The present approach to capacity planning in operations producing heavy metal shapes causes problems in fulfilling
the required financial and volume indexes in production, as well as in meeting the work order completion
dates. The article represents the methodology for optimal production scheduling in operations producing
heavy plate cut shapes, which significantly eliminates the above-mentioned problems. The methodology is based
on the application Generalized Assignment Problem (GAP).Postojeći pristup planiranju kapaciteta u proizvodnji
metalnih proizvoda od limova uzrokuje probleme kod ispunjavanja zahtjeva financijskih indeksa, razine produktivnosti
te kompletiranja isporuke u danim rokovima. Članak daje metodologiju za optimiziranje terminiranja
proizvodnje kod rezanja limova, koja značajno eliminira naprijed navedene probleme. Metodologija je
temeljena na primjeni Općeg problema pridruživanja (OPP)

Lenort, Radim

Samolejová, Andrea

Staš, David

The present approach to capacity planning in operations producing heavy metal shapes causes problems in fulfilling the required financial and volume indexes in production, as well as in meeting the work order completion dates. The article represents the methodology for optimal production scheduling in operations producing heavy plate cut shapes, which significantly eliminates the above-mentioned problems. The methodology is based on the application Generalized Assignment Problem (GAP).Postojeći pristup planiranju kapaciteta u proizvodnji metalnih proizvoda od limova uzrokuje probleme kod ispunjavanja zahtjeva financijskih indeksa, razine produktivnosti te kompletiranja isporuke u danim rokovima. Članak daje metodologiju za optimiziranje terminiranja proizvodnje kod rezanja limova, koja značajno eliminira naprijed navedene probleme. Metodologija je temeljena na primjeni Općeg problema pridruživanja (OPP)

R. Lenort

D. Staš

A. Samolejová

HRČAK - Portal of Croatian Scientific and Professional Journals

R. LENORT, D. STA[, A. SAMOLEJOVÁCAPACITY PLANNINGIN OPERATIONS PRODUCING HEAVY PLATE CUT SHAPESReceived – Prispjelo: 2008-04-04Accepted – Prihva}eno: 2009-01-10Proffesional paper – Strukovni radINTRODUCTIONA wide product range is typical for operations pro-ducing cut shapes from heavy plates. The production ca-pacity of these operations depends on the producingability of the basic technological machines, which arethe flame-cutting machines. The product range of theflame-cutting machines is primarily stated by a combi-nation of the sizes and shapes of the cut shapes, whichrepresent as many as numerous thousands of productitems. Based on the product range, the producing abilityof the single flame-cutting machines may, however,move within a very large range. The lower limit of thisrange is close to only few kilos per hour, whereas the up-per limit normally achieves several tons per hour. Pro-cessing of various product ranges thus significantly af-fects the production capacity of the operation.Capacity planning under these conditions is very dif-ficult. Based on authors’ detailed survey of availableprofessional publications it is possible to state that therehas not been proposed any comprehensive methodologyof capacity planning for operations producing cutshapes from heavy plates so far. With regard to the spe-cific characteristics of the monitored work places no sofar known solutions from other production areas can beapplied here.The present approach to the capacity planning, ap-plied in the mentioned operations, is based on veryrough heuristics. Therefore, situations when the produc-tion either does not fill the production capacity or makesit impossible to reach the required amount within the us-able production time are very often. Due to this fact, therequired financial and volume indexes are not achievedin the production, and as a result the work order comple-tion dates are not fulfilled too.The objective of the authors’ research is to find suchmethodology in the area of capacity planning in opera-tions for the production of cut shapes from heavy plateswhich would enable optimizing the schedule of work or-ders for single flame-cutting machines and eliminatingthe above-defined problems.EXPERIMENTAL WORKGeneralized Assignment Problem (GAP) was cho-sen as the basic approach to optimize the work orders’schedule for single flame-cutting machine. The GAPwas introduced by Ross and Soland 1. Ferland has pre-sented a general overview and discussion of assignmenttype problems and their generalizations 2.The aim of GAP is to find the optimal assignment ofn jobs into m agents where each of them are with a fixedcapacity availability. Each job must be assigned exactlyto one agent. The capacity of each agent has an up limit,METALURGIJA 48 (2009) 3, 209-211 209The present approach to capacity planning in operations producing heavy metal shapes causes problems in ful-filling the required financial and volume indexes in production, as well as in meeting the work order comple-tion dates. The article represents the methodology for optimal production scheduling in operations producingheavy plate cut shapes, which significantly eliminates the above-mentioned problems. The methodology is ba-sed on the application Generalized Assignment Problem (GAP).Key words: capacity planning, production of heavy plate cut shapes, production scheduling, optimization, Ge-neralized Assignment Problem (GAP)Planiranje kapaciteta kod rezanja te{kih limova. Postoje}i pristup planiranju kapaciteta u proizvodnjimetalnih proizvoda od limova uzrokuje probleme kod ispunjavanja zahtjeva financijskih indeksa, razine pro-duktivnosti te kompletiranja isporuke u danim rokovima. ^lanak daje metodologiju za optimiziranje termini-ranja proizvodnje kod rezanja limova, koja zna~ajno eliminira naprijed navedene probleme. Metodologija jetemeljena na primjeni Op}eg problema pridru`ivanja (OPP).Klju~ne rije~i: planiranje kapaciteta, proizvodnja rezanja te{kih limova, terminiranje proizvodnje, optimizacija,Op}i problem pridru`ivanja (OPP)ISSN 0543-5846METABK 48(3) 209-211 (2009)UDC – UDK UDC 669.013:658.5:519.673=111R. Lenort, D. Sta{, A. Samolejová, V[B – Technical University of Ostrava,Faculty of Metallurgy and Materials Engineering, Ostrava, Czech Republicwhich must not be exceeded. More than one job may beassigned to an agent. It is assumed that the capacity re-quirements of any job are known and that they dependon the agent the job is assigned to. In addition, it is as-sumed that the profit from assigning a job to an agent isalso known and it is required that the total profit from as-signing the n jobs is to be maximized.As a consequence of practical capacity-cost relation-ships, the profit from a job-agent assignment would beexpected to be increased if the corresponding capacityutilization is increased in practical instances of the prob-lem 4.Many applications of the GAP appear within thescheduling problems 5. However, in the frame of theapplication GAP the following problems had to be partic-ularly solved in the area of capacity planning in opera-tions for the production of cut shapes from heavy plates:– It has to be possible to assign work orders to sev-eral flame-cutting machines simultaneously whatenables to speed up the time of their processingsignificantly. Therefore, the constraints ensuringthat each job is assigned exactly to one agent mustbe changed.– In practice, the volume of production (in tons) isused as an optimization criterion but it does notensure maximization of the economic effect as forthe reached capacity plan. Therefore, an economiccriterion was proposed as the contribution to thesettlement.– The GAP does not take into account a fulfillmentof completion dates, i.e. the sequence of singlework orders on the given flame-cutting machine.Therefore, the proposed methodology has beenextended by the technique of planning by meansof buffers defined by Goldratt in his Theory ofConstraints 6.RESEARCH RESULTSThe methodology frame of the capacity planning inoperations for the production of cut shapes from heavyplates is the first result of authors’ research work:If the operation has m flame-cutting machines withvarious technical parameters or various cutting technol-ogies, we can recommend the following methodologyfor compiling an optimum capacity schedule:1. Compile a set of n work orders; the required com-pletion date which is within the given planning pe-riod, and the weight is given, i.e. we know theweight of the i-th work order planned to be com-pleted in the given planning period mzi (t). Thecompletion date is derived from the supplying dateby deduction, i.e. the supplying buffer. The sup-plying buffer determines such timing in advancewhich should sufficiently protect the supplyingdate against any eventual delay in production.When determining the above-mentioned supply-ing buffer, it is always required to consider thecosts of the buffer maintenance and the eventualconsequences of the “buffer over-utilization”.Within the work, it is suitable to express the size ofthe buffers in a certain number of days.2. Sort the obtained files by the required productioncompletion date in ascending work order.3. Determine the j-th flame-cutting machines whichare able to process the required work order (j = 1, 2,… m). The reasons for the non-completion of thegiven work order on a flame-cutting machine may,for instance, include cut shapes with a size and/orlength exceeding the usable surface of the cuttingtable or cut shapes with a thickness that cannot beprocessed by the stated cutting technology.4. Determine the processing time of the i-th workorder on the j-th flame-cutting machine related toa ton of product tij (min/ t). If the work order canbe implemented on the flame-cutting machine,the processing time of the i-th work order is cal-culated from the producing ability of the j-thflame-cutting machine Pij (t /h):tPijij60, (min/t) (1)i = 1, 2, … nj = 1, 2, … m.For determining Pij, we can recommend a proce-dure for analyzing floating bottlenecks in metal-lurgical production 7.5. Determine the usable time of the j-thflame-cutting machine Ctj:C Cbktj njjs 	1 100(min) (2)Cnj – nominal time of the j-th machine (min), i.e.calendar time Ck (min) reduced by the amount oftime resulting from breaks by virtue ofnonworking daysbj – time of planned downtimes of the j-th ma-chine in % of nominal timeks – shift working coefficientj = 1, 2, … m.6. Define the criteria for optimizing the schedule ofwork orders for flame-cutting machines in theplanned period. In the work, this mainly includesthe maximization of the economic result of thecompleted production measured by the operationmargin to the settlement of i-th work order ci:c p vi i i  , (€ /t) (3)pi – price of the i-th work order (€)vi – costs of the i-th work order (€)i = 1, 2, … n.7. Compile an analytical model of the schedulingtask, i.e. determine the basic variable of themodel, the set of constraints and the objectivefunctions:– basic model variable - volume of the i-th workorder assigned to the j-th flame-cutting machinexij (t),210 METALURGIJA 48 (2009) 3, 209-211R. LENORT et al.: CAPACITY PLANNING IN OPERATIONS PRODUCING HEAVY PLATE CUT SHAPES– constraint by the usable time of the flame-cu-tting machines – the total time consumed for theproduction of single work orders on the j-thflame-cutting machine must not exceed its usabletime Ctj:t x Cij ij tjin1, (min) (4)j = 1, 2, … m– constraint by work order weight – the sum ofthe partial weights of the i-th work order assignedto the single flame-cutting machines must not ex-ceed the total work order weight mzi:x mij zijm1, (t) (5)i = 1, 2, … n– conditions for non-negativity:xij 0, (6)– the objective function for cases is the criterionfor optimization of the total profit maximization:max c xi ijjmin11, (€) (7)8. Optimize the compiled model – the output of themodel solution is a schedule of the work ordersfor the single flame-cutting machines optimizedaccording to the selected criterion, including thesequence and time for processing on the relatedmachines tzij (processing time of the i-th work or-der on the j-th flame-cutting machine in min).9. Compile the time schedule for work using theflame-cutting machines based on the output ofthe model solution and verify the fulfilment ofthe supplying dates (not exceeding the supplyingbuffer).10.Carry out the capacity balancing in the case thatthe requirements of the production capacity –given by the volume of work orders with the re-quired completion date in the given planning pe-riod – are not compliant with the available opera-tional capacity.CONCLUSIONThe proposed methodology frame is the base for cre-ating of a comprehensive methodology of the capacityplanning in operations for the production of cut shapesfrom heavy plates. In the consequential scientific workthe following areas will be examined in particular:– The analytical model for work order assignment toflame-cutting machines can be considered asNP-Hard. There is no polynomial time algorithmwhich finds a feasible solution. The known exactsolution approaches can solve only small-sizetask. To overcome the limitation of exact meth-ods, a new heuristic or metaheuristic algorithmwill have to be proposed.– Heuristic algorithms for determining flame-cuttingmachine producing ability will be created.– The analytical model does not consider businesspriorities (completion of work orders for strategiccustomers). The set of constraints is assumed forfurther modification.– The methodology frame is fully deterministic.The next step will bring creation of its stochasticversion.REFERENCES1 G. T. Ross, R. M. Soland, Mathematical Programming, 8(1975), 91-105.2 J. A. Ferland, Generalized Assignment-Type Problems a Po-werful Modeling Scheme - The Practice and Theory of Auto-mated Timetabling II., E. K. Burke, M. W. Carter (ed.),Springer Lecture Notes in Computer Science, (1997), 53-77.3 T. Öncan, INFOR, 45 (2007) 3, 123-141.4 A. P. French, J. M. Wilson, Journal of Heuristics, 8 (2002),143-153.5 D. G. Cattryse, L. N. Van Wassenhove, European Journal ofOperational Research, 60 (1992), 260-272.6 E. M. Goldratt, Theory of Constraints, The North RiverPress Publishing Corporation Great Barrington, (1999).7 R. Lenort, A. Samolejová, Metalurgija, 46 (2007) 1, 61-66.List of symbolsm – Number of flame-cutting machinesn – Number of production work ordersmzi – Weight of the i-th work order (t)tij – Processing time of the i-th work order on the j-thflame-cutting machine related to a ton of product(min / t)Pij – Producing ability of the j-th flame-cuttingmachine (t/h)Ctj – Usable time of the j-th flame-cuttingmachine (min)Cnj – Nominal time of the j-th machine (min)Ck – Calendar time (min)bj – Time of planned downtimes of the j-th machinein % of nominal timeks – Shift working coefficientci – Operational margin of i-th work order (€/t)pi – Price of the i-th work order (€)vi – Costs of the i-th work order (€)xij – Weight of the i-th work order assigned to the j-thflame-cutting machine (t)tzij – Processing time of the i-th work order on the j-thflame-cutting machine (min)Acknowledgement – This work is supported by the re-search plan of the Ministry of Education, Youth andSports of the Czech Republic No. MSM 6198910015.Note: Responsible translator: the translation company, Aspena, s.r.o.METALURGIJA 48 (2009) 3, 209-211 211R. LENORT et al.: CAPACITY PLANNING IN OPERATIONS PRODUCING HEAVY PLATE CUT SHAPES

English

Planiranje kapaciteta kod rezanja teških limova

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DSpace at VSB Technical University of Ostrava

R. LENORT, D. STA[, A. SAMOLEJOVÁCAPACITY PLANNINGIN OPERATIONS PRODUCING HEAVY PLATE CUT SHAPESReceived – Prispjelo: 2008-04-04Accepted – Prihva}eno: 2009-01-10Proffesional paper – Strukovni radINTRODUCTIONA wide product range is typical for operations pro-ducing cut shapes from heavy plates. The production ca-pacity of these operations depends on the producingability of the basic technological machines, which arethe flame-cutting machines. The product range of theflame-cutting machines is primarily stated by a combi-nation of the sizes and shapes of the cut shapes, whichrepresent as many as numerous thousands of productitems. Based on the product range, the producing abilityof the single flame-cutting machines may, however,move within a very large range. The lower limit of thisrange is close to only few kilos per hour, whereas the up-per limit normally achieves several tons per hour. Pro-cessing of various product ranges thus significantly af-fects the production capacity of the operation.Capacity planning under these conditions is very dif-ficult. Based on authors’ detailed survey of availableprofessional publications it is possible to state that therehas not been proposed any comprehensive methodologyof capacity planning for operations producing cutshapes from heavy plates so far. With regard to the spe-cific characteristics of the monitored work places no sofar known solutions from other production areas can beapplied here.The present approach to the capacity planning, ap-plied in the mentioned operations, is based on veryrough heuristics. Therefore, situations when the produc-tion either does not fill the production capacity or makesit impossible to reach the required amount within the us-able production time are very often. Due to this fact, therequired financial and volume indexes are not achievedin the production, and as a result the work order comple-tion dates are not fulfilled too.The objective of the authors’ research is to find suchmethodology in the area of capacity planning in opera-tions for the production of cut shapes from heavy plateswhich would enable optimizing the schedule of work or-ders for single flame-cutting machines and eliminatingthe above-defined problems.EXPERIMENTAL WORKGeneralized Assignment Problem (GAP) was cho-sen as the basic approach to optimize the work orders’schedule for single flame-cutting machine. The GAPwas introduced by Ross and Soland 1. Ferland has pre-sented a general overview and discussion of assignmenttype problems and their generalizations 2.The aim of GAP is to find the optimal assignment ofn jobs into m agents where each of them are with a fixedcapacity availability. Each job must be assigned exactlyto one agent. The capacity of each agent has an up limit,METALURGIJA 48 (2009) 3, 209-211 209The present approach to capacity planning in operations producing heavy metal shapes causes problems in ful-filling the required financial and volume indexes in production, as well as in meeting the work order comple-tion dates. The article represents the methodology for optimal production scheduling in operations producingheavy plate cut shapes, which significantly eliminates the above-mentioned problems. The methodology is ba-sed on the application Generalized Assignment Problem (GAP).Key words: capacity planning, production of heavy plate cut shapes, production scheduling, optimization, Ge-neralized Assignment Problem (GAP)Planiranje kapaciteta kod rezanja te{kih limova. Postoje}i pristup planiranju kapaciteta u proizvodnjimetalnih proizvoda od limova uzrokuje probleme kod ispunjavanja zahtjeva financijskih indeksa, razine pro-duktivnosti te kompletiranja isporuke u danim rokovima. ^lanak daje metodologiju za optimiziranje termini-ranja proizvodnje kod rezanja limova, koja zna~ajno eliminira naprijed navedene probleme. Metodologija jetemeljena na primjeni Op}eg problema pridru`ivanja (OPP).Klju~ne rije~i: planiranje kapaciteta, proizvodnja rezanja te{kih limova, terminiranje proizvodnje, optimizacija,Op}i problem pridru`ivanja (OPP)ISSN 0543-5846METABK 48(3) 209-211 (2009)UDC – UDK UDC 669.013:658.5:519.673=111R. Lenort, D. Sta{, A. Samolejová, V[B – Technical University of Ostrava,Faculty of Metallurgy and Materials Engineering, Ostrava, Czech RepublicDSpace VŠB-TUO http://hdl.handle.net/10084/71273 18/06/2012which must not be exceeded. More than one job may beassigned to an agent. It is assumed that the capacity re-quirements of any job are known and that they dependon the agent the job is assigned to. In addition, it is as-sumed that the profit from assigning a job to an agent isalso known and it is required that the total profit from as-signing the n jobs is to be maximized.As a consequence of practical capacity-cost relation-ships, the profit from a job-agent assignment would beexpected to be increased if the corresponding capacityutilization is increased in practical instances of the prob-lem 4.Many applications of the GAP appear within thescheduling problems 5. However, in the frame of theapplication GAP the following problems had to be partic-ularly solved in the area of capacity planning in opera-tions for the production of cut shapes from heavy plates:– It has to be possible to assign work orders to sev-eral flame-cutting machines simultaneously whatenables to speed up the time of their processingsignificantly. Therefore, the constraints ensuringthat each job is assigned exactly to one agent mustbe changed.– In practice, the volume of production (in tons) isused as an optimization criterion but it does notensure maximization of the economic effect as forthe reached capacity plan. Therefore, an economiccriterion was proposed as the contribution to thesettlement.– The GAP does not take into account a fulfillmentof completion dates, i.e. the sequence of singlework orders on the given flame-cutting machine.Therefore, the proposed methodology has beenextended by the technique of planning by meansof buffers defined by Goldratt in his Theory ofConstraints 6.RESEARCH RESULTSThe methodology frame of the capacity planning inoperations for the production of cut shapes from heavyplates is the first result of authors’ research work:If the operation has m flame-cutting machines withvarious technical parameters or various cutting technol-ogies, we can recommend the following methodologyfor compiling an optimum capacity schedule:1. Compile a set of n work orders; the required com-pletion date which is within the given planning pe-riod, and the weight is given, i.e. we know theweight of the i-th work order planned to be com-pleted in the given planning period mzi (t). Thecompletion date is derived from the supplying dateby deduction, i.e. the supplying buffer. The sup-plying buffer determines such timing in advancewhich should sufficiently protect the supplyingdate against any eventual delay in production.When determining the above-mentioned supply-ing buffer, it is always required to consider thecosts of the buffer maintenance and the eventualconsequences of the “buffer over-utilization”.Within the work, it is suitable to express the size ofthe buffers in a certain number of days.2. Sort the obtained files by the required productioncompletion date in ascending work order.3. Determine the j-th flame-cutting machines whichare able to process the required work order (j = 1, 2,… m). The reasons for the non-completion of thegiven work order on a flame-cutting machine may,for instance, include cut shapes with a size and/orlength exceeding the usable surface of the cuttingtable or cut shapes with a thickness that cannot beprocessed by the stated cutting technology.4. Determine the processing time of the i-th workorder on the j-th flame-cutting machine related toa ton of product tij (min/ t). If the work order canbe implemented on the flame-cutting machine,the processing time of the i-th work order is cal-culated from the producing ability of the j-thflame-cutting machine Pij (t /h):tPijij60, (min/t) (1)i = 1, 2, … nj = 1, 2, … m.For determining Pij, we can recommend a proce-dure for analyzing floating bottlenecks in metal-lurgical production 7.5. Determine the usable time of the j-thflame-cutting machine Ctj:C Cbktj njjs 	1 100(min) (2)Cnj – nominal time of the j-th machine (min), i.e.calendar time Ck (min) reduced by the amount oftime resulting from breaks by virtue ofnonworking daysbj – time of planned downtimes of the j-th ma-chine in % of nominal timeks – shift working coefficientj = 1, 2, … m.6. Define the criteria for optimizing the schedule ofwork orders for flame-cutting machines in theplanned period. In the work, this mainly includesthe maximization of the economic result of thecompleted production measured by the operationmargin to the settlement of i-th work order ci:c p vi i i  , (€ /t) (3)pi – price of the i-th work order (€)vi – costs of the i-th work order (€)i = 1, 2, … n.7. Compile an analytical model of the schedulingtask, i.e. determine the basic variable of themodel, the set of constraints and the objectivefunctions:– basic model variable - volume of the i-th workorder assigned to the j-th flame-cutting machinexij (t),210 METALURGIJA 48 (2009) 3, 209-211R. LENORT et al.: CAPACITY PLANNING IN OPERATIONS PRODUCING HEAVY PLATE CUT SHAPESDSpace VŠB-TUO http://hdl.handle.net/10084/71273 18/06/2012– constraint by the usable time of the flame-cu-tting machines – the total time consumed for theproduction of single work orders on the j-thflame-cutting machine must not exceed its usabletime Ctj:t x Cij ij tjin1, (min) (4)j = 1, 2, … m– constraint by work order weight – the sum ofthe partial weights of the i-th work order assignedto the single flame-cutting machines must not ex-ceed the total work order weight mzi:x mij zijm1, (t) (5)i = 1, 2, … n– conditions for non-negativity:xij 0, (6)– the objective function for cases is the criterionfor optimization of the total profit maximization:max c xi ijjmin11, (€) (7)8. Optimize the compiled model – the output of themodel solution is a schedule of the work ordersfor the single flame-cutting machines optimizedaccording to the selected criterion, including thesequence and time for processing on the relatedmachines tzij (processing time of the i-th work or-der on the j-th flame-cutting machine in min).9. Compile the time schedule for work using theflame-cutting machines based on the output ofthe model solution and verify the fulfilment ofthe supplying dates (not exceeding the supplyingbuffer).10.Carry out the capacity balancing in the case thatthe requirements of the production capacity –given by the volume of work orders with the re-quired completion date in the given planning pe-riod – are not compliant with the available opera-tional capacity.CONCLUSIONThe proposed methodology frame is the base for cre-ating of a comprehensive methodology of the capacityplanning in operations for the production of cut shapesfrom heavy plates. In the consequential scientific workthe following areas will be examined in particular:– The analytical model for work order assignment toflame-cutting machines can be considered asNP-Hard. There is no polynomial time algorithmwhich finds a feasible solution. The known exactsolution approaches can solve only small-sizetask. To overcome the limitation of exact meth-ods, a new heuristic or metaheuristic algorithmwill have to be proposed.– Heuristic algorithms for determining flame-cuttingmachine producing ability will be created.– The analytical model does not consider businesspriorities (completion of work orders for strategiccustomers). The set of constraints is assumed forfurther modification.– The methodology frame is fully deterministic.The next step will bring creation of its stochasticversion.REFERENCES1 G. T. Ross, R. M. Soland, Mathematical Programming, 8(1975), 91-105.2 J. A. Ferland, Generalized Assignment-Type Problems a Po-werful Modeling Scheme - The Practice and Theory of Auto-mated Timetabling II., E. K. Burke, M. W. Carter (ed.),Springer Lecture Notes in Computer Science, (1997), 53-77.3 T. Öncan, INFOR, 45 (2007) 3, 123-141.4 A. P. French, J. M. Wilson, Journal of Heuristics, 8 (2002),143-153.5 D. G. Cattryse, L. N. Van Wassenhove, European Journal ofOperational Research, 60 (1992), 260-272.6 E. M. Goldratt, Theory of Constraints, The North RiverPress Publishing Corporation Great Barrington, (1999).7 R. Lenort, A. Samolejová, Metalurgija, 46 (2007) 1, 61-66.List of symbolsm – Number of flame-cutting machinesn – Number of production work ordersmzi – Weight of the i-th work order (t)tij – Processing time of the i-th work order on the j-thflame-cutting machine related to a ton of product(min / t)Pij – Producing ability of the j-th flame-cuttingmachine (t/h)Ctj – Usable time of the j-th flame-cuttingmachine (min)Cnj – Nominal time of the j-th machine (min)Ck – Calendar time (min)bj – Time of planned downtimes of the j-th machinein % of nominal timeks – Shift working coefficientci – Operational margin of i-th work order (€/t)pi – Price of the i-th work order (€)vi – Costs of the i-th work order (€)xij – Weight of the i-th work order assigned to the j-thflame-cutting machine (t)tzij – Processing time of the i-th work order on the j-thflame-cutting machine (min)Acknowledgement – This work is supported by the re-search plan of the Ministry of Education, Youth andSports of the Czech Republic No. MSM 6198910015.Note: Responsible translator: the translation company, Aspena, s.r.o.METALURGIJA 48 (2009) 3, 209-211 211R. LENORT et al.: CAPACITY PLANNING IN OPERATIONS PRODUCING HEAVY PLATE CUT SHAPESDSpace VŠB-TUO http://hdl.handle.net/10084/71273 18/06/2012

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Planiranje kapaciteta kod rezanja teških limova

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