ 0. Precisely said (18)
This inequality is equivalent to an inequality wi ðzÞ k by Proposition 1. Therefore x¯ i1 ¼ minfz 2 Zjwi ðzÞ kg.
6.2. Algorithm An algorithm is given under the condition DEC. On account of this, let us introduce the functions win ðzÞ
ði ¼ 1; 2; . . . ; N; n ¼ 1; 2; . . . ; N i þ 1Þ
inductively as follows: For z x¯ iþ1 put n z X ac wiþ1 ðbÞ k fi ðz bÞ, win ðzÞ ¼ wi ðzÞ þ a n1 hþp iþ1
D1 There is an integer R such that wi ðRÞ k þ ac=ðh þ pÞ for all i ð1 i NÞ. D2 If z is an integer, then Dwi ðzÞ 0 for all ið1 i NÞ.
x¯ i1 ¼ minfz 2 ZjDHi ðzÞ 0g.
273
(19)
Let x be the initial inventory level before a regular order is placed. By virtue of Theorem 4 the decision of the inventory problem which is one of single period i is as follows: If xo¯xi1 , then order x¯ i1 x, and if x x¯ , then do not order. In the inventory problem of period N the solution is obtained by finding an integer x¯ 1N and it is carried out by considering the inequality DF 1N1 ðzÞ 0. 6. The case decreasing demand If we find an integer x¯ 1N , then the inventory problem can be solved. Theorem 4 exhibits to us the way how to get it. However, in general it is very difficult to seek x¯ 1N as there are a lot of cases when it has to be considered. 6.1. Condition DEC In Sakaguchi (2007b) an algorithm is presented where demand is decreasing. That is a generalization of the research in Sakaguchi and Kodama (2002) when demand distribution is static. The condition DEC is designated during this state. DEC: If z is an integer with 0 z R, then
DH1 ðzÞ DH2 ðzÞ DHN ðzÞ. It is also written by Proposition 1. DEC: If z is an integer with 0 z R, then w1 ðzÞ w2 ðzÞ wN ðzÞ. Propositions 2 and 3 yield the following propositions. Proposition 5. If L1 L2 LN , then the inventory system with uniform distribution ffi ðbÞg at the three examples in Section 4 satisfies the condition DEC.
b¼¯xn
(20) where wi0 ðzÞ ¼ wi ðzÞ þ ac=ðh þ pÞ. Now we have the following Proposition 7. Let i be an integer with 1 i N and n an integer with 1 n N i þ 1. If conditions D1, D2 and DEC hold, then ( c p ac þ ðh þ pÞwi ðzÞ if 0 zo¯xiþ1 n ; DF in ðzÞ ¼ iþ1 i c p ac þ ðh þ pÞwn ðzÞ if x¯ n z: The following algorithm follows from this proposition. Algorithm. Assume the conditions D1, D2 and DEC hold. Then Step 1: Find an integer R with w1 ðRÞ k þ ac=ðh þ pÞ. Step 2: Let n ¼ 1. that is minimal among an Step 3: Find an integer x¯ Nnþ1 n ðzÞ k þ ac=ðh þ pÞ. integer z with wNnþ1 n1 Step 4: If noN; then n :¼ n þ 1 and go to Step 3. If n ¼ N, then we stop. 6.3. Numerical illustration of DEC We illustrate the use of this algorithm on three types of examples in Section 4. Calculations are done by using a computer software Mathematica. Let N ¼ 3;
c ¼ 100;
p ¼ 200;
h ¼ 5;
a ¼ 0:975.
Then
k¼
pc 0:487805 hþp
and
kþ
ac 0:963415. hþp
First in order to get R solve the inequality 1
w ðzÞ k þ ac=ðh þ pÞ. Next solve the inequality w30 ðzÞ k þ ac=ðh þ pÞ;
w3 ðzÞ k
x¯ 31.
and find an integer Then the function w21 ðzÞ is captured and consider the inequality w21 ðzÞ k þ ac=ðh þ pÞ to obtain x¯ 22 . Finally we may find an integer x¯ 13 by w12 ðzÞ k þ ac=ðh þ pÞ.
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Hence:
Table 1 Uniform demand with DEC. Type
R
x¯ 31
x¯ 22
x¯ 13
Example 1 Example 2 Example 3
223 243 196
36 57 16
119 143 91
153 177 161
Find the minimal integer x¯ 21 with w2 ðzÞ k. If w1 ð¯x21 1Þ k þ ac=ðh þ pÞ, then x¯ 12 is the minimal
integer with w1 ðx¯ 12 Þ k þ ac=ðh þ pÞ. If w1 ð¯x21 1Þok þ ac=ðh þ pÞ, then x¯ 12 is the mimimal integer such that w11 ð¯x12 Þ k þ ac=ðh þ pÞ.
7.2. Numerical illustration of the 2-period model
Table 2 Poisson demand with DEC. Type
R
x¯ 31
x¯ 22
x¯ 13
Example 1 Example 2 Example 3
292 302 279
97 139 47
244 253 232
292 302 279
Keep some constants of 6.3 in this section. Set N ¼ 2;
c ¼ 100;
h ¼ 5;
a ¼ 0:975.
First we show procedures in the case DEC. Put L1 ¼ 300;
Uniform demand: Let L1 ¼ 300; L2 ¼ 250; L3 ¼ 200. It follows from Proposition 5 that a system of uniform distribution ffi ðbÞg satisfies the condition DEC. Therefore the solution x¯ 13 is obtained in Table 1 by using the algorithm. Poisson demand: Set l1 ¼ 300; l2 ¼ 250; l3 ¼ 200. Then this is the case in the state of DEC by Proposition 6 and there are solutions x¯ 13 in Table 2. It is amazed that x¯ 13 ¼ R.
p ¼ 200;
L2 ¼ 250
and
l1 ¼ 300;
l2 ¼ 250.
It is quite similar to Section 6.3. Uniform demand: Table 3 shows us the required number x¯ 12. Poisson demand: Table 4 gives us the number x¯ 12 in the problem with a Poisson distribution. In the case INC, two examples in Section 4.1 are shown by numerical analysis. Set e ¼ w1 ðx¯ 21 1Þ k ac=ðh þ pÞ. Uniform demand: Let us consider the inventory problem in Example 1 with a uniform demand. There are integers x¯ 12 in Table 5. Poisson demand: There are integers x¯ 12 in Table 6 that yield our decision in the case a Poisson demand.
7. Model of a few periods In Cheaitou et al. (2009) there are researches on the inventory system of 2-period. In this article the following observation is valid. When a parameter of demand distribution varies from period to period in our inventory model, it seems that there is a property of the functions wi ðzÞ ð1 i NÞ such that either an inequality wi ðzÞ wj ðzÞ holds for all integers z with z 0 or an inequality wi ðzÞ wj ðzÞ holds for all integers z with z 0. Because Propositions 5 and 6 suggest these facts.
7.1. 2-Period model
( ¼
c p ac þ ðh þ pÞw1 ðzÞ c p ac þ ðh þ
Let us consider three functions w1 ðzÞ, w2 ðzÞ and w3 ðzÞ that satisfy the following condition: If 0 z R, then wi ðzÞ wj ðzÞor wi ðzÞ wj ðzÞ for all i and j Then there are six cases of relations among w1 ðzÞ, w2 ðzÞ and w3 ðzÞ. One of them is the case DEC, and there are numerical analyses in Section 6.3. At present there is no way to deal with them together. Therefore it is too complicated and it needs
Table 3 Uniform demand with DEC.
Let N ¼ 2. Then there are two functions w1 ðzÞ and w2 ðzÞ. First assume that w1 ðzÞ w2 ðzÞ for all integers z ðz 0Þ. That is a condition DEC and it happens in the case a uniform demand with L1 L2 or in the case a Poisson demand with l1 l2 in Section 4. The algorithm in 6.2 works usefully to obtain x¯ 21 . On the contrary suppose that w1 ðzÞ w2 ðzÞ for all integers z ðz 0Þ. Denote by INC this condition. The algorithm as above does not work in this case. However Proposition 7 yields again a method to catch the number x¯ 21 . Because if N ¼ 2, then Proposition 7 hold without the condition DEC, that is
DF 11 ðzÞ
7.3. 3-Period model
pÞw11 ðzÞ
if 0 zo¯x21 ; if
x¯ 21
z:
(21)
Type
R
x¯ 21
x¯ 12
Example 1 Example 2 Example 3
223 243 196
54 85 24
117 146 111
Table 4 Poisson demand with DEC. Type
R
x¯ 21
x¯ 12
Example 1 Example 2 Example 3
292 302 279
121 174 59
292 302 279
ARTICLE IN PRESS M. Sakaguchi / Int. J. Production Economics 122 (2009) 269–275
Table 5 Uniform demand with INC.
275
Table 7 x¯ i1 .
L1
L2
x¯ 21
e
x¯ 12
x¯ 11
x¯ 21
x¯ 31
250 100
300 300
223 223
0.030 0.037
186 75
54
36
45
Table 8 x¯ i2 .
Table 6 Poisson demand with INC.
l1
l2
x¯ 21
e
x¯ 12
250 100
300 300
146 146
0.379 0.037
244 100
considerable work. Our basis of getting the number x¯ 13 is shown the following inequalities in Sakaguchi (2007a): 8 > DH1 ðzÞ ac if zo¯x22 ; > < z 1 P DF 2 ðzÞ ¼ DH1 ðzÞ ac þ a DF 21 ðbÞf1 ðz bÞ if z x¯ 22 > > : b¼x¯ 2
x¯ 12
x¯ 22
119
93
Observe x¯ 22 x¯ 31 . It is able to gain a value of x¯ 13 by considering the inequality w12 ðzÞ k þ
ac . hþp
In fact we have x¯ 13 ¼ 133.
2
References
and
DF 21 ðzÞ
¼
8 > DH2 ðzÞ ac > <
if zo¯x31 ; z P
DH2 ðzÞ ac þ a DF 30 ðbÞf2 ðz bÞ > > : b¼x¯ 3
if z x¯ 31 :
1
7.4. Numerical illustration of 3-period model Let us examine x¯ 13 in the case Example 1 with uniform demand. Keep N ¼ 3; c ¼ 100; p ¼ 200; h ¼ 5 and a ¼ 0:975. Let L1 ¼ 300; L2 ¼ 200 and L3 ¼ 250: This is the case w1 ðzÞ w2 ðzÞ and w2 ðzÞ w3 ðzÞ. Then the numbers x¯ i1 ði ¼ 1; 2; 3Þ are obtained in Table 7 by solving the inequalities wi ðzÞ k ði ¼ 1; 2; 3Þ. Since L1 L2 , it is a problem of 2-period with DEC to get the numbers x¯ 12 . It is also a problem of 2-period with INC to obtain x¯ 22 . This is due to L2 L3 . As in Section 7.1 we have x¯ i2 ði ¼ 1; 2Þ in Table 8.
Cheaitou, A., Delft, C.V., Dallery, Y., Jemel, Z., 2009. Two-period production planning and inventory control. International Journal of Production Economics 118, 118–130. Sakaguchi, M., Kodama, M., 2002. On the dynamic probabilistic inventory problems with piecewise cost functions which may not be piecewise smooth. In: Bulletin of Informatics and Cybernetics, vol. 34. Research Association of Statistical Sciences, Kyushu University, pp. 75–90. Sakaguchi, M., Kodama, M., 2005. Inventory models with a varying demand distribution. In: Proceedings of the International Workshop on Recent Advances in Stochastic Operations Research, Canmore, August 25–26, pp. 237–244. Sakaguchi, M., 2007a. On the probabilistic inventory models with timevarying demand, system sciences for economics and informatics. Monograph series of Institute for Advanced Studies, vol. 39, Hiroshima Shudo University, Kyushu University Press, Japan. Sakaguchi, M., 2007b. Optimal ordering for an inventory system with decreasing demand. In: Proceeding of Stochastic Operations Research II, Well On, Nagoya, Japan, pp. 251–258. Sakaguchi, M., Kodama, M., Sensitivity analysis of an economic ordering quantity for dynamic inventory models with discrete demand. International Journal of Manufacturing Technology and Management, to appear. Silver, A.E., Pyke, F.D., Peterson, R., 1998. Inventory Management and Production Planning and Scheduling, third ed.. Wiley, New York.