LEVENTE SZÁSZ An MRP-based integer programming model for capacity planning...3

Transcription

1

2 LEVENTE SZÁSZ An MRP-based ineger programming model for capaciy planning...3 MELINDA ANTAL Reurn o schooling in Hungary before and afer he ransiion years...23 LEHEL GYÖRFY ANNAMÁRIA BENYOVSZKI ÁGNES NAGY ISTVÁN PETE1 TÜNDE PETRA PETRU Influencing facors of early-sage enrepreneurial aciviy in Romania...33 ILDIKÓ RÉKA CARDOª ISTVÁN PETE DUMITRU MATIª Tradiional or advanced cos calculaion sysems? This is he quesion in every organizaion...47

3

4 LEVENTE SZÁSZ 1, 2 Absrac Capaciy planning decisions represen an imporan opic in he operaions managemen lieraure, as hey grealy influence he performances of he firm and deermine if he firm is able o saisfy marke demand in each period. In order o be able o deal wih uncerainies and flucuaions of marke demand and oher elemens of he inernal and exernal environmen, capaciy planning decisions are broken down ino several levels of aggregaion. This paper builds a general ineger programming model for capaciy planning a he maser producion scheduling (MPS) level. Inpu daa and parameers for he model can be gahered from a ypical MRP/MRP II-based informaion sysem. Using he ineger programming model we can deermine he opimal level of producion capaciies, a which operaing coss are minimal. The las chaper of he aricle presens a case sudy, where we use he ineger programming model buil o invesigae wha cos reducions are possible by opimizing capaciy levels a a small-sized firm from he exile indusry. Keywords: capaciy planning, MRP/MRP II sysem, producion planning and conrol, ineger programming, maser producion schedule. 1. Inroducion Capaciy planning and conrol decisions are one of he mos fundamenal decisions in operaions managemen as hey provide he capabiliy of a company o saisfy curren and fuure marke demand. Addiionally, capaciy planning decisions provide a rigid framework and consrain for oher operaions and producion decisions. In a coninuously changing, dynamic environmen firms face many uncerainies, like variable cusomer demand, and he managemen of hese firms has o find a way o effecively deal wih hese uncerainy facors. In capaciy planning he main facor ha influences a firm s operaional efficiency is flucuaing marke demand. A firm can deal wih demand uncerainies basically in wo differen ways on aggregae level (Vörös 2007): acively aemp o change demand paerns o fi capaciy availabiliy (1), or reacively adjus producion capaciies o deal wih he flucuaions in demand 1 Inves in people! PhD scholarship in Projec financed from European Social Fund hrough he Secoral Operaional Program Human Resources Developmen Ph. D. candidae, Babeº Bolyai Universiy, Faculy of Economics and Business Adminisraion.

5 Levene Szász (2). In his paper we will invesigae he second opion o deal wih changes in demand. The second, reacive opion includes wo differen capaciy sraegies: he firm can predic fuure demand flucuaions and se producion capaciy o a consan level, which sill makes possible o fulfill marke demand in each period, or i can aemp o follow he flucuaion of marke demand by adjusing capaciy levels in each period. In his aricle we ry o build an ineger programming model which follows he laer sraegy and changes producion capaciy levels accordingly o marke demand flucuaions. In order o achieve his, in he second chaper we will demonsrae ha capaciy planning decisions play a crucial role in a firm s operaions sraegy and can have a major impac on a firm s overall producion efficiency. In he hird chaper we also briefly review hese imporan capaciy decisions are broken down ino muliple hierarchical levels in order o find he bes capaciy configuraion possible. The fourh chaper invesigaes he connecion beween producion planning and conrol (PPC) informaion sysems and he capaciy planning process, highlighing hose elemens of a PPC sysem which can offer a high qualiy inpu daa for capaciy planning on he maser producion scheduling (MPS) level. Based on hese inpu daa, in he fifh chaper, we build a general linear programming model wih ineger variables for capaciy planning on he MPS level. The sixh chaper conains a case sudy which uses he model buil in he previous chaper o demonsrae in a real case ha by deermining he opimal capaciy level he firm can improve he efficiency of is producion sysem and he firm s overall performance. 2. The sraegic role of capaciy planning decisions Capaciy planning is an imporan par of operaions sraegy as i plays a criical role in he success of an organizaion. In operaions managemen capaciy planning problems are one of he mos criical ones, since all oher operaions planning problems are managed wihin he framework of he capabiliies se by he capaciy plan. Producion capaciy decisions for new, expanding, and exising faciliies have a direc impac on a firm's compeiive posiion and resuling profiabiliy (Hammersfahr e al. 1993). There are several reasons, which make capaciy decisions one of he mos fundamenal decisions of sraegic operaions managemen (Sevenson 1996): Capaciy decisions have a major impac on a firm s abiliy o mee fuure demand for producs and services; There is a significan relaionship beween capaciy and operaing coss. Coss of over- and undersized capaciy should always be aken ino consideraion in sraegic planning.

6 An MRP-based ineger programming model for capaciy planning The sraegic imporance of capaciy decisions also lies in he iniial cos involved. Required producion capaciy is usually a major deerminan of invesmen coss. The imporance of capaciy decisions sems from he ofen required longerm commimen of resources and he fac ha, once hey are implemened, i may be difficul or impossible o modify hose decisions wihou incurring major coss. Referring o he imporan role of capaciy planning and conrolling problems, oher auhors argue ha he decisions operaions managers ake in devising heir capaciy plans will affec several differen aspecs of performance (Slack e al. 2007): Coss will be affeced by he balance beween capaciy and demand. Capaciy levels in excess of demand could mean under-uilizaion of capaciy and herefore high uni coss. Revenues will also be affeced by he balance beween capaciy and demand, bu in he opposie way. Capaciy levels equal o or higher han demand a any poin in ime will ensure ha all demand is saisfied and no revenue is los. Working capial will be affeced if an operaion decides o build up finished goods invenory prior o demand. This migh allow demand o be saisfied, bu he organizaion will have o fund he invenory unil i can be sold. Qualiy of goods or services migh be affeced by a capaciy plan, which involved large flucuaions in capaciy levels, by hiring emporary saff for example. The new saff and he disrupion o he operaion s rouine working could increase he probabiliy of errors being made. Speed of response o cusomer demand could be enhanced, eiher by he build-up of invenories (allowing cusomers o be saisfied direcly from he invenory raher han having o wai for iems o be manufacured) or by he deliberae provision of surplus capaciy o avoid queuing. Dependabiliy of supply will also be affeced by how close demand levels are o capaciy. The closer demand ges o operaion s capaciy ceiling, he less able i is o cope wih any unexpeced disrupions and he less dependable is deliveries of goods and services could be. Flexibiliy, especially volume flexibiliy, will be enhanced by surplus capaciy. If demand and capaciy are in balance, he operaion will no be able o respond o any unexpeced increase in demand. 3. Main objecive and srucure of capaciy planning decisions Capaciy planning is a very complex decisional process and, as we have seen, can have a major impac on a firm s compeiive posiion. As he mos

7 Levene Szász imporan goal of capaciy planning is o mee fuure cusomer demand, running a producion sysem wih sufficienly large capaciies seems o be a prioriy. However, having oo much capaciy leads o underuilizaion of capaciies and wased capial coss, which weaken he compeiive posiion of a firm. On he oher hand, having less capaciy han required in a cerain period of ime leads o unfulfilled marke demand, which can damage he firm s public image and incurs he coss of los sales. Therefore, over-uilizaion of capaciies has also a negaive impac on he firm s compeiive posiion. Thus, capaciy planning involves delicae decisions, which balance beween he wo exremiies illusraed above. Uncerainies regarding fuure marke demand increase he difficuly of his problem. However, finding he opimal soluion of capaciy-sizing problems and expanding/reducing capaciies a he righ momen of ime can significanly conribue o he efficiency of a producion sysem. In order o find an opimal, balanced soluion, capaciy planning decisions are usually made a muliple hierarchical levels in an organizaion, which involve differen planning horizons. Usually, we can idenify wo major levels of capaciy decisions, which are par of eiher he long-erm, or he shor-erm capaciy planning process (Krajewski e al. 2007, Worman e al. 1996). Bahl (Bahl 1991) describes managerial decisions in capaciy planning as a hierarchy of decisions, where capaciy planning involves longerm decisions wih a planning horizon of 3-10 years, while capaciy adjusmens and balancing involve medium-range decisions wih a planning horizon of 1-3 years. In oday s rapidly changing markes hese numbers may seem somewha rough and large, when flexibiliy and quick responsiveness o environmenal changes is considered o be a key facor o marke success. Sill, he hierarchisaion of hese decisions shows he complexiy of capaciy planning process, involving differen planning horizons. Long- erm capaciy planning decisions are also referred o as rough-cu capaciy check (RCCC) or simply capaciy planning, while shor-erm capaciy planning is also referred o as capaciy requiremens planning (CRP) or deailed capaciy planning (DCP) (Gupa e al. 2006, Worman e al. 1996). The firs caegory represens he planning process which decides how much of each capaciy resource o insall, while he second caegory allocaes available capaciy among differen producs or jobs (Balachandran e al. 1997). Acually, RCCC is a capaciy availabiliy check applied a he maser producion schedule level for longer-erm planning purposes, while DCP is a capaciy availabiliy check applied o planned order releases for shorer-erm planning purposes (Ding e al. 2007).

8 An MRP-based ineger programming model for capaciy planning Anoher approach o hese conceps suggess ha he enire planning process is sared from he highes level of produc aggregaion and i is coninued unil he lowes possible level of deail is reached. The ypical hierarchy of plans involves differen approaches o capaciy planning a every level: Table 1. Typical hierarchy of capaciy planning decisions Aggregaion level Aggregae planning Maser Producion Scheduling (MPS) Maerial Requiremens Planning (MRP) Shop floor conrol Type of capaciy planning Resource Requiremens Planning (RRP) Rough-Cu Capaciy Planning (RCCP) Capaciy Requiremens Planning (CRP) Inpu-oupu analysis Source: Diaz - Laguna 1996 According o he same auhors, subdividing he capaciy planning problem ino hierarchical levels according o differen degrees of aggregaion has he advanage of increasing he racabiliy of he corresponding subproblems. In addiion, hese subproblems usually map o differen responsibiliy levels of he organizaional srucures of mos companies. This paper focuses mainly on long-erm capaciy planning (Rough-cu capaciy planning) and on he iming of capaciy expansions/reducions, bu in he case-sudy presened we include also some shor-erm consideraions regarding capaciy adjusmens ino he capaciy planning model. Looking a he aggregaion level we focus basically on he Maser Producion Scheduling level by performing a rough-cu capaciy check. However, by including in our model he possibiliy o use overime, we reach down in he hierarchy unil he Maerial Requiremens Planning level, and idenify some adjusmen possibiliies on he level of differen produc ypes. 4. The role of informaion sysems in capaciy planning decisions For an efficien capaciy planning process here is a need for accurae informaion regarding (Naghi e al. 2009): 1. Quaniies of expeced fuure demand for differen produc ypes; 2. Duraion and ype of manufacuring processes needed (or machine imes ) in order o produce he produc ypes demanded;

9 Levene Szász 3. Curren capaciies available regarding every ype of manufacuring processes (or number of differen machine ypes available); 4. Manufacuring coss on process level (monhly cos of mainaining and operaing differen ypes of machines, cos of labor force, cos of overime ec.). These informaion are needed o develop a capaciy plan which enables he producion of differen produc ypes - in he demanded quaniy and a he lowes possible cos. This requires a capaciy plan, which minimizes he cos of over- and underuilizaion of producion capaciies, bu does no incur he cos of los sales. Informaion sysems for producion planning and conrol usually posses a daabase, which conains a major par of he informaion, needed as inpu for an efficien capaciy planning process. As informaion soluions evolved saring from Maerials Requiremen Planning (MRP) sysems, hrough Manufacuring Resource Planning (MRP II) sysems, reaching he Enerprise Resource Planning (ERP) sysem s complexiy, capaciy planning echniques were ried o be included as par of hese sysems, wih greaer or lesser success. Original MRP sysems aimed a efficien scheduling of producion requiremens so ha raw maerials, componens and subassemblies can be provided in he righ amoun and a he righ ime (Ram e al. 2006). ERP sysems oday are he laes and he mos significan developmen of he original MRP philosophy. The hisorical developmen of ERP sysems can be followed on figure 1. Increasing impac on he whole supply nework Web-inegraed enerprise resource planning (collaboraive commerce, e-commerce) Enerprise resource planning (ERP) Manufacuring resource planning (MRP II) Maerial requiremens planning (MRP) Increasing inegraion of informaion sysems Source: Slack e al Figure 1. The evoluion of MRP-based producion planning and conrol sysems

10 An MRP-based ineger programming model for capaciy planning Many auhors sae (Du - Wolfe 2000, Gould 1998), ha one major problem of MRP-based informaion sysems is ha hey mainly ignore capaciy consrains, assuming infinie resources (machine capaciy and labor). Capaciy Requiremens Planning (CRP) and is complemenary sysems a differen levels of aggregaion were added o MRP II sysems and hey ried o correc his siuaion by idenifying under-uilizaion and overload condiions a a machine or manufacuring uni. Some auhors repor ha even CRP sysems in an MRP II environmen provide only informaion regarding a capaciy problem, bu no a viable soluion o he planner (Wuipornpun - Yenradee 2004). In his paper we develop a simple capaciy planning sysem based on he 4 ypes of informaion, lised a he beginning of his chaper. Looking a an MRP/MRP II based informaion sysem as a se of algorihms and objecs, he needed informaion is provided by he following pars (unis, objecs, daa ables) of he sysem: 1. Fully funcional MRP II sysems include a uni, which suppors he sales and markeing funcion of he company. This uni nex o order enry and billing is also responsible for projecing, forecasing produc demand based on hisorical daa (Duchessi e al. 1989). Using powerful saisical ools hey can provide vial inpu informaion for capaciy planning regarding he expeced demand of differen produc ypes. Having he expeced demand we can build he fuure maser producion schedule (MPS), which is a prerequisie o perform he rough-cu capaciy check (Diaz - Laguna 1996). 2. Machine ypes and machine imes needed o produce he demanded ypes of producs is included in he operaions lis. This describes he duraion and sequence of asks o be performed o produce a finished good or a fabricaed componen. 3. As MRP sysems evolved o MRP II sysems, a larger daabase was also creaed which included daa no only referring o he ype and quaniy of maerials needed in producion, bu also regarding he number of machines, ypes of capaciies and labor force needed (Segerean 2002). This informaion provides us he acual capaciy consrains, which have o be aken ino consideraion during he capaciy planning process. 4. The evoluionary process of MRP sysems, illusraed on figure 1., also mean, ha MRP II sysems ook over he role o ie he basic MRP sysem o he company s financial sysem and oher core processes (Krajewski e al. 2007). This enables us o calculae he exac coss o mainain a uni of capaciy (a machine), including he cos of capial, he cos of labor force and oher incidenal expenses.

11 5. An ineger programming model for capaciy planning Using he informaion lised above we can formulae a general linear programming model, which raionalizes capaciy uilizaion. We build his model based on a paper presened by he auhor and wo co-auhors on an inernaional conference (Naghi e al. 2009), and we also add some minor modificaions. The following model aims o minimize he oal operaing cos of he producion capaciy used, while saisfying marke demand (los sales are no allowed). For a general linear programming model we firs need o deermine he values of he wo parameers lised below: c he average number of days of order cycle imes h he average number of days afer which he company can make decisions regarding capaciy dimensioning. Wih he help of hese o parameers, we can find ou how many orders occur over one capaciy planning horizon (T), which is he average number of days in a capaciy planning horizon (h) divided by he order cycle ime (c): T = h / c For example in he case sudy presened in his paper we use he following ime periods: Average order cycle ime is one week afer receiving he placed order from he cusomer he company has one week o produce he demanded quaniy Capaciy planning ime horizon is one monh he company can make capaciy resizing decisions on a monhly basis Having his informaion, he objecive funcion of he linear programming model can be formulaed as follows: T =1 Levene Szász n i=1 n i i i i i=1 where he elemens of he objecive funcion denoe he following quaniies: z objecive funcion value ha has o be minimized; z represens he oal cos needed o operae he producion capaciies (machines) per monh (boh in regular ime and overime); curren index of order cycle ime T number of order cycle imes in one capaciy planning ime-horizon i index number denoing he curren machine ype n number of machine ypes used by he company i

12 An MRP-based ineger programming model for capaciy planning x i represens he vecor of he decision variables (X), which conains he curren number available of every i h ype producion equipmen needed in a monh (i = 1, n): X = [ x 1, x 2,..., x n ]; c i sands for he operaing cos of he i h machine ype in regular ime during one order cycle ime. Here we include all of he expenses which would no become due in case of he machine was no used. If he i h machine ype is rened or leased by he company over he planning horizon c i can be calculaed as follows: c i = ren or lease paymen per week + cos of labor force needed o operae he machine per week + average incidenal coss per week (mainenance coss, repairing coss ec.). In case he i h machine ype is no rened or leased, we assume ha i was purchased by he company o be used for a cerain period of ime (useful lifeime of he machine). In his case he price of he machine can be convered ino equivalen paymens in every order cycle ime, using an annuiy formula, dispersing he invesmen coss over he lifeime of he machine. We can deermine hese paymens by using he annuiy formula, given below (Brealey - Myers 2005): PV = C 1 1, where r r (1 + r) PV presen value of he annuiy (C yearly paymens), in our case he amoun of he iniial invesmen (he price of he machine) C yearly paymens r average cos of capial used by he company for evaluaing invesmen opions useful lifeime of he machine Expressing he value of C from he formula above and dividing i by he number of order cycle imes in a year (denoed by cy), we ge he amoun of weekly paymen (c i ) equivalen wih he iniial purchase price of he machine: c 1 i = cy 1 r PV 1 r (1 + r).

13 Levene Szász The remaining elemens of he objecive funcion denoe he following quaniies: o i average number of hours worked in overime by he i h machine ype in period. Overime decisions during he capaciy planning horizon can be represened by he following TO marix ( = 1, T i = 1, n): TO = o 1 1 o 1 n o T 1 o T n co i cos of operaing he i h machine ype in overime per hour, including only he cos of labor and incidenal expenses, excluding ren or lease paymen. We wan o minimize he objecive funcion, subjec o he following consrains: 5.1. Monhly demand consrain m j=1 j ji i i i m number of produc ypes manufacured by he company D j quaniy demanded of he j h produc ype in period. The forecasing module of he producion planning and conrol sysem should provide he projecion of demand for he following capaciy planning period for each produc ype, which can be wrien in he following marix (D) form ( = 1, T j = 1, m): D = D 1 1 D 1 m D T 1 D T m h ji manufacuring hours needed on he i h machine o produce he j h produc ype (j = 1, m i = 1, n) H = h 11 h 1n h m1 h mn Every row of he marix above represens he operaions lis for differen

14 An MRP-based ineger programming model for capaciy planning produc ypes, indicaing how much operaion ime a cerain produc requires on differen machines. r i oal regular operaing ime available for machine i in a week, which remains consan during one capaciy planning horizon (i = 1, n) TR = [r 1, r 2,..., r n ] Regular operaing ime per week can be calculaed as: r i = (number of working days in one order cycle period) (available working hours for machine i per day). The demand consrain expresses ha here has o be enough capaciy during every period o fulfill he demand by producing in regular ime and in overime Overime consrain o i r i ol (%) i = 1, n ; = 1, T, where ol(%) he maximum limi of overime per period, expressed as a percenage of regular operaing ime per period. The value of his maximum limi can be deermined according o he regulaions of he curren legal sysem regarding labor force or can be found in a conracual framework wih he workers or syndicae Non-negaiviy consrains x i 0 i = 1, n o i 0 i = 1, n = 1, T The value of every decision variable should be greaer han or equal o zero Ineger consrain x i ineger i = 1, n The value of he x i decision variable, represening he number of ype i machines should have an ineger value. By inroducing his laer consrain we ransform he linear programming problem ino an ineger programming model.

15 Levene Szász Summarizing he ineger programming model buil in his chaper we ge: Decision variables: X = [x 1, x 2,..., x n ] number of machines of differen ypes o i - overime decisions (hours) for each machine ype and each period of he capaciy planning horizon ( = 1, 4 i = 1, n): TO = o 1 1 o 1 n Parameers: c i one period s operaing cos for machine i in regular ime co i one period s operaing cos for machine i in overime D j projeced demand for produc j in period h ji manufacuring ime for produc j on machine i r i oal regular operaing ime available for machine i per period ol(%) maximum limi of overime per period, expressed as a percenage of regular ime available curren number of period wihin he capaciy planning horizon n number of differen machine ypes used by he firm m number of differen produc ypes produced by he company Objecive funcion: T =1 n i=1 o T 1 o T n n i i i i i=1 i Subjec o: m D j h ji j=1 o i x i (r i + o i ) i = 1, n; = 1, T r i ol (%) i = 1, n; = 1, T x i 0 i = 1, n o i 0 i = 1, n; = 1, T x i ineger 6. Case sudy In he presen case sudy we analyze some aspecs of he capaciy planning process a a Romanian small-sized enerprise from he exile indusry, based

16 An MRP-based ineger programming model for capaciy planning on a paper presened on an inernaional conference (Naghi e al. 2009). The firm execues exile operaions on a make-o-order basis, working in lohn sysem. The lohn producion sysem means ha maerials, pars, subassemblies and semi-finie producs needed o execue he producion process are delivered by he cusomer, who places he order. In such a sysem basic MRP asks like maerial handling, invenory conrol and procuremen are handled by he cusomer. Afer only six monhs of operaions he firm was forced o declare bankrupcy, due o he exremely high operaing expenses. In he presen case sudy we analyze if he firm using he capaciy planning model presened in he previous chaper could have improved he efficiency of he producion sysem by raionalizing capaciies and decrease operaing expenses Producion resources Looking a he maerial resources used by he firm hese were specific o he exile and clohing indusry. All ypes of maerial resources were delivered by he clien, who was responsible for shipping he righ quaniy a he righ ime for he firm. The effec of he lohn sysem on he cos-srucure of such a firm can be easily undersood if we look a he percenage of maerial coss relaed o he oal coss of he firm in one period. Using daa from he monhly income saemen of he firm we can calculae he average value of his raio over he lifeime of he firm, which was equal o 0.545%, a percenage ha can be considered insignifican. Having such low maerial coss, machine operaing coss and he cos of labor accouned for a major par of he firm s oal expenses. Consequenly, capaciy decisions were a major deerminan of he firm s overall performance. The firm was using 10 differen ypes of machines (for simpliciy le us denoe hem wih capial leers: A, B,..., J), each of hem requiring one qualified worker. The machines were rened on a monhly basis (for a fee of 50 RON/equipmen/monh). The acual number of differen machine ypes is presened in he able below: Table 2. Number of differen machine ypes used by he firm A B C D E F G H I J No. of machines Source: own calculaions

17 Levene Szász 6.2. Parameers of he ineger programming model As capaciies were rened on a monhly basis, he lengh of he planning horizon is se o one monh, and average order cycle ime was one week. Therefore, he value of T (number of weeks in a monh) is se o 4. The number of differen machine ypes: n = 10. The number if differen produc ypes produced by he company: m = 6. Furher on, we will denoe he differen produc ypes wih T1, T2, T3, T4, T5 and T6. To consruc he maser producion schedule (MPS) we use order enries from he monh wih he maximum demand. Having he order enries of his monh we consruc he MPS, indicaing produced quaniies for each week and each produc ype: Table 3. Maser producion schedule for a monh MPS T1 T2 T3 T4 T5 T6 Week Week Week Week Source: own calculaions Daa from he able above represen he values of he D marix, conaining he produced quaniies of every produc j in week. To consruc he operaions lis, which conains he duraion and ype of operaions needed for every ype of produc, we use he echnological descripions of he producion process for every produc ype. The able below conains manufacuring imes in minues for one uni of each produc ype on each machine. If a cerain machine is no needed o produce a cerain produc ype, manufacuring ime is se o zero. Daa from he 4. able represen he values of he H marix conaining values of h ji (j = 1, m i = 1, n) manufacuring ime needed for produc j (in columns) on machine i (in rows). Toal regular operaing ime in a week for each machine (r i ), expressed in minues is equal o: Days in a week: 5 Effecive operaing hours: 7 Minues/hour: min.

18 An MRP-based ineger programming model for capaciy planning Table 4. Manufacuring imes (minues) Produc ype Machine ype A B C D E F G H I J T1 T2 T3 T4 T5 T Source: own calculaions Noe ha he effecive operaing hours are less han 8 hours per day, due o 30 minues mainenance ime and a 30 minue lunch-break. Regular operaing ime is equal o 2100 minues for every ype of machine: r i = 2100 min i = 1, n The maximum limi of overime is se o 10% of regular operaing ime per week (ol(%)=10%), which is equal o a maximum of 3.5 hours overime per week. Calculaing weekly operaing coss for he differen ypes of machines includes he cos of capial, cos of labor force needed o operae a machine and he cos of mainenance and breakdowns. Supposing ha monhly ren is 50 RON/machine, monhly gross salary of one qualified worker is 900 RON and mainenance coss are 50 RON/machine, oal operaing cos per machine in regular ime can be calculaed as follows: Ren per week: Labor cos per week: 225 Mainenance cos per week: RON

19 Levene Szász In our case oal weekly operaing cos is he same for each ype of machine. Therefore, regular operaing coss are equal o 250 RON for every machine: c i = 250 RON i = 1, n Overime operaing coss for each machine ype will include only he cos of labor (which is 25% higher han in regular ime) and he addiional cos of mainenance. We calculae overime operaing cos per minue, since values of o i oal overime of one uni of machine i in week are also expressed in minues. Labor cos per minue: Mainenance cos per minue: RON Noe ha labor cos per minue in overime is 25% higher han in regular ime. Overime operaing coss are consan for every machine ype. c i = 0,122 RON i = 1, n 6.3. Building and solving he ineger programming model Having all he parameers calculaed as seen above we can build an ineger programming model o search for he opimal capaciy planning decision a our firm. Objecive funcion: 4 =1 10 i=1 10 i i i i=1 Subjec o: 6 D j j=1 h ji x i ( o i ) i = 1, 10; = 1, 4 o i % i = 1, 10; = 1, 4 x i o i 0 i = 1, 10, x i ineger 0 i = 1, 10; = 1, 4

20 An MRP-based ineger programming model for capaciy planning Lef-hand side parameers and he opimal righ-hand side values (for each machine i and each week ) of he firs consrain can be found in he Appendix. By calculaing he opimal soluion o his linear programming model (for example wih Microsof Excel s Solver funcion) we ge he opimal values of he decision variables (number of machines, quaniy of overime used) a which he value of oal operaing cos per monh (z) is minimal. The able below conains he opimal ineger values for each machine ype o be used (x i ): Table 5. Opimal number of differen machine ypes A B C D E F G H I J No. of machines Source: own calculaions This opimal soluion requires an overime of minues in week 3 only for machine E, which means ha: o 5 3 = 95.57, and o i = 0 i = 1, 10 ; = 1, 4 where i 5 and 3 Using he opimal values of he decision variables we calculae he opimal value of he objecive funcion: z = RON Hence, oal operaing cos of capaciies in a monh (including regular and overime) is equal o 4558,49 RON, from which: Regular ime cos = 4500 RON Overime cos = RON To evaluae wha improvemens his model could have brough o he firm, we calculae he oal operaing cos per monh based on he iniial capaciy configuraion (see Table 2) in which case he z value equals 6500 RON. This value is much higher hen he opimal objecive funcion value, meaning ha an almos 30% cos decrease would have been achievable (-29.87%). I is no sure if his cos reducion could have been he mos criical facor o avoid bankrupcy, bu i subsanially improves he efficiency of he producion sysem, increasing he compeiiveness of he firm.

21 Levene Szász The model can also be used o make opimal decisions abou he iming of capaciy expansions or reducions. By running his linear programming model for several consecuive monhs he decision maker can follow he opimal level of capaciies for each period and decide abou expanding i or as in our case reducing i. 7. Conclusions In he presen paper we reviewed he main challenges of he capaciy planning process and argued ha capaciy decisions can grealy influence he overall performance of a firm. We analyzed he srucure and hierarchy of capaciy planning decision and idenified informaion (inpu daa) needed o perform he capaciy planning process. Based on he informaion provided by an MRP/MRP II based producion planning and conrol sysem we developed a linear programming model wih ineger variables, which deermines he opimal dimension of producion capaciies by minimizing oal operaing coss. As demonsraed in he case sudy, his model can be used o bring subsanial improvemens o he efficiency of a producion sysem and can also help decision makers o make opimal decisions abou capaciy expansions or reducions a he righ momen of ime. In his laer case he ineger linear programming model has o be run using as inpu daa he maser producion schedules of several fuure ime periods. The model indicaes he opimal capaciy levels for each period; however, i does no deermine he opimal capaciy dimension for a longer period of ime. Sill, one of he model s mos imporan limiaions is ha i execues capaciy planning only on an aggregae level, by making a rough-cu capaciy check based on he maser producion schedule. The model does no ake ino accoun oher parameers from a lower level of aggregaion, like lo sizing rules, seup imes or boleneck capaciies. However, i akes ino consideraion he possibiliy of a firm o use overime producion. The model can be furher refined by adding oher consrains, reaching down o lower levels of aggregaion and including oher special coss like seup coss.

22 An MRP-based ineger programming model for capaciy planning Appendix 1. The marix of he necessary machine imes for each machine ype in each week o saisfy marke demand (Lef-hand side values of he firs consrain, L): Machine A B C D E F G H I J Week Week Week Week Every elemen of he marix above is calculaed as follows: L i = 6 D j j=1 h ji i = 1, 10; = 1, 4 2. Available machine imes per machine ype in each week, in case of opimal capaciy configuraion - assuming opimal number of differen machine ypes (see Table 5) (Righ-hand side of he firs consrain, R): Machine A B C D E F G H I J Week Week Week Week Every elemen of he marix above is calculaed as follows: R i = x i ( o i ) i = 1, 10; = 1, 4 References Bahl, H.C. - Taj, S. - Corcoran, W A linear-programming model formulaion for opimal produc-mix decisions in maerial-requiremens-planning environmens. Inernaional Journal of Producion Research, 29(5), Balachandran, B.V. - Balakrishnan, R. - Sivaramakrishnan, K Capaciy planning wih demand uncerainy. The Engineering Economis, 43(1),

23 Levene Szász Brealy, R. - Myers, S Principles of corporae finance. Hungarian ediion, Panem, Budapes. Diaz, B. - Laguna, M Modelling he load levelling problem in maser producion scheduling for MRP sysems. Inernaional Journal of Producion Research. 34(2), Ding, F.-Y. - Raghavan, B. - Pollard, S Supplier capaciy analisys for a manufacuring firm wih case sudy. Producion Planning & Conrol, 18(6), Du, T.C. - Wolfe, P.M Building an acive maerial requiremens planning sysem. Inernaional Journal of Producion Research. 38(2), Duchessi, P. - Schaninger, C.M. - Hobbs, D.R Implemening a Manufacuring Planning and Conrol Informaion Sysem. California Managemen Review. 31(3), Gould, L.S Inroducing APS: geing producion in lock sep wih cusomer demand. Auomoive Manufacuring & Producion. 110(5), Gupa, A. - Lödding, H. - Tseng, M.M An approach of capabiliy represenaion for improving capaciy planning. Inernaional Journal of Producion Research, 44(17), Hammersfahr, R.D. - Pope, J.A. - Ardalan, A Sraegic planning for producion capaciy. Inernaional Journal of Operaions & Producions Managemen, 13(5), Krajewski, L. - Rizman, L. - Malhora, M Operaions managemen. Processes and value chains. Prenice Hall, New Jersey. Naghi, M. - Segerean, R. - Szász, L Reorganizing he producion sysem for increasing compeiiveness. Managerial Challenges of he Conemporary Sociey inernaional conference, , Cluj-Napoca, Romania. Ram, B. - Naghshineh-Pour, M. R. - Yu, X Maerial requiremens planning wih flexible bills-of-maerial. Inernaional Journal of Producion Research, 44(2), Slack, N., Chambers, S., Johnson, R Operaions managemen. Pearson Educaion Limied, Harlow, Essex, England. Segerean, R Siseme moderne de conducere a producþiei. Ediura Dacia, Cluj-Napoca. Sevenson, W.J Producion/operaions managemen. Irwin, Chicago. Vörös, J Termelési-szolgálaási rendszerek vezeése. Janus Pannonius Egyeemi Kiadó, Pécs, Hungary. Worman, J.C. - Euwe, M.J. - Taal, M. - Wiers, V.C.S A review of capaciy planning echniques wihin sandard sofware packages. Producion Planning & Conrol, 7(2), Wuipornpun, T. - Yenradee, P Developmen of finie capaciy maerial requiremen planning sysem for assembly operaions. Producion Planning & Conrol, 15(5),