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A decision support model for the initial design of FMS
Computers ind. Engng Vol. 33, Nos 3-4, pp. 549-552, 1997 © 1997 EIscvi©r Science Ltd Printed in Great Britain. All rights reserved 0360-8352/97 $17.00 + 0.00
Pergmon
PII: S0360-8352(9"/)00190-3
A Decision Support Model for the Initial Design of FMS Hosub Shin', Jeongho Park", Choonghwa Lee- and Jinwoo Park" * Dept. of Industrial Engineering, Seoul National University, Seoul, Korea ** Dept. of System Technology, Tong, Heaw Industry Co., Changwon, Korea *** CIM Business Unit, Daewoo Information Systems Co., Kwacheon, Korea Abstract This study proposes a decision support model for the design and evaluation of FMS. The model optimizes the profit by analyzing the interaction between candidate part types and FMS configuration under the constraints of production requirement and cost effectiveness. The model is solved by using the closed queueing network model and optimization model iteratlvely. The throughput of current FMS configuration is evaluated by the closed queueing network model, and then its results are fed tothe optimization model for the selection of best combination of pert types and production resources. The resulting solution gives enswers to a sedes of decision problems at the initial design stage of FMS. An example problem is presented to show the superiority of the proposed approach. @ 1997 Elsevier Science Ltd Key words : candidate part types, FMS configuration, economic feasibility
and we don't consider detailed layout problems explicitly. There have been numerous previous researches on FMS design problem[3,5,6,7,8,12,13,14]. Unfortunately though, most previous researches assumed that the part types to be produced at FMS are given, and tried to determine an optimal configuration for given part types. However, we will have better and more realistic solution if we consider in our decision process the interaction between candidate pert types and configurations explicitly. Our current research is motivated from that concept, and a nonlinear integer optimization model is developed. Since solving such optimization model directly is not an easy job, an lteratlve procedure is proposed to find the best solution of the proposed model. The resulting solution gives answers to a series of decision problems raised at the initialdesign stage of FMS. In Section 2, a more detailed descril~on of the problem is provided, and a mathematical model is presented. The solution procedure is provided in section 3. A case study is given in section 4, and finally some concluding remarks are presented.
1. Introduction An FMS isan automated cellular manufacturing system consisting of a group of NC machine tools and automated matedal handling equipment, so that a group of parts can be prooessed randomly under the control of a central computer, adapting automatically to changes in product mix or output level. The advantages of fkedble marring technology are improved productivity, reduced work-in-process inventory and lead time, fester response to customer request and so on. Despite these a c k t a t ~ , theimplementation of FMS has not been popular in Korea. Them are two main reaSons for such phenomenon. One is the difficulties encountered in opera6ng the Inataiiod FMS, and the other is the lack of structured approach that can help top management to decide whether an FMS is a viable alternative to their production environment or not. This paper deals with the latter subject. In order to introduce an FMS successfully, a preliminary feasibility study should be performed to enhance introduction and installation of FMS. Draper Lab. in reference [1] proposed following three steps for evetuat]ngfeasibility of FMS. 1. Select candidate pert types and machine tools for FMS. 2. Develop a number of possible configurations. 3. Evaluate the alternative designs and their variations based on technical and economic feasibility. The selection of part types and machine tools is a subtle decision ixoblem because any decision to allocate some production resources for the production of one pert type necessarily affects the part types' that can be produced by the selected resources[10]. In other words, these three sub-steps necessarily interact and revolve around until the prescribed feasibility criterion is satisfied. In the following, we are going to introduce a decision support model for such decision problem. However, our model works at aggregate planning level
2. Problem Formulation
Description
and
Mathematical
We assume that there are n different candidate pert types and M different stations in FMS. X=(x,,...,x,) represents the eligible pad types for the production in FMS. Each integer 0-1 decision variable xj represents the selection of part type i for current production, x~ is equal to 1 if all the pert of type i are processed in FMS and 0, otherwise. In this paper an FMS configuration is characterized by a specific combination of (S,N). In S=(s,,...,s~), each nonnegatlve integer decision vedable s,{k=-l,...,M) represents the number of machines, setup areas, transportation vehicles, etc. at station k depending on the type of work station. We assume that each station consists of identical production resources. And index M is reserved for the modeling of transportation vehicles. N is the number of jigs end 549
Proceedings of 1996 ICC&IC
550
fixtures that conefituta an FMS configuration. The objective of this study is to find and assign values to (X,S,N). That is, we determine the best combination of part types and configuration which maximizes profit Model 1 n
Max ~ c#,x,- z~(X,S,N)
(I)
s.t. z~,~,t(X,S.N)
(2)
n
T(X,S,N) > ~T~x,
i should be processed within annual operation time AOT of FMa, the throughput requirement T~of pert type i is given to d/AOT. The sum of each Tj of selected part type is the system throughput requested to FMS. Due to the nonlinear property of the throughput function given, the problem is formulated into a nonlinear integer programming formulation. The notations not referred to in detail are summadzed in Table I below. Table 1. Summary of Notations
(3)
X=(x,...,x~)
Ct.o
(4) (5)
s = (~....,s~)
pert type i
X, e {0.1}.$, E { 0 } ~ J Z + . N e Z + (6)
As a cdtedon of economic feasibility of an FMS, payback method is widely used in real application. Thus, we adopt a p-year payback pedod as a chtedon of economic feesibity and it is reflected in the o b ~ function of Model 1. Equation (1) mprasents the objective function of one year profit which is obtained by subtracting the yearly production cost z~(X,S,N) from total annual sales volume. The positive yeady profit(positive value of objective function) means that all the initial investment for the introductfon and implementation of an FMS can be returned within p years, cj and dj represent the selling pdce and annual demand of each part type i respectively. The cost function zp(X,S,N) is assumed to be a monotonic nondecreesing linear function. As shown below, the annual production cost zp(X,S,N) is composed of three parts: dedicated cutting tools, equipment types and jigs and fixtures. Each component consists of annual operation cost and amortization should be paid dudng p years. Subscription o, A means operation and amortization respectively.
z p ( X S N ) = T (Ct~o +ct~ A)x~ +
+l
k=l
(,)
Since there may exist some common cutting louis(e.g. rough milling cutter) shared by various pad types, without loss of generality, we assume that all part types share some common cutting tools and have their own dedicated cuffing tools which does not overlap between different pert types. Equation (2) of Model 1 implies that the initial investment needed to install an FM8 is constrained by the available budget BU, and the details of the cost function is given below in equation (8).
xI
H
+ ~ cs, j.~.,
s, + cnj...,
N
ctil,kwo=t Cpo
cp^
c~^ Cak,L~,~t
t~
amortized cost of dedicated tools to process part type i initial investment of dedicated cutting tools to process part type i operation cost of a jig and fixture amortized cost of a jig and fixture initial investment needed for procurement of a jig and fixture operation cost of equipment type k amortized cost of equipment type k initial investment needed for procurement of equipment type k processing time of operation j of pert type i at equipment type k
3. Solution Procedure Initially all the part types of X=(x~,...,x,) are selected as candidate part types. Then, we calculate the required system throughput T,w, which is the sum of each T,, selected. Next, the minimum number of each machine should be decided to process all the part types selected. The required number of each machine tool is the smallest integer greater than the value
d,x,~ t~, / AOT (total assigned workload divided by
,o
zi.~o.j ( X S N ) = ~ ctj j.~.,
operation cost of dedicated tools to process
(8)
k=t
Equation (3) represents the throughput requirement of the FMS configuration we are going to build. For the evaluation of FMS throughput, we referred to the dosed quaueing network theory. The left hand side of equation (3),T(X,S,N), represents the throughput of the FMS configuration in terms of aggregate part type. An aggregate part type is the r e p r e s e ~ e part type of candidate part types. Thus, for current candidate pert types, the throughput req~ to FMS is also represented in terms of an aggregate part type, Since yearly demand dlof part type
annual operation time of FM8). If we just assign the workloads of candidate part types to the processing capacities of current FMS configuration, the whole workloads cannot be processed within planned operation time by the dynamic effect of actual syatem[2]. To evaluate the throughput of current FMS configuration, dosed quoueing network model .is widely used. In our case, we chose to use CANQ developed by Solberg[9]. Initial part types and the corresponding FMS configuration are fed to CANQ as input data. By the asymptotic bound[4] of dosed queuing network theory, the throughput of FMS goes to a limiting value as the number of jigs and &xturee goes to infinity. Here we assume that there are enough jigs and fixtures. After running CANQ, the throughput of current FMS configuration Tm is obtained. But it may happen that Tm is less than T~. This means that the FMS configuration previously detarmined cannot fulfill the demands of part types selected. To overcome this drawback, two approaches may be possible. One is increasing the system throughput by expanding the capacities of machine tools. The other is decreasing the workload by dropping certain part types from current candidate part types. Model 2 is for the latter approach. The former approach will be described in the latter part of this section. Model 2 is a lineadzed revision of Model 1 whose nonlinear throughput constraint is converted to two linear constraints (10) and (11).
Proceedings of 1996 ICC&IC Mode/2
n
Max ~c~,.x,-z.(X,S,N),.= s.t.
n
(9)
I
~d,.x,.~t~,
(10)
~T~x, < T=~
(11)
z,,,,=r (X,S.N) ~ B U X=(x,,...,x,,),
(12) (13) (14)
,st
S = (s;,...,s~), x, e {0,1} i=l....,n
another approach whk:h increases procemng cape=ty from current best cont~umtJon. One server is raided to the station whose uBizatk)n per server U, is the higheat. Again the iterative procedure is r e a p ~ ~ a better solution is found or budget constlldn is violated. The entire solution procedure is shown in [Figure 1].
Sot CB to IS J li'~ioitialsolution
I
s, E{0}~JZ~ k=l,...,M.
NO
N~Z* (15) In equation (10), the number of machine tools required and the workload imposed by candidate part types are determined simultaneously. To reflect the dynamic effect of FMS previously mentioned, we add another constraint (11). This enables Model 2 to determine the pert types satisfying the throughput of current FMS configuration. After execution of Model 2, we can obtain another candidate pert types that satisfy Txe. If the pert type(s) obtained from the solution of Model 2 is different from the previous pert type(s) selected, the current T,w, T,= are no longer valid. Because the pert mix ratio has been changed, T,= is sought again and it is fed to Model 2. This iterative procedure is applied until T,= of current FMS configuration exceeds T,w of current candidate part types. Because current values T ~ , T~q do not assume the same configuration, the difference lies in the number of jigs and fixtures. Lastly, what we had to decide was the minimum number of jigs and fixtures N which satisfy T,,q. From the viewpoint of FMS operation policy, there can be two different number of jigs and fixtures. If unmanned operation is considered, the minimum number of jigs and fixtures required can be simply represented by equation (16). Nml n =Tre q " U M T
(16)
UMT is the unmanned production time determined by the operation policy. This equation means that at least N,,, number of jigs and fixtures are required to preserve the system throughput T~ during the unmanned operation time. If unmanned operation is not considered, the minimum number of jigs and fixtures is sought by Littie's formula. N,,,=n = 1",,,7 " M F T
551
(1 7 )
T,,~ is determined by current candidate pert types, but we will not know the mean flow time M F T in advance. Thus we decrease the number N until the prescribed throughput Trw is satisfied Then we have to change the constraint N ~ Z* to N > N,,~nin Model 2. But we don't have to execute the revised Mode 2. Though the jigs and fixtures are shared by part types there is no interaction between N end X S This property does not change the solution except N. Setting N to N ~ makes the solution fulfill both optimization model and dosed queuelng network model. If the current combination (X,S,N) yields positive profit, this becomes the current best solution. Until now we used an approach dropping some part types from current candidate part types. To improve the current best solution, the solution procedure adopts
°°
Td ~ r r o n t I
l olution
Mod'l 2 Yes I
I
J. Oeter"n"" N I
CB currentbest solution IS initialsolution NS now solution
Yel STOP NO
I
NO
SetCBt°NS I
J ..............
{u I1_
[Figure 1] Solution Procedure 4. A Case Study The Tongil Heavy Ind. Co. is located in Chang-Won, Korea. The Machine Tool Business Unit of the company produces a vadety of machine tools including lathes, CNC turning centers and machining centers, etc. When they planned to introduce an FMS, the first questions raised were what would be the FMS configuration, what pert types will be processed in the FMS, how long will it take to recover the capital invested, etc. As candidate part types, they selected some components of machine tools(e.g, gear box, spindle box, tail stock, cross slide, etc.). Since these are prismatic part types, 3-axis hodzontal machining center and 5-axis vertical machining center were selected as processing equipment. For material handling, a stacker crane was selected. It feeds work pieces mounted on jigs and fixtures from setup station/control buffer to the input buffer of processing equipment directly. When an operation of the workplace is finished, it is moved to setup station or central buffer directly. Among the components of machine tools, 21 pert types were selected as the eligible part types X. What we had to decide was the pert types to be processed in FMS and the number of horizontal and vertical machining centers which maximizes the profit. Table 2 shows the expodmentel results of our solution procedure applied to this problem. At iteration 0, all part types were selected as candidate part types. The required machine tools were 2 hodzontel machining centers and 1 vertical machining center. BUt the processing capacities of current FMS configuration could not fulfill the throughput required by ell the pert types selected. Thus the proposed solution procedure
Proceedings of 1996 ICC&IC
552
[Table2] Experimental result Iteration Selected Pert Types Hor. MC Vert.MC T,w T== 0 All 2 1 0.0402174 _ 1..(C_.A_N_Q) All 2 1 0.036384 2(Model 2) 1,2,4,5,6,9,10,11,14,14,16,17,18,19,20 2 1 0.0300725 _3(CA_N_QI _ 1,2.4.5.L6.~9.!O ,11,1._4j4 ,_1~ 17 j _8,19,_2._0- ..... --2.... 1 0.02934504 4(Model 2) 1,4,5,6,7,9,10,11,12,13,14,15,16,17,18,19,20,21 2 1 0.0284420 _5_(C_.ANQI _ 1_,4_,5_,6_, 7.~9~.10,11,1..2.13 ,_14~1_5,_.16,!7_,1_8,_19_,2 0_,2_1_ _2 ..... 1 ............. 0.02644582 6(Model 2) 1,4,5,8,7,9,10,11,13,17,20,21 2 1 0.0262681 _7(C_A_N_Q1 _ 1 ,_4,5.,_6,7,_9,10,11,1_3,17,20,21 ........... -2 ..... _1 . . . . . . . . . . . . __.0:02572442 8(Model 2) 1,4,5,6,7,10,11,13,14,16,17,18,19,20,21 2 1 0.0258039 _9(C_A_N_Q)_ 1,4_, 5_L6_, 7,1_0,_11_,13j.14,_16~17=.1_8,1~2_0,._2! . . . . . - 2. . . . 1 0.02547082 10(Model2) 1,4,5,6,7,10,11,13,14,16,17,18,20,21 2 1 0.0252415 _ I_I(C_A_NQ_)_ __1_4_,5_,6_.,7_,1_0,_11,13_14,_1~17,_18,20_,2_1. . . . . . . - 2. . . . 1 0.0254637 12 All 3 1 budget violated
was applied and the result is shown Table 2. As the iteration progressed, we reached the resulting solution (X,S,N) that maximized profit.
3.
5. Charaotedetics of the Proposed DSS and Concluding Remarks
4.
We have proposed an approach to find the bast combination of candidate part types and FMS configuration that maximize profit simultaneously. The solution of the presented problem leads FMS users/vendors' staff to a successful introduction of FMS at the initial design stage that the following questions may be raised. • What kinds of work pieces will be processed in FMS? • What is the type and size of each production resource needed to process the work pieces? • How much is the total investment size? • And how long will it take to return the invested capital? An integrated decision support system named FMS Designer supports the interfaces to FMS ueera/Vendors' staff between optimization model and dosed queueing network model to solve the decision problems. It is running on Windows 95 and data needed to execute both models are collected through the interface to the database containing the information of work pieces, routinga, tools and processing equipment. Current approach works well to • relatively small problem. If the size of eligible part types and FMS configuration becomes large, we cannot foretell the running time of Model 2. Thus an implicit enumeration method substituting the role of Model 2 is being developed.
5.
6.
7.
8.
9.
10.
11.
12.
References 1.
2.
Automation and Management Systems Divisions Charles Stark Draper Lab., Inc., FLEX/B~ MANUFACTURING SYSTEMS HANDBOOK. Noyes Publications,1984. Avgnts,L. H. and Wu=Nmhove L. N.," The Part Mix and Routing Mix Problem in FMS: A Coupling between an LP model end a ~ queoelng network," Int. J. Prod. Res, Vol. 26, No. 12, 1988, pp.1891-1902.
13.
14.
Dallery,Y. and Frain, Y.," An Efficient Method to Determine the Optimal Configuration of a Flexible Manufectudng System," Proc. Of 2r~ ORSA/rlMS Conference on Flexible Manufacturing Systems, 1986, 269-282. Kleinrock, L. queueino Svsterns Volume fl: ~ , ~ , John Wiley and Sons, 1976. Lee, H. F., Sdnivesen, M. M. and Yano, C. A., " Some Characteristics of Optimal Workload Allocation for Closed Queueing Network," Performance Evaluation, 13(1991), pp. 255-268. Lee, H. F., Srinivesan, M. M. and Yano, C. A., " Algorithms for the Minimum Cost Configuration Problem in Flexible Manufacturing System," Int. J. Flexible Manufacturing Systems, Vol. 3, Nos. 3/4, 1991, pp. 1213-230. Shantlkumar,J. G. and Yeo, D. D.," Optimal Server Allocation in a System of Multi-server Stations," Management Science, Vol. 33, No. 9, 1987, pp.1173- 1180. Shantikumar,J. G. and Yeo, D. D.," On Server Allocation in Multiple Center Manufacturing Systems," Operations Research, Vol. 36, No.2, 1988, pp. 333-342. Solberg, J. J., "A Mathematical Model of Computerized Manufactudng Systems', Proc. 4~ International Conference on Production Research, Tokyo, 1977. Suri, R. and Whitney, C. K., " Decision Support Requirements in Flexible Manufactudng,' J. Manufacturing Systems, Vol. 3, No.l, 1984,pp.6169. Tempelmeier, H. and Kuhn, H. F~XIB~E MANUFACTURING STYTEMS Decision Suooort for Design and Ooeration, John Wiley and Sons,1993. Tetzleff, U. A. W.," Capacity Optimization of Flexible Manufacturing Systems under Budget Constraints," Int. J. Flexible Manufacturing Systems, 6(1994), pp. 55-67. Tetzlaff, U. A. W.,"A Model for the minimum Cost Configuration Problem in Flexible Manufacturing Systems," Int. J. Flexible Manufactudng Systems, 7(1995), pp. 127-146. VInod, B. and Solberg, J. J., "The Optimal Design of Flexible Manufactudng Systems," Vol. 23, No. 6, 1985, pp.1141-1151.
Pergmon
PII: S0360-8352(9"/)00190-3
A Decision Support Model for the Initial Design of FMS Hosub Shin', Jeongho Park", Choonghwa Lee- and Jinwoo Park" * Dept. of Industrial Engineering, Seoul National University, Seoul, Korea ** Dept. of System Technology, Tong, Heaw Industry Co., Changwon, Korea *** CIM Business Unit, Daewoo Information Systems Co., Kwacheon, Korea Abstract This study proposes a decision support model for the design and evaluation of FMS. The model optimizes the profit by analyzing the interaction between candidate part types and FMS configuration under the constraints of production requirement and cost effectiveness. The model is solved by using the closed queueing network model and optimization model iteratlvely. The throughput of current FMS configuration is evaluated by the closed queueing network model, and then its results are fed tothe optimization model for the selection of best combination of pert types and production resources. The resulting solution gives enswers to a sedes of decision problems at the initial design stage of FMS. An example problem is presented to show the superiority of the proposed approach. @ 1997 Elsevier Science Ltd Key words : candidate part types, FMS configuration, economic feasibility
and we don't consider detailed layout problems explicitly. There have been numerous previous researches on FMS design problem[3,5,6,7,8,12,13,14]. Unfortunately though, most previous researches assumed that the part types to be produced at FMS are given, and tried to determine an optimal configuration for given part types. However, we will have better and more realistic solution if we consider in our decision process the interaction between candidate pert types and configurations explicitly. Our current research is motivated from that concept, and a nonlinear integer optimization model is developed. Since solving such optimization model directly is not an easy job, an lteratlve procedure is proposed to find the best solution of the proposed model. The resulting solution gives answers to a series of decision problems raised at the initialdesign stage of FMS. In Section 2, a more detailed descril~on of the problem is provided, and a mathematical model is presented. The solution procedure is provided in section 3. A case study is given in section 4, and finally some concluding remarks are presented.
1. Introduction An FMS isan automated cellular manufacturing system consisting of a group of NC machine tools and automated matedal handling equipment, so that a group of parts can be prooessed randomly under the control of a central computer, adapting automatically to changes in product mix or output level. The advantages of fkedble marring technology are improved productivity, reduced work-in-process inventory and lead time, fester response to customer request and so on. Despite these a c k t a t ~ , theimplementation of FMS has not been popular in Korea. Them are two main reaSons for such phenomenon. One is the difficulties encountered in opera6ng the Inataiiod FMS, and the other is the lack of structured approach that can help top management to decide whether an FMS is a viable alternative to their production environment or not. This paper deals with the latter subject. In order to introduce an FMS successfully, a preliminary feasibility study should be performed to enhance introduction and installation of FMS. Draper Lab. in reference [1] proposed following three steps for evetuat]ngfeasibility of FMS. 1. Select candidate pert types and machine tools for FMS. 2. Develop a number of possible configurations. 3. Evaluate the alternative designs and their variations based on technical and economic feasibility. The selection of part types and machine tools is a subtle decision ixoblem because any decision to allocate some production resources for the production of one pert type necessarily affects the part types' that can be produced by the selected resources[10]. In other words, these three sub-steps necessarily interact and revolve around until the prescribed feasibility criterion is satisfied. In the following, we are going to introduce a decision support model for such decision problem. However, our model works at aggregate planning level
2. Problem Formulation
Description
and
Mathematical
We assume that there are n different candidate pert types and M different stations in FMS. X=(x,,...,x,) represents the eligible pad types for the production in FMS. Each integer 0-1 decision variable xj represents the selection of part type i for current production, x~ is equal to 1 if all the pert of type i are processed in FMS and 0, otherwise. In this paper an FMS configuration is characterized by a specific combination of (S,N). In S=(s,,...,s~), each nonnegatlve integer decision vedable s,{k=-l,...,M) represents the number of machines, setup areas, transportation vehicles, etc. at station k depending on the type of work station. We assume that each station consists of identical production resources. And index M is reserved for the modeling of transportation vehicles. N is the number of jigs end 549
Proceedings of 1996 ICC&IC
550
fixtures that conefituta an FMS configuration. The objective of this study is to find and assign values to (X,S,N). That is, we determine the best combination of part types and configuration which maximizes profit Model 1 n
Max ~ c#,x,- z~(X,S,N)
(I)
s.t. z~,~,t(X,S.N)
(2)
n
T(X,S,N) > ~T~x,
i should be processed within annual operation time AOT of FMa, the throughput requirement T~of pert type i is given to d/AOT. The sum of each Tj of selected part type is the system throughput requested to FMS. Due to the nonlinear property of the throughput function given, the problem is formulated into a nonlinear integer programming formulation. The notations not referred to in detail are summadzed in Table I below. Table 1. Summary of Notations
(3)
X=(x,...,x~)
Ct.o
(4) (5)
s = (~....,s~)
pert type i
X, e {0.1}.$, E { 0 } ~ J Z + . N e Z + (6)
As a cdtedon of economic feasibility of an FMS, payback method is widely used in real application. Thus, we adopt a p-year payback pedod as a chtedon of economic feesibity and it is reflected in the o b ~ function of Model 1. Equation (1) mprasents the objective function of one year profit which is obtained by subtracting the yearly production cost z~(X,S,N) from total annual sales volume. The positive yeady profit(positive value of objective function) means that all the initial investment for the introductfon and implementation of an FMS can be returned within p years, cj and dj represent the selling pdce and annual demand of each part type i respectively. The cost function zp(X,S,N) is assumed to be a monotonic nondecreesing linear function. As shown below, the annual production cost zp(X,S,N) is composed of three parts: dedicated cutting tools, equipment types and jigs and fixtures. Each component consists of annual operation cost and amortization should be paid dudng p years. Subscription o, A means operation and amortization respectively.
z p ( X S N ) = T (Ct~o +ct~ A)x~ +
+l
k=l
(,)
Since there may exist some common cutting louis(e.g. rough milling cutter) shared by various pad types, without loss of generality, we assume that all part types share some common cutting tools and have their own dedicated cuffing tools which does not overlap between different pert types. Equation (2) of Model 1 implies that the initial investment needed to install an FM8 is constrained by the available budget BU, and the details of the cost function is given below in equation (8).
xI
H
+ ~ cs, j.~.,
s, + cnj...,
N
ctil,kwo=t Cpo
cp^
c~^ Cak,L~,~t
t~
amortized cost of dedicated tools to process part type i initial investment of dedicated cutting tools to process part type i operation cost of a jig and fixture amortized cost of a jig and fixture initial investment needed for procurement of a jig and fixture operation cost of equipment type k amortized cost of equipment type k initial investment needed for procurement of equipment type k processing time of operation j of pert type i at equipment type k
3. Solution Procedure Initially all the part types of X=(x~,...,x,) are selected as candidate part types. Then, we calculate the required system throughput T,w, which is the sum of each T,, selected. Next, the minimum number of each machine should be decided to process all the part types selected. The required number of each machine tool is the smallest integer greater than the value
d,x,~ t~, / AOT (total assigned workload divided by
,o
zi.~o.j ( X S N ) = ~ ctj j.~.,
operation cost of dedicated tools to process
(8)
k=t
Equation (3) represents the throughput requirement of the FMS configuration we are going to build. For the evaluation of FMS throughput, we referred to the dosed quaueing network theory. The left hand side of equation (3),T(X,S,N), represents the throughput of the FMS configuration in terms of aggregate part type. An aggregate part type is the r e p r e s e ~ e part type of candidate part types. Thus, for current candidate pert types, the throughput req~ to FMS is also represented in terms of an aggregate part type, Since yearly demand dlof part type
annual operation time of FM8). If we just assign the workloads of candidate part types to the processing capacities of current FMS configuration, the whole workloads cannot be processed within planned operation time by the dynamic effect of actual syatem[2]. To evaluate the throughput of current FMS configuration, dosed quoueing network model .is widely used. In our case, we chose to use CANQ developed by Solberg[9]. Initial part types and the corresponding FMS configuration are fed to CANQ as input data. By the asymptotic bound[4] of dosed queuing network theory, the throughput of FMS goes to a limiting value as the number of jigs and &xturee goes to infinity. Here we assume that there are enough jigs and fixtures. After running CANQ, the throughput of current FMS configuration Tm is obtained. But it may happen that Tm is less than T~. This means that the FMS configuration previously detarmined cannot fulfill the demands of part types selected. To overcome this drawback, two approaches may be possible. One is increasing the system throughput by expanding the capacities of machine tools. The other is decreasing the workload by dropping certain part types from current candidate part types. Model 2 is for the latter approach. The former approach will be described in the latter part of this section. Model 2 is a lineadzed revision of Model 1 whose nonlinear throughput constraint is converted to two linear constraints (10) and (11).
Proceedings of 1996 ICC&IC Mode/2
n
Max ~c~,.x,-z.(X,S,N),.= s.t.
n
(9)
I
~d,.x,.~t~,
(10)
~T~x, < T=~
(11)
z,,,,=r (X,S.N) ~ B U X=(x,,...,x,,),
(12) (13) (14)
,st
S = (s;,...,s~), x, e {0,1} i=l....,n
another approach whk:h increases procemng cape=ty from current best cont~umtJon. One server is raided to the station whose uBizatk)n per server U, is the higheat. Again the iterative procedure is r e a p ~ ~ a better solution is found or budget constlldn is violated. The entire solution procedure is shown in [Figure 1].
Sot CB to IS J li'~ioitialsolution
I
s, E{0}~JZ~ k=l,...,M.
NO
N~Z* (15) In equation (10), the number of machine tools required and the workload imposed by candidate part types are determined simultaneously. To reflect the dynamic effect of FMS previously mentioned, we add another constraint (11). This enables Model 2 to determine the pert types satisfying the throughput of current FMS configuration. After execution of Model 2, we can obtain another candidate pert types that satisfy Txe. If the pert type(s) obtained from the solution of Model 2 is different from the previous pert type(s) selected, the current T,w, T,= are no longer valid. Because the pert mix ratio has been changed, T,= is sought again and it is fed to Model 2. This iterative procedure is applied until T,= of current FMS configuration exceeds T,w of current candidate part types. Because current values T ~ , T~q do not assume the same configuration, the difference lies in the number of jigs and fixtures. Lastly, what we had to decide was the minimum number of jigs and fixtures N which satisfy T,,q. From the viewpoint of FMS operation policy, there can be two different number of jigs and fixtures. If unmanned operation is considered, the minimum number of jigs and fixtures required can be simply represented by equation (16). Nml n =Tre q " U M T
(16)
UMT is the unmanned production time determined by the operation policy. This equation means that at least N,,, number of jigs and fixtures are required to preserve the system throughput T~ during the unmanned operation time. If unmanned operation is not considered, the minimum number of jigs and fixtures is sought by Littie's formula. N,,,=n = 1",,,7 " M F T
551
(1 7 )
T,,~ is determined by current candidate pert types, but we will not know the mean flow time M F T in advance. Thus we decrease the number N until the prescribed throughput Trw is satisfied Then we have to change the constraint N ~ Z* to N > N,,~nin Model 2. But we don't have to execute the revised Mode 2. Though the jigs and fixtures are shared by part types there is no interaction between N end X S This property does not change the solution except N. Setting N to N ~ makes the solution fulfill both optimization model and dosed queuelng network model. If the current combination (X,S,N) yields positive profit, this becomes the current best solution. Until now we used an approach dropping some part types from current candidate part types. To improve the current best solution, the solution procedure adopts
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I
J. Oeter"n"" N I
CB currentbest solution IS initialsolution NS now solution
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[Figure 1] Solution Procedure 4. A Case Study The Tongil Heavy Ind. Co. is located in Chang-Won, Korea. The Machine Tool Business Unit of the company produces a vadety of machine tools including lathes, CNC turning centers and machining centers, etc. When they planned to introduce an FMS, the first questions raised were what would be the FMS configuration, what pert types will be processed in the FMS, how long will it take to recover the capital invested, etc. As candidate part types, they selected some components of machine tools(e.g, gear box, spindle box, tail stock, cross slide, etc.). Since these are prismatic part types, 3-axis hodzontal machining center and 5-axis vertical machining center were selected as processing equipment. For material handling, a stacker crane was selected. It feeds work pieces mounted on jigs and fixtures from setup station/control buffer to the input buffer of processing equipment directly. When an operation of the workplace is finished, it is moved to setup station or central buffer directly. Among the components of machine tools, 21 pert types were selected as the eligible part types X. What we had to decide was the pert types to be processed in FMS and the number of horizontal and vertical machining centers which maximizes the profit. Table 2 shows the expodmentel results of our solution procedure applied to this problem. At iteration 0, all part types were selected as candidate part types. The required machine tools were 2 hodzontel machining centers and 1 vertical machining center. BUt the processing capacities of current FMS configuration could not fulfill the throughput required by ell the pert types selected. Thus the proposed solution procedure
Proceedings of 1996 ICC&IC
552
[Table2] Experimental result Iteration Selected Pert Types Hor. MC Vert.MC T,w T== 0 All 2 1 0.0402174 _ 1..(C_.A_N_Q) All 2 1 0.036384 2(Model 2) 1,2,4,5,6,9,10,11,14,14,16,17,18,19,20 2 1 0.0300725 _3(CA_N_QI _ 1,2.4.5.L6.~9.!O ,11,1._4j4 ,_1~ 17 j _8,19,_2._0- ..... --2.... 1 0.02934504 4(Model 2) 1,4,5,6,7,9,10,11,12,13,14,15,16,17,18,19,20,21 2 1 0.0284420 _5_(C_.ANQI _ 1_,4_,5_,6_, 7.~9~.10,11,1..2.13 ,_14~1_5,_.16,!7_,1_8,_19_,2 0_,2_1_ _2 ..... 1 ............. 0.02644582 6(Model 2) 1,4,5,8,7,9,10,11,13,17,20,21 2 1 0.0262681 _7(C_A_N_Q1 _ 1 ,_4,5.,_6,7,_9,10,11,1_3,17,20,21 ........... -2 ..... _1 . . . . . . . . . . . . __.0:02572442 8(Model 2) 1,4,5,6,7,10,11,13,14,16,17,18,19,20,21 2 1 0.0258039 _9(C_A_N_Q)_ 1,4_, 5_L6_, 7,1_0,_11_,13j.14,_16~17=.1_8,1~2_0,._2! . . . . . - 2. . . . 1 0.02547082 10(Model2) 1,4,5,6,7,10,11,13,14,16,17,18,20,21 2 1 0.0252415 _ I_I(C_A_NQ_)_ __1_4_,5_,6_.,7_,1_0,_11,13_14,_1~17,_18,20_,2_1. . . . . . . - 2. . . . 1 0.0254637 12 All 3 1 budget violated
was applied and the result is shown Table 2. As the iteration progressed, we reached the resulting solution (X,S,N) that maximized profit.
3.
5. Charaotedetics of the Proposed DSS and Concluding Remarks
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We have proposed an approach to find the bast combination of candidate part types and FMS configuration that maximize profit simultaneously. The solution of the presented problem leads FMS users/vendors' staff to a successful introduction of FMS at the initial design stage that the following questions may be raised. • What kinds of work pieces will be processed in FMS? • What is the type and size of each production resource needed to process the work pieces? • How much is the total investment size? • And how long will it take to return the invested capital? An integrated decision support system named FMS Designer supports the interfaces to FMS ueera/Vendors' staff between optimization model and dosed queueing network model to solve the decision problems. It is running on Windows 95 and data needed to execute both models are collected through the interface to the database containing the information of work pieces, routinga, tools and processing equipment. Current approach works well to • relatively small problem. If the size of eligible part types and FMS configuration becomes large, we cannot foretell the running time of Model 2. Thus an implicit enumeration method substituting the role of Model 2 is being developed.
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References 1.
2.
Automation and Management Systems Divisions Charles Stark Draper Lab., Inc., FLEX/B~ MANUFACTURING SYSTEMS HANDBOOK. Noyes Publications,1984. Avgnts,L. H. and Wu=Nmhove L. N.," The Part Mix and Routing Mix Problem in FMS: A Coupling between an LP model end a ~ queoelng network," Int. J. Prod. Res, Vol. 26, No. 12, 1988, pp.1891-1902.
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Dallery,Y. and Frain, Y.," An Efficient Method to Determine the Optimal Configuration of a Flexible Manufectudng System," Proc. Of 2r~ ORSA/rlMS Conference on Flexible Manufacturing Systems, 1986, 269-282. Kleinrock, L. queueino Svsterns Volume fl: ~ , ~ , John Wiley and Sons, 1976. Lee, H. F., Sdnivesen, M. M. and Yano, C. A., " Some Characteristics of Optimal Workload Allocation for Closed Queueing Network," Performance Evaluation, 13(1991), pp. 255-268. Lee, H. F., Srinivesan, M. M. and Yano, C. A., " Algorithms for the Minimum Cost Configuration Problem in Flexible Manufacturing System," Int. J. Flexible Manufacturing Systems, Vol. 3, Nos. 3/4, 1991, pp. 1213-230. Shantlkumar,J. G. and Yeo, D. D.," Optimal Server Allocation in a System of Multi-server Stations," Management Science, Vol. 33, No. 9, 1987, pp.1173- 1180. Shantikumar,J. G. and Yeo, D. D.," On Server Allocation in Multiple Center Manufacturing Systems," Operations Research, Vol. 36, No.2, 1988, pp. 333-342. Solberg, J. J., "A Mathematical Model of Computerized Manufactudng Systems', Proc. 4~ International Conference on Production Research, Tokyo, 1977. Suri, R. and Whitney, C. K., " Decision Support Requirements in Flexible Manufactudng,' J. Manufacturing Systems, Vol. 3, No.l, 1984,pp.6169. Tempelmeier, H. and Kuhn, H. F~XIB~E MANUFACTURING STYTEMS Decision Suooort for Design and Ooeration, John Wiley and Sons,1993. Tetzleff, U. A. W.," Capacity Optimization of Flexible Manufacturing Systems under Budget Constraints," Int. J. Flexible Manufacturing Systems, 6(1994), pp. 55-67. Tetzlaff, U. A. W.,"A Model for the minimum Cost Configuration Problem in Flexible Manufacturing Systems," Int. J. Flexible Manufactudng Systems, 7(1995), pp. 127-146. VInod, B. and Solberg, J. J., "The Optimal Design of Flexible Manufactudng Systems," Vol. 23, No. 6, 1985, pp.1141-1151.