FUZZY PARAMETRIC GEOMETRIC PROGRAMMING WITH APPLICATION IN FUZZY EPQ MODEL UNDER FLEXIBILITY AND RELIABILITY CONSIDERATION

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1 ISSN , England, UK Journal of Inforation and Coputing Science Vol. 7, No. 3, 1, pp FUZZY PARAMERIC GEOMERIC PROGRAMMING WIH APPLICAION IN FUZZY EPQ MODEL UNDER FLEXIBILIY AND RELIABILIY CONSIDERAION G.S. Mahapatra 1.K. Mandal and G.P. Saanta 1 Departent of Matheatics,Siliguri Institute of echnology, P.O.-Sukna, Dist.-Dareeling, Siliguri-7349, West Bengal, India. Departent of Matheatics,Bengal Engineering and Science University, Shibpur Howrah, West Bengal, India (Received Noveber9, 11, accepted Junly, 1) Abstract. An econoic production quantity odel with deand dependent unit production cost in fuzzy environent has been developed. Fleibility and reliability consideration are introduced in the production process. he odels are developed under fuzzy goal and fuzzy restrictions on budgetary cost. he inventory related costs and other paraeters are taken as fuzzy in nature. he proble is solved by paraetric geoetric prograing technique. he odel is illustrated through nuerical eaple. he sensitivity analyses of the cost function due to different easures are perfored and presented graphically. Keywords: paraetric geoetric prograing technique; production process 1. Introduction Since late 196 s Geoetric Prograing (GP) has been very popular in various fields of science and engineering. Duffin, Peterson and Zener [1] discussed the basic theories on GP with engineering application in their books. Peterson [], Rickaert [3], Jefferson and Scott [4] have presented inforative surveys on GP. he paraeter used in the GP proble ay not be fied. It is ore fruitful to use fuzzy paraeter instead of crisp paraeter. In that case we introduced the concept of fuzzy paraetric GP technique, where the paraeters are fuzzy. Application of GP can be observed in any aspects of inventory/production, there appears only few papers concerned with the solution of inventory probles using GP (Cheng [5, 6, 7]; Jung and Klein [8]; Kochenberger [9]; Lee [1]; Worrall and Hall [11]). he deterination of the ost cost-effective production quantity under rather stable conditions is coonly known as classical econoic production quantity (EPQ) inventory proble. Fabulous aount of research effort has been epended on topic leading to the publication of any interesting results in the literature ([Clark [1], Urgelleti [13], Velnott [14]). A basic assuption in the classical EPQ odel is that the production set-up cost is fied. In addition the odel also iplicitly assues that ites produced are of perfect quality (Ha and Canadea [15]). However, in reality product quality is not always perfect but directly affected by the reliability of the production process eployed to anufacturer the product. hus a high-level of product quality can only be consistently achieved with substantial investent in iproving the reliability of production process. Furtherore, while the set-up tie, hence set-up cost, will be fied in short ter, it will tend to decrease in the long ter because of the possibility of investent in new achineries that are highly fleible, e.g. fleible anufacturing syste. Van and Putten [16] have addressed etensively the issue of fleibility iproveent production and inventory anageent under various scenarios, while the issues of process reliability, quality iproveent and set-up tie reduction have been discussed by Porteus [17, 18], Rosenblatt and Lee [19] Corresponding author. el.: E-ail address: g_s_ahapatra@yahoo.co. Published by World Acadeic Press, World Acadeic Union

2 4 G.S. Mahapatra, et al: FUZZY PARAMERIC GEOMERIC PROGRAMMING WIH APPLICAION IN FUZZY EPQ MODEL UNDER FLEXIBILIY AND RELIABILIY CONSIDERAION and Zangwill []. Cheng [5] proposed a general equation to odel the relationship between production setup cost and process reliability and fleibility. Cheng [6] also introduced deand dependent unit production cost in an EOQ odel. ripathy et al. [1] developed an EOQ odel with iperfect production process and the unit production cost is directly related to process reliability and inversely related to the deand rate. Isla and Roy [] developed an EPQ odel with fleibility and reliability consideration in fuzzy environent and the odel is solved by fuzzy geoetric prograing technique. Leung [3] considered an EPQ odel with fleibility and reliability considerations using GP based on the arithetic-geoetric ean inequality. In this paper we introduced the concept of fuzzy paraetric GP technique. Here we have considered the coefficients of the proble are fuzzy and taken these in paraetric for and solve it by GP technique which is fored as a fuzzy paraetric GP. An econoic production quantity odel with deand dependent unit production cost in fuzzy environent has been developed. Fleibility and reliability consideration are introduced in the production process. he odels are developed under fuzzy goal and fuzzy restrictions on budgetary cost. he inventory related costs and other paraeters are taken as fuzzy in nature. he proble is solved by paraetric geoetric prograing technique.. Matheatical odel Let a copany produces a single product using a conventional production process with a certain level of reliability. he process reliability depends on a nuber of factors such as achine capability, use of on-line onitoring devices, skill level of the operating personal and aintenance and replaceent policies. he process, thereby reducing the costs of scrap and rework of substandard products, wasted aterial and labor hours, ore consistently produces higher reliability eans products with acceptable quality. However, high reliability can only be achieved with substantial capital investent that will increase the cost of interest and depreciation of the production process. A odern fleible production process that substantially reduces the production set-up tie can produce the product ore efficiently. It is thus econoical to produce in saller batch sizes with fleible process, thereby reducing the inventory holding cost. Also, substantial capital ependiture due to illustration of the new production process will give rise to ight interest changes and great depreciation cost..1. Notations o construct a odel for this proble, we define the following variables and paraeters: S set-up cost per batch (a decision variable), h inventory carrying cost per ite per unit tie, D deand rate (a decision variable), q production quantity per batch (a decision variable), r production process reliability (a decision variable), f(s, r) total cost of interest and depreciation for a production process per production cycle, C(D, S, q, r) total average cost, P unit production cost, B total budgetary cost... Assuptions he following assuptions are ade for developing the atheatical production quantity odel: 1) he rate of deand D is unifor over tie ) Shortages are not allowed 3) he tie horizon is infinite 4) otal cost of interest and depreciation per production cycle is inversely related to a set-up cost and directly related to process reliability according to the following equation JIC eail for contribution: editor@ic.org.uk

3 Journal of Inforation and Coputing Science, Vol. 7 (1) No.3, pp f(s, r) = as -b r c (1.1) where a, b, c > are constant real nubers chosen to provide the best fit of the estiated cost function. 5) he unit production cost (p) is a continuous function of deand (D) and takes the following for p = γ D - (1.) where (>1) is called price elasticity and γ (>) is a scaling constant. he first three assuptions are the basic assuptions used in the classical EPQ odel. he fourth assuption is based on the fact that to reduce the costs of production set-up and scrap and rework on shoddy protects substantial investent in iproving the fleibility and reliability of the production process is necessary. he fifth assuption is ainly based on the unit variable production is deand dependent. When the deand of an ite increases then the production/purchasing cost spread all over the ites and hence the unit purchasing cost reduces and varies inversely with deand. he process reliability level r eans of all the ites produced in a production run only r % are acceptable quality that can be used to eet deand. he situation of the inventory odel is illustrated in figure below: Figure.1 Scheatic of the situation of the econoic production quantity odel.3. Crisp Model If q(t) is the inventory level at tie t over the tie period (, ), then dq( t) D for t (1.3) dt with initial and boundary conditions q() = rq, q() =. he solution of this differential equation is obtained as q(t) = rq Dt (1.4) and = (rq) / D (1.5) Now, the inventory-carrying cost is given by JIC eail for subscription: publishing@wau.org.uk

4 6 G.S. Mahapatra, et al: FUZZY PARAMERIC GEOMERIC PROGRAMMING WIH APPLICAION IN FUZZY EPQ MODEL UNDER FLEXIBILIY AND RELIABILIY CONSIDERAION h q( t) dt h ( rq Dt) dt hr q (1.6) D It is evident that the length of the production cycle is the su of set-up, production, inventory carrying and interest and depreciation costs, that is total cost per cycle is S + pq + hq r / D + f(s, r) (1.7) Our obective is to iniize the total cost per unit tie under liited budgetary cost. So C (D, S, q, r) = (total cost per cycle) / (qr / D) (1.8) After substituting (1.1), (1.) and (1.7) in (1.8) which becoes Hqr C( D, S, q, r) DSq r D r ads q r b 1 c1 (1.9) It is natural to epect the cost a product to be ore if it is ore sophisticated and reliable (ecept possible in the event of soe aor technological breakthrough). So, one can consider the budgetary function as an increasing function of reliability. Let the production cost per unit is Pr and the total budgetary cost of the process is less or equal to B. Here we have considered budgetary cost as a constraint function as follows: Pr q B,1 (1.1) Hence the inventory odel can be written as follows b 1 Miniize C( D, S, q, r) DSq r D r ads q r subect to Pr q B D, S, q, r >,1 Hqr c1 (1.11) he above proble (1.11) can be treated as a Posynoial Geoetric Prograing proble with zero Degree of Difficulty..4. Fuzzy Model If the coefficients of obective function and constraint goal of (1.11) are fuzzy [4] in nature then crisp odel (1.11) transfored into a fuzzy odel as follows Hqr b 1 c1 Miniize C( D, S, q, r) DSq r D r ads q r subect to Pr q B,1 (1.1) where, h, a, P and B are fuzzy in nature. D, S, q, r>. 3. Prerequisite Matheatics Definition 3.1 Fuzzy Set: A fuzzy set A in a universe of discourse X is defined as the following set JIC eail for contribution: editor@ic.org.uk

5 Journal of Inforation and Coputing Science, Vol. 7 (1) No.3, pp A, : X of pairs A the fuzzy set A and A : X,1. Here A is a apping called the ebership function of is called the ebership value or degree of ebership of X in the fuzzy set A. he larger A is the stronger the grade of ebership in A. Definition 3. Noral Fuzzy Set: A fuzzy set A of the universe of discourse X is called a noral fuzzy set iplying that there eists at least one X such that A 1. Otherwise the fuzzy set is subnoral. Definition 3.3 -Level Set or -cut of a Fuzzy Set: he -level set (or interval of confidence at level or -cut) of the fuzzy set A of X is a crisp set A that contains all the eleents of X that have ebership values in A greater than or equal to i.e. A :, X, [,1] A Definition 3.4 Fuzzy Nuber: A fuzzy nuber A is a fuzzy set of the real line whose ebership function ( ) has the following characteristics with a 1 a a 3 a 4 A L ( ) for a1 a 1 for a a3 ( ) A R ( ) for a3 a4 for otherwise where L ( ) :[a 1,a ][,1] is continuous and strictly increasing; R ( ) :[a 3, a 4 ][, 1] is continuous and strictly decreasing. 4. Matheatical analysis Consider a particular non-linear prograing proble Min g ( ) s. t. gi( ) 1 (1 i n) >. Its obective and constraint functions are of the for i ik i ik k1 1 g ( ) c ( i n) where > ; and c ik, ρ ik are real nubers. he constraint in (4.1) needs softening and considering the proble of fuzzy obective and constraint with fuzzy coefficients, we transfor (4.1) into a fuzzy geoetric prograing [5] as follows: Min g ( ) (4.) subect to gi ( ) 1 (1 i n), >, (4.1) JIC eail for subscription: publishing@wau.org.uk

6 8 G.S. Mahapatra, et al: FUZZY PARAMERIC GEOMERIC PROGRAMMING WIH APPLICAION IN FUZZY EPQ MODEL UNDER FLEXIBILIY AND RELIABILIY CONSIDERAION where,,..., nubers. 1 is a variable vector, i ik i ik k1 1 are all posynoial of in which coefficients ik g ( ) c ( i n) c are fuzzy Here for fuzzy nubers c c, c, c containing the coefficients c i n;1 k ik 1ik ik 3ik ik i, with the ebership function as follow cik t for c1 ik t cik cik c1 ik t cik c t for c ik ik t c3ik (4.3) c3ik cik otherwise c i n;1 k is given by Here α-cut of ik i c c, c c c c, c c c (4.4) ik ikl ikr 1ik ik 1ik 3ik 3ik ik Proposition 4.1 If the coefficients of the fuzzy geoetric prograing proble are taken as fuzzy nubers then the proble (4.) reduces to k kl (4.5) Min c Subect to k 1 1 i ik 1, 1 ikl k 1 1. c i n If the coefficients are taken as fuzzy nubers then the fuzzy geoetric prograing proble (4.) will take the for: Subect to i Min c k k1 1 i ik k 1 1 k ik c 1, 1 i n. Using α-cut of the fuzzy nubers coefficients, the above proble is reduces to Which is equivalent to k kl, kr (4.6) Min c c k 1 1 i ik Subect to c, c 1, 1 i n ikl ikr k 1 1. k kl (4.7) Min c k 1 1 JIC eail for contribution: editor@ic.org.uk

7 Journal of Inforation and Coputing Science, Vol. 7 (1) No.3, pp Subect to i ik 1, 1 ikl k 1 1. c i n 4.1. Solution procedure of fuzzy paraetric geoetric prograing Solution of paraetric proble (4.7) using fuzzy paraetric geoetric prograing proble is discussed here. Proble (4.7) is a constrained posynoial GP proble. he nuber of ters in each posynoial constraint function varies and it is denoted by r for each r=,1,,,l. Let = l be the total nuber of ters in the prial progra. he Degree of Difficulty =-(+1). he dual proble of the prial proble (4.7) is as follows rk l r r crkl Maiize d( ) rs (4.8) r k 1 rk s1r 1 subect to k 1, (Norality condition) k 1 l r rk rk, (=1,,...,) (Orthogonality conditions) r k 1 rk, (r=,1,,,l; k=1,,.., r ). (Positivity conditions) Case I. For +1, the dual progra presents a syste of linear equations for the dual variables. A solution vector eists for the dual variables. Case II. For <+1, in this case generally no solution vector eists for the dual variables. he solution of the GP proble is obtained by solving the syste of linear equations of dual proble (4.8). Ones optial dual variable vector are known, the corresponding values of the prial variable vector t is found fro the following relations: ik ikl i 1 c d (i=,1,,, ) (4.9) aking logariths in (4.9), log-linear siultaneous equations are transfored as i d ik (log ) log (i=,1,,., ) (4.1) 1 cikl It is a syste of linear equations in t (=log ) for =1,,...,n. Since there are ore prial variables than the nuber of ters, any solutions (=1,,...,n) ay eist. So, to find the optial prial variables (=1,,...,n), it reains to iniize the prial obective function with respect to reduced - () variables. When - = i.e. nuber of prial variables = nuber of log-linear equations, prial variables can be deterined uniquely fro log-linear equations. For different value of α [,1], equ.(4.1) will return different solution set of i. And hence different solution set for dual as well as prial proble will be obtained. hese solutions sets will help decisionaker to take apt decision. 5. Paraetric Geoetric Prograing echnique on EPQ Model According to section 4, the fuzzy EPQ odel (1.11) reduces to a fuzzy paraetric geoetric prograing by replacing (1 ) 1, H H (1 ) H, a a 1 (1 ) a1, P P (1 ) P1 where,1 and B B (1 ) B1 in (1.11). he odel takes the reduces for as follows rk JIC eail for subscription: publishing@wau.org.uk

8 3 G.S. Mahapatra, et al: FUZZY PARAMERIC GEOMERIC PROGRAMMING WIH APPLICAION IN FUZZY EPQ MODEL UNDER FLEXIBILIY AND RELIABILIY CONSIDERAION Miniize C ( D, S, q, r) DSq r (1 ) D r b 1 c1 a (1 ) a1 DS q r (1 ),1 subect to r q W W1 D, S, q, r>. Applying GP technique the dual prograing of the proble (5.1) is (1 ) H (1 ) H a (1 ) a H (1 ) H qr 1 (5.1) Ma d( ) W (1 ) W subect to (1 ) 1 4 b ( c 1) (5.) (5.3) his is a syste of five linear equations in five 5 unknowns. Solving we get the optial values as follows b( 1)( 1) 1 (1 b)( ) c( 1) (5.4) ( 1)(1 b) (1 b)( ) c( 1) (5.5) (1 b) ( 1)( c) 3 (1 b)( ) c( 1) ( 1)( 1) 4 (1 b)( ) c( 1) c(1 ) (1 b) ( 1) 5 ( 1) (1 b)( ) c( 1) (5.6) (5.7) (5.8) Putting these values in (5.1) we get the optial solution of dual proble. he values of D, S, q, r is obtained by using the prial dual relation as follows Fro prial dual relation we get 1 1 DSq r 1 d 1 1 (1 ) D r d 1 H (1 ) H qr d 1 3 b 1 c1 (1 ) 1 4 a a DS q r d 1 W (1 ) W 1 r q he optiu solution of the odel (1.1) through paraetric approach is given by 5 JIC eail for contribution: editor@ic.org.uk

9 Journal of Inforation and Coputing Science, Vol. 7 (1) No.3, pp (1 ) H (1 ) H a (1 ) a 1 d ( ) W (1 ) W c( H (1 ) H1) 5 q 5 W (1 ) W1 d ( ) 3 d ( ) 3 c( H (1 ) H1) 5 r D S 1 1 ( d ( )) 3 ( (1 ) 1)( H (1 ) H1) c 5 d ( ) ( d ( )) 1 ( H (1 ) H1) ( (1 ) 1)( H (1 ) H1) c Note that optial solution of GP technique in paraetric approach is depends on α. 6. An illustrative eaple of the EPQ odel A anufacturing copany produces a achine. It is given that the inventory carrying cost of the achine is $1.5 per unit per year. he production cost of the achine varies inversely with the deand. Fro the past eperience, the production cost of the achine is 15D-3.6, where D is the deand rate. he total cost of interest and depreciation per production cycle is 15S-1.6r, where S and r are set-up cost per batch and production process reliability respectively. Let production cost per unit is 1.6 8r and total budgetary cost (C) is $ 58. Deterine the deand rate (D), set-up cost (S), production quantity (q), production process reliability (r), and optiu total average cost (C) of the production syste. Forulation of the said odel is presented as follows: 1.5qr subect to r q Min C( D, S, q, r) DSq r 15 D r 15DS q r ,1 he optiu solution of the proble (6.1) by Non-Linear prograing (NLP) and Geoetric Prograing (GP) using LINGO [6] are presented in able 1. able 1 Optial solution of EPQ odel (6.1) Method C ($) D S ($) q r GP NLP When the coefficient are taken as fuzzy nuber i.e ,, and ,, fuzzy odel by fuzzy paraetric geoetric prograing is presented in table (6.1) , the optial solutions of the JIC eail for subscription: publishing@wau.org.uk

10 3 G.S. Mahapatra, et al: FUZZY PARAMERIC GEOMERIC PROGRAMMING WIH APPLICAION IN FUZZY EPQ MODEL UNDER FLEXIBILIY AND RELIABILIY CONSIDERAION able Optial solution of fuzzy EPQ odel of (6.1) α C ($) D S ($) q r he solution of obective function and decision variables for different value of α is shown by graphical presentation in figure. 7. Sensitivity analysis Figure Optial obective value, decision variables D, S, q and r vs. α he change of optial solutions of the proble for fuzzy odel with sall change of tolerance of constraint goal when α change is, given in able 3. able 3 shows that as B changes increasingly the total average cost of the given proble slightly decreases, which is epected. So it is clear fro the sensitivity analysis that if the anageent traced on proceed reliability, the D and q will be less, so they should not the over 1 deand. Another fact is that if production anageent decided on the fact they should try to fulfill the deand then the fact is on to be relation on the production process reliability and setup cost with changing of tolerance of constraint goal, decision variables are also changed. It is noted that the deand and order quantity are increasing with increasing tolerance of constraint goal but setup cost and process reliability are decreasing when tolerance increasing. JIC eail for contribution: editor@ic.org.uk

11 Journal of Inforation and Coputing Science, Vol. 7 (1) No.3, pp able 3 Change of value of obective function and decision variables for change of α α olerance C ($) D S ($) Q r of B Conclusion In this paper, an econoic production quantity odel with investent costs for set-up reduction and quality iproveent is forulated. he odel has involved one budgetary constraint. he proble is solved by Fuzzy paraetric GP ethod. he Fuzzy paraetric GP ethod provides an alternative approach to this proble. he ethod, as illustrated, is efficient and reliable. Here decision aker ay obtain the optial results according to his epectation. he ethod presented is quite general and can be applied to the odel in other areas of operation research and other field of optiization involveent. 9. References [1] R.J. Duffin, E.L. Peterson, C. Zener, Geoetric Prograing theory and application, John wiley, New York, [] E.L. Peterson, Geoetric prograing, Sia Review. 18 (1974) [3] M.J. Rickaert, Survey of progras in geoetric prograing, C.C.E.R.O. 16 (1974) [4].R. Jefferson and C. H. Scott, Avenues of geoetric prograing, New Zealand Operational Research. 6 (1978) [5].C.E. Chang, An econoic production quantity odel with fleibility and reliability considerations, European Journal of Operational Research 39 (1989) [6].C.E. Chang, An econoic production quantity odel with deand-dependent unit cost, European Journal of Operational Research 4 (1989) [7].C.E. Cheng, An econoic order quantity odel with deand-dependent unit production cost and iperfect production processes. IIE ransactions 3, (1991a), 3 8. [8] ih. Jung, C.M. Klein, Optial inventory policies for an econoic order quantity odel with decreasing cost functions, European Journal of Operational Research 165, (5) JIC eail for subscription: publishing@wau.org.uk

12 34 G.S. Mahapatra, et al: FUZZY PARAMERIC GEOMERIC PROGRAMMING WIH APPLICAION IN FUZZY EPQ MODEL UNDER FLEXIBILIY AND RELIABILIY CONSIDERAION [9] G.A. Kochenberger, Inventory odels: Optiization by geoetric prograing. Decision Sciences, (1971) [1] W.J. Lee, Deterining order quantity and selling price by geoetric prograing: Optial solution, bounds, and sensitivity. Decision Science 4, (1993) [11] B.M. Worrall, M.A. Hall, he analysis of an inventory control odel using posynoial geoetric prograing. International Journal of Production Research, (198) [1] [A.J. Clark, An inforal survey of ulti-echelon inventory theory. Naval Research Logistics Quarterly 19, (197) [13] G. Urgeletti, Inventory control: odels and probles, European Journal of Operational Research 14 (1983) 1-1. [14] A.F. Veinott, he status of atheatical inventory theory. Manageent Science 1, (1966) [15] A.C. Ha, D. Canada, Production and Inventory Manageent. Prentice-Hall, New Jersey [16] B.P. Van, C. Putten, OR contributions to fleibility iproveent in production/inventory systes. European Journal of Operational Research 31, (1987) 5 6. [17] E.L. Porteus, Investing in reduced set-ups in the EOQ odel. Manageent Science 31, (1985) [18] E.L. Porteus, Optial lot sizing, process quality iproveent and set-up cost reduction. Operations Research 34, (1986) [19] M.J. Rosenblatt, H.L. Lee, Econoic production cycles with iperfect production processes. IIE ransactions 18, (1986) [] W.I. Zangwill, Fro EOQ to ZI. Manageent Science 33, (1987) [1] P.K. ripathy, W.M. Wee, P.R. Mahi, An EOQ odel with process reliability considerations. Journal of the Operational Research Society 54, (3) [] S Isla,.K. Roy, A fuzzy EPQ odel with fleibility and reliability consideration and deand dependent unit production cost a space constraint : A fuzzy geoetric prograing approach, Applied Matheatics and coputation 176 (6) [3] K.N.F. Leung, A generalized geoetric-prograing solution to An econoic production quantity odel with fleibility and reliability considerations European Journal of Operational Research 176 (7) [4] L.A. Zadeh, Fuzzy sets, Inforation and Control 8 (1965) [5] B.Y. Cao, Fuzzy geoetric prograing. Series: Applied Optiization Vol.76: Kluwer Acadeic Publishers, Dordrecht,. [6] LINGO: the odeling language and optiizer (1999). Lindo syste Inc., Chicago,IL 66, USA. JIC eail for contribution: editor@ic.org.uk