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A fuzzy EOQ model with demand-dependent unit cost under limited storage capacity
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER
European Journal of OperationalResearch 99 (1997) 425-432
Theory and Methodology
A fuzzy EOQ model with demand-dependent unit cost under limited storage capacity T.K, Roy, M. Maiti * Department of Appged Mathematics, Vidyasagar University, Midnapore 721 102, West Bengal, India
Received 28 March 1995; accepted22 April 1996
Abstract
A fuzzy EOQ model is developed with limited storage capacity where demand is related to the unit price and the setup cost varies with the quantity produced/purchased. Here fuzziness is introduced in both objective function and storage area. It is solved by both fuzzy nonlinear and geometric programming techniques for linear membership functions. The model is illustrated with a numerical example and a sensivity analysis is made. Generalisation to a multi-item problem is also presented and its numerical results are compared with those of the crisp model. © 1997 Elsevier Science B.V. Keywords: Inventory;Fuzzy sets; Nonlinear programming;Geometricprogramming
1. Introduction
For several years, classical economic order quantity (EOQ) problems with different variations were solved by many researchers and had been reported in the reference books and survey papers (e.g. Hadley and Whitin, 1963; Whitin, 1954; Taha, 1976; Clark, 1972; Urgeletti Tinnarelli, 1983). Recently, for a single product with demand related to unit price Cheng (1989) and for multi-products with several constraints, Worrall and Hall (1982) have solved the EOQ model by geometric programming method. All these models are developed in the crisp environment which does not depict the real production situation fully. In 1965, the first publication in fuzzy set theory by Zadeh (1965) showed the intention to accommo* Corresponding author.
date uncertainty in the nonstochastic sense rather than the presence of random variables. After that, the fuzzy set theory has been applied in many fields including production related areas. Sommer (1981) applied fuzzy dynamic programming to an inventory and production scheduling problem in which the management wishes to fullfil a contract for providing a product and then withdraw from the market. Kacprzyk and Staniewski (1982) considered the determination of optimal of firms from a global view point of top management in a fuzzy environment with fuzzy constraints imposed on replenishments and a fuzzy goal for preferable inventory levels to be attained. Instead of minimizing the average inventory cost, they reduced it to a multi-stage fuzzy decision-making problem and used a branch and bound algorithm in which a fuzzy optimal decision is the intersection of fuzzy constraints and fuzzy goal.
0377-2217/97/$17.00 Copyright © 1997 Elsevier Science All rights reserved. Pll S0377-221 7(96)001 63 -4
426
T.K. Roy, M. Maiti / European Journal of Operational Research 99 (1997) 425-432
Park (1987) examined the EOQ formula in the fuzzy set theoretic perspective associating the fuzziness with the cost data. Here, inventory costs were represented by trapezoidal fuzzy numbers (TrFN) and the EOQ model was transformed to a fuzzy optimization problem. Insted of solving it, he gave an equivalent crisp expression of average inventory cost. But none has considered the space constraint with the objective goal in fuzzy environment and attacked the fuzzy optimization problem directly using either fuzzy nonlinear or fuzzy geometric programming technique. Till now, no literature is available for the inventory models of multi-items in fuzzy environment with or without constraints. In this paper, an EOQ model is developed where unit price varies inversely with the demand and setup cost increases with the increase of production. In industry, total expenditure for production and storage area are normally limited but imprecise, uncertain and flexible. These are defined within some ranges. Here, statistical analysis also is not applied as these parameters can not be estimated due to the lack of statistical observations. However, these nonstochastic and illformed inventory models can be realistically represented in the fuzzy environment. The problem is reduced to a fuzzy optimization problem associating fuzziness with the storage area and total expenditure. The optimum order quantity is evaluated by both fuzzy nonlinear programming (FNLP) and fuzzy geometric programming (FGP) methods and the results are obtained for linear membership functions. The model is illustrated with a numerial example and with the variation in tolerance limits for both storage area and total expenditure. A sensitivity analysis is presented. Fuzzy formulation for the multi-item inventory model is also presented and the numerical results for fuzzy and crisp models are compared.
Min
1 C( D, q) = Co3qe- lD + KDl-13 + ~Clq
s.t.
Aq < B, D,q>0. (1)
where
q = number of order quantity, D = demand per unit time; C~ = holding cost per item per unit time, C 3 = setup cost = Co3q~, (C03(> 0) and 3'(0 < 3' < 1) are constants), p = unit production cost = K D - ~ ( K ( > 0) and /3(> 1) are constants).
Here lead time is zero, no back order is permitted and replenishment rate is infinite. The above model in a fuzzy environment is Min
1 C( D, q) = Co3q ~- ID + KD 1-~ + 2Clq
s.t.
Aq < IB, D,q>O.
(2) (A wavy bar ( ~ ) parameters.)
represents fuzzification of the
3. Mathematical analysis
3.1. Fuzzy nonlinear programming (FNLP) We consider a fuzzy nonlinear programming problem with fuzzy resources and objective as Min
go(x)
(3)
2. Mathematical model
s.t.
gi( X) ~_ hi
A single item inventory model with demand-dependent unit price and variable setup cost under storage constraint is formulated as
In fuzzy set theory, the fuzzy objective and fuzzy resources are obtained by their membership functions, which may be linear or nonlinear. Here/x 0 and /z i (i = 1,2,3 . . . . . m) are assumed to be nonincreas-
i = 1,2 . . . . . m.
427
T.K. Roy, M. Maiti / European Journal of Operational Research 99 (1997) 425-432
ing continous linear membership functions for objective and resources respectively such as
OL --=0, Oxj
I&i( gi( x) ) if gi( x) < h i ,
1
gi( X) -- b i =
1
Pi 0
plies that optimal values x I , x 2 . . . . . A~, A~ . . . . . A,~ should satisfy j=l,2
x~, a o , a I ,
. . . . . n,
OL if b i <_ gi( x) <_ b i + Pi,
0~ - o,
(5)
A i ( g i ( x ) - b i - (1 - a ) Pi) = O,
if gi( x) > b i + Pi,
i = 0 , ,2 . . . . . m. In this formulation, the fuzzy objective goal is b 0 and its corresponding tolerance is P0 and for the fuzzy containts, the goals are bg's and their corresponding tolerances are Pi's (i = 1,2, 3 . . . . . m). To solve the problem (3), we use the max-min operator of Bellman and Zadeh (1970) and the appraoch of Zimmermann (1976). The membership function of the decision set,
/xv(x), is
gi( X) <_ b i + (1 - a ) P / , hi_<0, i = 0 , 1 , 2 . . . . . m. Moreover, Kuhn-Tucker's sufficient condition demands that the objective function for maximization and the constraints should be respectively concave and convex. In this formulation, it can be shown that both objective function and constraints satisfy the required sufficient conditions. Now, solving (5), the optimal solution for the FNLP problem is obtained.
/xD(x) = min { / x 0 ( x ) , / * l ( x ) , / x 2 ( x ) . . . . . /Xm(X)} for all x ~ X The min operator is used here to model the intersection of the fuzzy sets of objective and constraints. Since the decision maker wants to have a crisp decision proposal, the maximizing decision will correspond to the value of x, Xmax that has the highest degree of membership in the decision set. J~D(Xmax)
= max [ m i n { / x 0 ( x ) , / x , ( x ) . . . . . /*m( x)}].
3.2. Fuzzy geometric programming (FGP) If the objective function and the constraints, g0(x) and gi( x) (i = 1, 2 . . . . . m) are of posynomial form, then the problem (3) reduces to a fuzzy geometric programming (FGP) problem. Proceeding as before, the expression (4) is obtained in an alternative form as Min
a- 1
s.t.
gi( X) Pi - + a_< 1, bi + Pi bi + Pi
x_>0
It is equivalent to solving the following crisp nonlinear programming problem Max s.t.
x_>0,
a /*0(x) >_ a ,
txi( x) >_ a ,
i = 1,2 . . . . . m
x___0,
o ~ (0,1).
(4)
A new function, i.e. the Lagrangian function L( a , x, A) is formed by introducing (m + 1) Lagrangian multipliers A = ()t 0, )tl, )t2 . . . . . h m) as
(6)
a~ (0,1).
where x = (x 1, X 2 . . . . . Xn )T. Now we solve the problem (6) by the usual crisp geometric programming algorithm.
4. Solution o f the proposed inventory model
L ( f f , x,)t) = a - ~ ) t i ( g i ( x ) - b i - ( 1 - a ) P i ) . i=0
4.1. By FNLP
The necessary condition of Kuhn and Tucker (1951) for the optimal solution to this problem im-
The proposed inventory model depicted by Eq. (2),
T.K.Roy,M. Maiti/ EuropeanJournalof OperationalResearch99 (1997)425-432
428
Min
1 C( D, q) = Co3qY- lD + KDl-~ + TClq
s.t.
Aq <_B, D,q>O,
4.2. By FGP From Eq. (2), we have, as per Eq. (6),
reduces to, following Eq. (4), Max
Min
a- 1
s.t.
BjDq ~-1 + B 2 D l - B + B 3 q + B 4 o t < _ Bsq+ B6a< l, D,q>O, a ~ (0,1),
a Co3 qr- 1D + KD l - 13 + 71 C~ q
s.t.
_ C O + (1 - a ) P o ,
(7)
AQ<_B+(1-a)P, D,q>O, a e (0,1).
(8) where B l = Co3/(C o + Po), B2 = K/(Co + Po), B3 = C 1 / 2 ( C o + Po), B4 = Po/(Co + Po), B5 = A / ( B
Here, the objective goal is C O with tolerance P0 and the space constraint goal is B with tolerance P. So, the corresponding Lagrangian function is
+ P), B6 = P / ( B
+ P).
Here the number of degrees of difficulty = 3 and the dual of Eq. (8) is given by Max d ( w )
L ( a , D , q, a l, a2) = a-
I-
al(Co3q ~- ~D + KD 1-~
+½C, q - C o - ( 1 - a ) P o ) -Az( Aq-B-(1-a)P). From K u h n - T u c k e r ' s necessary conditions, we have
Co3q'/-1D + KD I-8 + ~I C l q _ Co_ (1 - a ) P o < O, Aq-B(1 - a ) P < O, AI(Co3qy-ID + KD '-~ + ½ C l q - CO- (1 - a ) P o )
O, A2(Aq-B-
(1-or)P)
x
ti~lWi )
x(%
I-W55) ~-~6 ]
+ w6) W,+w0 '
where W 0 = 1,
Wo -~ W4-~-W5 = l, W l "l- ( l -- ~ ) W 2 = 0 ,
=0,
( ' ~ - - 1 ) W 1 -b'W 3 --]-W5 = 0 .
/~'1, A2 ~ 0, 1 -- A l P o -
1,
A2P = 0,
A , ( C o 3 ( Y - 1 ) q v - e D + ½Ci) + AzA = 0 , hl(C03q y-I q-K(1
-- ] 3 ) D - / 3 ) = 0.
Solving these equations, optimum quantities are B+(1--a*)e q = A
Let W I = t~, W 3 = t 2, W4= t 3. solving the above equation, we get tl 1412- 1( 13 - -~ ' Ws=tl(1-y)-t2' W6= 1 - t 3 and then, the above dual expression becomes Max d( tl, t2, t3)
=(~l )t'(Bff-lt----~-)~-1(B-~3)12(~3 t2 where a * is a root of
×
C03
(l_y)tl_t2 13
H(B+(1-.)P) -~
2A
so C * ( D * , q * ) = l * + ~Clq •
and
-(1-
a)Po=O
Co3q*~'-lD * + K D *l-~
×
t~ + t 2 + t 3
X(1 + ( 1 -- T ) t , + t 2 + t 3 ) '+(l-y)q+t2+t3
T.K. Roy, M. Maiti / European Journal of Operational Research 99 (l 997) 425-432
429
Table 1 Optimal values for proposed inventory model Model
Method
Quantity q *
Demand D *
Average total cost C * ( D *, q * )
a *
Aq *
Fuzzy model
FNLP FGP
6.0449 6.043
9.8115 9.8068
53.9324 53.9328
0.3033 0.3043
60.94 60.94
Crisp model
NLP
5
9.21
54.43
1
50
Solving the equations Od/Ot~ = O, Od/Ot 2 = O, Od/Ot 3 = O, t I , t 2 , t~ are evaluated and hence Wo*,
5. N u m e r i c a l
Wl*, W2*, W3*, W4*, Ws*, W6* are also determined. Therefore, optimum values are
For a particular EOQ problem, let C03 = $4, K = 100, C = $ 2 , y = 0 . 5 , /3 = 1.5, A = 10 units, B = 50 units, C o = $ 4 0 , P0 = $ 2 0 , P = 15 units. For these values, the optimal value of production batch quantity q *, optimal demand rate D* and minimum average total cost C* ( D * , q * ) and O/* obtained by FNLP and FGP are given in Table 1. Total average cost is less in the case of the fuzzy model than that obtained for the crisp model.
q*=
W3* B"'~ ( W I * + W 2 * + W 3 * " ] - W 4 * )
1'
1 D* =
2 (Wl,
nt. W2, " t W 3 , _ t _ W 4 , ) - i
1 /3
example
W4*
O/* =
B;
( w l * Jl- W2* '~- W3* 't- W4*)
1
and C*(D
6. S e n s i t i v i t y
*
,q
*
)=Co3q*Z'-lD
*
1 +KD*l-~+~Ciq
analysis
*
There are three cases for overall achivement of
So, by both FNLP and FGP techniques, the optimal values of q and D and the corresponding minimum cost are evaluated for the known values of other parameters.
O/: 1. O/max ~ 1, 2. O~max = 0,
3. 0 < Otmax < 1.
Table 2 Effect of variations in Po PO
a*
q*
D*
To
T
C * ( O * , q* )
20 25 50 100 200 500 1 000 5 000 10 000 50 000 100 000 500 000
0.3033 0.4398 0.7163 0.857 0.9282 0.9712 0.9856 0.9971 0.3985 0.9997 0.9998 0. 99997
6.04 5.84 5.42 5.21 5.12 5.04 5.02 5.00 5.00 5.00 5.00 5.00
9.81 9.70 9.46 9.34 9.27 9.24 9.23 9.21 9.21 9.21 9.21 9.21
13.93 t 4.00 14.18 14.50 14.36 14.40 14.41 14.42 14.42 14.42 14.42 14.42
10.45 8.40 4.25 2.14 1.08 0.43 0.22 0.04 0.00 0.00 0.00 0.00
53.93 54.00 54.18 54.30 54.36 54.40 54.41 54.42 54.42 54.42 54.42 54.42
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T.K. Roy, M. Maiti/ European Journal of Operational Research 99 (1997) 425-432
Table 3 Effect of variations in P P
a*
q~
D*
To
T
C*(D*, q*)
15 16 20 23 36 38 40 41 42 44 156 160 161 162
0.30 0.30 0.30 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.30 oo oo - 0.005
6.04 6.11 6.38 6.59 7.46 7.60 7.73 7.80 7.87 8.01 19.99 20.84 21.06 21.29
9.81 9.85 9.99 10.10 10.53 10.59 10.65 10.68 10.72 10.78 14.61 14.82 14.87 14.93
13.93 13.91 13.84 13.79 13.70 13.69 13.69 13.69 13.70 13.70 19.22 19.80 19.96 20.116
10.45 11.13 13.84 15.86 24.65 26.02 27.39 28.08 28.76 30.14 152.02 158.43 160.66 162.94
53.93 53.91 53.84 53.79 53.70 53.69 53.69 53.69 53.70 53.70 59.23 59.80 59.95 60.116
Following Hamacher et al. (1978), we perform sensitivity analysis in these three cases for the given numerical example. 1. O/max 1. The optimal solution is q * = 5, D * = 9.21, C * ( D *, q* ) = $54.43, which is also the optimal solution for the crisp nonlinear model without any tolerance. Here the optimal solution is invariant for the variation of P0 to P0 + A P0 o r / a n d P to p+Ap. =
2. O/max 0. In this case, the optimal solution is q* = 6.5, D* = 15.74, C * ( D * , q * ) = $56.40, which is the optimum solution for the crisp problem with m a x i m u m tolerance. =
3 . 0 < O/max< 1. For P0 = 20, the optimal solution of the problem is q * = 6.04, D * = 9.81, O/* = 0.3033. Now, changing to P0 = 21 (i.e. A P 0 = 1) we get q* = 6.00, D* = 9.78, O/* = 0.3358. The difference in the optimal values of the objective function O/ is 0.0324. Therefore, for the change of P0 by 5%, O/ changes by 10.7%.
Following Dutta et al. (1993) we study the effect of tolerance in the said EOQ model with the earlier numerical values and construct Tables 2 and 3 for the degrees of violation To ( = ( 1 - O/)P0) and r ( = (1 - O/)P) for two constraints given by Eq. (8). From Table 2, we see that: (i) For higher tolerances of P0, the value of O/max does not achieve 1 though it is very near to 1. (ii) For higher acceptable variations of P0, the optimal solutions remain invariant and the optimal solutions are very close to the solutions (q* = 5, D* = 9.21, C * ( D * , q * ) = 54.43) of the crisp model without tolerance (O/= 1). From Table 3 we observe that: (i) For different values of P, degrees of violations TO and T are never zero, i.e. different optimal solutions are obtained. (ii) As P increases from 15, the minimum average cost C * ( D * , q *) decreases and becomes m i n i m u m at P = 41 with maximum q * ( = 7.80). (iii) When P is more than 41, average cost C * ( D * , q * ) increases with P and at P = 160 or more, O/ becomes 0 or less than 0. In this case, optimal solutions are not basic.
Table 4 Input data i
C03 i
1
4
2 3
5 5
"~i 0.5 0.7 0.6
Ki 100 110 115
~i
Cli
Ai
1.5
2
5
1.6 1.8
2.3 2.5
6 5
ui 0.5 0.3 0.4
Table 5 Otherinput data Co
Po
B
P
U
Pl
160
10
90
15
3.5
2
T.K. Roy, M. Maiti / European Journal of Operational Research 99 (1997)425-432
431
Table 6 Optimal values for proposed multi-item inventory model Model
B
P
Crisp
90 105 105 90
Fuzzy
90
U
PI
3.5 3.5 5.5 5.5 15
7. G e n e r a l i s a t i o n
3.5
2
Total average cost C*(D 7 , q i * )
Totalspace required
Totalcapitalinvestment
164.12 164.12 164.01 164.12
90 90.40 104.89 90
3.5 3.5 4.09 3.5
164
to m u l t i - i t e m m o d e l
An inventory for N items under two constraints total capital investment for the quantities and total storage area is mathematically expressed as
--
Min
C( D i, qi) N = E (Co31Diq~ i - t + K i D ) -f3i+ ½Cliqi) i=1 N
s.t.
1
7 E uiqi <- U, i=1 N
E Aiqi <- B , i=1
Di, qi > O,
where u i is the cost of the ith unit product, U the total capital investment for the quantities and other parameters have the usual meanings. After fuzzification the above problem reduces to Max
c~ N
~_~ (Co3iDiq7 ~-' "+-KiD~-~i-t- lCliqi)
s.t.
i=l
96
3.74
is solved and the numerical results are presented below. Input data for the model are shown in Tables 4 and 5 and the results are shown in Table 6. In Table 6, the fuzzy result is better than the crisp model with different combinations of the constraints goals. 8. Conclusion In this paper, we solve real life inventory models for single and multi-items in fuzzy environment by FNLP and FGP techniques. Some sensivity analysis on the tolerance limits have been presented. The results of the fuzzy models are compared with those of crisp model which reveals that fuzzy models give better result than the usual crisp models. This method is quite general and can be extended to other similar inventory models including the ones with shortages. Acknowledgements The authors are thankful to the reviewers for their valuable suggestion and comments for the improvement of the paper.
_< C o + (1 - c~)P o, N |
7 ~ uiqi <- U + (1 - a ) P , , i=1 N
~_, Aiq i <_B + (1 - a ) P , i=1
Di,qi > 0 ,
ae(0,1),
where P~ is the allowable tolerance of u and other parameters are as explained earlier. Following the procedure of fuzzy nonlinear programming method out line earlier, the above problem
References Bellman, R.E. and Zadeh, L.A. (1970), "Decision-making in a fuzzy environment", Management Science 17, B141-B164. Cheng, T.C.E. (1989), " A n economic order quantity model with demand-dependent unit cost", European Journal of Operational Research 40, 252-256. Clark, AJ. (1972), " A n informal survey of multi-echelon inventory theory", Naval Research Logistics Quarterly 19, 621650. Dutta, D., Rao, J.R. and Tiwari, R.N. (1993), "Effect of tolerance in fuzzy linear fractional programming", Fuzzy Sets and Systems 55, 133-142.
432
T.K. Roy, M. Maiti / European Journal of Operational Research 99 (1997)425-432
Hadley, G. and Whitin, T.M. (1963), Analysis of Inventory Systems, Prentice-Hall, Englewood Clipps, NJ. Hamacher, H., Leberling, H. and Zimmermann, H.J. (1978), "Sensitivity analysis in fuzzy linear programming", Fuzzy Sets and Systems 1,269-281. Kacprzyk, J. and Staniewski, P. (1982), "Long term inventory policy-making through fuzzy decision making models", Fuzzy Sets and Systems 8, 117-132. Kuhn, H.W. and Tucker, A.W. (1951), "Nonlinear programming", in: J. Neyman (ed.), Proceedings Second Berkely Symposium on Mathematical Statistics and Probability, University of California Press, 481-494. Park, K.S. (1987), "Fuzzy set theoretic interpretation of economic order quantity", IEEE Transactions on Systems, Man, and Cybernetics SMC-17/6, 1082-1084. Sommer, G. (1981), "Fuzzy inventory scheduling", in: G. Lasker (ed.), Applied Systems and Cybernetics, VI, Acdameic Press, New York.
Taha, H.A. (1976), Operations Research - - An Introduction, 2nd edn., Macmillan, New York. Urgeletti Tinarelli, G. (1983), "Inventory control models and problems", European Journal of Operational Research 14, 1-12. Whitin, T.M. (1954), "Inventory control research & survey", Management Science 1, 32-40. Worrall, D.M. and Hall, M.A. (1982), "The analysis of an inventory control model using posynomial geometric programming", International Journal of Production Research 20, 657-667. Zadeh, L.A. (1965), "Fuzzy sets", Information and Control 8, 338-353. Zimmermann, H.J. (1976), "Description and optimization of fuzzy systems", International Journal of General Systems 2, 209215.
European Journal of OperationalResearch 99 (1997) 425-432
Theory and Methodology
A fuzzy EOQ model with demand-dependent unit cost under limited storage capacity T.K, Roy, M. Maiti * Department of Appged Mathematics, Vidyasagar University, Midnapore 721 102, West Bengal, India
Received 28 March 1995; accepted22 April 1996
Abstract
A fuzzy EOQ model is developed with limited storage capacity where demand is related to the unit price and the setup cost varies with the quantity produced/purchased. Here fuzziness is introduced in both objective function and storage area. It is solved by both fuzzy nonlinear and geometric programming techniques for linear membership functions. The model is illustrated with a numerical example and a sensivity analysis is made. Generalisation to a multi-item problem is also presented and its numerical results are compared with those of the crisp model. © 1997 Elsevier Science B.V. Keywords: Inventory;Fuzzy sets; Nonlinear programming;Geometricprogramming
1. Introduction
For several years, classical economic order quantity (EOQ) problems with different variations were solved by many researchers and had been reported in the reference books and survey papers (e.g. Hadley and Whitin, 1963; Whitin, 1954; Taha, 1976; Clark, 1972; Urgeletti Tinnarelli, 1983). Recently, for a single product with demand related to unit price Cheng (1989) and for multi-products with several constraints, Worrall and Hall (1982) have solved the EOQ model by geometric programming method. All these models are developed in the crisp environment which does not depict the real production situation fully. In 1965, the first publication in fuzzy set theory by Zadeh (1965) showed the intention to accommo* Corresponding author.
date uncertainty in the nonstochastic sense rather than the presence of random variables. After that, the fuzzy set theory has been applied in many fields including production related areas. Sommer (1981) applied fuzzy dynamic programming to an inventory and production scheduling problem in which the management wishes to fullfil a contract for providing a product and then withdraw from the market. Kacprzyk and Staniewski (1982) considered the determination of optimal of firms from a global view point of top management in a fuzzy environment with fuzzy constraints imposed on replenishments and a fuzzy goal for preferable inventory levels to be attained. Instead of minimizing the average inventory cost, they reduced it to a multi-stage fuzzy decision-making problem and used a branch and bound algorithm in which a fuzzy optimal decision is the intersection of fuzzy constraints and fuzzy goal.
0377-2217/97/$17.00 Copyright © 1997 Elsevier Science All rights reserved. Pll S0377-221 7(96)001 63 -4
426
T.K. Roy, M. Maiti / European Journal of Operational Research 99 (1997) 425-432
Park (1987) examined the EOQ formula in the fuzzy set theoretic perspective associating the fuzziness with the cost data. Here, inventory costs were represented by trapezoidal fuzzy numbers (TrFN) and the EOQ model was transformed to a fuzzy optimization problem. Insted of solving it, he gave an equivalent crisp expression of average inventory cost. But none has considered the space constraint with the objective goal in fuzzy environment and attacked the fuzzy optimization problem directly using either fuzzy nonlinear or fuzzy geometric programming technique. Till now, no literature is available for the inventory models of multi-items in fuzzy environment with or without constraints. In this paper, an EOQ model is developed where unit price varies inversely with the demand and setup cost increases with the increase of production. In industry, total expenditure for production and storage area are normally limited but imprecise, uncertain and flexible. These are defined within some ranges. Here, statistical analysis also is not applied as these parameters can not be estimated due to the lack of statistical observations. However, these nonstochastic and illformed inventory models can be realistically represented in the fuzzy environment. The problem is reduced to a fuzzy optimization problem associating fuzziness with the storage area and total expenditure. The optimum order quantity is evaluated by both fuzzy nonlinear programming (FNLP) and fuzzy geometric programming (FGP) methods and the results are obtained for linear membership functions. The model is illustrated with a numerial example and with the variation in tolerance limits for both storage area and total expenditure. A sensitivity analysis is presented. Fuzzy formulation for the multi-item inventory model is also presented and the numerical results for fuzzy and crisp models are compared.
Min
1 C( D, q) = Co3qe- lD + KDl-13 + ~Clq
s.t.
Aq < B, D,q>0. (1)
where
q = number of order quantity, D = demand per unit time; C~ = holding cost per item per unit time, C 3 = setup cost = Co3q~, (C03(> 0) and 3'(0 < 3' < 1) are constants), p = unit production cost = K D - ~ ( K ( > 0) and /3(> 1) are constants).
Here lead time is zero, no back order is permitted and replenishment rate is infinite. The above model in a fuzzy environment is Min
1 C( D, q) = Co3q ~- ID + KD 1-~ + 2Clq
s.t.
Aq < IB, D,q>O.
(2) (A wavy bar ( ~ ) parameters.)
represents fuzzification of the
3. Mathematical analysis
3.1. Fuzzy nonlinear programming (FNLP) We consider a fuzzy nonlinear programming problem with fuzzy resources and objective as Min
go(x)
(3)
2. Mathematical model
s.t.
gi( X) ~_ hi
A single item inventory model with demand-dependent unit price and variable setup cost under storage constraint is formulated as
In fuzzy set theory, the fuzzy objective and fuzzy resources are obtained by their membership functions, which may be linear or nonlinear. Here/x 0 and /z i (i = 1,2,3 . . . . . m) are assumed to be nonincreas-
i = 1,2 . . . . . m.
427
T.K. Roy, M. Maiti / European Journal of Operational Research 99 (1997) 425-432
ing continous linear membership functions for objective and resources respectively such as
OL --=0, Oxj
I&i( gi( x) ) if gi( x) < h i ,
1
gi( X) -- b i =
1
Pi 0
plies that optimal values x I , x 2 . . . . . A~, A~ . . . . . A,~ should satisfy j=l,2
x~, a o , a I ,
. . . . . n,
OL if b i <_ gi( x) <_ b i + Pi,
0~ - o,
(5)
A i ( g i ( x ) - b i - (1 - a ) Pi) = O,
if gi( x) > b i + Pi,
i = 0 , ,2 . . . . . m. In this formulation, the fuzzy objective goal is b 0 and its corresponding tolerance is P0 and for the fuzzy containts, the goals are bg's and their corresponding tolerances are Pi's (i = 1,2, 3 . . . . . m). To solve the problem (3), we use the max-min operator of Bellman and Zadeh (1970) and the appraoch of Zimmermann (1976). The membership function of the decision set,
/xv(x), is
gi( X) <_ b i + (1 - a ) P / , hi_<0, i = 0 , 1 , 2 . . . . . m. Moreover, Kuhn-Tucker's sufficient condition demands that the objective function for maximization and the constraints should be respectively concave and convex. In this formulation, it can be shown that both objective function and constraints satisfy the required sufficient conditions. Now, solving (5), the optimal solution for the FNLP problem is obtained.
/xD(x) = min { / x 0 ( x ) , / * l ( x ) , / x 2 ( x ) . . . . . /Xm(X)} for all x ~ X The min operator is used here to model the intersection of the fuzzy sets of objective and constraints. Since the decision maker wants to have a crisp decision proposal, the maximizing decision will correspond to the value of x, Xmax that has the highest degree of membership in the decision set. J~D(Xmax)
= max [ m i n { / x 0 ( x ) , / x , ( x ) . . . . . /*m( x)}].
3.2. Fuzzy geometric programming (FGP) If the objective function and the constraints, g0(x) and gi( x) (i = 1, 2 . . . . . m) are of posynomial form, then the problem (3) reduces to a fuzzy geometric programming (FGP) problem. Proceeding as before, the expression (4) is obtained in an alternative form as Min
a- 1
s.t.
gi( X) Pi - + a_< 1, bi + Pi bi + Pi
x_>0
It is equivalent to solving the following crisp nonlinear programming problem Max s.t.
x_>0,
a /*0(x) >_ a ,
txi( x) >_ a ,
i = 1,2 . . . . . m
x___0,
o ~ (0,1).
(4)
A new function, i.e. the Lagrangian function L( a , x, A) is formed by introducing (m + 1) Lagrangian multipliers A = ()t 0, )tl, )t2 . . . . . h m) as
(6)
a~ (0,1).
where x = (x 1, X 2 . . . . . Xn )T. Now we solve the problem (6) by the usual crisp geometric programming algorithm.
4. Solution o f the proposed inventory model
L ( f f , x,)t) = a - ~ ) t i ( g i ( x ) - b i - ( 1 - a ) P i ) . i=0
4.1. By FNLP
The necessary condition of Kuhn and Tucker (1951) for the optimal solution to this problem im-
The proposed inventory model depicted by Eq. (2),
T.K.Roy,M. Maiti/ EuropeanJournalof OperationalResearch99 (1997)425-432
428
Min
1 C( D, q) = Co3qY- lD + KDl-~ + TClq
s.t.
Aq <_B, D,q>O,
4.2. By FGP From Eq. (2), we have, as per Eq. (6),
reduces to, following Eq. (4), Max
Min
a- 1
s.t.
BjDq ~-1 + B 2 D l - B + B 3 q + B 4 o t < _ Bsq+ B6a< l, D,q>O, a ~ (0,1),
a Co3 qr- 1D + KD l - 13 + 71 C~ q
s.t.
_ C O + (1 - a ) P o ,
(7)
AQ<_B+(1-a)P, D,q>O, a e (0,1).
(8) where B l = Co3/(C o + Po), B2 = K/(Co + Po), B3 = C 1 / 2 ( C o + Po), B4 = Po/(Co + Po), B5 = A / ( B
Here, the objective goal is C O with tolerance P0 and the space constraint goal is B with tolerance P. So, the corresponding Lagrangian function is
+ P), B6 = P / ( B
+ P).
Here the number of degrees of difficulty = 3 and the dual of Eq. (8) is given by Max d ( w )
L ( a , D , q, a l, a2) = a-
I-
al(Co3q ~- ~D + KD 1-~
+½C, q - C o - ( 1 - a ) P o ) -Az( Aq-B-(1-a)P). From K u h n - T u c k e r ' s necessary conditions, we have
Co3q'/-1D + KD I-8 + ~I C l q _ Co_ (1 - a ) P o < O, Aq-B(1 - a ) P < O, AI(Co3qy-ID + KD '-~ + ½ C l q - CO- (1 - a ) P o )
O, A2(Aq-B-
(1-or)P)
x
ti~lWi )
x(%
I-W55) ~-~6 ]
+ w6) W,+w0 '
where W 0 = 1,
Wo -~ W4-~-W5 = l, W l "l- ( l -- ~ ) W 2 = 0 ,
=0,
( ' ~ - - 1 ) W 1 -b'W 3 --]-W5 = 0 .
/~'1, A2 ~ 0, 1 -- A l P o -
1,
A2P = 0,
A , ( C o 3 ( Y - 1 ) q v - e D + ½Ci) + AzA = 0 , hl(C03q y-I q-K(1
-- ] 3 ) D - / 3 ) = 0.
Solving these equations, optimum quantities are B+(1--a*)e q = A
Let W I = t~, W 3 = t 2, W4= t 3. solving the above equation, we get tl 1412- 1( 13 - -~ ' Ws=tl(1-y)-t2' W6= 1 - t 3 and then, the above dual expression becomes Max d( tl, t2, t3)
=(~l )t'(Bff-lt----~-)~-1(B-~3)12(~3 t2 where a * is a root of
×
C03
(l_y)tl_t2 13
H(B+(1-.)P) -~
2A
so C * ( D * , q * ) = l * + ~Clq •
and
-(1-
a)Po=O
Co3q*~'-lD * + K D *l-~
×
t~ + t 2 + t 3
X(1 + ( 1 -- T ) t , + t 2 + t 3 ) '+(l-y)q+t2+t3
T.K. Roy, M. Maiti / European Journal of Operational Research 99 (l 997) 425-432
429
Table 1 Optimal values for proposed inventory model Model
Method
Quantity q *
Demand D *
Average total cost C * ( D *, q * )
a *
Aq *
Fuzzy model
FNLP FGP
6.0449 6.043
9.8115 9.8068
53.9324 53.9328
0.3033 0.3043
60.94 60.94
Crisp model
NLP
5
9.21
54.43
1
50
Solving the equations Od/Ot~ = O, Od/Ot 2 = O, Od/Ot 3 = O, t I , t 2 , t~ are evaluated and hence Wo*,
5. N u m e r i c a l
Wl*, W2*, W3*, W4*, Ws*, W6* are also determined. Therefore, optimum values are
For a particular EOQ problem, let C03 = $4, K = 100, C = $ 2 , y = 0 . 5 , /3 = 1.5, A = 10 units, B = 50 units, C o = $ 4 0 , P0 = $ 2 0 , P = 15 units. For these values, the optimal value of production batch quantity q *, optimal demand rate D* and minimum average total cost C* ( D * , q * ) and O/* obtained by FNLP and FGP are given in Table 1. Total average cost is less in the case of the fuzzy model than that obtained for the crisp model.
q*=
W3* B"'~ ( W I * + W 2 * + W 3 * " ] - W 4 * )
1'
1 D* =
2 (Wl,
nt. W2, " t W 3 , _ t _ W 4 , ) - i
1 /3
example
W4*
O/* =
B;
( w l * Jl- W2* '~- W3* 't- W4*)
1
and C*(D
6. S e n s i t i v i t y
*
,q
*
)=Co3q*Z'-lD
*
1 +KD*l-~+~Ciq
analysis
*
There are three cases for overall achivement of
So, by both FNLP and FGP techniques, the optimal values of q and D and the corresponding minimum cost are evaluated for the known values of other parameters.
O/: 1. O/max ~ 1, 2. O~max = 0,
3. 0 < Otmax < 1.
Table 2 Effect of variations in Po PO
a*
q*
D*
To
T
C * ( O * , q* )
20 25 50 100 200 500 1 000 5 000 10 000 50 000 100 000 500 000
0.3033 0.4398 0.7163 0.857 0.9282 0.9712 0.9856 0.9971 0.3985 0.9997 0.9998 0. 99997
6.04 5.84 5.42 5.21 5.12 5.04 5.02 5.00 5.00 5.00 5.00 5.00
9.81 9.70 9.46 9.34 9.27 9.24 9.23 9.21 9.21 9.21 9.21 9.21
13.93 t 4.00 14.18 14.50 14.36 14.40 14.41 14.42 14.42 14.42 14.42 14.42
10.45 8.40 4.25 2.14 1.08 0.43 0.22 0.04 0.00 0.00 0.00 0.00
53.93 54.00 54.18 54.30 54.36 54.40 54.41 54.42 54.42 54.42 54.42 54.42
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T.K. Roy, M. Maiti/ European Journal of Operational Research 99 (1997) 425-432
Table 3 Effect of variations in P P
a*
q~
D*
To
T
C*(D*, q*)
15 16 20 23 36 38 40 41 42 44 156 160 161 162
0.30 0.30 0.30 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.30 oo oo - 0.005
6.04 6.11 6.38 6.59 7.46 7.60 7.73 7.80 7.87 8.01 19.99 20.84 21.06 21.29
9.81 9.85 9.99 10.10 10.53 10.59 10.65 10.68 10.72 10.78 14.61 14.82 14.87 14.93
13.93 13.91 13.84 13.79 13.70 13.69 13.69 13.69 13.70 13.70 19.22 19.80 19.96 20.116
10.45 11.13 13.84 15.86 24.65 26.02 27.39 28.08 28.76 30.14 152.02 158.43 160.66 162.94
53.93 53.91 53.84 53.79 53.70 53.69 53.69 53.69 53.70 53.70 59.23 59.80 59.95 60.116
Following Hamacher et al. (1978), we perform sensitivity analysis in these three cases for the given numerical example. 1. O/max 1. The optimal solution is q * = 5, D * = 9.21, C * ( D *, q* ) = $54.43, which is also the optimal solution for the crisp nonlinear model without any tolerance. Here the optimal solution is invariant for the variation of P0 to P0 + A P0 o r / a n d P to p+Ap. =
2. O/max 0. In this case, the optimal solution is q* = 6.5, D* = 15.74, C * ( D * , q * ) = $56.40, which is the optimum solution for the crisp problem with m a x i m u m tolerance. =
3 . 0 < O/max< 1. For P0 = 20, the optimal solution of the problem is q * = 6.04, D * = 9.81, O/* = 0.3033. Now, changing to P0 = 21 (i.e. A P 0 = 1) we get q* = 6.00, D* = 9.78, O/* = 0.3358. The difference in the optimal values of the objective function O/ is 0.0324. Therefore, for the change of P0 by 5%, O/ changes by 10.7%.
Following Dutta et al. (1993) we study the effect of tolerance in the said EOQ model with the earlier numerical values and construct Tables 2 and 3 for the degrees of violation To ( = ( 1 - O/)P0) and r ( = (1 - O/)P) for two constraints given by Eq. (8). From Table 2, we see that: (i) For higher tolerances of P0, the value of O/max does not achieve 1 though it is very near to 1. (ii) For higher acceptable variations of P0, the optimal solutions remain invariant and the optimal solutions are very close to the solutions (q* = 5, D* = 9.21, C * ( D * , q * ) = 54.43) of the crisp model without tolerance (O/= 1). From Table 3 we observe that: (i) For different values of P, degrees of violations TO and T are never zero, i.e. different optimal solutions are obtained. (ii) As P increases from 15, the minimum average cost C * ( D * , q *) decreases and becomes m i n i m u m at P = 41 with maximum q * ( = 7.80). (iii) When P is more than 41, average cost C * ( D * , q * ) increases with P and at P = 160 or more, O/ becomes 0 or less than 0. In this case, optimal solutions are not basic.
Table 4 Input data i
C03 i
1
4
2 3
5 5
"~i 0.5 0.7 0.6
Ki 100 110 115
~i
Cli
Ai
1.5
2
5
1.6 1.8
2.3 2.5
6 5
ui 0.5 0.3 0.4
Table 5 Otherinput data Co
Po
B
P
U
Pl
160
10
90
15
3.5
2
T.K. Roy, M. Maiti / European Journal of Operational Research 99 (1997)425-432
431
Table 6 Optimal values for proposed multi-item inventory model Model
B
P
Crisp
90 105 105 90
Fuzzy
90
U
PI
3.5 3.5 5.5 5.5 15
7. G e n e r a l i s a t i o n
3.5
2
Total average cost C*(D 7 , q i * )
Totalspace required
Totalcapitalinvestment
164.12 164.12 164.01 164.12
90 90.40 104.89 90
3.5 3.5 4.09 3.5
164
to m u l t i - i t e m m o d e l
An inventory for N items under two constraints total capital investment for the quantities and total storage area is mathematically expressed as
--
Min
C( D i, qi) N = E (Co31Diq~ i - t + K i D ) -f3i+ ½Cliqi) i=1 N
s.t.
1
7 E uiqi <- U, i=1 N
E Aiqi <- B , i=1
Di, qi > O,
where u i is the cost of the ith unit product, U the total capital investment for the quantities and other parameters have the usual meanings. After fuzzification the above problem reduces to Max
c~ N
~_~ (Co3iDiq7 ~-' "+-KiD~-~i-t- lCliqi)
s.t.
i=l
96
3.74
is solved and the numerical results are presented below. Input data for the model are shown in Tables 4 and 5 and the results are shown in Table 6. In Table 6, the fuzzy result is better than the crisp model with different combinations of the constraints goals. 8. Conclusion In this paper, we solve real life inventory models for single and multi-items in fuzzy environment by FNLP and FGP techniques. Some sensivity analysis on the tolerance limits have been presented. The results of the fuzzy models are compared with those of crisp model which reveals that fuzzy models give better result than the usual crisp models. This method is quite general and can be extended to other similar inventory models including the ones with shortages. Acknowledgements The authors are thankful to the reviewers for their valuable suggestion and comments for the improvement of the paper.
_< C o + (1 - c~)P o, N |
7 ~ uiqi <- U + (1 - a ) P , , i=1 N
~_, Aiq i <_B + (1 - a ) P , i=1
Di,qi > 0 ,
ae(0,1),
where P~ is the allowable tolerance of u and other parameters are as explained earlier. Following the procedure of fuzzy nonlinear programming method out line earlier, the above problem
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T.K. Roy, M. Maiti / European Journal of Operational Research 99 (1997)425-432
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