OPTIMIZATION OF MULTI-OBJECTVE
FACILITY LAYOUT USING
NON-TRADITIONAL OPTIMIZATION
TECHNIQUE
S. NARAYANA REDDY
Brindavan Institute of Technology and Science Department of Mechanical Engineering
Kurnool (A.P) – 518002 [email protected]
V.VARAPRASAD
Assistant ProfessorDepartment of Mechanical Engineering G.Pulla Reddy Engineering College
Kurnool (A.P) – 518002
DR. V.VEERANNA
Dean (R&D)
Brindavan Institute of Technology and Science Department of Mechanical Engineering
Kurnool (A.P) – 518002 [email protected]
Abstract:
Increased global competition in manufacturing and increased consciousness towards reducing manufacturing costs has renewed interest in the efficient design of facility layout. The success of a manufacturing organization depends on the proper design of various systems required in the production cycle. One such system is design of the facility that is able to adapt quickly and effectively, the technological changes and market requirements. Facility layout problem deals with the physical arrangement of a given number of machines or departments within a given configuration. The assignment of facilities to locations is one of the most important issues that must be resolved in manufacturing systems. The facilities layout focuses on the organization of a company’s physical facilities to promote the efficient use of resources such as equipment, material, people and energy. Well-studied combinatorial optimization problem arises in a variety of problems such as schools, airports, etc. but the focus of our work is on solving the facility layout of manufacturing plants. Several researchers have used different methodologies for getting better solutions of facility layout problems. The methodologies that are used to solve the facility layout problems are discussed briefly in this paper work. Alternative layouts are generated for the arrangement of facilities using a software program written in Turbo C++, taking qualitative and quantitative inputs in the form of flow matrix, distance matrix and closeness rating matrix. Scores are computed for each of the alternative layout generated. Based on their scores, the alternative layouts are evaluated and the best among them is selected. The results are obtained by running the program. The best layout among the alternatives is evaluated based on lowest objective function value.
Keywords: Facility Layout, particle swarm optimization, Heuristic procedure
1. Facility Layout Planning
system design. The material handling cost contributes a major portion of the nonproductive cost. Reducing the travel distances of components can reduce the material handling cost. Layout optimization focuses on travel distances and arrangement of facilities.
1.1Objectives Of The Facility Layout Problem
The following are the general objective functions by which optimality of facility layout problems are measured: 1. Minimize material handling cost
2. Minimize frequency of handling
3. Minimize capital and operating cost in equipment and plant. 4. Increase effective and economical use of space
5. Minimize overall production time
6. Minimize variation in types of material handling equipment 7. Maintain flexibility of arrangement and operation
8. Provide safe and efficient construction
9. Maximize the market share in multi competitive facilities. 10. Maximize the closeness rating.
1.2 Classifications of Layout Problems
1. Single-row machine layout problem 2. Multi-row equal area facility layout problem 3. Multi-row unequal area facility layout problem 4. Multi-objective facility layout problem
1.2.1 Single-Row Machine Layout Problem
Among the various flow path configurations, linear single-row path is still popular due to its simple structure and easy flow control. In a single-row layout pattern, facilities are arranged linearly in one row. During the manufacturing process, there are occasions where not all parts will pass through all machines for processing in the same sequence. The handling of such parts is normally done using Automated Guided Vehicle (AGV) or an operator, assuming the flow movement is unidirectional.
Example For The Single Row Machine Layout
A B C D E F
Fig1. The multi row equal area machine layout
1.2.2 Multi-Row Machine Layout Problem
The multi-row equal area layout has facilities arranged linearly in two or more rows. The multi-row layout is usually associated with job-shop and Flexible Manufacturing System (FMS) environments, where the parts can be processed in any sequence. The equal area layout problem is to determine how to allocate a set of discrete facilities, to a set of discrete locations, in such a way that each facility is assigned to a single location.
Example For The Multi Row Equal Area Machine Layout
A B C D E F
1.2.3 Multi-Row Unequal Area Machine Layout Problem
In the layout optimization of unequal area facilities, the area of each facility is divided into a number of unit cells. The dimensions of the unit cells are equal to those of the discredited grid locations on which the facilities are to be placed. It is also difficult to control the shapes of the facilities during the optimization process. The unequal area layout problem is to determine how to allocate all facilities within a block plan or available area.
Example For The Multi Row Unequal Area Machine Layout
A B C
D E F
Fig3. The multirow unequal area machine layout
1.2.4 Multi-Objective Facility Layout Problem
In real life, the facility layout problem must consider quantitative and qualitative criteria and this falls into the category of the Multi-Objective Facility Layout (MOFL) problem. The primary purpose in solving the MOFL is to generate efficient alternatives that can be presented to the decision maker for the selection and implementation.
These types of problems are classified into congruent and conflicting objectives. Conflicting objectives aim at minimization of total flow cost and maximization of total closeness rating. Congruent objectives aim at minimization of distances based on cost of several attributes namely flow, closeness rating, hazardous movements, safety, etc. The generation and evaluation of the various efficient solutions to the MOFL problem are difficult because of the lack of a suitable measure for effectiveness with respect to multiple objectives.
2. Traditional Optimization Techniques
Different researchers have applied various traditional techniques for optimization of layout design. The following traditional techniques were used in facility layout optimization problems.
1. Quadratic Assignment Formulation 2. Integer Programming Formulation 2.1 Quadratic Assignment Formulation
Traditionally, the layout problem has been modeled as Quadratic Assignment Problem (QAP) and Koopmans and Beckman (1957) introduced the QAP to model the problem of locating interacting plants of equal areas. The QAP has been applied to a wide range of applications, including urban planning, control panel layout and wiring design. Bazaraa (1975) noted that the QAP is a special case of the facility layout problem because it assumes that all departments have equal areas and that all locations are fixed and a known priori. The name was so given because the objective function is a second-degree function of the variables and the constraints are linear functions of the variables.
The QAP formulation assigns every department to one location and at the most one department to each location. The cost of placing a department at a particular location is dependent on the location of the interacting departments. Kaku et al (1991) have proposed heuristics for the QAP. A recent alternative formulation of the QAP considers assigning interdepartmental distances to department pairs.
2.2 Integer Programming Formulation
Lawler (1963) was the first to formulate the facility layout problem as a linear integer-programming problem. He also proved that the above integer programming problem and the QAP are equivalent. Love and Wong (1976) proposed a simple integer programming formulations for the QAP in which the locations are given as points on a two-dimensional plane and transportation costs are proportional to weighted rectangular distances. In reality, all the decision variables of an integer programming problem need not be integers.
When some variables are restricted to take integer values, the optimization problem is called a 'mixed integer programming problem'. Kaufman and Broeckx (1978) developed a linear mixed integer program which has the smallest number of variables and constraints amongst all integer programming formulations of the QAP. Three major algorithms have been developed to solve integer programming problems as listed below.
1. Branch and Bound Algorithms 2. Cutting Plane algorithms 3. Dynamic Programming
2.2.1 Branch And Bound Algorithms
The Branch and Bound Algorithms proceed on the basis of stage by stage assignment of facilities to locations. At each stage backtracking occurs and certain assignments are excluded and the forward search process is resumed. This method can be considered a refined enumeration method in which most of the non promising integer points are discarded without testing them. The first two Branch and Bound Algorithms were developed by Gilmore and Lawler. The main difference between the independent work of Gilmore (1962) and Lawler (1963) was in computing the lower bounds. Both the algorithms implicitly evaluate all potential solutions. If no bounds are considered for pruning the decisions tree in the above two methods, then the procedure complete enumeration technique is used. Mans et al (1995) have described parallel depth first branch and bound algorithm for the QAP. Lee et al (2001) have made an approach to describe properties of loop network layout problem and dominance theorems. They adopted a heuristic approach and a branch and bound method. But both the problem size and the material flow density have been found to affect the solution quality and the computational efficiency.
2.2.2 Cutting Plane Algorithms
The cutting plane method begins with a user - defined search space. At every iteration, some part of that search space is cut (or eliminated) by constructing linear hyper planes from the most violated constraint at the current point. The objective function is minimized in the resulting search space and a new point is found. Depending on the obtained point, a certain portion of the search space is further eliminated. Bazaraa and Sherali (1980) have developed cutting plane algorithm based on Bender's partitioning scheme to solved FLP. The optimal cutting plane algorithms have a high time and storage complexity. For example, the largest facility layout problem solved optimally by a cutting plane algorithm is the layout problem with eight facilities. A common experience with the optimal algorithms is that the optimal solution is found early in the branching process but is not verified until a substantially high number of solutions have been enumerated.
2.2.3 Dynamic Programming
Dynamic programming is a mathematical technique which is useful in many types of decision problems. The problems in which the decisions are to be made at a number of stages, are called multi-stage decision problems. Dynamic programming is a mathematical technique well suited for the optimization of multi stage decision problems. This type of problem focus on the dynamic case or the problem of facility layout over time (multi-period). The problem is broken into sub-problems or stages. States are associated with each stage and states represent various possible conditions of the system. The lesser the number of states, the easier it is to solve the problem.
3. Heuristic Procedure
problem, the number of feasible layouts will grow exponentially, even for small increment in problem size. As a result, it is impossible to solve large size problem optimally. Hence, we should resort to heuristic approach to get near optimal solution. Using particle swarm optimization method multi objective facility layout problem is solved in this paper work.
4. Particle Swarm Optimization Program Structure:
The systematic procedure followed in PSO algorithm is given below. Step 1 Generate initial solution randomly for all particles.
Step 2 Assign pbest[i] = initial solution where e = 1,2, np (np: no of particles). Step 3 Find the best among all particles and assign this to the best.
Step 4 Generate initial velocities randomly for all particles.
Step 5 Add velocities to the corresponding particles i.e. present(t) = present[t] (old) + V[t]. Step 6 Update velocity as follows:
New Value = present value + Velocity
Velocity = CI * rand 1 * (pbest - present value) + C2 * rand 2 *(gbest-present value)
Step 7 If number of iterations < tmax (termination criteria) Go to step 5. Step 8 Choose the best among all the best.
Step 9 Write the solution. PROBLEM FORMULATION
The mathematical model which is based on two relationships, one for material handling cost and the other for closeness rating is adopted.
n-1 n
Min TFC = ∑ ∑ fij dij i=1 j=i+1 n-1 n
Min TFC = ∑ ∑ rij dij i=1 j=i+1
Combined Objective Function = (W1) TNR + (W2) TFC Where TFC= Total Flow Cost
TNR=Total Numerical Rating
dij -Minimum distance by which facilities 1 and j are to be separated or Distance matrix element fij -Number of trips to be made between facilities and j or Flow matrix element
rij- Closeness rating between departments i and j. xij -Position of facility 'i' at lh location.·· WI - Weightage given to qualitative data
W 2 - Weightage given to quantitative data
5. Case Study Problems
Table1. Flow matrix
Departments 1 2 3 4 5 6
1 0 2 4 1 3 5
2 2 0 2 3 1 3
3 4 2 0 5 3 1
4 1 3 5 0 2 4
5 3 1 3 2 0 2
6 5 3 1 4 2 0
Table2. Distance matrix
Departments 1 2 3 4 5 6
1 0 1 2 3 1 2
2 1 0 1 2 2 1
3 2 1 0 1 3 2
4 3 2 1 0 4 3
5 1 2 3 4 0 1
6 2 1 2 3 1 0
Table3. Closeness Rating matrix
Departments 1 2 3 4 5 6
1 0 4 -1 0 2 0
2 4 0 1 0 1 3
3 -1 1 0 2 0 4
4 0 0 2 0 0 1
5 2 1 0 0 0 3
6 0 3 4 1 3 0
Where
Absolutely necessary=4, Especially important =3, Important=2, Ordinary=1, Unimportant =0, Undesirable = -1 6. Results
Optimum layout is selected depending upon the combined objective function value. To get the best optimal solution the program should be run many times. Here it is run ten times and the best layout is shown for six departments arranged in single row and multi row equal area layout. The software developed can be used for a maximum of 12 departments.
6.1 Single Row Machine Layout
6 4 3 5 2 1
Fig4. The single row equal area machine layout
6.2 Multi Row Equal Area Machine Layout
6 4 3
1 2 5
7. Conclusion
Designing of facility layouts can be done either by taking qualitative or quantitative approaches into consideration. Most of the researchers have concentrated on single objective function models but in real life both quantitative and qualitative factors are to be considered. In this paper work, a multi objective criterion is taken into consideration while designing the layout. By assigning weights on trial basis to objectives ranging from 0-1, the facility layout score has been found as combined objective value.
Metaheuristics Technique Particle Swarm Optimization has been implemented for the problem and the optimal layouts are obtained. For generating the optimal layouts, software has been developed using C++. The alternative layouts are compared based on their combined objective function values .The lower the objective function better is the layout. The software frame work which has been developed can be used as a practical tool for other types of layout also.
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