hp://dx.doi.org/.474/apem.. Advances in Producion Engineering & Managemen 7 (), 6-7 ISSN 854-65 Scienific paper MULTI-PERIOD PRODUCTION PLANNING UNDER FUY CONDITIONS Naadimuhu, G.; Liu, Y.L. & Lee, E.S. Fairleigh Dickinson Universiy, Madison, NJ 794, USA Kansas Sae Universiy, Manhaan, KS 6656, USA E-mail: naadi@fdu.edu Absrac: In he real-world muli-period producion/operaions managemen (MP-POM) problems, he parameers mus be esimaed and hey are frequenly given by inerval esimaes. Bu for mos POM models, hese inerval esimaes mus be ranslaed ino single numbers. This ofen resuls in errors and in he loss of a considerable amoun of informaion. The purpose of his paper is o develop, apply, and illusrae a new fuzzy approach using fuzzy numbers o solve he inerval MP-POM problem. I consiss of employing appropriae fuzzy numbers o represen he inerval esimaes in he muli-sage decision problems; using he operaions of fuzzy numbers combined wih dynamic programming o solve he problem; and deermining he required minimum/maximum fuzzy number hrough ranking echniques. To demonsrae he applicaion of his approach, hree MP-POM problems wih fuzzy coss and/or fuzzy demands are solved. The main advanages of his approach are fuzzy represenaive soluions for he opimal producion schedule and he minimum oal cos in erms of inerval unis raher han single numbers. This enables he producion engineers and operaions managers o manage he producion flexibly and conrol he coss effecively. A significan original conribuion of his research is he developmen of an effecive echnique o solve he fuzzy MP-POM problems ha have no been addressed hus far. Suggesions for fuure research include exending he proposed general fuzzy approach o solve large scale mulisage fuzzy problems using compuers; and o solve problems wih fuzzy goals and consrains defined in differen spaces. Key Words: Muli-period Producion Planning, Muli-period Producion/Operaions Managemen (MP-POM), Fuzzy Condiions, Scalar Demands, Fuzzy Demands, Scalar Coss, Fuzzy Coss, Crisp Dynamic Programming (CDP), Fuzzy Dynamic Programming (FDP). INTRODUCTION Producion/operaions managemen (POM) has a significan impac on he economy of mos firms. Because of is muli-period naure, POM mus be planned before he beginning of producion. However, he exac informaion needed is no available unil he producion evens occur. So in muli-period POM (MP-POM) problems, he fuure parameers mus be esimaed. A frequenly used mehod is o give hem inerval (range) esimaes. For almos all models, hese inerval esimaes mus be ranslaed ino single numbers. This may resul no only in errors, bu also in he loss of a considerable amoun of informaion. Since fuzzy numbers can be used o overcome hese difficulies, hey offer an ideal approach for solving real-world MP-POM problems. The main purpose of his sudy is o find an effecive mehod o solve he inerval MP- POM problems, while reaining all he original informaion. The proposed echnique is a general fuzzy muli-sage decision approach based on fuzzy numbers using a problemoriened poin of view. The effeciveness of his approach is illusraed hrough is applicaion o solve hree differen fuzzy MP-POM problems fuzzy coss and scalar demands; fuzzy demands and scalar coss; and fuzzy coss and fuzzy demands. The hree MP-POM models 6
are solved by boh he proposed fuzzy approach and he curren crisp mehods. The resuls are compared and conrased o show he advanages of he proposed procedure.. LITERATURE SURVEY There have been many publicaions in lieraure on MP-POM. Corsano e al. [] presened a muli-period mixed ineger nonlinear opimizaion model for muliproduc bach plans under seasonal and marke flucuaions. Neiro and Pino [] opimized muli-period producion planning models by mixed ineger nonlinear programs under uncerain condiions of prices and demands. Safaei and Tavakkoli-Moghaddam [] proposed an inegraed mahemaical model of he muli-period cell formaion and producion planning in a dynamic cellular manufacuring sysem o minimize coss hrough a mixed ineger programming echnique. Porkka e al. [4] used a mixed ineger linear programming based capaciaed lo sizing models ha included carryovers incorporaing se-up imes wih associaed coss. Moreno e al. [5] employed a general muli-period opimizaion model for muli-produc bach plans o maximize an economic funcion consising of incomes and coss using mixed ineger linear programming. Moreno and Monagna [6] uilized a linear general disjuncive programming model for muli-period producion planning in a muli-produc bach environmen o maximize he expeced ne presen value of he benefi under uncerainy in demands. El Hafsi and Bai [7] deermined an opimal muli-period producion plan for a single produc over a finie planning horizon o minimize he oal invenory and backlog coss by solving a nonlinear programming problem. Feylizadeh [8] inegraed projec managemen nework and mahemaical programming echniques o decrease he oal cos by conrolling he compleion ime in a muli-period muli-produc producion planning problem. Li e al. [9] derived an opimal soluion srucure by he dynamic programming approach for a join manufacuring and remanufacuring sysem in a muli-period horizon. Yildirim e al. [] devised a rolling-horizon approach based on solving he saic problem a each ime period for a sochasic muli-produc producion planning and sourcing problem. Filho [] provided a sochasic opimizaion model wih consrains on he producion and invenory variables for a muli-produc muli-period long-erm producion planning problem hrough Gaussian approximaion. Kaminsky and Swaminahan [] developed heurisics ha uilize knowledge of demand forecas evoluion for capaciaed muli-period producion planning. Kazancioglu and Saiou [] applied a simulaion based mehod o aid muli-period producion capaciy planning using a muli-objecive geneic algorihm. Sox and Mucksad [4] formulaed an algorihm using Lagrangian relaxaion for he finie horizon capaciaed muli-period producion planning problem wih random demand for muliple producs. Nagasawa e al. [5] inroduced wo algorihms, one wih saionary demand and he oher wih seasonal demand, for deermining he value of generalized planning horizon based upon a propery of an analyical soluion o a muli-period producion planning problem. Kogan and Porougal [6] focused on he conrol decisions for muli-period aggregae producion planning o minimize he expeced oal coss. Balakrishnan and Cheng [7] addressed cellular manufacuring under condiions of muli-period planning horizons wih demand and resource uncerainies. Ryu [8] illusraed hrough numerical examples muliperiod planning sraegies wih simulaneous consideraion of demand flucuaions and capaciy expansion. The above publicaions on MP-POM required precise or sochasic daa. However, in real life he daa are usually available in imprecise erms. A realisic and beer way o represen hese imprecise daa is o use fuzzy numbers. Thus, his siuaion in realiy is a fuzzy problem. This paper aemps o solve he fuzzy MP-POM problem hrough he applicaion of fuzzy muli-sage decision making processes. 6
. PROBLEM DESCRIPTION AND SOLUTION METHODOLOGY Le he producion cos in period be if P K P () A c P if P and he invenory holding cos in period be K I h I () Then he general MP-POM problem becomes: Minimize C T Subjec o I I T A c P h I P P D P Max I I Max,,..., T, T where C = oal cos of producion and invenory wih he managemen horizon = o T A = fixed producion cos in period c = variable uni producion cos in period h = uni invenory holding cos in period P = producion in period P Max = maximum producion capaciy D = forecased demand in period I = invenory a he end of period I Max = maximum invenory capaciy This problem can be solved by dynamic programming (DP). Based on he following properies of opimal soluion I P P, D, D D, D D D,..., D Wagner and Whiin (958) developed a more effecive MP-POM model using DP. In his model he oal cos of producion and invenory over he period j o k is: k Pjk k I jk Aj c jpj j C k h I (4) jk where C jk = cos of producion and invenory in period j+ o saisfy demand in j+, j+,, k. The global opima can be obained by he following DP recursive equaion: k min j C jk k,,..., T (5) j k A each sage of he recursion, we seek o minimize he combinaion of he cos of producion and invenory beween wo regeneraion poins (j and k) and he opimal program up o j. The recursion is compued for k = o T. In he real world MP-POM problem, he amouns of A, c, h, and D are esimaed based on experience, expeced value or oher saisical echniques. These amouns mus be approximaed by single numbers, when we solve he problem by crisp DP. This ype of approximaion always loses some informaion. A beer choice is o represen he esimaed amouns by he use of fuzzy numbers. In his projec, we solve hree MP-POM problems wih () 6
esimaed coss and demands by applying he general fuzzy approach based on he use of fuzzy numbers. Specifically, we develop fuzzy MP-POM models by: (a) using appropriae riangular fuzzy numbers o represen he inerval esimaes A, c, h, D in equaion (4); (b) employing he fuzzy number operaions (+) and (.) o carry ou he corresponding operaions + and. in equaions (4) and (5); and (c) applying he fuzzy number ranking mehod o complee he min problems by boh he crisp and fuzzy models. Then we compare he resuls o evaluae he advanages of he proposed fuzzy MP-POM approach over he curren crisp approaches. 4. MP-POM PROBLEM WITH FUY COSTS AND SCALAR DEMANDS Le us consider a firm making a producion plan for a produc over a managemen horizon of hree periods wih zero iniial invenory. The informaion for his problem is given in Table I. Table I: The informaion of his problem. period demand D seup cos A uni cos c holding cos (unis) ($) ($/uni) ($/uni/period) h (5,,5) (,,6) (,4,5) (l,, 4) (,,6) (,, 5) In his able, he seup coss and variable uni coss are given by riangular fuzzy numbers represening he inerval esimaes. 4. Soluion by Crisp Dynamic Programming (CDP) To obain he opimal producion schedule by CDP, we ranslae he esimaed numbers ino single numbers as shown in Table II. Table II: To obain he opimal producion schedule by CDP. D (unis) A ($,) c ($) h ($) 4 In he above able, he mos possible numbers are chosen as he approximaed single numbers of he esimaed inervals; all he oher informaion is los. Following he Wagner and Whiin [9] approach, we obain: C A c D 5 5 C A c C A c D 5 4 7 C A c D D D h D D hz D D D h D 7 Or 8 Or C A cd D h D 5 4 Or C A c D 7 () 9 9 The minimum oal cos is $9,. The producion schedule (Table III) is found by racing he soluion backwards. 64
9 Produce unis in period for period. 7 Produce 4 unis in period for boh periods and. Table III: The producion schedule. Period Producion 4 4. Soluion by Fuzzy Dynamic Programming (FDP) Now he fuzzy numbers are employed o replace he ranslaed single numbers in he above problem; o obain he opimal producion schedule by FDP, we use he general fuzzy approach and he Wagner and Whiin [9] approach. Using he daa of Table I, we obain: C A c 5,, 5,, 6. 5, 5, 5, 5, C A c D D h 5,, 5,,6. 4 () 5,7,. D. D C ) A c. ( D Or 5, 5,, 4, 45,, 4. 85,8, 75 (5, 7, ) = 5 (85, 8, 75) = 8 8 < 5; (85, 8, 75) < (5, 7, ) (+) C = (85, 8, 75) C A c. D D D h D D h D 5,, 5,, 6. 7 6 45,, 56 C c D D h D Or. 5, 5,, 4, 45,, 4. 45,, 45 Or C A c. D 85,8, 75,, 6,, 5. 65,, 485 C 45,, 56 75 C 45,, 45 9 C 65,, 485 6 = (45,, 45) The producion schedule (Table IV) is obained by racing he opimal fuzzy soluion backwards. = (45,, 45) Produce 6 unis in period for periods and. = (5, 5, ) Produce unis in period for period. 65
4. Comparison of he Two Soluions Table IV: The producion schedule. Period Producion 6 The producion schedules obained by CDP are differen from hose obained by FDP. The oal cos (5, 9, 5) by CDP is more han ha (45,, 45) by FDP (Figure ). This difference in he above example is predicaed upon wheher a single crisp number or a fuzzy inerval is used for A. Obviously i is more pracical o specify ha he esimaed amoun A is abou $, wih an upper limi of $5, and a lower limi of $5,, as indicaed by he fuzzy number A = (5,, 5), raher han by a single number A =,. Thus he producion schedule obained by FDP is more realisic han ha obained by CDP. Also, he producion engineers or operaions managers would require exacly $9,, as per he schedule obained by he crisp model. However, his is no he case, since he minimum oal cos should be (5, 9, 5) housand dollars in order o saisfy he forecased demand of he managemen horizon. This fuzzy cos indicaes ha he oal producion cos is abou $ 9,, wih an upper limi of $59, and a lower limi of $6,. This will enable he producion manager o give appropriae leeway for he oal cos in he budge. Furhermore, he schedules obained by he crisp and he fuzzy models are no necessarily differen. In his example, he fuzzy number operaions used are (i) addiion of a scalar and a riangular fuzzy number; (ii) addiion of wo riangular fuzzy numbers; and (iii) muliplicaion of a riangular fuzzy number and a scalar. If all he fuzzy coss in he problem are symmerical riangular fuzzy numbers and he demands are scalars, hen he resuls afer applying he above hree operaions are sill symmerical riangular fuzzy numbers, as proven by Kaufman and Gupa []. The main par of fuzzy number ranking mehod is he generalized mean, and he generalized mean of any symmerical riangular fuzzy number, M = (a,b,c), is c; hence, if all coss in he POM problem are represened by symmerical riangular fuzzy numbers or scalars, hen he schedule obained by he fuzzy model should be he same as ha obained by he crisp model. Thus he fuzzy model is an alernaive o having o use single numbers in muli-period producion planning. I provides a comparable or beer schedule han he crisp model. Figure : Cos Comparison of Two Differen Schedules. 5. MP-POM PROBLEM WITH FUY DEMANDS AND SCALAR COSTS The daa for his problem are given in Table V. In his example, he forecased demands are fuzzy numbers, while he coss c, A and h are scalars. 66
Table V: The daa for his problem. D A ($) c ($) h ($) (5,, ) (5,, 4) 4 (,, 4) 5. Soluion by Crisp Dynamic Programming (CDP) If we ranslae he esimaed demands ino single numbers and solve he problem by he Wagner and Whiin [9] approach, he daa remain he same as shown in Table II, and he opimal schedule also remains he same as shown in Table III. 5. Soluion by Fuzzy Dynamic Programming (FDP) Here again, we combine he general fuzzy approach wih Wagner and Whiin [9] approach and use all of he informaion given in Table 5 o solve his MP-POM problem. C A c. D. 5,, 5, 5, 8 5, 5, 8 C A c. D D h D D D. 5,, 5,, 4 5,, 5,, 4 5,,,7, 55. C A c. D 5, 5, 8 4. 5,, 4 C 545/ Or 5,8, 4 C 57/ C, 7, 55 C A c. D D D h. D D D D h. D D D D D. 5,, 5,, 4,, 4 5,, 5,, 4,, 4 5,, 5,, 5,, 4,, 4 5,, 5,, 4 9,, 485 Or C A c. D D ( ) h D D. 5, 5, 8 4. 5,, 4,, 4 5,, 4,, 4 5,, 4 5,, 45 C A c. D,7, 55.,, 4, 9, 45 Or D 67
C 5/ C 9/ C 95/ C, 9, 45 The producion schedule (Table VI) is obained by racing he opimal fuzzy soluion backwards. Table VI: The producion schedule. Period Producion (, 4, 6) (,, 4) = ( (+) C ) = (, 9, 45); Produce D = (,, 4) unis in period for period. = (+) C ) = (, 7, 55); Produce D (+) D unis in period for boh periods and. 5. Comparison of he Two Soluions In his example, he resuls obained by CDP and FDP are quie differen. The producion schedule obained by he fuzzy model is represened by fuzzy numbers, which reain all he original informaion; bu he schedule obained by he crisp model is represened by a single number, which los mos of he original informaion. Also, he minimum oal coss obained by he crisp and he fuzzy models are similarly differen. The above wo differences are imporan and useful in pracice. According o he schedule obained by he crisp model, he firm mus produce exacly 4 unis in period and unis in period. However, in pracice, his may no be necessary; also, i may require addiional producion ime leading o increase in cos. 6. MP-POM PROBLEM WITH FUY COSTS AND FUY DEMANDS For his case, we consider a wo-period producion managemen problem as shown in Table VII. In his example, all coss and demands are given by inerval esimaes. Table VII: A wo-period producion managemen problem. D (unis) A ($) c ($) h ($) (,,6) (,,4) (,4,6) (,,) (,5,6) (,4,5) (,,5) (,,4) 6. Soluion by Crisp Dynamic Programming (CDP) The ranslaed scalar coss and demands are given in Table VIII. 68
Table VIII: The ranslaed scalar coss and demands. C 5 A c D 4 D A c h 4 5 4 5 C A c D D h 4 5 5 45 D Or C A c D 5 4 5 4 The opimal schedule is shown in Table IX. Table IX: The opimal schedule. Period Producion 5 6. Soluion by Fuzzy Dynamic Programming (FDP) C A c. D,, 4, 4, 6.,, 6 Or, 8 6 x 4x, C A.5.5 / 6, /, x 4 4 x 5 5 x 4 x 4 c. D D h D D D,,4, 4, 6.,, 6, 5, 6,,.,, 6, 5, 6,, 6., 4 6 6x 74 6 5x, C A.5 /.5 /, 6, x 6 6 x 45 45 x 6 x 6 c D, 4, 5,, 5., 5, 6.,.5 7 49 x /,.5 49 x /6,, C C C x x 4 4 x 75 x 75 The opimal producion schedule is shown in Table X. 69
Table X: The opimal producion schedule. Period Producion D = (,,6) D = (,5,6) 6. Comparison of he Two Soluions The soluions obained by CDP and FDP are compleely differen. The former is represened by single numbers and has los much of he original informaion. The laer is represened by fuzzy numbers and sill reains he inerval naure. As discussed earlier, his difference is very imporan and useful in pracice. These examples demonsrae he procedures and advanages of he general fuzzy approach using fuzzy numbers. Their resuls indicae ha his proposed approach is an effecive echnique o solve he fuzzy MP-POM problems. 7. DISCUSSION In his secion hree fuzzy MP-POM mehods are compared and conrased. They are: Sommer s mehod, Kacprzyk-Saniewski s (K-S s) mehod, and he mehod developed in his work. 7. Sommer s Mehod Sommer [] solved a numerical example of a fuzzy MP-POM problem o demonsrae his mehod. Mahemaically his problem is described as follows: The membership funcion of producion level is:, if Pi 6 i.5i Pi /, if 6 i Pi 8 i C Pi (6) 5.5i Pi /, if 8 i Pi i, if 95 i Pi The membership funcion of invenory level a he end of he managemen horizon is: xn / if xn xn (7) elsewhere The managemen horizon is N = 4, he iniial sock level is x =, and he demands are: D = 45, D = 5, D = 45, D 4 = 6. In his siuaion, boh demand and invenory are crisp numbers, bu he wo objecives - producion level shall decrease as seadily as possible and he ending invenory level shall be as low as possible - are fuzzy. I seems ha he fuzzy environmen of his producion managemen problem is provided arificially raher han naurally. The reason for our employing a fuzzy se or a fuzzy number o denoe a goal is ha he goal canno be precisely defined due o lack of accurae informaion. However, when exac daa are available, such as demand and invenory in he above example, he decision maker should no se up a fuzzy objecive, since i is invariably eiher subopimal or infeasible. In Sommer s example, if we hink of he objecive he ending invenory level shall be as low as possible as a consrain x N+ =, and his oher objecive he producion level shall decrease as seadily as possible as a funcion of ime, dp/d = f(), hen he opimal producion schedule can be easily obained by crisp dynamic programming echniques. 7. Kacprzyk - Saniewski s (K-S s) Mehod The convenional approaches o producion and invenory conrol usually involve opimizaion of a performance index consising of some cos-relaed erms. Kacprzyk and Saniewski [] ransformed he cos opimizaion problem ino one of mainaining some desired invenory 7
level. Tha is, hey believe he average coss usually are known o be some funcions of invenory and hey are mosly isomorphic; he same applies o replenishmen or producion. In heir example, i is assumed ha X = (,, l); u, Y = l,...,5). Given are he following: he reference fuzzy invenory level S,.,S 5 ; he reference fuzzy replenishmen C,.,C ; he fuzzy demand D and fuzzy goal G; he fuzzy consrains C(S ),.,C(S 5 ). The above problem differs from Sommer s problem in ha boh is demand and invenory are fuzzy. This problem is more fuzzy and more pracical han Sommer s problem in he sense of fuzzy ses heory. Bu from he pracical poin of view, i is oo hard for he decision maker o provide he necessary and consisen daa for deermining he membership funcions of he S i s, C j s and C(S i ) s. This is especially rue when i and j are big numbers. As an alernaive o using Kacprzyk and Saniewski s mehod, he problem can be solved by a procedure similar o our proposed general fuzzy approach as follows. (a) use an appropriae fuzzy number, I, o indicae he desired invenory level ; (b) use he ransacion funcion I = I - (+) P (-) D and he given I and D o deermine P, = l,,...,n, where P is he fuzzy replenishmen or producion o mainain he desired invenory level and saisfy he fuzzy demand; (c) consider boh P and he consrains on he replenishmen or producion simulaneously o deermine he opimal replenishmen policy; (d) if he cos funcions relaed o invenory level, f(i ) and replenishmen level, f(r ) are known, we can use he oal cos funcion F = f(i ) (+) f(r ) o deermine he opimal policy. Following he above procedure, all he decision maker has o do is providing esimaes for he desired invenory level, he replenishmen consrains, and he cos funcions. 7. Our Proposed Mehod As shown in secions 4, 5, and 6 above, he fuzzy environmen provided in our hree examples is a naural one. We can obain he necessary informaion on fuzzy coss and fuzzy demands by requiring he decision maker o provide he corresponding esimaes. Our mehod no only gives he opimal fuzzy schedule bu also he minimum fuzzy oal cos. From a pracical view poin, hese are very imporan and useful for producion engineers and operaions managers. Thus, we can summarize he essenial feaures of he hree fuzzy MP- POM mehods in Table XI. 7
Table XI: The hree fuzzy MP-POM mehods. Sommer s K-S s Ours Goal Fuzzy Fuzzy Fuzzy Consrain Fuzzy Fuzzy Fuzzy Demand Crisp Fuzzy Fuzzy Invenory Crisp Fuzzy Fuzzy Coss -------- -------- Fuzzy Oher Condiions Same Same Same Fuzzy Environmen Arificial Naural Naural Transiion Equaion I I P D I I P D I I P D Opimizaion Funcion Max. G C Max. G C Min. Toal Cos Schedule Mos Saisfacion of Decision Maker Mos Saisfacion of Decision Maker Minimizaion of Toal Cos of Firm Daa Collecion Medium Difficul Easy Compuaion Simple Tedious Medium Fuzzy Mulisage Decision Bellman & adeh s Bellman & adeh s Process Model Model 8. CONCLUSIONS General Fuzzy Approach Using Fuzzy Numbers This work achieved he purpose of finding an effecive mehod o solve he inerval MP-POM problem while reaining all he original informaion hrough he general fuzzy approach, namely employing appropriae fuzzy numbers o represen he inerval esimaes in he mulisage decision problems; using he operaions of fuzzy numbers combined wih dynamic programming o solve he problem; and deermining he required minimum/maximum fuzzy number hrough fuzzy number ranking echniques. Three MP-POM problems wih fuzzy coss and/or fuzzy demands were solved by his approach. The opimal soluions of producion schedule and oal cos were indicaed by fuzzy numbers. The main advanages of his approach are fuzzy represenaive soluions for he opimal producion schedule as well as he minimum oal cos in erms of inerval unis raher han single numbers. These soluions give he producion engineers and he operaions managers a clear picure abou he inerval unis of producion a each period and he inerval unis of he required minimum oal cos, hus enabling hem o manage he producion flexibly and conrol he cos effecively. On he oher hand, he crisp model gave soluions by single numbers and los a significan amoun of useful informaion. In such a siuaion, he only opion available o he engineers and managers is o produce he exac number of unis given by he schedule and hope o incur he so called minimum oal cos. Obviously, he proposed fuzzy approach is more realisic han he radiional crisp approach. Anoher advanage of he proposed fuzzy approach is i parially overcomes he dimensionaliy difficuly of large scale problems. Thus a poenial area for fuure research is o represen he daa needed o solve real life mulisage problems by fuzzy numbers and carry ou he compuaions by dynamic programming using compuers. In his paper, he proposed approach was applied o solve problems wih fuzzy goals and fuzzy consrains defined in he same space. Hence anoher promising area for furher research is o exend his approach o solve problems wih fuzzy goals and consrains defined in differen spaces. REFERENCES [] Corsano, G.; Aguirre, P.A.; Monagna, J.M. (9). Muli-period Design and Planning of Muliproduc Bach Plans wih Mixed-Produc Campaigns, American Insiue of Chemical Engineers (AIChE) Journal, Vol. 55, Issue 9, 56-69 7
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