NONLINEAR MODEL: CELL FORMATION

Transcription

1 ational Institute of Tehnology Caliut Deartment of Mehanial Engineering OLIEAR MODEL: CELL FORMATIO System Requirements and Symbols 0-Integer Programming Formulation α i - umber of interell transfers due to art i β i - Demand of art i γ i - Unit interell transfer ost of art i M ik - Mahine tye required for the oeration k of the art i i inde of arts, j Inde of ells, i,2, j,2, k - Inde of oeration, k,2, i M inde of mahines, M,2, m i - umber of oerations required to omlete the roessing requirement of art i m min - Minimum number of mahine tyes er ell m ma - Maimum number of mahine tyes er ell θ - minimum number of arts er family Z Interell material handling ost if art i is assigned to ell j ij M ik, j 0 if mahine tye M ik is assigned to ell Otherwise j umber of art families umber of mahine ells. A mahine tye is assigned to only one ell. When a art is assigned to a ell and some of the mahines required for omleting the oeration required for this art is available in another ell, then the interell transfer alulation involves the following: Conseutive oerations of a art in a non-assigned ell reate intraell moves. Eah suh intraell move is onsidered as an interell move. onlinear Model: Cell Formation 57 Marh 203

2 ational Institute of Tehnology Caliut Deartment of Mehanial Engineering Formulation Minimize This may be interreted as that any movement of a art other than in the assigned ell has to be enalized. (That is, if a art is assigned to a ell then all its oeration should be omleted in the ell in whih it is assigned. The interell alulation equation below takes into aount this fat Z α iβiγi i [] αi i ij, j k Mik j Mi ( k, j i [2] Subjeted to j ij i [3] i ij θ j [4] i M, j i i [5] ik k j m M m M Mj mmin Mj mma { 0,} i, k j M ik, j, ij, j [6] j [7] onlinear Model: Cell Formation 58 Marh 203

3 ational Institute of Tehnology Caliut Deartment of Mehanial Engineering Eamle Problem Minimize the interell movement for the 5 mahine 4 art roblem given below. m min (Minimum number of mahine tyes er ell m ma 3 (Maimum number of mahine tyes er ell θ (Minimum number of arts er family 2 (umber of ells γ i (Unit interell transfer ost Mahines β i Part Table: Oerations required for arts The above table rovides the mahine sequene required to omlete the various oerations of arts. For eamle, first oeration of art is in mahine 3, seond oeration is in mahine 2, third oeration is in mahine 5 and so on. Hene, the mahine sequene for the art is Formulation tis ( umber of terms in the objetive funtion i umber of onstraints umber of deision variables i m onlinear Model: Cell Formation 59 Marh 203

4 ational Institute of Tehnology Caliut Deartment of Mehanial Engineering onlinear Model: Cell Formation 60 Marh 203 Formulation for the above roblem Objetive funtion 0 ( 20 ( 5 ( 0 ( Z Subjeted to Constraints due to eqn Constraints due to eqn Constraints due to eqn Constraints due to eqn Constraints due to eqn Integrality onstraints { },0, Mj ij

5 ational Institute of Tehnology Caliut Deartment of Mehanial Engineering A feasible solution to this roblem is,, 3,, 2,, 3, All other deision variables have value equal to zero This has an objetive funtion value of 55 The orresonding ells and art families are as follows: Cell {, 2, 3} & Part-family {2, 3} Cell 2 {4, 5} & Part-family 2 {, 4} onlinear Model: Cell Formation 6 Marh 203

6 ational Institute of Tehnology Caliut Deartment of Mehanial Engineering QUESTIOS:. Interell moves reate long flow aths for art roessing, why? Are interell moves desirable? Substantiate your answer. 2. Write down the integer rogramming formulation of the ell formation roblem. The manufaturing environment onsists of a deterministi rodut mi for the lanning horizon of the design. Minimum number of tyes of mahines and maimum number of tyes of mahines in a ell are onstrained. The number of arts in a art family is also onstrained. The route sheet and demand information of the art mi are available. 3. Define a deision variable of a mathematial rogramming roblem whih shows a art assignment to a ell (art family. A art an be assigned only to a art family. Generate all suh onstraints. α i [ ] 4. i ij M ik, j M i( k, j j k i This equation is used for interell movement alulation for a single eriod ellular manufaturing system design roblem. The notations and arameters used are as follows: α is the number of interell movements due to art i. i 2 (umber of ells i is number of oerations for art i. β i is the demand of art i. M ik, j if mahine tye M ik is assigned to ell j ij if art i is assigned to ell j i inde of arts, i,2, j inde of ells, k inde of oerations j,2, The oeration sequene for 4 arts whih are roessed in 5 mahines is rovided in the table given below. Mahine onlinear Model: Cell Formation 62 Marh 203

7 ational Institute of Tehnology Caliut Deartment of Mehanial Engineering Part β i A feasible solution for the above roblem is given below. 2,,,, 3 3,,, Determine the interell movements using the above equation. 5. ij θ j... [] i This equation shows a onstraint for a single eriod ellular manufaturing system design roblem. The notations and arameters used are as follows: θ (Minimum number of arts er family, 2 (umber of ells 4 (umber of arts ij if art i isassigned toell j i inde of arts, i,2, j Inde of ells, j,2, Write down all the onstraints generated by the equation [] for the given data. 6. The matri below shows the oeration sequene for 8 roduts rodued in a rodution system. For eamle, the oeration sequene of art is and the orresonding mahine sequene is C-D-H-F. That is, the mahine tye used for the first oeration of art is C. Similarly, mahine tyes required for oerations 2, 3 and 4 an be established. Part Mahine A B C D E 4 F 4 3 G 4 4 H 3 5 onlinear Model: Cell Formation 63 Marh 203

8 ational Institute of Tehnology Caliut Deartment of Mehanial Engineering A ell formation is onsidered for this rodution system with the following features: θ 3 (Minimum number of arts er family, 2 (umber of ells A mathematial model for ell formation of the above rodution system is develoed whih ontains the following onstraints. ij i i k j θ M ik j..., j i i [] [2] The notations used in the above onstraints are as follows: M ik, j ij 0 if art i isassigned toell j - umber of arts if mahine tye Otherwise M ik isassigned toell i - umber of oerations required to omlete the roessing requirement of art i M - Mahine tye required for the oeration k of the art i ik k - Inde of oeration, k,2, i M inde of mahines, M,2, m (M is orresonding mahine A, 2 orresonds to B, and so on j. i inde of arts, i,2, j Inde of ells, j,2, Write down all the onstraints generated by [] and [2] for the data given above. onlinear Model: Cell Formation 64 Marh 203