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Some inventory problems with fuzzy shortage cost
Fuzzy Sets and Systems 111 (2000) 87–97
www.elsevier.com/locate/fss
Some inventory problems with fuzzy shortage cost Hideki Katagiri ∗ , Hiroaki Ishii Graduate School of Engineering, Osaka University, 2-1 Yamada-oka, Suita, Osaka 565-0871, Japan Received October 1998
Abstract This paper investigates an allocation problem of perishable commodities with fuzzy shortage cost based on LIFO issuing policy. Under the rotation allocation policy, commodities are distributed from a regional center to n locations in the region. Costs are charged at each location for every unit short, outdated and transported. But the unit shortage cost is considered to be ambiguous and so estimated as an L fuzzy number. The purpose of this paper is to clarify the di erence between optimal solution of nonfuzzy shortage cost case and that of the fuzzy shortage cost case. Furthermore, possibilities of c 2000 Elsevier Science B.V. All rights reserved. inventory problems with other fuzzy costs are discussed. Keywords: Inventory; Fuzzy shortage cost; Minimal rotation allocation policy; Mathematical programming
1. Introduction We consider perishable commodities that become obsolete and cannot be used after a certain period of time [2, 3, 5 – 8]. Blood, photographic lm and fresh foods are typical examples of perishable commodities. Optimal rotation allocation policies for perishable commodities were analyzed rst by Prastacos [9 – 11]. These models were discussed under charging costs only for shortage and outdating. Nose et al. [8] considered a single-period allocation problem of perishable commodities based on a LIFO issuing and a rotation allocation policy taking account of shortage, outdating and transportation costs and proposed an algorithm to nd an optimal rotation allocation policy. But in all models so far considered, the ambiguous nature of shortage cost has been ignored. For this reason, we consider a fuzzy version of the model by Nose et al. [8]. Section 2 gives some assumptions and notations used throughout this paper. Section 3 formulates the problem and introduces fuzzy minimal order. Section 4 de nes minimal rotation allocation policies and presents a procedure to obtain some of them by using a solution procedure of corresponding nonfuzzy problem in [8]. Section 5 discusses fuzzifying possibilities of other costs such as outdating cost and investigates the in uence when some other costs are also fuzzi ed. ∗
Corresponding author. Tel.: +81-6-6877-5111/3641; fax: +81-6-6879-7871. E-mail address: [email protected] (H. Katagiri)
c 2000 Elsevier Science B.V. All rights reserved. 0165-0114/00/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 9 ) 0 0 4 5 6 - 4
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2. Assumptions and notations A periodic review inventory model is considered for one planning horizon and single item. Ordering takes place at the start of a period and costs are incurred during a period. The following are assumed to construct the model: 1. Maximum lifetime of the perishable commodities is xed and equal to M periods. 2. Inventory is depleted by demand at the start of each period according to a LIFO issuing policy. 3. The stock remaining at the end of each period ages one period monotonically and is returned from each location to regional center (rotation policy). 4. If the commodities have not been depleted by demand until the period it reaches age M , then it outdates and must be discarded. 5. The following costs are charged at each location k; k = 1; : : : ; n, for every unit short, outdated and transported. • s˜k : shortage cost per unit of demand un lled at location k; k = 1; : : : ; n. • wk : outdating cost per unit perished at location k; k = 1; : : : ; n. • uk : transportation cost per unit shipped from regional center to location k; k = 1; : : : ; n, or returned from location k; k = 1; : : : ; n, to regional center. (We assume that uk ¿wk ; k = 1; : : : ; n, since otherwise we return the unit perished at location to the center with outdating.) • s˜k is ambiguous and assumed to be an L fuzzy number with the following membership function s˜k (t) = max{L((t − mk )= k ); 0}; where L is a shape function from R to R satisfying the following conditions. (a) L(− t) = L(t) for any t ∈ R, (b) L(t) = 1 i t = 0, (c) L(·) is nonincreasing on [0; ∞), (d) Let t0 = inf {t¿0 | L(t) = 0}, then 0¡t0 ¡∞ (t0 is called the zero point of L). Without loss of generality, we assume that mk − k t0 ¿0 since shortage cost may be positive in any case. Though the above condition (b) is slightly strong, this condition is required in order to utilize the good property described in Theorem 1 in the next section. 6. Demands at each location k; Dk , are independent random variables with distribution function Fk (·) and density function fk (·). The following notations are used in this paper. • Nk : allocated quantity not subject to outdating at location k (age 0 to age M − 2) • Bk : allocated quantity subject to outdating at location k (age M − 1): N = (N1 ; : : : ; Nn ); N=
n X k=1
Nk ;
B=
B = (B1 ; : : : ; Bn ); n X
Bk :
k=1
For the sake of simplicity, L fuzzy number A˜ with membership function A˜ (t) = max{L((t − m)= ); 0}; t ∈ R ˜ For two L fuzzy numbers is denoted simply by A˜ = (m; )L (the number m is called the center of A). A˜ = (mA ; A )L ; B˜ = (mB ; B )L and ordinary positive number p, the following hold: A˜ + B˜ = (mA + mB ; A + B )L ;
A˜ + p = (mA + p; A )L
and
pA˜ = (pmA ; p A )L :
(1)
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89
3. Problem formulation ˜ ; B) when fresher units N and older units B are This section formulates the expected cost function C(N prepared. Shortage, outdating and transportation quantity at each location k are the following, respectively: 1. Shortage quantity: Dk − Nk − Bk ; Dk ¿Nk + Bk ; 0 otherwise: 2. Outdating quantity: Nk + Bk − Dk ; Nk ¡Dk ¡Nk + Bk ; 0¡Dk ¡Nk ; Bk ; 0 otherwise: 3. Transportation quantity (a) From regional center to location k: Nk + Bk . (b) From location k to regional center: Nk − Dk ; 0¡Dk ¡Nk ; 0 otherwise: ˜ ; B) is given by Consequently, the total expected cost C(N Z n Z ∞ X ˜ ; B) = s˜k (x − Nk − Bk ) dFk (x) + wk C(N Nk +Bk
k=1
Z
+ wk Bk Fk (Nk ) + (Nk + Bk )uk + uk
Nk +Bk
Nk
Nk
0
(Nk + Bk − x) dFk (x)
(Nk − x) dFk (x) :
(2)
˜ ; B) is an L fuzzy number with the membership function Due to the fuzziness of the shortage cost s˜k ; C(N C(N ˜ ; B) (t) = max{L((t − m(N ; B))= (N ; B)); 0}; as is easily calculated from (1) (see [8]) where Z ∞ Z n X mk (x − Nk − Bk ) dFk (x) + wk m(N ; B) = NK +Bk
k=1
+ wk Bk Fk (Nk ) + (Nk + Bk )uk + uk and (N ; B) =
n X k=1
Z k
∞
Nk +Bk
Z 0
(x − Nk − Bk ) dFk (x):
Now we de ne an ordering of L fuzzy numbers (see [1]). De nition 1. For two L fuzzy numbers A˜ = (mA ; A )L
and
B˜ = (mB ; B )L ;
Nk +Bk
Nk
Nk
(Nk + Bk − x) dFK (x)
(Nk − x) dFk (x)
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A˜ 4 B˜ i it holds that 1. mA 6mB , 2. there exists a real number c such that (a) mA 6c6mB ; (b) A˜(x)¿B˜ (x); for all x¡c; (c) A˜(x)6B˜ (x); for all x¿c. Then the following theorem holds. Theorem 1 (Furukawa [1]). For two L fuzzy numbers A˜ = (m; )L and B˜ = (n; )L ; we have A˜ 4 B˜ ⇔ t0 | − | 6n − m: ˜ ; B) Since this order is not linear, usually the unique optimal rotation allocation policy minimizing C(N does not exist and so we seek for minimal rotation allocation policies (N1 ; N2 ; : : : ; Nn ; B1 ; B2 ; : : : ; Bn ) under the condition that n X
Nk = N;
k=1
n X
Bk = B;
N1 ; N2 ; : : : ; Nn ; B1 ; B2 ; : : : ; Bn ¿0
k=1
by fully utilizing Theorem 1. 4. Minimal policies De nition 2. For two rotation allocation policies (N ; B) and (N 0 ; B 0 ), we say that (N ; B) dominates (N 0 ; B 0 ) ˜ ; B) 4 C(N ˜ 0 ; B 0 ) and C(N ˜ ; B) 6= C(N ˜ 0 ; B 0 ). Rotation allocation policy (N ; B) is de ned to be minimal i C(N if there exists no rotation allocation policy that dominates (N ; B). (Hereafter we simply call a minimal rotation allocation policy a minimal policy.) ˜ ; B) 4 C(N ˜ 0 ; B 0 ) is equivalent to the We give two candidates of minimal policies. First note that C(N condition t0 | (N 0 ; B 0 ) − (N ; B)|6m(N 0 ; B 0 ) − m(N ; B) from Theorem 1. Further, this is equivalent to conditions m(N ; B) − t0 (N ; B)6m(N 0 ; B 0 ) − t0 (N 0 ; B 0 )
(3)
m(N ; B) + t0 (N ; B)6m(N 0 ; B 0 ) + t0 (N 0 ; B 0 ):
(4)
and
That is, n X
Z (mk − t0 k )
k=1
+
n X k=1
∞
Nk +Bk
(x − Nk − Bk ) dFk (x) +
wk Bk Fk (Nk ) +
n X
Z wk
k=1 n X k=1
uk (Nk + Bk ) +
n X k=1
Z uk
0
Nk
Nk +Bk
Nk
(Nk + Bk − x) dFk (x)
(Nk − x) dFk (x)
H. Katagiri, H. Ishii / Fuzzy Sets and Systems 111 (2000) 87–97
Z n X 6 (mk − t0 k )
Nk0 +Bk0
k=1
+
∞
n X
wk Bk0 Fk (Nk0 ) +
k=1
Z (mk + t0 k )
k=1
+
n X
∞
Nk +Bk
wk Bk Fk (Nk ) +
uk (Nk0
Z (mk + t0 k )
uk (Nk + Bk ) +
n X
n X
∞
wk Bk0 Fk (Nk0 ) +
k=1
n X
Z uk
wk Nk
0
Nk0
0
Z
Z uk
k=1
Nk0 +Bk0
k=1 n X
+
Bk0 ) +
k=1
k=1
n X
+
n X
Z wk
Nk0
Nk +Bk
n X k=1
uk (Nk0 + Bk0 ) +
k=1
n X
Z uk
k=1
(Nk0 + Bk0 − x) dFk (x)
(5)
(Nk + Bk − x) dFk (x)
(Nk − x) dFk (x)
(x − Nk0 − Bk0 ) dFk (x) +
n X
Nk0 +Bk0
(Nk0 − x) dFk (x)
Nk
k=1 n X
n X k=1
(x − Nk − Bk ) dFk (x) +
k=1
6
−
Bk0 ) dFk (x) +
k=1
and n X
(x −
Nk0
91
0
Nk0
Z wk
Nk0 +Bk0
Nk0
(Nk0 + Bk0 − x) dFk (x)
(Nk0 − x) dFk (x):
(6)
From (3) and (4), if there exists a rotation allocation policy minimizing both m(N ; B) − t0 (N ; B) and m(N ; B) + t0 (N ; B), it is a minimal policy. If there exists such a policy, it minimizes also m(N ; B) though usually there does not exist such a policy. Theorem 2. A rotation allocation policy (N ∗ ; B ∗ ) that minimizes m(N ; B) is a minimal policy. Proof. If (N ∗ ; B ∗ ) is not a minimal policy, then there exists a rotation allocation policy (N o ; B o ) such that t0 | (N ∗ ; B ∗ )− (N o ; B o )|6m(N ∗ ; B ∗ )−m(N o ; B o ) and (m(N ∗ ; B ∗ ); (N ∗ ; B ∗ )) 6= (m(N o ; B o ); (N o ; B o )) from Theorem 1. However, this is impossible since in this case t0 | (N ∗ ; B ∗ ) − (N o ; B o )|¿0¿m(N ∗ ; B ∗ ) − m(N o ; B o ) holds from m(N ∗ ; B ∗ ) being minimum. Minimization of m(N ; B) can be done by using the procedure of Nose et al. [8] since the problem is the very same as that of Nose et al. [8] if the ordinary shortage cost sk in [8] is set to mk (center of our fuzzy shortage cost s˜k ). This solution is the same as that of the ordinary one in [8]. But in some cases we pursue a minimal policy with the least ambiguity. Then the solution becomes di erent to the ordinary one. We show this minimal solution next. First, we consider the minimization of Z ∞ n X k (x − Nk − Bk ) dFk (x); (N ; B) = k=1
PA: Minimize
Nk +Bk
n X k=1
subject to
n X k=1
Z k
∞
Nk +Bk
Nk = N;
(x − Nk − Bk ) dFk (x) n X
Bk = B;
k=1
N1 ; N2 ; : : : ; Nn ; B1 ; B2 ; : : : ; Bn ¿0:
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R∞ Let us de ne k (Nk ; Bk ) ≡ k Nk +Bk (x − Nk − Bk ) dFk (x); k = 1; : : : ; n, where we assume that each k ¿0 for a while since otherwise k (Nk ; Bk ) = 0 and we omit this term from (N ; B). Then, since @ k (Nk ; Bk ) = k Fk (Nk + Bk ) − k ; @Nk @ 2 k (Nk ; Bk ) = k fk (Nk + Bk ); @Nk2
@ k (Nk ; Bk ) = k Fk (Nk + Bk ) − k ; @Bk @ 2 k (Nk ; Bk ) = k fk (Nk + Bk ) @Bk2
and @ 2 k (Nk ; Bk ) = k fk (Nk + Bk ); @Nk @Bk it holds that @ 2 k (Nk ; Bk ) @ 2 k (Nk ; Bk ) − @Nk2 @Bk2
@ 2 k (Nk ; Bk ) @Nk @Bk
2 =0
and this implies that the function k (Nk ; Bk ) is convex. Thus, (N ; B) is convex since it is the sum of convex function and hence PA is a convex programming problem. In order to solve PA, we construct the following Lagrange function: ! ! Z ∞ n n n X X X k (x − Nk − Bk ) dFk (x) + Nk − N + Bk − B : L(N ; B) ≡ k=1
Nk +Bk
k=1
k=1
Then an optimal solution of PA is that of the following Kuhn–Tucker condition KT: KT :
@L(N ; B) = k Fk (Nk + Bk ) − k + ¿0; @Nk @L(N ; B) = k Fk (Nk + Bk ) − k + ¿0; @Bk n n X X Nk = N; Bk = B; k=1
@L · Nk = 0; @Nk
k = 1; : : : ; n;
k=1
@L · Bk = 0; @Bk
k = 1; : : : ; n:
We show the solution procedure of PA in case of B = 0 (N = 0) and in case that both N and B are positive, respectively. Case (a): B = 0 (N = 0). If B = 0 (N = 0), then all Bk = 0, k = 1; : : : ; n (Nk = 0, k = 1; : : : ; n). And if both N and B = 0, the optimal solution of PA is Nk = 0; Bk = 0; k = 1; : : : ; n. If either N or B = 0, we nd a solution of KT as follows. We only show the case of B = 0 since the case N = 0 is the very same as B = 0. Since B = 0, i.e., Bk = 0; k = 1; : : : ; n, KT becomes as follows: KT0 :
@L(N ) = k Fk (Nk ) − k + ¿0; @Nk @L(N ) · Nk = 0; @Nk
k = 1; : : : ; n;
n X k=1
Nk = N:
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93
The following procedure PKT nds a solution of KT0 . PKT: 1. First n locations are arranged and renumbered so as to be the following order, 1 6 · · · 6 n . Further let us de ne Ii ≡ ( i ; i+1 ); i = 0; : : : ; n − 1 where 0 ≡ 1 (1 − F1−1 (N )). Go to 2. 2. Set i = 0 and go to 3. 3. Set ∈ Ii . Then the following are obtained: Nk = 0;
k = 1; : : : ; i; ; k = i + 1; : : : ; n: Nk = Fk−1 1 − k Pn Pn If k=i+2 Fk−1 (1 − ( i+1 = k ))¿N , then go to 4. Otherwise, nd o such that k=i+1 Fk−1 (1 − (o = k )) = N and set Nk = Fk−1 (1 − (o = k )); k = i + 1; : : : ; n, and Nk = 0; k = 1; : : : ; i. Terminate. 4. Set i = i + 1 and return to 3. Theorem 3. The above procedure PKT nds an optimal solution of PA in case B = 0. Proof. From KT0 ; Nk is positive if and only if ¡ k and so the validity of the above procedure is deduced. Case (b): N ¿0 and B¿0. Next we consider a solution of KT when both N and B are positive. First we de ne Tk ≡ Nk + Bk ; k = 1; : : : ; n, and T ≡ N + B. In this case, it may be cumbersome to nd a solution of PA from that of KT. Noting that the objective function of PA is a function of T1 ; T2 ; : : : ; Tn , we transform PA into the following problem: Z ∞ n X k (x − Tk ) dFk (x) PAT: Minimize subject to
k=1 n X
Tk
Tk = T;
Tk ¿0; k = 1; 2; : : : ; n:
k=1
An optimal solution of PA is obtained from an optimal solution Tko ; k = 1; 2; : : : ; n, of PAT by solving the following simultaneous linear equations SLP: SLP: Nk + Bk = Tko ; n X Nk = N; k=1
k = 1; : : : ; n; n X Bk = B; N1 ; : : : ; Nn ; B1 ; : : : ; Bn ¿0: k=1
Since SLP has 2n unknowns and n + 2 equations (one of them is redundant), its solution exists except for the trivial case n = 1 for which we need not make any e orts to nd a solution of PA. Further it has another merit from the following reason. We seek a minimal solution and so minimization of m(N ; B) among optimal solutions of PA should be done since there exists n − 1 free variables in SLP. Therefore, we consider the following problem: ) ( Z o Z Nk n Tk X o o (Tk − x) dFk (x) + wk (Tk − Nk )Fk (Nk ) + uk (Nk − x) dFk (x) wk PAM: Minimize C + k=1
subject to
n X k=1
Nk = N;
Nk
06Nk 6Tko ; k = 1; : : : ; n;
0
94
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R∞ Pn where C ≡ k=1 {mk T o (x − Tko ) dfk (x) + uk Tko } is the constant part of m(N ; B) when Tk = Tko ; k = 1; : : : ; n. k Note that constraints of PAM is equivalent to those of SLP since we can derive the constraint 06Nk 6Tko ; k = 1; : : : ; n of SLP by simple calculations. By solving PAT, SLP and PAM, an optimal solution of PA in case that both N and B are positive is obtained. It is the solution that minimizes m(N ; B) under the condition that the ambiguity is minimized. Now we show the solution procedure of PAT. Again we construct the following Lagrangian L(T1 ; : : : ; Tn ; ) corresponding to the problem PAT: ! Z ∞ n n X X L(T1 ; T2 ; : : : ; Tn ; ) ≡ k (x − Tk ) dFk (x) + Tk − T : k=1
Tk
k=1
From L(T1 ; T2 ; : : : ; Tn ; ), we obtain the following Kuhn–Tucker condition: KTT:
@L = k (Fk (Tk ) − 1) + ¿0; @Tk n X @L · Tk = 0; k = 1; : : : ; n; Tk = T: @Tk k=1
Since PAT is a convex programming problem, a solution of KTT is an optimal solution of PAT. Further KTT is the same as KT0 if Nk ; k = 1; : : : ; n, and N are replaced by Tk ; k = 1; : : : ; n and T , respectively. So we omit the solution procedure of KTT and show only the solution procedure of PAM supposing that an optimal solution of PAT is found. Note that the objective function of PAT is transformed into the following m(N ˆ ) where the constant term C is deleted: ) ( Z o Z Nk n Tk X m(N ˆ )≡ Fk (x) dx + uk (Nk − x) dFk (x) : wk k=1
Nk
0
Here we assume that all Tko ; k = 1; : : : ; n, are positive without loss of generality since otherwise if Tko0 = 0 for some k 0 , we can set Nk 0 = 0, renumber k by skipping such k 0 and reset n = n0 where n0 is the number of ˆ ) is convex since positive Tko . Further this function m(N @2 mˆ = (uk − wk )fk (Nk )¿0 @Nk2 from the assumption uk ¿wk ; k = 1; : : : ; n (i.e., each term of the above sum in m(N ˆ ) is convex) and the sum of convex functions is also convex. Let us construct a Lagrangian to solve PAM. Then a solution of the following Kuhn–Tucker condition KTM is an optimal solution of PAM: ) ( Z o Z Nk n Tk X L(N ; ; 1 ; : : : ; n ) = Fk (x) dx + uk (Nk − x) dFk (x) wk k=1
+ N −
Nk
n X k=1
KTM: Pn
0
! Nk
+
n X
k (Nk − Tko ) :
k=1
k=1 Nk = N . For each k; k ¿0 and exactly one of following conditions holds: 1. Nk = 0; (uk − wk )Fk (0) − ¿0.
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95
2. Nk = Tko ; (uk − wk )Fk (Tko ) − + k 60. 3. Tko ¿Nk ¿0; (uk − wk )Fk (Nk ) − = 0. Thus, we obtain the following procedure: PKTM: 1. Let a1 = (u1 −w1 )F1 (T1o ); : : : ; an = (un −wn )Fn (Tno ) and b1 = (u1 −w1 )F1 (0); : : : ; bn = (un −wn )Fn (0). Sorting these ak ; bk ; k = 1; : : : ; n, let the result be c1 ¡c2 ¡ · · · ¡c nˆ where nˆ is the number of di erent ak ; bk ; k = 1; : : : ; n. Further let Jk = (ck ; ck+1 ); k = 1; : : : ; nˆ − 1. Set i = 1 and go to 2. 2. Let Ai = {k | ak 6ci }; Ei = {k | bk ¿ci+1 } and Gi = {1; 2; : : : ; n} − Ai − Ei . Further set Nk = Tko for k ∈ Ai ; P Nk = 0 for k ∈ Ei and Nk = Fk−1 (=(uk −wk )) for k ∈ Gi . If there exists = i ∈ Ji such that k∈Gi Nk = N − P −1 i o o k∈Ai Tk , set an optimal solution of PAM as Nk = Fk ( =(uk − wk )) for k ∈ Gi ; Nk = Tk for k ∈ Ai and Nk = 0 for k ∈ Ei and terminate. Otherwise, go to 3. 3. Set i = i + 1 and return to 2. We denote an optimal solution of PAM by N o . Then the following B o together with this N o constitutes an optimal solution of PA: Bko = Tko − Nko ;
k = 1; 2; : : : ; n:
Theorem 4. The above (N o ; B o ) becomes an optimal solution of PA minimizing (N ; B). Proof. If 6 all bk ; k = 1; : : : ; n, then all Nk = 0 from Kuhn–Tucker condition KTM. But this contradicts the assumption that N 6= 0. Further from the monotonicity of the distribution function Fk ; c1 must be some bk . So we can start to check the region including a solution of KTM. From alternative conditions 1–3 and the condition of k ¿0 in KTM, if ¿ak , then Nk must be Tko and if 6bk , then Nk must be 0. Further, if exists between bk and ak , then we can set Nk = Fk−1 (=(uk − wk )). Consequently, Pn the validity of setting of iteration i in 2 of the above procedure PKTM is derived. Since B¿0; N 6T ( k=1 Tko ); must be less than c nˆ . Therefore PKTM terminates before i = Nˆ . The above discussion assures the validity of PKTM to nd an optimal solution N of PAM. Thus (N o ; B o ) gives an optimal solution of PA minimizing (N ; B). In order to check the minimality of (N o ; B o ), we consider another problem: PMA: Minimize
m(N ; B) + t0 (N ; B) Z n X = (mk + t0 k ) k=1
Z
+ wk
NK +Bk
Nk +Bk
Nk
subject to
k=1
Nk = N;
(x − Nk − Bk ) dFk (x)
(Nk + Bk − x) dFk (x) + wk Bk Fk (Nk ) Z
+ (Nk + Bk )uk + uk n X
∞
n X
Nk
0
Bk = B;
(Nk − x) dFk (x) N1 ; : : : ; Nn ; B1 ; : : : ; Bn ¿0:
k=1
PMA is solved by using the procedure of Nose et al. [8] since again this problem is the same as that of the problem in [8] if sk is replaced by mk + t0 k . Let a solution with minimum m(N ; B) among optimal solutions of PMA be (N u ; B u ).
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Theorem 5. (N u ; B u ) is a minimal solution of our problem and if m(N u ; B u ) = m(N o ; B o ) and (N u ; B u ) = (N o ; B o ); then (N o ; B o ) is also a minimal solution. Proof. If (N o ; B o ) is not a minimal solution, then from Theorem 1, there exists another solution of our problem (N d ; B d ) such that m(N o ; B o ) + t0 (N o ; B o )¿m(N d ; B d ) + t0 (N d ; B d ); since (N o ; B o ) is an optimal solution of PA. This implies that (N o ; B o ) is also an optimal solution of PMA. As for minimality of (N u ; B u ), rst note that sup{t¿0 | L((t − m(N ; B))= (N ; B))¿0} = m(N ; B) + t0 (N ; B): If (N u ; B u ) is not minimal, then there exists a solution (N C ; B C ) such that m(N u ; B u ) + t0 (N u ; B u )¿m(N C ; B C ) + t0 (N C ; B C ): But this is impossible since (N u ; B u ) is an optimal solution of PMA.
5. Conclusion We have shown some minimal solutions of our problem. Note that Theorem 5 shows that (N o ; B o ) may be a minimal solution with respect to our ordering but this includes the least “ambiguity”. If we also need a solution with the least “ambiguity” among alternatives, then (N o ; B o ) is a suitable solution. Together with shortage cost, outdating cost may be ambiguous and so we must consider perishable inventory models with both costs as fuzzy numbers. We are now tackling this problem with more general models than our model in this paper in a sense that they include other fuzzy costs though analysis becomes very complicated. Even in the case where only the shortage cost is ambiguous, we must consider the policy not only minimizing the most possible total cost m(N ; B) but also minimizing the total cost in the worst case m(N ; B) + t0 (N ; B) (further in the most optimistic case m(N ; B) − t0 (N ; B)). More generally, we can derive minimal solutions from minimizing m(N ; B) + (2 − 1)t0 (N ; B) for various between 0 and 1. (Note that minimizing m(N ; B) corresponds to the case = 1=2; = 1 corresponds to minimization of the worst case and = 0 to the most optimistic case.) Finally note that in more general models with fuzzy costs, in uence of the variation on these costs may change “optimal” policies drastically. References [1] N. Furukawa, A parametric total order on fuzzy numbers and a fuzzy shortest route problem, Optimization 10 (1994) 367–377. [2] H. Ishii, Perishable inventory problem with two types of customers and di erent selling prices, J. Oper. Res. Soc. Jpn. 36 (1993) 199 –205. [3] H. Ishii, T. Nose, S. Shiode, T. Nishida, Perishable inventory management subject to stochastic lead-time, European J. Oper. Res. 8 (1981) 76 – 85. [4] A. Kaufmann, M.M. Gupta, Introduction to Fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold, New York, 1985. [5] S. Nahmias, Optimal and approximate ordering policies for a perishable product subject to stochastic demand, Doctoral Dissertation, University of Northwestern, 1972. [6] S. Nahmias, Perishable inventory theory: a review, Oper. Res. 30 (1982) 680 –708. [7] T. Nose, H. Ishii, T. Nishida, Some properties of perishable inventory control subject to stochastic lead-time, J. Oper. Res. Soc. Jpn. 24 (1981) 110 –135. [8] T. Nose, H. Ishii, T. Nishida, LIFO allocation problem for perishable commodities, J. Oper. Res. Soc. Jpn. 26 (1983) 135 –145.
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G.P. Prastacos, Optimal myopic allocation of a product with xed lifetime, J. Oper. Res. Soc. 29 (1978) 905 – 913. G.P. Prastacos, LIFO distribution systems, J. Oper. Res. Soc. 30 (1979) 539 – 546. G.P. Prastacos, Allocation of perishable product inventory, Oper. Res. 29 (1981) 95 –107. H. Tanaka, H. Ichihashi, K. Asai, A formulation of fuzzy linear programming problem based on comparison of fuzzy numbers, Control Cybernet. 30 (1984) 185 –194.
www.elsevier.com/locate/fss
Some inventory problems with fuzzy shortage cost Hideki Katagiri ∗ , Hiroaki Ishii Graduate School of Engineering, Osaka University, 2-1 Yamada-oka, Suita, Osaka 565-0871, Japan Received October 1998
Abstract This paper investigates an allocation problem of perishable commodities with fuzzy shortage cost based on LIFO issuing policy. Under the rotation allocation policy, commodities are distributed from a regional center to n locations in the region. Costs are charged at each location for every unit short, outdated and transported. But the unit shortage cost is considered to be ambiguous and so estimated as an L fuzzy number. The purpose of this paper is to clarify the di erence between optimal solution of nonfuzzy shortage cost case and that of the fuzzy shortage cost case. Furthermore, possibilities of c 2000 Elsevier Science B.V. All rights reserved. inventory problems with other fuzzy costs are discussed. Keywords: Inventory; Fuzzy shortage cost; Minimal rotation allocation policy; Mathematical programming
1. Introduction We consider perishable commodities that become obsolete and cannot be used after a certain period of time [2, 3, 5 – 8]. Blood, photographic lm and fresh foods are typical examples of perishable commodities. Optimal rotation allocation policies for perishable commodities were analyzed rst by Prastacos [9 – 11]. These models were discussed under charging costs only for shortage and outdating. Nose et al. [8] considered a single-period allocation problem of perishable commodities based on a LIFO issuing and a rotation allocation policy taking account of shortage, outdating and transportation costs and proposed an algorithm to nd an optimal rotation allocation policy. But in all models so far considered, the ambiguous nature of shortage cost has been ignored. For this reason, we consider a fuzzy version of the model by Nose et al. [8]. Section 2 gives some assumptions and notations used throughout this paper. Section 3 formulates the problem and introduces fuzzy minimal order. Section 4 de nes minimal rotation allocation policies and presents a procedure to obtain some of them by using a solution procedure of corresponding nonfuzzy problem in [8]. Section 5 discusses fuzzifying possibilities of other costs such as outdating cost and investigates the in uence when some other costs are also fuzzi ed. ∗
Corresponding author. Tel.: +81-6-6877-5111/3641; fax: +81-6-6879-7871. E-mail address: [email protected] (H. Katagiri)
c 2000 Elsevier Science B.V. All rights reserved. 0165-0114/00/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 9 ) 0 0 4 5 6 - 4
88
H. Katagiri, H. Ishii / Fuzzy Sets and Systems 111 (2000) 87–97
2. Assumptions and notations A periodic review inventory model is considered for one planning horizon and single item. Ordering takes place at the start of a period and costs are incurred during a period. The following are assumed to construct the model: 1. Maximum lifetime of the perishable commodities is xed and equal to M periods. 2. Inventory is depleted by demand at the start of each period according to a LIFO issuing policy. 3. The stock remaining at the end of each period ages one period monotonically and is returned from each location to regional center (rotation policy). 4. If the commodities have not been depleted by demand until the period it reaches age M , then it outdates and must be discarded. 5. The following costs are charged at each location k; k = 1; : : : ; n, for every unit short, outdated and transported. • s˜k : shortage cost per unit of demand un lled at location k; k = 1; : : : ; n. • wk : outdating cost per unit perished at location k; k = 1; : : : ; n. • uk : transportation cost per unit shipped from regional center to location k; k = 1; : : : ; n, or returned from location k; k = 1; : : : ; n, to regional center. (We assume that uk ¿wk ; k = 1; : : : ; n, since otherwise we return the unit perished at location to the center with outdating.) • s˜k is ambiguous and assumed to be an L fuzzy number with the following membership function s˜k (t) = max{L((t − mk )= k ); 0}; where L is a shape function from R to R satisfying the following conditions. (a) L(− t) = L(t) for any t ∈ R, (b) L(t) = 1 i t = 0, (c) L(·) is nonincreasing on [0; ∞), (d) Let t0 = inf {t¿0 | L(t) = 0}, then 0¡t0 ¡∞ (t0 is called the zero point of L). Without loss of generality, we assume that mk − k t0 ¿0 since shortage cost may be positive in any case. Though the above condition (b) is slightly strong, this condition is required in order to utilize the good property described in Theorem 1 in the next section. 6. Demands at each location k; Dk , are independent random variables with distribution function Fk (·) and density function fk (·). The following notations are used in this paper. • Nk : allocated quantity not subject to outdating at location k (age 0 to age M − 2) • Bk : allocated quantity subject to outdating at location k (age M − 1): N = (N1 ; : : : ; Nn ); N=
n X k=1
Nk ;
B=
B = (B1 ; : : : ; Bn ); n X
Bk :
k=1
For the sake of simplicity, L fuzzy number A˜ with membership function A˜ (t) = max{L((t − m)= ); 0}; t ∈ R ˜ For two L fuzzy numbers is denoted simply by A˜ = (m; )L (the number m is called the center of A). A˜ = (mA ; A )L ; B˜ = (mB ; B )L and ordinary positive number p, the following hold: A˜ + B˜ = (mA + mB ; A + B )L ;
A˜ + p = (mA + p; A )L
and
pA˜ = (pmA ; p A )L :
(1)
H. Katagiri, H. Ishii / Fuzzy Sets and Systems 111 (2000) 87–97
89
3. Problem formulation ˜ ; B) when fresher units N and older units B are This section formulates the expected cost function C(N prepared. Shortage, outdating and transportation quantity at each location k are the following, respectively: 1. Shortage quantity: Dk − Nk − Bk ; Dk ¿Nk + Bk ; 0 otherwise: 2. Outdating quantity: Nk + Bk − Dk ; Nk ¡Dk ¡Nk + Bk ; 0¡Dk ¡Nk ; Bk ; 0 otherwise: 3. Transportation quantity (a) From regional center to location k: Nk + Bk . (b) From location k to regional center: Nk − Dk ; 0¡Dk ¡Nk ; 0 otherwise: ˜ ; B) is given by Consequently, the total expected cost C(N Z n Z ∞ X ˜ ; B) = s˜k (x − Nk − Bk ) dFk (x) + wk C(N Nk +Bk
k=1
Z
+ wk Bk Fk (Nk ) + (Nk + Bk )uk + uk
Nk +Bk
Nk
Nk
0
(Nk + Bk − x) dFk (x)
(Nk − x) dFk (x) :
(2)
˜ ; B) is an L fuzzy number with the membership function Due to the fuzziness of the shortage cost s˜k ; C(N C(N ˜ ; B) (t) = max{L((t − m(N ; B))= (N ; B)); 0}; as is easily calculated from (1) (see [8]) where Z ∞ Z n X mk (x − Nk − Bk ) dFk (x) + wk m(N ; B) = NK +Bk
k=1
+ wk Bk Fk (Nk ) + (Nk + Bk )uk + uk and (N ; B) =
n X k=1
Z k
∞
Nk +Bk
Z 0
(x − Nk − Bk ) dFk (x):
Now we de ne an ordering of L fuzzy numbers (see [1]). De nition 1. For two L fuzzy numbers A˜ = (mA ; A )L
and
B˜ = (mB ; B )L ;
Nk +Bk
Nk
Nk
(Nk + Bk − x) dFK (x)
(Nk − x) dFk (x)
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A˜ 4 B˜ i it holds that 1. mA 6mB , 2. there exists a real number c such that (a) mA 6c6mB ; (b) A˜(x)¿B˜ (x); for all x¡c; (c) A˜(x)6B˜ (x); for all x¿c. Then the following theorem holds. Theorem 1 (Furukawa [1]). For two L fuzzy numbers A˜ = (m; )L and B˜ = (n; )L ; we have A˜ 4 B˜ ⇔ t0 | − | 6n − m: ˜ ; B) Since this order is not linear, usually the unique optimal rotation allocation policy minimizing C(N does not exist and so we seek for minimal rotation allocation policies (N1 ; N2 ; : : : ; Nn ; B1 ; B2 ; : : : ; Bn ) under the condition that n X
Nk = N;
k=1
n X
Bk = B;
N1 ; N2 ; : : : ; Nn ; B1 ; B2 ; : : : ; Bn ¿0
k=1
by fully utilizing Theorem 1. 4. Minimal policies De nition 2. For two rotation allocation policies (N ; B) and (N 0 ; B 0 ), we say that (N ; B) dominates (N 0 ; B 0 ) ˜ ; B) 4 C(N ˜ 0 ; B 0 ) and C(N ˜ ; B) 6= C(N ˜ 0 ; B 0 ). Rotation allocation policy (N ; B) is de ned to be minimal i C(N if there exists no rotation allocation policy that dominates (N ; B). (Hereafter we simply call a minimal rotation allocation policy a minimal policy.) ˜ ; B) 4 C(N ˜ 0 ; B 0 ) is equivalent to the We give two candidates of minimal policies. First note that C(N condition t0 | (N 0 ; B 0 ) − (N ; B)|6m(N 0 ; B 0 ) − m(N ; B) from Theorem 1. Further, this is equivalent to conditions m(N ; B) − t0 (N ; B)6m(N 0 ; B 0 ) − t0 (N 0 ; B 0 )
(3)
m(N ; B) + t0 (N ; B)6m(N 0 ; B 0 ) + t0 (N 0 ; B 0 ):
(4)
and
That is, n X
Z (mk − t0 k )
k=1
+
n X k=1
∞
Nk +Bk
(x − Nk − Bk ) dFk (x) +
wk Bk Fk (Nk ) +
n X
Z wk
k=1 n X k=1
uk (Nk + Bk ) +
n X k=1
Z uk
0
Nk
Nk +Bk
Nk
(Nk + Bk − x) dFk (x)
(Nk − x) dFk (x)
H. Katagiri, H. Ishii / Fuzzy Sets and Systems 111 (2000) 87–97
Z n X 6 (mk − t0 k )
Nk0 +Bk0
k=1
+
∞
n X
wk Bk0 Fk (Nk0 ) +
k=1
Z (mk + t0 k )
k=1
+
n X
∞
Nk +Bk
wk Bk Fk (Nk ) +
uk (Nk0
Z (mk + t0 k )
uk (Nk + Bk ) +
n X
n X
∞
wk Bk0 Fk (Nk0 ) +
k=1
n X
Z uk
wk Nk
0
Nk0
0
Z
Z uk
k=1
Nk0 +Bk0
k=1 n X
+
Bk0 ) +
k=1
k=1
n X
+
n X
Z wk
Nk0
Nk +Bk
n X k=1
uk (Nk0 + Bk0 ) +
k=1
n X
Z uk
k=1
(Nk0 + Bk0 − x) dFk (x)
(5)
(Nk + Bk − x) dFk (x)
(Nk − x) dFk (x)
(x − Nk0 − Bk0 ) dFk (x) +
n X
Nk0 +Bk0
(Nk0 − x) dFk (x)
Nk
k=1 n X
n X k=1
(x − Nk − Bk ) dFk (x) +
k=1
6
−
Bk0 ) dFk (x) +
k=1
and n X
(x −
Nk0
91
0
Nk0
Z wk
Nk0 +Bk0
Nk0
(Nk0 + Bk0 − x) dFk (x)
(Nk0 − x) dFk (x):
(6)
From (3) and (4), if there exists a rotation allocation policy minimizing both m(N ; B) − t0 (N ; B) and m(N ; B) + t0 (N ; B), it is a minimal policy. If there exists such a policy, it minimizes also m(N ; B) though usually there does not exist such a policy. Theorem 2. A rotation allocation policy (N ∗ ; B ∗ ) that minimizes m(N ; B) is a minimal policy. Proof. If (N ∗ ; B ∗ ) is not a minimal policy, then there exists a rotation allocation policy (N o ; B o ) such that t0 | (N ∗ ; B ∗ )− (N o ; B o )|6m(N ∗ ; B ∗ )−m(N o ; B o ) and (m(N ∗ ; B ∗ ); (N ∗ ; B ∗ )) 6= (m(N o ; B o ); (N o ; B o )) from Theorem 1. However, this is impossible since in this case t0 | (N ∗ ; B ∗ ) − (N o ; B o )|¿0¿m(N ∗ ; B ∗ ) − m(N o ; B o ) holds from m(N ∗ ; B ∗ ) being minimum. Minimization of m(N ; B) can be done by using the procedure of Nose et al. [8] since the problem is the very same as that of Nose et al. [8] if the ordinary shortage cost sk in [8] is set to mk (center of our fuzzy shortage cost s˜k ). This solution is the same as that of the ordinary one in [8]. But in some cases we pursue a minimal policy with the least ambiguity. Then the solution becomes di erent to the ordinary one. We show this minimal solution next. First, we consider the minimization of Z ∞ n X k (x − Nk − Bk ) dFk (x); (N ; B) = k=1
PA: Minimize
Nk +Bk
n X k=1
subject to
n X k=1
Z k
∞
Nk +Bk
Nk = N;
(x − Nk − Bk ) dFk (x) n X
Bk = B;
k=1
N1 ; N2 ; : : : ; Nn ; B1 ; B2 ; : : : ; Bn ¿0:
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H. Katagiri, H. Ishii / Fuzzy Sets and Systems 111 (2000) 87–97
R∞ Let us de ne k (Nk ; Bk ) ≡ k Nk +Bk (x − Nk − Bk ) dFk (x); k = 1; : : : ; n, where we assume that each k ¿0 for a while since otherwise k (Nk ; Bk ) = 0 and we omit this term from (N ; B). Then, since @ k (Nk ; Bk ) = k Fk (Nk + Bk ) − k ; @Nk @ 2 k (Nk ; Bk ) = k fk (Nk + Bk ); @Nk2
@ k (Nk ; Bk ) = k Fk (Nk + Bk ) − k ; @Bk @ 2 k (Nk ; Bk ) = k fk (Nk + Bk ) @Bk2
and @ 2 k (Nk ; Bk ) = k fk (Nk + Bk ); @Nk @Bk it holds that @ 2 k (Nk ; Bk ) @ 2 k (Nk ; Bk ) − @Nk2 @Bk2
@ 2 k (Nk ; Bk ) @Nk @Bk
2 =0
and this implies that the function k (Nk ; Bk ) is convex. Thus, (N ; B) is convex since it is the sum of convex function and hence PA is a convex programming problem. In order to solve PA, we construct the following Lagrange function: ! ! Z ∞ n n n X X X k (x − Nk − Bk ) dFk (x) + Nk − N + Bk − B : L(N ; B) ≡ k=1
Nk +Bk
k=1
k=1
Then an optimal solution of PA is that of the following Kuhn–Tucker condition KT: KT :
@L(N ; B) = k Fk (Nk + Bk ) − k + ¿0; @Nk @L(N ; B) = k Fk (Nk + Bk ) − k + ¿0; @Bk n n X X Nk = N; Bk = B; k=1
@L · Nk = 0; @Nk
k = 1; : : : ; n;
k=1
@L · Bk = 0; @Bk
k = 1; : : : ; n:
We show the solution procedure of PA in case of B = 0 (N = 0) and in case that both N and B are positive, respectively. Case (a): B = 0 (N = 0). If B = 0 (N = 0), then all Bk = 0, k = 1; : : : ; n (Nk = 0, k = 1; : : : ; n). And if both N and B = 0, the optimal solution of PA is Nk = 0; Bk = 0; k = 1; : : : ; n. If either N or B = 0, we nd a solution of KT as follows. We only show the case of B = 0 since the case N = 0 is the very same as B = 0. Since B = 0, i.e., Bk = 0; k = 1; : : : ; n, KT becomes as follows: KT0 :
@L(N ) = k Fk (Nk ) − k + ¿0; @Nk @L(N ) · Nk = 0; @Nk
k = 1; : : : ; n;
n X k=1
Nk = N:
H. Katagiri, H. Ishii / Fuzzy Sets and Systems 111 (2000) 87–97
93
The following procedure PKT nds a solution of KT0 . PKT: 1. First n locations are arranged and renumbered so as to be the following order, 1 6 · · · 6 n . Further let us de ne Ii ≡ ( i ; i+1 ); i = 0; : : : ; n − 1 where 0 ≡ 1 (1 − F1−1 (N )). Go to 2. 2. Set i = 0 and go to 3. 3. Set ∈ Ii . Then the following are obtained: Nk = 0;
k = 1; : : : ; i; ; k = i + 1; : : : ; n: Nk = Fk−1 1 − k Pn Pn If k=i+2 Fk−1 (1 − ( i+1 = k ))¿N , then go to 4. Otherwise, nd o such that k=i+1 Fk−1 (1 − (o = k )) = N and set Nk = Fk−1 (1 − (o = k )); k = i + 1; : : : ; n, and Nk = 0; k = 1; : : : ; i. Terminate. 4. Set i = i + 1 and return to 3. Theorem 3. The above procedure PKT nds an optimal solution of PA in case B = 0. Proof. From KT0 ; Nk is positive if and only if ¡ k and so the validity of the above procedure is deduced. Case (b): N ¿0 and B¿0. Next we consider a solution of KT when both N and B are positive. First we de ne Tk ≡ Nk + Bk ; k = 1; : : : ; n, and T ≡ N + B. In this case, it may be cumbersome to nd a solution of PA from that of KT. Noting that the objective function of PA is a function of T1 ; T2 ; : : : ; Tn , we transform PA into the following problem: Z ∞ n X k (x − Tk ) dFk (x) PAT: Minimize subject to
k=1 n X
Tk
Tk = T;
Tk ¿0; k = 1; 2; : : : ; n:
k=1
An optimal solution of PA is obtained from an optimal solution Tko ; k = 1; 2; : : : ; n, of PAT by solving the following simultaneous linear equations SLP: SLP: Nk + Bk = Tko ; n X Nk = N; k=1
k = 1; : : : ; n; n X Bk = B; N1 ; : : : ; Nn ; B1 ; : : : ; Bn ¿0: k=1
Since SLP has 2n unknowns and n + 2 equations (one of them is redundant), its solution exists except for the trivial case n = 1 for which we need not make any e orts to nd a solution of PA. Further it has another merit from the following reason. We seek a minimal solution and so minimization of m(N ; B) among optimal solutions of PA should be done since there exists n − 1 free variables in SLP. Therefore, we consider the following problem: ) ( Z o Z Nk n Tk X o o (Tk − x) dFk (x) + wk (Tk − Nk )Fk (Nk ) + uk (Nk − x) dFk (x) wk PAM: Minimize C + k=1
subject to
n X k=1
Nk = N;
Nk
06Nk 6Tko ; k = 1; : : : ; n;
0
94
H. Katagiri, H. Ishii / Fuzzy Sets and Systems 111 (2000) 87–97
R∞ Pn where C ≡ k=1 {mk T o (x − Tko ) dfk (x) + uk Tko } is the constant part of m(N ; B) when Tk = Tko ; k = 1; : : : ; n. k Note that constraints of PAM is equivalent to those of SLP since we can derive the constraint 06Nk 6Tko ; k = 1; : : : ; n of SLP by simple calculations. By solving PAT, SLP and PAM, an optimal solution of PA in case that both N and B are positive is obtained. It is the solution that minimizes m(N ; B) under the condition that the ambiguity is minimized. Now we show the solution procedure of PAT. Again we construct the following Lagrangian L(T1 ; : : : ; Tn ; ) corresponding to the problem PAT: ! Z ∞ n n X X L(T1 ; T2 ; : : : ; Tn ; ) ≡ k (x − Tk ) dFk (x) + Tk − T : k=1
Tk
k=1
From L(T1 ; T2 ; : : : ; Tn ; ), we obtain the following Kuhn–Tucker condition: KTT:
@L = k (Fk (Tk ) − 1) + ¿0; @Tk n X @L · Tk = 0; k = 1; : : : ; n; Tk = T: @Tk k=1
Since PAT is a convex programming problem, a solution of KTT is an optimal solution of PAT. Further KTT is the same as KT0 if Nk ; k = 1; : : : ; n, and N are replaced by Tk ; k = 1; : : : ; n and T , respectively. So we omit the solution procedure of KTT and show only the solution procedure of PAM supposing that an optimal solution of PAT is found. Note that the objective function of PAT is transformed into the following m(N ˆ ) where the constant term C is deleted: ) ( Z o Z Nk n Tk X m(N ˆ )≡ Fk (x) dx + uk (Nk − x) dFk (x) : wk k=1
Nk
0
Here we assume that all Tko ; k = 1; : : : ; n, are positive without loss of generality since otherwise if Tko0 = 0 for some k 0 , we can set Nk 0 = 0, renumber k by skipping such k 0 and reset n = n0 where n0 is the number of ˆ ) is convex since positive Tko . Further this function m(N @2 mˆ = (uk − wk )fk (Nk )¿0 @Nk2 from the assumption uk ¿wk ; k = 1; : : : ; n (i.e., each term of the above sum in m(N ˆ ) is convex) and the sum of convex functions is also convex. Let us construct a Lagrangian to solve PAM. Then a solution of the following Kuhn–Tucker condition KTM is an optimal solution of PAM: ) ( Z o Z Nk n Tk X L(N ; ; 1 ; : : : ; n ) = Fk (x) dx + uk (Nk − x) dFk (x) wk k=1
+ N −
Nk
n X k=1
KTM: Pn
0
! Nk
+
n X
k (Nk − Tko ) :
k=1
k=1 Nk = N . For each k; k ¿0 and exactly one of following conditions holds: 1. Nk = 0; (uk − wk )Fk (0) − ¿0.
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2. Nk = Tko ; (uk − wk )Fk (Tko ) − + k 60. 3. Tko ¿Nk ¿0; (uk − wk )Fk (Nk ) − = 0. Thus, we obtain the following procedure: PKTM: 1. Let a1 = (u1 −w1 )F1 (T1o ); : : : ; an = (un −wn )Fn (Tno ) and b1 = (u1 −w1 )F1 (0); : : : ; bn = (un −wn )Fn (0). Sorting these ak ; bk ; k = 1; : : : ; n, let the result be c1 ¡c2 ¡ · · · ¡c nˆ where nˆ is the number of di erent ak ; bk ; k = 1; : : : ; n. Further let Jk = (ck ; ck+1 ); k = 1; : : : ; nˆ − 1. Set i = 1 and go to 2. 2. Let Ai = {k | ak 6ci }; Ei = {k | bk ¿ci+1 } and Gi = {1; 2; : : : ; n} − Ai − Ei . Further set Nk = Tko for k ∈ Ai ; P Nk = 0 for k ∈ Ei and Nk = Fk−1 (=(uk −wk )) for k ∈ Gi . If there exists = i ∈ Ji such that k∈Gi Nk = N − P −1 i o o k∈Ai Tk , set an optimal solution of PAM as Nk = Fk ( =(uk − wk )) for k ∈ Gi ; Nk = Tk for k ∈ Ai and Nk = 0 for k ∈ Ei and terminate. Otherwise, go to 3. 3. Set i = i + 1 and return to 2. We denote an optimal solution of PAM by N o . Then the following B o together with this N o constitutes an optimal solution of PA: Bko = Tko − Nko ;
k = 1; 2; : : : ; n:
Theorem 4. The above (N o ; B o ) becomes an optimal solution of PA minimizing (N ; B). Proof. If 6 all bk ; k = 1; : : : ; n, then all Nk = 0 from Kuhn–Tucker condition KTM. But this contradicts the assumption that N 6= 0. Further from the monotonicity of the distribution function Fk ; c1 must be some bk . So we can start to check the region including a solution of KTM. From alternative conditions 1–3 and the condition of k ¿0 in KTM, if ¿ak , then Nk must be Tko and if 6bk , then Nk must be 0. Further, if exists between bk and ak , then we can set Nk = Fk−1 (=(uk − wk )). Consequently, Pn the validity of setting of iteration i in 2 of the above procedure PKTM is derived. Since B¿0; N 6T ( k=1 Tko ); must be less than c nˆ . Therefore PKTM terminates before i = Nˆ . The above discussion assures the validity of PKTM to nd an optimal solution N of PAM. Thus (N o ; B o ) gives an optimal solution of PA minimizing (N ; B). In order to check the minimality of (N o ; B o ), we consider another problem: PMA: Minimize
m(N ; B) + t0 (N ; B) Z n X = (mk + t0 k ) k=1
Z
+ wk
NK +Bk
Nk +Bk
Nk
subject to
k=1
Nk = N;
(x − Nk − Bk ) dFk (x)
(Nk + Bk − x) dFk (x) + wk Bk Fk (Nk ) Z
+ (Nk + Bk )uk + uk n X
∞
n X
Nk
0
Bk = B;
(Nk − x) dFk (x) N1 ; : : : ; Nn ; B1 ; : : : ; Bn ¿0:
k=1
PMA is solved by using the procedure of Nose et al. [8] since again this problem is the same as that of the problem in [8] if sk is replaced by mk + t0 k . Let a solution with minimum m(N ; B) among optimal solutions of PMA be (N u ; B u ).
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Theorem 5. (N u ; B u ) is a minimal solution of our problem and if m(N u ; B u ) = m(N o ; B o ) and (N u ; B u ) = (N o ; B o ); then (N o ; B o ) is also a minimal solution. Proof. If (N o ; B o ) is not a minimal solution, then from Theorem 1, there exists another solution of our problem (N d ; B d ) such that m(N o ; B o ) + t0 (N o ; B o )¿m(N d ; B d ) + t0 (N d ; B d ); since (N o ; B o ) is an optimal solution of PA. This implies that (N o ; B o ) is also an optimal solution of PMA. As for minimality of (N u ; B u ), rst note that sup{t¿0 | L((t − m(N ; B))= (N ; B))¿0} = m(N ; B) + t0 (N ; B): If (N u ; B u ) is not minimal, then there exists a solution (N C ; B C ) such that m(N u ; B u ) + t0 (N u ; B u )¿m(N C ; B C ) + t0 (N C ; B C ): But this is impossible since (N u ; B u ) is an optimal solution of PMA.
5. Conclusion We have shown some minimal solutions of our problem. Note that Theorem 5 shows that (N o ; B o ) may be a minimal solution with respect to our ordering but this includes the least “ambiguity”. If we also need a solution with the least “ambiguity” among alternatives, then (N o ; B o ) is a suitable solution. Together with shortage cost, outdating cost may be ambiguous and so we must consider perishable inventory models with both costs as fuzzy numbers. We are now tackling this problem with more general models than our model in this paper in a sense that they include other fuzzy costs though analysis becomes very complicated. Even in the case where only the shortage cost is ambiguous, we must consider the policy not only minimizing the most possible total cost m(N ; B) but also minimizing the total cost in the worst case m(N ; B) + t0 (N ; B) (further in the most optimistic case m(N ; B) − t0 (N ; B)). More generally, we can derive minimal solutions from minimizing m(N ; B) + (2 − 1)t0 (N ; B) for various between 0 and 1. (Note that minimizing m(N ; B) corresponds to the case = 1=2; = 1 corresponds to minimization of the worst case and = 0 to the most optimistic case.) Finally note that in more general models with fuzzy costs, in uence of the variation on these costs may change “optimal” policies drastically. References [1] N. Furukawa, A parametric total order on fuzzy numbers and a fuzzy shortest route problem, Optimization 10 (1994) 367–377. [2] H. Ishii, Perishable inventory problem with two types of customers and di erent selling prices, J. Oper. Res. Soc. Jpn. 36 (1993) 199 –205. [3] H. Ishii, T. Nose, S. Shiode, T. Nishida, Perishable inventory management subject to stochastic lead-time, European J. Oper. Res. 8 (1981) 76 – 85. [4] A. Kaufmann, M.M. Gupta, Introduction to Fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold, New York, 1985. [5] S. Nahmias, Optimal and approximate ordering policies for a perishable product subject to stochastic demand, Doctoral Dissertation, University of Northwestern, 1972. [6] S. Nahmias, Perishable inventory theory: a review, Oper. Res. 30 (1982) 680 –708. [7] T. Nose, H. Ishii, T. Nishida, Some properties of perishable inventory control subject to stochastic lead-time, J. Oper. Res. Soc. Jpn. 24 (1981) 110 –135. [8] T. Nose, H. Ishii, T. Nishida, LIFO allocation problem for perishable commodities, J. Oper. Res. Soc. Jpn. 26 (1983) 135 –145.
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