Manufacturing Planning and Control

Transcription

1 Manufacuring Planning and Conrol Sephen C. Graves Massachuses nsiue of echnology November 999 Manufacuring planning and conrol enails he acquisiion and allocaion of limied resources o producion aciviies so as o saisfy cusomer demand over a specified ime horizon. As such, planning and conrol problems are inherenly opimizaion problems, where he obecive is o develop a plan ha mees demand a minimum cos or ha fills he demand ha maximizes profi. he underlying opimizaion problem will vary due o differences in he manufacuring and marke conex. his chaper provides a framework for discree-pars manufacuring planning and conrol and provides an overview of applicable model formulaions.

2 Manufacuring Planning and Conrol Sephen C. Graves Massachuses nsiue of echnology November 999 Manufacuring planning and conrol address decisions on he acquisiion, uilizaion and allocaion of producion resources o saisfy cusomer requiremens in he mos efficien and effecive way. ypical decisions include work force level, producion lo sizes, assignmen of overime and sequencing of producion runs. Opimizaion models are widely applicable for providing decision suppor in his conex. n his aricle we focus on opimizaion models for producion planning for discree-pars, bach manufacuring environmens. We do no cover deailed scheduling or sequencing models (e. g., Graves, 98), nor do we address producion planning for coninuous processes (e. g., Shapiro, 993). We consider only discree-ime models, and do no include coninuous-ime models such as developed by Hackman and Leachman (989). Our inen is o provide an overview of applicable opimizaion models; we presen he mos generic formulaions and briefly describe how hese models are solved. here is an enormous range of problem conexs and model formulaions, as well as soluion mehods. We make no effor o be exhausive in he reamen herein. Raher, we have made choices of wha o include based on personal udgmen and preferences. We have organized he aricle ino four maor secions. n he firs secion we presen a framework for he decisions, issues and radeoffs involved in implemening an opimizaion model for discree-par producion planning. he remaining hree secions presen and discuss hree disinc ypes of models. n he second secion we discuss linear programming models for producion planning, in which we have linear coss. his caegory is of grea pracical ineres, as many imporan problem feaures can be capured wih hese models and powerful soluion mehods for linear programs are readily available. n he hird secion, we presen a producion-planning model for a single aggregae produc wih quadraic coss; his model is of hisorical significance as i represens one of he earlies applicaions of opimizaion o manufacuring planning. n he final secion we inroduce he muli-iem capaciaed lo-size problem, which is modeled as a mixed ineger linear program. his is an imporan model as i inroduces economies of scale in producion, due o he presence of producion seups. Framework here are a variey of consideraions ha go ino he developmen and implemenaion of an opimizaion model for manufacuring planning and conrol. n his secion we highligh and commen upon a number of key issues and quesions ha should be addressed. Excellen general references on producion planning are homas and McClain (993), Shapiro (993) and Silver e al. (998). 2

3 Any planning problem sars wih a specificaion of cusomer demand ha is o be me by he producion plan. n mos conexs, fuure demand is a bes only parially known, and ofen is no known a all. Consequenly, one relies on a forecas for he fuure demand. o he exen ha any forecas is ineviably inaccurae, one mus decide how o accoun for or reac o his demand uncerainy. he opimizaion models described in his aricle rea demand as being known; as such hey mus be periodically revised and rerun o accoun for forecas updaes. A key choice is wha planning decisions o include in he model. By definiion, producion-planning models include decisions on producion and invenory quaniies. Bu in addiion, here migh be resource acquisiion and allocaion decision, such as adding o he work force and upgrading he raining of he curren work force. n many planning conexs, an imporan consruc is o se a planning hierarchy. Namely, one srucures he planning process in a hierarchical way by ordering he decisions according o heir relaive imporance. Hax and Meal (975) inroduced he noion of hierarchical producion planning and provide a specific framework for his, whereby here is an opimizaion model wih each level of he hierarchy. Each opimizaion model imposes a consrain on he model a he nex level of he hierarchy. Biran and irupai (993) provide a comprehensive survey of hierarchical planning mehods and models. he idenificaion of he relevan coss is also an imporan issue. For producion planning, one ypically needs o deermine he variable producion coss, including seuprelaed coss, invenory holding coss, and any relevan resource acquisiion coss. here migh also be coss associaed wih imperfec cusomer service, such as when demand is backordered. A planning problem exiss because here are limied producion resources ha canno be sored from period o period. Choices mus be made as o which resources o include and how o model heir capaciy and behavior, and heir coss. Also, here may be uncerainy associaed wih he producion funcion, such as uncerain yields or lead imes. One migh only include he mos criical or limiing resource in he planning problem, e. g., a boleneck. Alernaively, when here is no a dominan resource, hen one mus model he resources ha could limi producion. We describe in his aricle wo ypes of producion funcions. he firs assumes a linear relaionship beween he producion quaniy and he resource consumpion. he second assumes ha here is a required fixed charge or seup o iniiae producion and hen a linear relaionship beween he producion quaniy and resource usage. Relaed o hese choices is he selecion of he ime period and planning horizon. he planning lieraure disinguishes beween big bucke and small bucke ime periods. A ime period is a big bucke if muliple iems are ypically produced wihin a ime period; a small bucke is such ha a mos one iem would be produced in he ime period. For big bucke models, one has o worry abou how o schedule or sequence he producion runs assigned o any ime period. he choice of planning horizon is dicaed 3

4 by he lead imes o enac producion and resource-relaed decisions, as well as he qualiy of knowledge abou fuure demand. Planning is ypically done in a rolling horizon fashion. A plan is creaed for he planning horizon, bu only he decisions in he firs few periods are implemened before a revised plan is issued. ndeed, as noed above, he plan mus be periodically revised due o he uncerainies in he demand forecass and producion. For insance a firm migh plan for he nex 26 weeks, bu hen revise his once a monh o incorporae new informaion on demand and producion. Producion planning is usually done a an aggregae level, for boh producs and resources. Disinc bu similar producs are combined ino aggregae produc families ha can be planned ogeher so as o reduce planning complexiy. Similarly producion resources, such as disinc machines or labor pools, are aggregaed ino an aggregae machine or labor resource. Care is required when specifying hese aggregaes o assure ha he resuling aggregae plan can be reasonably disaggregaed ino feasible producion schedules. Finally for complex producs, one mus decide he level and exen of he produc srucure o include in he planning process. For insance, in some conexs i is sufficien o us plan he producion of end iems; he producion plan for componens and subassemblies is subservien o he maser producion schedule for end iems. n oher conexs, planning us he end iems is sub-opimal, as here are criical resource consrains applicable o muliple levels of he produc srucure. n his insance, a mulisage planning model allows for he simulaneous planning of end iems and componens or subassemblies. Of course, his produces a much larger model. 4

5 Producion Planning: Linear Programming Models n his secion we develop and sae he mos basic opimizaion model for producion planning for he following conex: muliple iems wih independen demand muliple shared resources big-bucke ime periods linear coss. We define he following noaion decision variables p i producion of iem i during ime period invenory of iem i a end of ime period q i parameers,, K number of ime periods, iems, resources, respecively a ik b k d i cp i cq i amoun of resource k required per uni of producion of iem i amoun of resource k available in period demand for iem i in period uni variable cos of producion for iem i in ime period uni invenory holding cos for iem i in ime period We now formulae he linear program P: P: Min cp p + cq q () s.. = i= q + p q = d i, (2) i, i i i i= a p b k, (3) ik i k p, q 0 i, i i i i i i he obecive funcion () minimizes he variable producion coss plus he invenory holding coss for all iems over he planning horizon of periods. Equaion (2) is a se of invenory balance consrains ha equae he supply of an iem in a period wih is demand or usage. n any period, he supply for an iem is he invenory from he prior period q i,-, plus he producion in he period p i. his supply can be used o mee demand in he period d i, or held in invenory as q i. As we require he invenory 5

6 o be non-negaive, hese consrains assure ha demand is saisfied for each iem in each period. We are given as inpu he iniial invenory for each iem, namely q i0. Equaion (3) is a se of resource consrains. Producion in each period is limied by he availabiliy of a se of shared resources, where producion of one uni of iem i requires a ik unis of resource k, for k =, 2,... K. ypical resources are various ypes of labor, process and maerial handling equipmen, and ransporaion modes. he number of decision variables is 2, and he number of consrains is + K. For any realisic problem size, we can solve P by any good linear-programming algorihm, such as he simplex mehod. We briefly describe nex a number of imporan exensions o his basic model. We inroduce hese as if hey were independen; however, we noe ha many conexs require a combinaion of hese exensions. Demand Planning: Los Sales For some problems we have he opion of no meeing all demand in each ime period. ndeed, here migh no be sufficien resources o mee all demand. n effec, he demand parameers represen he demand poenial, and he opimizaion problem is o decide wha demand o mee and how. We assume ha demand ha canno be me in a period is los, hus reducing revenue. n addiion, a firm migh incur a loss of cusomer goodwill ha would manifes iself in erms of reduced fuure sales. his los sales cos is very difficul o quanify as i represens he fuure unknown impac from poor service oday. We pose a new planning problem o maximize revenues ne of he producion, invenory and los sales coss. We inroduce addiional noaion and hen sae he model: decision variables u i unme demand of iem i during ime period parameers r i uni revenue for iem i in period uni cos of no meeing demand for iem i in ime period cu i P2: Max r ( d u ) cp p cq q cu u s.. (3) q + p q + u = d i, p, q, u 0 i, = i= i, i i i i i i i i i i i i i i i i 6

7 he obecive funcion has been modified o include revenue as well as he cos of los sales. he poenial revenue, Σ Σ r i d i, is a consan and could be dropped in he obecive funcion. n his case, we can resae he problem as a cos minimizaion problem, where he cos of los sales includes he los revenue. Also, in P2, he invenory balance consrain has been modified o permi he opion of no meeing demand; hus demand in a period can be me from producion or invenory, or no saisfied a all. he resource consrain (3) remains unchanged. Demand Planning: Backorders A relaed problem variaion is when i is possible o reschedule or backorder demand. ha is, we migh defer curren demand unil a laer period, when i can be served from producion. Of course here is a cos for doing his, which we erm he backorder cos. Like he los sales cos, he backorder cos includes hard-o-quanify coss due o loss of cusomer goodwill, as well as reduced revenue and addiional processing or expediing coss due o he deferral of he demand fulfillmen. We assume ha his cos is linear in he number of backorders in each period. We define addiional noaion and hen sae he model. decision variables v i backorder level for iem i a end of ime period parameers cv i uni cos of backorder for iem i in ime period P3: Min s.. (3) = i= q v + p q + v = d i, p, q, v 0 i, cp p + cq q + cv v i i i i i i i, i, i i i i i i i n comparison wih P, we now include a backorder cos on he obecive funcion for P3. he invenory balance equaion is modified o accoun for he backorders, which in effec behave like negaive invenory. We ypically would add a erminal consrain on he backorders a he end of he planning horizon; for insance, we migh require v i = 0, so ha over he -period planning horizon all demand is evenually me by he producion plan. Any iniial backorders, namely v i0, can be dropped by adding hem o he firsperiod demand; ha is, we resae he demand as d i : = d i + v i0, and hen drop v i0 from he formulaion. 7

8 n his formulaion, when demand is backordered, he cos of his even is linear in he size and duraion of he backorder. ha is, if i akes n ime periods o fill he backorder, he cos is proporional o n. n conras, in some cases, he backorder cos migh no depend on he duraion bu only on he occurrence and size of he backorder. We can modify his formulaion for his case by defining a variable o represen new backorders in period, given by max [0, v i - v i,- ]; hen we would apply he backorder cos o his variable in he obecive funcion. his modificaion assumes ha we fill backorders in a las-in, las-ou fashion, as here is no incenive o do oherwise for his cos assumpion. Piecewise Linear Producion Cos Funcions n P he relevan producion cos is linear in he producion quaniy. n many conexs he acual cos funcion is non-linear. n his secion, we consider a convex cos funcion, and assume ha i is well modeled as a piecewise linear funcion. We model concave cos funcions ha exhibi economies of scale in aler secion. Le Cos(p i ) denoe he cos funcion for iem i in period ; we presen he case where his cos funcion is he same in each period, and inroduce he following noaion: decision variables p is producion of iem i during ime period, ha falls in cos segmen s parameers S number of segmens in cos funcion cp is uni variable cos of producion for iem i in ime period in cos segmen s upper bound on cos segmen s for iem i P is hus, we assume ha Cos(p i ) is given by: Cos( p ) = where p i = S i s= p is s= cp p is 0 p P s is is S is n order for Cos(p i ) o be convex, we require ha he uni variable coss be sricly increasing from one segmen o he nex: 0 < cp i < cp i2 <... < cp is his cos funcion applies when here are muliple opions or sources for producion, and hese opions can be ranked by heir variable coss. A common example is when one 8

9 models regular ime and overime producion. We have wo cos segmens (S=2) where he firs segmen corresponds o producion during regular ime, and he second is overime producion. he variable producion cos is usually more during overime, as workers earn a rae premium. Anoher example is when he firm works muliple shifs and he variable coss differ beween hese shifs. A hird example is when here are subconracing or ousourcing opions; here are muliple coss segmens, one o represen in-house producion and he ohers o represen he ousourcing opions ranked by cos. We model he planning problem wih convex, piecewise linear producion cos funcions by replacing he producion cos in P wih he above formulaion for Cos(p i ): P4: Min ( cq q + cp p ) s.. (2) i= = i= a p b k, iks is k p p = 0 i, i S s= is 0 p P i, s is is p, p, q 0 i, s, i is i S i i is s= is n P4 we have modified he resource consrains (3) o accommodae he possibiliy ha he usage of he shared resources depends on he producion quaniy by source or cos segmen, i. e., p is, raher han us on p i. n his case, a iks denoes he amoun of resource k required per uni of iem i produced a source or cos segmen s. his form permis grea flexibiliy in modeling producion coss as well as resource consrains. n one of he firs papers on producion planning, Bowman (956) formulaes his problem as a ransporaion problem, when here are muliple ime periods and muliple producion opions, bu only one iem and one resource ype. Resource Planning Up o now we have assumed ha he resource levels are fixed and given. n some cases, an imporan elemen of he planning problem is o decide how o adus he resource levels over he planning horizon. For insance, one migh be able o change he work force level, by means of hiring and firing decisions. Hansmann and Hess (960) provide an early example of his ype of model. Suppose for ease of noaion ha we have us one ype of resource, namely he work force. We inroduce addiional noaion and hen sae he model: decision variables w work force level in ime period 9

10 h f change o work force level by hiring in ime period change o work force level by firing in ime period parameers a i amoun of work force (labor) required per uni of producion of iem i cw variable uni cos of work force in ime period ch variable hiring cos in ime period cf variable firing cos in ime period P5: Min s.. (2) i= = a p w 0 i i cw w + ch h + cf f + cp p + cq q w + h f w = 0 p, q, w, h, f 0 i, i i i i i i = i= We add he variable cos for he work force o he obecive funcion, along wih coss for hiring and firing workers. he hiring cos includes coss for finding and aracing applicans as well as raining coss. he firing cos includes coss of ouplacemen and reraining of displaced workers, as well as severance coss; here migh also be a cos of lower produciviy due o lower work-force morale, when firings or layoffs occur. he invenory balance consrain (2) remains he same as for P, and we resae he resource consrain, reflecing he work force as a decision variable and as he sole resource. We hen add a new se of balance consrains for planning he work force: he work force in period is ha from he prior period plus new hires minus he number fired. We have saed P5 for a single resource, represening he work force. he model exends immediaely o include oher resources ha migh be managed in a similar fashion over he planning horizon. n addiion, here migh be oher consideraions o model such as ime lags when adusing a resource level. here migh be limis on how quickly new workers can be added due o raining requiremens. f here were limied raining resources, hen his imposes a consrain on h. Alernaively, new hires migh be less producive unil hey have acquired some experience. n his case, we modify he formulaion o model differen caegories of workers, depending on heir enure and experience level. Anoher common variaion of his model is when here are wo labor classes, say, permanen employees and emporary employees. hese classes differ in erms of heir cos coefficiens, and possibly heir efficiency facors. Permanen employees have higher hiring and firing cos, as hey receive more raining and have more righs and proecion from layoffs. Bu heir variable producion cos, normalized by heir produciviy, should be lower han ha for emporary workers. he planning problem hen enails he 0

11 managemen and planning of boh work classes over he planning horizon. For compleeness, we revise P5 for wo work force classes: decision variables w work force level in ime period h change o work force level by hiring in ime period f change o work force level by firing in ime period producion of iem i during ime period, using labor class p i parameers a i amoun of labor required per uni of producion of iem i, using labor class cw variable uni cos of labor class in ime period ch variable hiring cos for labor class in ime period variable firing cos for labor class in ime period cf P6: Min s.. (2) 2 = = p p = 0 i, i i= = i a p w 0, i i 2 w + h f w = 0,, p, q, w, h, f 0 i,, i i cw w + ch h + cf f + cp p + cq q i i i i = i= n comparison o P5, we have decision variables for boh labor classes, as well as for heir hiring and firing decisions, in order o model he cos differences. We also inroduce producion decision variables, by labor class, o capure he differences in produciviy. Muliple Locaions n P, here is a single supply locaion or producion faciliy ha serves demand for all iems. Ofen here are muliple producion faciliies ha are geographically dispersed and ha supply muliple disribuion channels. he planning problem is o deermine he producion, invenory and disribuion plans for each faciliy o mee demand, which is now modeled by geographic region. here are many ways o formulae his ype of problem. We provide an example, and hen commen on some varians. We define he noaion and hen sae he model.

12 decision variables p i producion of iem i a faciliy during ime period q i invenory of iem i a faciliy a end of ime period shipmen of iem i from faciliy o demand locaion m in ime period x im parameers,, K number of ime periods, iems, resources, respecively J, M number of faciliy locaions, demand locaions, respecively a ik b k d im cp i cq i cx im amoun of resource k required per uni of producion of iem i a faciliy amoun of resource k available a faciliy in period demand for iem i a locaion m in period uni variable cos of producion for iem i a faciliy in ime period uni invenory holding cos for iem i a faciliy in ime period uni ransporaion cos o ship iem i from faciliy o demand locaion m in ime period P7: Min s.. = i= = q + p q x = i,, x = d i, m, a p b, k, p, q, x 0 i,, m, M i, i i im 0 m= J = i= im im ik i k i i im J [ cp p + cq q + cx x ] i i i i im im m= M he decision variables for producion and invenory are now specified by locaion, where invenory is held a he producion locaions. n addiion, we have aggregaed demand ino a se of demand regions or locaions, and inroduce a new se of decision variables o denoe ransporaion from he producion faciliies o he demand locaions. he obecive funcion capures producion and invenory holding cos, which depends on he faciliy, plus ransporaion or disribuion coss for moving he produc from a faciliy o he demand locaion. he invenory balance consrains assure ha he supply of an iem a each faciliy is eiher held in invenory or shipped o a demand locaion o mee demand. he second se of consrains assures ha he shipmens saisfy he demand each period a each locaion. he resource consrains are srucurally he same as in P, bu for muliple producion locaions. A key varian of his model occurs when here are addiional socking locaions, such as a nework of warehouses or disribuion ceners. hese socking locaions no only provide 2

13 addiional space o sore invenory close o he demand locaions, bu also permi economies of scale in ransporaion from he producion sies o hese socking locaions. n his case, one defines shipmens o and from each socking locaion, and has invenory balance consrains for each socking locaion. he shipmen coss would capure any differences in ransporaion modes ha migh be employed. he size of P7 creaes a challenge for implemening and mainaining such a model. here are now J(2+M) decision variables and (J + M + JK) consrains. A ypical problem migh have aggregae iem families, 5-0 faciliies, demand locaions, 2 20 ime periods, and 5 resource ypes. hus, he model migh have on he order of 00,000 o million decision variables, and on he order of 00,000 consrains. As he problem is sill a linear program, such problems are readily solved by commercial opimizaion packages. However, he real difficuly in such implemenaions is he developmen and mainenance of he parameers. here can be on he order of one million demand forecass, 00,000 o million cos coefficiens, and 0,000 resource coefficiens. he success of many applicaions ofen ress on wheher his daa can be obained and kep accurae. Dependen Demand ems So far we have considered producion plans for end iems or finished goods, which see independen demand. Bu hese end iems are usually comprised of many fabricaed componens and subassemblies, for which here needs o be a producion plan oo. hese iems differ from he end iems, in ha heir demand depends on he end-iem producion plan. Maerial requiremens planning (MRP) sysems are designed o characerize his dependen demand and o faciliae he planning for hese dependen-demand iems in a coordinaed and sysemaic way (Vollman e al. 992). Neverheless, i is quie easy o incorporae dependen-demand iems ino he opimizaion models presened here. W firs need o define he goes ino marix A = {α i }, where α i is he number of unis of iem i required per uni of producion of iem. hen for he model P we replace he invenory balance consrains (2) wih he following: i, i i α i i = q + p q p = d i, n his general form, he demand for iem i has wo pars: exogenous demand given by d i, and endogenous or induced demand given by Σα i p. For end iems we expec here o be no induced demand, whereas for componens, here will ypically be no or limied exogenous demand. One variaion o his model is when here are manufacuring lead imes, whereby producion of iem in ime period requires ha componen i be available a ime L, where L is he lead ime for producing. he invenory balance consrains can be easily modified o accommodae his, given ha he lead imes are known and deerminisic. 3

14 Billingon e al. (983) provide a comprehensive reamen of his model and problem, and presen mehods for reducing he size of he problem so as o faciliae is soluion. n he lieraure, his problem is referred o as a muliple sage problem, where end iems, subassemblies, and componens migh represen disinc sages in a manufacuring process. 4

15 Quadraic Cos Models and Linear Decision Rule One of he earlies producion-planning modeling effors was ha of Hol, Modigliani, Muh and Simon (960), who developed a producion-planning model for he Pisburgh Pain Company. hey assume a single aggregae produc, and hen define hree decision variables: p q w producion of he aggregae iem during ime period invenory of he aggregae iem a end of ime period work force level in ime period Hol, Modigliani, Muh and Simon (HMMS) assume ha he cos funcion in each period has four componens. he firs componen is he regular payroll coss ha is a linear funcion of he work force level. he second componen is he hiring and layoff coss, which were assumed o be a quadraic funcion in he change in work force from one period o he nex. he nex cos componen is for overime and idle-ime coss. HMMS assume here is an ideal producion arge ha is a linear funcion of he work-force level. f producion is greaer han his arge here is an overime cos, while if producion is less han his arge here is an idle-ime cos. Again, HMMS assume ha he cos is quadraic abou he variance beween he acual producion and he producion arge for he work-force level. he final componen is invenory and backorder coss. Similar o he overime and idleime coss, here is an invenory arge each period, which is a linear funcion of he demand in he period. he invenory and backorder cos is a quadraic funcion of he deviaion beween he invenory and he invenory arge. he HMMS opimizaion is o minimize he sum of he expeced coss over a fixed horizon, subec o an invenory balance consrain. he expecaion is over he demand random variables, where we are given an unbiased forecas of demand over he planning horizon. he analysis of his opimizaion yields wo key resuls. Firs, he opimal soluion can be characerized as a linear decision rule, whereby he aggregae producion rae in each period is a linear funcion of he fuure demand forecass, as well as he work force and invenory level in he prior period. here is a similar linear funcion for specifying he work force level in each period. Second, he opimal decision rule is derived for he case of sochasic demand, bu only depends on he mean of he demand random variables. ha is, we only need o know (or assume) ha he demand forecass are unbiased in order o apply he linear decision rule. 5

16 Producion Planning: Lo-Size Models n his secion we consider producion-planning problems for which here are economies of scale associaed wih he producion aciviy or funcion. he mos common example occurs when here is a required seup o iniiae he producion of an iem. For insance, o iniiae he producion of an iem migh require a change in ooling, or dies, or raw maerial; he seup migh also require a change in he producion conrol seings, as well as an iniial run o assure ha he producion oupu mees qualiy specificaions. here migh be a seup cos, corresponding o labor coss for performing he seup, plus direc expendiures for maerials and ools. here migh also be resource requiremens for he seup, usually referred o as he seup ime. he producion resource canno produce unil he seup is compleed; hus he seup consumes producion capaciy, equal o is duraion, namely he seup ime. Given he presence of seups, once an iem is seup o produce, we may wan o produce a large bach or lo size so as o cover demand over a number of fuure periods and hence defer he nex ime when he iem will be seup and produced. Whereas producing large baches will reduce he seup coss, his also increases invenory as more demand is produced earlier in ime. he lo-sizing problem, as described here, is o deermine he relaive frequency of seups so as o minimize he seup and invenory coss, wihin he resource and service consrains of he producion-planning problem. We sar wih he simples model and hen briefly discuss varians o i. We develop and sae his model for he following conex: muliple iems wih independen demand a single shared resource big-bucke ime periods linear coss, excep for seup coss. For ease of presenaion we assume a single resource; he exension o consider muliple resources is sraighforward. We define he following noaion decision variables p i producion of iem i during ime period q i invenory of iem i a end of ime period binary decision variable o denoe seup of iem i in ime period y i parameers, number of ime periods, iems, respecively a i a i2 amoun of resource required per uni of producion of iem i amoun of resource required for seup of producion of iem i 6

17 b d i B cp i cq i cy i amoun of resource available in period demand for iem i in period a large consan uni variable cos of producion for iem i in ime period uni invenory holding cos for iem i in ime period seup cos for producion for iem i in ime period We now formulae he mixed-ineger linear program P8: P8: Min cp p + cq q + cy y (4) s.. = i= q + p q = d i, (5) i, i i i i= a p + a y b i i i2 i (6) p By i, (7) i i p, q 0; y = 0, i, i i i i i i i i i he obecive funcion (4) minimizes he sum of variable producion coss, he invenory holding coss and he seup coss for all iems over he planning horizon of periods. Equaion (5) is he same as (2), he invenory balance consrains ha equae he supply of an iem in a period wih is demand or usage. he resource consrains (6) reflec he resource consumpion boh due o he producion quaniy for each iem, and due o he seup. Producion of one uni of iem i consumes a i unis of he shared resource, while he seup requires a i2 unis. he consrain se (7) is for he so-called forcing consrains. hese consrains relae he producion variables o he seup variables. For each iem and ime period, if here is no seup (y i =0), hen his consrain assures ha here can be no producion (p i =0). Conversely, if here is producion in a period (p i >0), hen here mus also be a seup (y i =). n (7), B is any large posiive consan ha exceeds he maximum possible value for p i ; for insance, one migh se B equal o he sum of all demand. his problem is now a mixed-ineger linear program, wih binary decision variables. For modes size problems wih, say, a few hundred binary decision variable, his problem can be reliably solved by commercial opimizaion packages. Bu specialized approaches are warraned for increasing problem size and complexiy. We discuss one of hese approaches nex. 7

18 Dual-Based Soluions n his secion we develop a dual problem for P8. We idenify a generalized linear program for solving his dual problem, which is equivalen o a convexificaion of P8. We can solve his problem by column generaion o obain a lower bound on P8; we also discuss how his soluion can be used o idenify near opimal soluions o P8. he approach is based on he original work of Manne (958) who suggesed he generalized linear program given below. Dzielinski and Gomory (965) exended he model of Manne o include resource planning decisions (i. e., hiring and firing labor) and applied he Danzig-Wolfe decomposiion mehod o inroduce column generaion. Lasdon and erung (97) reformulaed he linear program so as o provide a more efficien and effecive column generaion approach for solving he linear program; hey also address a varian of P8, where here are small ime buckes for planning producion and muliple resources corresponding o scarce machines and dies. o develop he dual problem, we firs define a Lagrangean funcion L(π) by dualizing he resource consrains (6): L( π) = Min - b π + ( cp + a π ) p + cq q + ( cy + a π ) y s..( 5),( 7) p, q 0; y = 0, i, i i i = i i i i i i i2 i = i= where π = (π, π 2,... π ) 0 is a vecor of dual variables. We sae he following observaions: he Lagrangean separaes by iem, where we have a single-iem dynamic lo-size problem wih no capaciy consrains for each iem. his is he so-called Wagner- Whiin (958) problem and can be solved by dynamic programming. f q i0 = 0, he exreme poins o L(π), namely he single-iem dynamic lo-size problem, have he propery whereby p i q i- =0. As a consequence he opimal producion quaniies are a sum of consecuive demands. ha is, if p i > 0, hen p i = d i d is for s. We will refer o his as a Wagner-Whiin schedule. Wihou loss of generaliy, we will assume ha q i0 = 0, and he above propery applies o opimal soluions. f q i0 > 0, hen we use his iniial invenory o reduce he iem s demand. ha is, we resae he demand as follows: d i : = 0 for =, 2, s- d is : = d i + d i2 + + d i, s- - q i0 where s such ha d i + d i2 + + d i, s- q i0 < d i + d i2 + + d is. We can use he Lagrangean funcion o define a dual problem o P8: 8

19 D8: Max L( π ) s.. π 0 he dual soluion need no and usually will no idenify a primal feasible soluion o P8. n such insances, a dualiy gap exiss and he dual soluion provides a lower bound o P8. One migh consider wo procedures for resolving he dualiy gap. Firs, one could use he dual problem for generaing bounds in a branch and bound procedure; he effeciveness of his depends on he ighness of he bounds and he number of ineger variables in P8. he second approach incorporaes he soluion of he dual problem ino a heurisic procedure. For each ieraion in he soluion of he dual, we migh generae, somehow, a corresponding feasible soluion o P8. he bes such feasible soluion can be compared wih he soluion of he dual o assess is near opimaliy; he procedure sops once he bes feasible soluion is sufficienly close o he opimum or afer a predeermined number of ieraions. We migh solve he dual D8 direcly by means of a mehod such as subgradien opimizaion, or a dual-ascen procedure. Alernaively, we can follow he general derivaion provided by Magnani e al. (976) o reformulae he dual problem as a generalized linear program. We denoe he exreme poins of he convex hull defined by consrain ses (5) and (7), he non-negaiviy consrains and he binary consrains by u = ( u, u,... u ) = (( p, q, y ),...( p, q, y )), 2,,, where for each iem i, u i is a Wagner-Whiin schedule. We define he cos for he h exreme poin and resource usage for he h exreme poin in ime period as cu = cp p + cq q + cy y a = a p + a y i= = i= i i i i i2 i. i i We can rewrie D8 in erms of he exreme poins as he following equivalen linear program: Max z s.. z + ( b a ) π cu = π 0, where z is unconsrained in sign. he dual of his problem is: i i 9

20 P9: Min cu x s.. = x J J = x a x b = 0, = J where J is he number of exreme poins; he decision variable x indicaes he fracion of he schedule given by he h exreme poin. Problem P9 is a convexificaion of he primal problem P8 in which we replace he feasible region defined by consrains (5), (7), he non-negaiviy and binary consrains by he convex hull of his region. he soluion of P9 provides a lower bound for P8. However, P9 is no all ha useful due o he large number of variables, on he order of 2 ; and he soluion o P9 will ypically be fracional, and no suggesive of good, near-opimal feasible soluions. We can reformulae P9 by noing ha we can express he se of exreme poins U = {u } in erms of exreme poins for he individual iems. ha is, U = U U 2... U where U i k = { u i } is he se of exreme poins or Wagner-Whiin schedules for iem i. For he k h exreme poin for iem i, we define is cos and resource usage parameers: k k k cu = cp p + cq q + cy y i k k a = a p + a y i = i i k i i i2 i. i i i k i We now can rewrie P9 in erms of he Wagner-Whiin schedules for he iems: 20

21 k P0: Min cu x s.. Ki k= x ik x ik Ki i= k= = i k a x b i ik 0, i, k Ki i= k= i ik where K i denoes he number of Wagner-Whiin schedules for iem i and he decision variable x ik is he fracion of he k h Wagner-Whiin schedule for iem i in he soluion. he firs consrain assures ha hese fracions sum o one, so ha he demand for each iem is me. he second consrain enforces he resource consrain. he opimal soluion o P0 solves he dual D8, and provides a lower bound o P8. f he opimal soluion o P0 is all ineger (x ik = 0 or for all i, k ), hen his soluion is opimal o P8. When he soluion o P0 is no ineger, i provides he basis for finding near opimal soluions. We describe his soluion sraegy in wo pars: firs how we solve P0 and second how we use he soluions o P0 o obain good feasible soluions o P8. Manne (958) firs proposed solving P0 as an approximaion o solving P8. However, as wih P9, here can be an enormous number of decision variables, on he order of 2 in his case. Raher han generae all of hese decision variables and heir parameers, we solve P0 by means of column generaion [Dzielinski and Gomory (965), Lasdon and erung (97)]. A each ieraion, we solve a maser problem, namely a reduced version of P0 wih a subse of columns. hen, using he dual values from he maser problem, we solve a Wagner-Whiin problem for each iem o find a new candidae schedule o ener ino he maser problem. he procedure ieraes beween he maser problem and he iem sub-problems unil he soluions o he Wagner-Whiin sub-problems yield no new iem schedules. One would ypically erminae his procedure when he gap beween he lower bound provided by he maser problem and a known feasible soluion o P8 is suiably small. Manne (958) noed ha alhough soluions o P0 are usually fracional, here is an ineger soluion for mos iems. An opimal soluion o P0, as well as he maser problem in he column generaion procedure, has + basic variables, as here are + consrains. hus, here are a mos + fracional variables in he soluion. Each iem mus have a leas one basic variable, in order o saisfy he convexiy consrain in P0. As a consequence, no more han iems can have wo or more basic variables. hus, a leas - iems have a single basic variable, which mus be ineger due o he convexiy consrain. For mos planning problems he number of iems would be much greaer han he number of ime periods. For insance, we migh be planning for iems, wih 2 or 3 2

22 ime periods. For =00 and =3, a soluion o P0 provides a single Wagner-Whiin schedule for a leas 87 iems. For he remaining iems, he soluion suggess a convex combinaion of wo or more Wagner-Whiin schedules. hese schedules saisfy he demand consrains in P8, bu require more seup ime and cos han assumed by P0. [P0 assumes ha we only incur a fracion of he seup cos and ime, for a schedule a a fracional level of aciviy.] hus, he se of schedules from P0 migh violae he resource consrains in P8. Manne noes ha in some conexs, he resource consrains are sof consrains and minor violaions can be ignored. n oher cases, one migh apply a heurisic o modify he soluion from P0 so as o make i saisfy he resource consrains. One expecs ha i is relaively easy o find a feasible soluion o P8 from he soluion o P0, as mos of he iems have a single schedule. Furhermore, one expecs ha he feasible soluions are near opimal, again due o he fac ha he soluion o P0 is near ineger. Compuaional experience in Manne, Dzielinski and Gomory and Lasdon and erung suppors hese claims. Also, rigeiro e al. (989) describe and es a heurisic smoohing procedure for consrucing feasible soluions. Variable Redefiniion Eppen and Marin (987) repor on an alernaive soluion approach o P8, based on variable redefiniion. hey reformulae P8 so ha is LP relaxaion provides a very igh bound, comparable o ha from he dual-based approaches of Lagrangean relaxaion or column generaion. hey hen solve he mixed ineger program using general-purpose opimizaion codes o obain opimal or near-opimal soluions for problems wih up o 200 producs and en ime periods. heir approach is based on he opimal propery of Wagner-Whiin schedules: when a producion aciviy occurs, he producion quaniy covers he demand in an ineger number of consecuive periods beginning wih he period of producion. hey hen define decision variables for each such producion opporuniy for each iem. Wih hese new variables, he schedule for each iem is a shores pah problem hrough a nework of nodes. hese shores pah problems are coupled by resource consrains ha cu across he individual iems. For compleeness we presen he Eppen-Marin model for muli-iem capaciaed losizing. decision variables z ik binary decision variable o denoe producion of iem i during ime period, where he producion quaniy is o saisfy demand for periods hrough k binary decision variable o denoe seup of iem i in ime period y i parameers, number of ime periods, iems, respecively a ik amoun of resource required in period, if z ik = 22

23 b cz ik cy i amoun of resource available in period variable producion and invenory holding cos for iem i, for producion in period o saisfy demand from period o k for iem i. seup cos for producion for iem i in ime period We now formulae he mixed-ineger linear program P: P: Min cz z + cy y s.. i= k= k= k= z ik a z b ik ik = i z z = 0 i, ik ik, z y i, ik i z 0, y = 0, i,, k ik k= - k= i ik ik i i i= = he firs se of consrains corresponds o he resource consrain (6) in P8. he nex wo ses model he flow balance for he underlying shores pah problem for each iem on nodes, 2,..., where z ik corresponds o he flow on he arc from node o node k+. he las se is he forcing consrain, equivalen o (7) in P8. Noe ha he demand parameers d i do no appear in P. Raher hey are embedded in he definiion of he cos parameers cz ik, and he resource coefficiens a ik. Also, whereas we define z ik o be a binary variable, we do no make his an explici requiremen when solving P. Eppen and Marin find ha P provides a igh lower bound o P8 and also idenifies nearopimal feasible soluions. Exensions o he Lo Size Model i= = k= We can exend formulaion P8 o incorporae all of he problem variaions ha were inroduced for he linear-programming producion-planning model. n addiion we menion here hree oher common exensions. n P8 we assume ha any quaniy could be produced, subec o he resource consrain. Furhermore, he resource usage for a producion quaniy consiss of a fixed seup ime and a variable amoun linear in he producion quaniy. n some conexs, producion occurs as a bach process, e. g., a hea rea or diffusion process. Each bach produces a fixed amoun of produc and consumes a fixed amoun of he limied resource. We can 23

24 model his by inroducing an ineger decision variable for he number of baches produced of an iem in a ime period. A second variaion is when he seups are sequence dependen; ha is, he seup for an iem will depend upon wha was us previously processed. here is no an easy way o modify P8 o accommodae his feaure. ndeed, in general, he sandard represenaion of sequence-dependen seups is o map his ino a raveling salesman problem, which resuls in a new level of complexiy. he hird variaion is when seups can be carried over from one period o he nex. n P8 we assume ha his is no possible; ha is, every period ha we produce an iem we incur a seup. n some conexs, hough, we migh be able o preserve he las seup in he period. hus, we would incur only one seup if we produce iem i las in one period and firs in he nex period. Karmarkar e al. (987) examine a single-produc version of his problem. Anoher variaion of his problem is when here are muliple producs and we assume small ime buckes, so ha a mos one iem is produced in a period. Lasdon and erung formulae and address his problem by means of column generaion. and Eppen and Marin show how o solve boh of hese problems efficienly by variable redefiniion. Word Coun:

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26 Manne, A. S., Programming of Economic Lo Sizes, Managemen Science, Vol. 4, No. 2 (January 958), pp Shapiro, J. F., Mahemaical Programming Models and Mehods for Producion Planning and Scheduling, n Handbooks in Operaions Research and Managemen Science, Volume 4, Logisics of Producion and nvenory, edied by S. C. Graves, A. H. G. Rinnooy Kan and P. H. Zipkin, Amserdam, Elsevier Science Publishers B. V., 993, pp Silver, E. A., D. F. Pyke, and R. Peerson, nvenory Managemen and Producion Planning and Scheduling, 3 rd Ediion, New York, John Wiley nc., 998. homas. L. J. and J. O. McClain, An Overview of Producion Planning, n Handbooks in Operaions Research and Managemen Science, Volume 4, Logisics of Producion and nvenory, edied by S. C. Graves, A. H. G. Rinnooy Kan and P. H. Zipkin, Amserdam, Elsevier Science Publishers B. V., 993, pp rigeiro, W. W., L. J. homas and J. O. McClain, Capaciaed Lo Sizing wih Seup imes, Managemen Science, Vol. 35, No. 3 (March 989), pp Vollman,. E., W. L. Berry and D. C. Whybark, Manufacuring Planning and Conrol Sysems, 3 rd ediion, Burr Ridge ll., Richard D. rwin nc., 992. Wagner, H. M. and. Whiin, Dynamic Version of he Economic Lo Size Model, Managemen Science, Vol. 5, No. (Ocober 958), pp