Determining Optimal Plan of Fabric Cutting with the Multiple Criteria Programming Methods

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1 Proceedings o the International MultiConerence o Engineers and Computer Scientists 008 Vol II Determining Optimal Plan o Fabric Cutting with the Multiple Criteria Programming Methods Tuno Perić and Zoran Babić Member IAENG Abstract: In this paper authors indicate by means o a concrete eample that it is possible to apply the method o multiple criteria integer linear programming method in dealing with the problem o determining an optimal plan or abric cutting optimization. The procedure involves the ollowing stages: ) Selection o the criteria or optimization o abric cutting ) Setting the problem o determining an optimal plan o abric cutting 3) Setting the model o multiple criteria linear integer programming in inding solution to a given problem (4) The solutions are obtained applying the method o satisactory goals. Inde Terms: abric cutting integer programming multiple criteria programming. I. INTRODUCTION Fabric cutting represents a very comple area o work which is present in many industrial branches or eample in wood and timber industry tetile industry paper industry leather industry metal-working industry etc. By abric cutting basic abric is transormed into desired orms and the result o cutting is a cut-out and the waste o basic material. Fabric cutting should provide the required number o cut-outs high productivity o work simplicity o machine work as well as the saety in the process o work. Industrial abric cutting presupposes the eistence o a plan o cutting. The plan o cutting implies a set o cutting schemes which determine the way certain units o material should be cut out in order to obtain the requested number o cut-outs. The selection o cutting schemes might be empiric and ormally mathematical. The applicability o certain cutting schemes depends on the character o the material cutting means as well as technical and organizational principles. Rational selection o cutting schemes is the one which enables the application o the eisting technology organization o work materials at disposal and cutting means. In addition each o the cutting schemes needs to be technologically achievable. The task o abric cutting consists o a demand or certain assortment o cut-outs and the best possible usage o basic abric means o work and labor with the respect or given technological constraints. Manuscript received December 007. Tuno Perić was with the Faculty o Economics Dubrovnik University o Split Republic o Croatia. He is now with the Bakery Sunce 0437 Bestove Rakite Rakitska cesta 98 Republic o Croatia. Phone: ; a: ; tuno.peric@zg.tcom.hr. Zoran Babić is with the Faculty o Economics University o Split 000 Split Matice hrvatske 3 Republic o Croatia. babic@est.hr. Accordingly abric cutting is a very serious and responsible phase o work in many industrial branches and thus involves the engagement o broad circle o specialists. Since in the process o production direct abric appears in terms o costs it is important to observe its course as in this way cost rate and the price o inished product can be inluenced. In this work we proceed rom the ollowing hypothesis: In the process o industrial abric cutting disposable means and labor are not being used in a rational way and thereore companies do not achieve optimal business results. The aim o this work is to: () point to the act that determining plan o abric cutting or a certain period is essentially a multiple criteria problem () develop the multiple criteria linear programming model by means o which the plan o abric cutting would be optimized (3) apply the appropriate method o multiple criteria linear and integer programming in inding a solution to determine optimal plan o abric cutting in a concrete company. II. MULTIPLE CRITERIA PROGRAMMING MODEL FOR DETERMINING THE OPTIMAL PLAN OF FABRIC CUTTING In the process o selecting the criteria or optimization o the plan o abric cutting it is necessary to have in mind the ollowing: - abric cutting according to dierent variants gives dierent amounts o waste - the time (costs) o abric cutting is dierent according to dierent variants. It would be ideal i the abric was cut out according to variants which give a minimal waste and i the time o abric cutting (also the costs) were minimal. Accordingly the ollowing criteria should be considered in the process o optimizing the plan o abric cutting:. the cut-out as a result o abric cutting in square meter terms. the time o abric cutting in minutes or costs o cutting in currency units. The construction o multiple criteria programming (MCP) model is possible by two-dimensional abric cutting o even Criteria is a direct measure or goal achievement. In order to make practical way o determining the level o goal achievement easier the criterion or a number o criteria or each goal achievement at the lowest level is determined. The criterion is quantitatively measurable whereby its value relects the level o goal achievement whose criterion is proper [4]. ISBN: IMECS 008

2 Proceedings o the International MultiConerence o Engineers and Computer Scientists 008 Vol II cutting properties (compactness and thickness) on the whole surace and in the case o abric o regular geometrical shape (circle rectangle square etc...). In the process o determining the optimal plan o abric cutting using MCP methods it is necessary to begin with the ollowing:. the criteria or optimization o abric cutting are given. the variants o abric cutting are amiliar 3. the quantity o abric or cutting is limited 4. the need or quantity o certain cut-outs is given. Let us introduce the ollowing signs: - l b = need or obect type l ( l = L m) - ail α = quantity o obects o type l contained in i variant o cutting completed on abric o dimension α ( i = L n ; α = L r ; l = L m) - i α = quantity o abric o dimension α which is to be cut out according to i variant ( i = L n ; α = L r) - Q α = disposable quantity o abric o dimension α ( α = L r) - oi α = abric waste o dimension α according to i variant ( i = L n ; α = L r) - hi α = the time necessary or machine work or abric cutting o dimension α according to i variant ( i = L n ; α = L r). On the basis o the above the MCP model or abric cutting has got the ollowing orm: n r n r (min) = oiα iα hiαiα i = α = i = α = s. t. () n r ailαiα bl i= α = ( l = L m) n iα Qα ( α = L r) i= i α 0 ( i = L m; α = L r). The setting o such a model is conditioned by the knowledge o obect placement variants. Cutting variants are generated by computation whereby it is usually understood that the appropriate arrangement variant is the one which provides that the abric waste is smaller than the smallest obect o the arrangement. However even though the problem is relatively slight (cutting as many as 0 dierent elements o two to three kinds o abric) in this way a great number o cutting variants are generated. All the obtained abric cutting variants should be eamined About the problem o abric cutting variants generating see in [3]. rom the view o technological possibilities o cutting and or technologically possible variants cutting machine work hours should be determined. This would be a tedious task. However even i we eamine all these variants we would orm an oversized model by introducing the «appropriate» variants into the model which might make the problem too comple to solve. Thereore it is proposed to condense the model regarding the number o the variants by inding all the variants o substitution and its inclusion into the basic model (a unit or more obect units o smaller dimensions could be received rom one obect o arrangement). Our suggestion is to condense the number o variants by adding the limitation to the program or cutting variants generating according to which the variant is satisying only i the entire uncovered surace (waste) is smaller than a certain percentage o the entire surace o the dimensions o the cut abric. The appropriate percentage would be reached in the interaction with the computer. At that point only the best variants remain which should be eamined rom the view o technological possibilities o cutting as well as rom the view o the necessary work hours o cutting machines or each variant. Technologically appropriate variants and the variants which do not demand an overwork o cutting machines are being integrated into the MCP model. By solving the given model in the interaction with the decision maker the most suitable plan or cutting abric would be reached. By solving the model above dierent methods o multiple criteria integer linear programming can be applied [4]. III. CASE STUDY In this section we set the problem o determining an optimal plan o cutting material in a concrete company. It is necessary here on the basis o a deined plan o production or a certain period o time to determine an optimal amount o certain cutting schemes or the production o the needed amount o cut-outs. Nineteen dierent technically eecutable variants or cutting chipboard are chosen o dimensions: mm (Dimension ) and mm (Dimension ) or the production o 0 dierent cut-outs. Regarding chipboard o dimensions mm 9 dierent cutting variants are chosen and regarding chipboard o dimensions mm 0 dierent cutting variants are chosen. The company has got 500 chipboards o dimension and 800 chipboards o dimension at its disposal. Acceptable variants o cutting o 0 dierent obects rom two chipboards o dierent dimensions necessary amounts o certain cut outs waste by variants in square meters as well as necessary work hours o cutting machine in «normal» conditions epressed in seconds are shown in the ollowing table: 3 3 Variants o cutting o given abric are being ormed by the usage o algorithm shown in the works: [3] and [7]. ISBN: IMECS 008

3 Proceedings o the International MultiConerence o Engineers and Computer Scientists 008 Vol II Table 3. Problem data Ord. Board : mm / Variants Dimensions o number obects in mm Waste in m Time o production in seconds Continuation o Table 3. Ord. Board : mm / Variant Necess. number amount Waste in m Time o prod. in sec From the previous table it can be noticed that it is necessary to minimize the unction o waste and the unction o work hours o the cutting machine. Let i α = be the amount o abric o dimension α which is to be cut according to i - variant ( i = L 9) ( α = ). Then the multiple criteria integer linear programming model or solving the concrete problem o determining an optimal plan o cutting abric is as ollows: a) Criteria unctions Waste: (min) = (min) = b) Constraints Necessary elements () to (0): () () (3) Time o production: ISBN: IMECS 008

4 Proceedings o the International MultiConerence o Engineers and Computer Scientists 008 Vol II (4) (5) (6) 00 6 (7) (8) (9) Fabric provision - () to (): () () (3) L 9 0 L 9 ; 0. IV. SOLVING THE MODEL OF DETERMINING AN OPTIMAL PLAN OF FABRIC CUTTING The model rom the previous chapter was irst solved by using the linear integer programming minimizing each o the two criteria unctions on the given set o constraints. The ollowing solutions are obtained: (0) Table 4. Optimal and marginal solutions o criteria unctions Variable value Criteria unctions value =6 6 =6 8 =44 9 =35 0 =80 = 3 =46 4 =50 6 = (00%) (40% o ) =96 6 =37 8 =8 9 =4 0 =835 =69 3 =7 5 =3 6 = (% o ) 3865 (00%) From the table 4 it can be noticed that by minimizing the unction the received minimal value or that unction is considerably dierent rom the value o that unction when unction is minimized. The implication is the lack o application o the linear integer programming in determining an optimal plan o abric cutting as well as the necessity o application o the multiple criteria integer linear programming (MCILP) methods. Namely i the MCILP methods are not being used the selection o the plan o abric cutting is reduced to one o the marginal solutions. By solving the given model using the MCILP methods the solution that would give more acceptable values to the criteria unctions would be received. A. Solving the Optimization Model o the Plan o Fabric Cutting by the Method o Satisactory Goals By application o the Method o Satisactory Goals the ollowing model is solved: (min) ( ) LS s.t. () ( ) 0 i = L m gi q ( ) L = L k ; LS whereby ( ) is the criteria unction with which the LS q L decision maker is the least satisied and ( = L k ) LS are the values o remained criteria unctions are determined by an analyst in step q q ( L < ). In our eample we deined: ( ) = LS ( ) and ( ) < ISBN: IMECS 008

5 Proceedings o the International MultiConerence o Engineers and Computer Scientists 008 Vol II The ollowing solution is obtained: Table 5. Optimal solution Variable value = 4; 6 = ; 8 = 55 ; 9 = 34 ; 0 = 800 ; = 54 ; = 54 ; 3 = 8 ; 4 = 6 ; 5 = 35 ; 6 = (04 % o Criteria unctions value ) (6 % o ) Since the decision maker wanted to decrease the value o unction we solved the ollowing model: (min) ( ) s.t. (3) X ( ) The ollowing solution is obtained: Table 6. Optimal (preerred) solution Variable values = 34 ; = 5 ; 6 = 0 ; 8 = 7 ; 9 = 3 ; 0 = 80; = 07 ; 4 = 93 ; 5 = 03 ; 6 = 96 ; Criteria values (07% o ) 375 (06 % o ) The decision maker accepted the presented solution since rom his point o view it provides acceptable values o criteria unction. V. CONCLUSION On the basis o the previously outlined we come to conclusion that the application o the method o integer MCP in solving problems o determining the optimal plan o abric cutting represents a «necessity» which is a result o the act that in a given problem there is a multiplicity o conlicting goals thus the optimization o criteria unction that suits one goal leads to insuicient actualization o another goal. This adversely aects actualization o total business results o a company. Developed model o multiple criteria integer decision making creates an opportunity or optimization o business in industrial companies dealing with abric cutting. REFERENCES [] D. ABEL Programmable Controllers Instruments & Control System 974 p [] C. BERG Getting the most rom your PC 978. [3] B. BOJANIĆ i M. BOJANIĆ Generirane varianti kroena materiala pomoću elektroničkog računala Praksa Beograd No str [4] V. CHANKONG and Y. Y. HAIMES Multiobective Decision Making: Theory and Methodology North Holland New York 983. [5] D. L. HWANG and A. S. M. MASUD Multiple Obective Decision Making: Methods and Applications Springer Verlag New York 979. [6] P. C. FISHBURN Leicographic Orders Utilities and Decision Rules: a Survey Management ScienceVol. 0 No. 974 pp [7] Panel Cutting Optimization Sotware - Plus D or Woodworking. Avaliable: ISBN: IMECS 008