fill - Semantic Scholar
TP1 - 3:45 NETWORK FLOW OPTIMIZATION IN FLEXIBLE MANUFACTURING SYSTEMS
J. Kimemia Laboratory for Information and Decision Systems Massachusetts Institute of Technology Cambridge, Massachusetts 02139
S.B. Gershwin Laboratory for Information and Decision Systems Massachusetts Institute of Technology Cambridge, Massachusetts 02139
Abstract
manufacturing systems have been made [Hutchinson, 1977] [Horev 1978]. They allow detailed investigation of the effects of parameter variations and strategy assignment on system performance. Solberg [1977] and Ward [19781 model the system as a closed network of queues. Steady state results which are in good agreement with simulation results and observed performance of an actual system are obtained. The use of the closed networki of queues model as an analytic method of strategy assignment has been suggested by Secco-Suardo [1978]. In this report, a n'etwork flow approach is used. Rather than analyze the movement of inthe system, the aggregated dividual pieces through the system, the aggregated Network of queues flow of pieces is analyzed. Network of queues m models are used to account for congestion effects
The problem of choosing an optimal mix of operating strategies in a flexible manufacturing system is solved by a network flow optimization approach. Mathematical methods which exploit the structure of the problem to generate manufacturing strategies are outlined. Numerical results show that the method produces results which agree with intuition for a two-workstation system. 1.
Introduction
A large proportion of manufacturing activity is at a level which does not justify dedicated automation in the form of assembly lines. In order to increase productivity in this sector of industry, flexible manufacturing systems are being designed built.
~~~~~~~~and
~at
the workstations. In Section 2, the model is presented and the optimization problem is formulated. Systems where there are non-deterministic arrivals and proces there are non-deterministic arrivals and processing times give rise to non-linear optimization problems. The production rate of the system should be maximized but the build of queues Alterwithin the system becomes a constraint. Altera price can be put on the number of pieces in the system (the in-process inventory)
A flexible manufacturing system consists of workstations capable of performing a number of different tasks, interconnected by a transportation loading station, undergo a sequence of operations at the workstations before being finally unloaded at an unloading station. The processes at the atX an . unladngstaio. heroesenatively, workstation are mostly automatic. At certain stations like the loading station for example, there may be some manual operations [Hughes 1977]. Sevoermal differenatkinds ofpiees aremanufacSeveral different kinds of pieces are manufactured simultaneously in the system. Each piece, has a given number of operations necessary for its manufacture. There is a choice in the system as to
[Kimemia and Gershwin 1978]. Deterministic systers or systems where the processing and interarrival times have small variances, give rise to linear programs. Assymptotic results for closed queueing models [Baskett et al 1975 and work-rate queueing models [Baskett et al 1975] and work-rate
formed. Any entering workpiece therefore has the choice of several different routes or manufacturing strategies available. A manufacturing strategy for a piece assigns each operation to a workstation and also specifies the sequence of workstation visits. In order to gain maximum output and utilization at minimum cost, the overall behavior of the system should be studied. Furthermore mathematical modelsole and algorithms are needed which will enable controllers to make decisions affecting the system with minimum human intervention. An important problem, which has a fundamental effect on the production rate and the utilization of the system, is the assignment of strategies to the workpieces. Given a flexible manufacturing system with a specified production mix of pieces and given the location at which all the operations can be performed in the system, one wishes to pick the optimal steady-state mix of manufacturing strategies for all the pieces being produced. Extensiv
simulation
studies of
pe
}
#l
~ iI ia
:
i i II ii !!
i
t the linear programs are valid for maximizing the production rate in systems with general service ibutions time distributions. Mathematical methods which exploit the struc of the problem order in to solve the optimization robleps of section 2 are discussed in section 3. Decomposition metod [Dantzig and 1963 are used to break he inear programs into a set of strategy-generatingminimum proces into a set of strategy-generating minimum processing time sub-problems each involving only one piece type. A master problem then finds the optima combinatin f tr the timal combination of strategies for all the The lagrange multiplier method of estenes and Powell [1968] converts the non-linear [1969] and Powell non-linear programming problem[1968] into converts a series the of optimization problem where a non-linear laranian function is minimized subject to lir an minimized subject to linear flow and resource coni [Cantor and Gerla 1974]
flexible
[ Defenderfer 1977]
i
iT_ a
i-i
can
This research has been supported by the National Science Foundation under NSF/RANN Grant No. APR76-12036.
633 X CH1392-0/78/0000-0633s00..15
Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors, and do not necessarily reflect the views of the National Science Foundation.
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0197b
IEE
i .
k
then be applied to minimize the lagrangian function. Numerical results for a two workstation systen are presented in section 4. The effect of changing system parameters on the optimal strat-l egy assignment, production rate and work-station utilization is investigated. 2.
... p k=l,.k. j=1 where R. is the production rate of production rate The total production rate is is given given P M P R R
xij = R. i=l
i~i
Modelling and Optimization of Flexible Manufacturing Systems
A flexible manufacturing system consists of M workstations connected by a transportation system. There may be P different types of pieces in the system simultaneously. Each piece has S systm strategies available for its manufacture. A stratassigned egy is a sequence of operations at workstations which are required to complete a workpiece. Thus strategies are different sequences of the same operations. The total number of strategies S=jSi may be large if there is a large number of options available in the system. It might not be worthwhile in such a case to identify in advance all possible
type i pieces. by by (2.3)
i=l j=l
The summation is carried out with k=l for convenience. The production ratio requirement states that Dieces of type i comprise a fraction a (Om 3 iJ
j
(2.9)
By applying asymptotic results for closed networks of queues [Cordon and Newell 19681[SeccoSuardo 1978], results for general queueing networks [Baskettet al 19751, the operational results
The average queue length q (x) at workstation j is evaluated by applying queueing network theory. The average number of pieces on the transportation system depends on the time t. to traverse each arc in~ .the~ network.z~ jrate
theorems of Chang and Lavenberg [1972], it is found that LP 2.1 also maximizes the production for a general class of systems [Kimemia and Gershwin 1978]. The program remains unchanged as '
An optimization problem can now be formulated, which maximizes the total production rate subject . . to a constraint imposed on the average level of in-process inventory, which is required to be less than a certain given value Q. In operating flexible manufacturing system, the average level of in-process inventory is an important quantity. In general the in-process inventory is a nonlinear function of the production rate. Although it is desirable to have the maximum possible production rate, the average level of in-process inventory should not as a consequence become too high. In this formulation this is controlled by setting a level Q and maximizing the production rate subject to the constraint that the average
•;~~~~~~~~~~
~~The
deterministic case can thus be viewed as a limiting case of a class of stochastic systems. ptimization Techniques for Flexible Manufacturing systems Manufacturing systems The general optimization problem can be stated as NLP 3.1 minimize f(x) (3.1) subject to T.
< 1
(3.2)
c.x.
= C.R 1i
(3.3)
A.x
= 0
i=l
level of in-process inventory does not exceed Q. NLP 2.1 P1 xil
Maximize
(2.10)
i=l,... ,p
(3.4)
i=l subject to (2.2), (2.4), (2.7), (2.8) x. and
u
J. Kimemia Laboratory for Information and Decision Systems Massachusetts Institute of Technology Cambridge, Massachusetts 02139
S.B. Gershwin Laboratory for Information and Decision Systems Massachusetts Institute of Technology Cambridge, Massachusetts 02139
Abstract
manufacturing systems have been made [Hutchinson, 1977] [Horev 1978]. They allow detailed investigation of the effects of parameter variations and strategy assignment on system performance. Solberg [1977] and Ward [19781 model the system as a closed network of queues. Steady state results which are in good agreement with simulation results and observed performance of an actual system are obtained. The use of the closed networki of queues model as an analytic method of strategy assignment has been suggested by Secco-Suardo [1978]. In this report, a n'etwork flow approach is used. Rather than analyze the movement of inthe system, the aggregated dividual pieces through the system, the aggregated Network of queues flow of pieces is analyzed. Network of queues m models are used to account for congestion effects
The problem of choosing an optimal mix of operating strategies in a flexible manufacturing system is solved by a network flow optimization approach. Mathematical methods which exploit the structure of the problem to generate manufacturing strategies are outlined. Numerical results show that the method produces results which agree with intuition for a two-workstation system. 1.
Introduction
A large proportion of manufacturing activity is at a level which does not justify dedicated automation in the form of assembly lines. In order to increase productivity in this sector of industry, flexible manufacturing systems are being designed built.
~~~~~~~~and
~at
the workstations. In Section 2, the model is presented and the optimization problem is formulated. Systems where there are non-deterministic arrivals and proces there are non-deterministic arrivals and processing times give rise to non-linear optimization problems. The production rate of the system should be maximized but the build of queues Alterwithin the system becomes a constraint. Altera price can be put on the number of pieces in the system (the in-process inventory)
A flexible manufacturing system consists of workstations capable of performing a number of different tasks, interconnected by a transportation loading station, undergo a sequence of operations at the workstations before being finally unloaded at an unloading station. The processes at the atX an . unladngstaio. heroesenatively, workstation are mostly automatic. At certain stations like the loading station for example, there may be some manual operations [Hughes 1977]. Sevoermal differenatkinds ofpiees aremanufacSeveral different kinds of pieces are manufactured simultaneously in the system. Each piece, has a given number of operations necessary for its manufacture. There is a choice in the system as to
[Kimemia and Gershwin 1978]. Deterministic systers or systems where the processing and interarrival times have small variances, give rise to linear programs. Assymptotic results for closed queueing models [Baskett et al 1975 and work-rate queueing models [Baskett et al 1975] and work-rate
formed. Any entering workpiece therefore has the choice of several different routes or manufacturing strategies available. A manufacturing strategy for a piece assigns each operation to a workstation and also specifies the sequence of workstation visits. In order to gain maximum output and utilization at minimum cost, the overall behavior of the system should be studied. Furthermore mathematical modelsole and algorithms are needed which will enable controllers to make decisions affecting the system with minimum human intervention. An important problem, which has a fundamental effect on the production rate and the utilization of the system, is the assignment of strategies to the workpieces. Given a flexible manufacturing system with a specified production mix of pieces and given the location at which all the operations can be performed in the system, one wishes to pick the optimal steady-state mix of manufacturing strategies for all the pieces being produced. Extensiv
simulation
studies of
pe
}
#l
~ iI ia
:
i i II ii !!
i
t the linear programs are valid for maximizing the production rate in systems with general service ibutions time distributions. Mathematical methods which exploit the struc of the problem order in to solve the optimization robleps of section 2 are discussed in section 3. Decomposition metod [Dantzig and 1963 are used to break he inear programs into a set of strategy-generatingminimum proces into a set of strategy-generating minimum processing time sub-problems each involving only one piece type. A master problem then finds the optima combinatin f tr the timal combination of strategies for all the The lagrange multiplier method of estenes and Powell [1968] converts the non-linear [1969] and Powell non-linear programming problem[1968] into converts a series the of optimization problem where a non-linear laranian function is minimized subject to lir an minimized subject to linear flow and resource coni [Cantor and Gerla 1974]
flexible
[ Defenderfer 1977]
i
iT_ a
i-i
can
This research has been supported by the National Science Foundation under NSF/RANN Grant No. APR76-12036.
633 X CH1392-0/78/0000-0633s00..15
Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors, and do not necessarily reflect the views of the National Science Foundation.
.,fill
0197b
IEE
i .
k
then be applied to minimize the lagrangian function. Numerical results for a two workstation systen are presented in section 4. The effect of changing system parameters on the optimal strat-l egy assignment, production rate and work-station utilization is investigated. 2.
... p k=l,.k. j=1 where R. is the production rate of production rate The total production rate is is given given P M P R R
xij = R. i=l
i~i
Modelling and Optimization of Flexible Manufacturing Systems
A flexible manufacturing system consists of M workstations connected by a transportation system. There may be P different types of pieces in the system simultaneously. Each piece has S systm strategies available for its manufacture. A stratassigned egy is a sequence of operations at workstations which are required to complete a workpiece. Thus strategies are different sequences of the same operations. The total number of strategies S=jSi may be large if there is a large number of options available in the system. It might not be worthwhile in such a case to identify in advance all possible
type i pieces. by by (2.3)
i=l j=l
The summation is carried out with k=l for convenience. The production ratio requirement states that Dieces of type i comprise a fraction a (Om 3 iJ
j
(2.9)
By applying asymptotic results for closed networks of queues [Cordon and Newell 19681[SeccoSuardo 1978], results for general queueing networks [Baskettet al 19751, the operational results
The average queue length q (x) at workstation j is evaluated by applying queueing network theory. The average number of pieces on the transportation system depends on the time t. to traverse each arc in~ .the~ network.z~ jrate
theorems of Chang and Lavenberg [1972], it is found that LP 2.1 also maximizes the production for a general class of systems [Kimemia and Gershwin 1978]. The program remains unchanged as '
An optimization problem can now be formulated, which maximizes the total production rate subject . . to a constraint imposed on the average level of in-process inventory, which is required to be less than a certain given value Q. In operating flexible manufacturing system, the average level of in-process inventory is an important quantity. In general the in-process inventory is a nonlinear function of the production rate. Although it is desirable to have the maximum possible production rate, the average level of in-process inventory should not as a consequence become too high. In this formulation this is controlled by setting a level Q and maximizing the production rate subject to the constraint that the average
•;~~~~~~~~~~
~~The
deterministic case can thus be viewed as a limiting case of a class of stochastic systems. ptimization Techniques for Flexible Manufacturing systems Manufacturing systems The general optimization problem can be stated as NLP 3.1 minimize f(x) (3.1) subject to T.
< 1
(3.2)
c.x.
= C.R 1i
(3.3)
A.x
= 0
i=l
level of in-process inventory does not exceed Q. NLP 2.1 P1 xil
Maximize
(2.10)
i=l,... ,p
(3.4)
i=l subject to (2.2), (2.4), (2.7), (2.8) x. and
u
