A simulated annealing algorithm for solving the bi-objective facility layout problem

Introduction• The optimal design of physical departments is one of the most important

issues to be considered in the design of a manufacturing system.

• The material handling cost which comprises between 20% and 50% of the total operating expenses within manufacturing is the most significant measure for determining the efficiency of a layout.

Traditionally, there are two approaches for FLP.

• Quantitative approach aiming at minimizing the total material handling cost between the departments

• Qualitative approach aiming at maximizing closeness rating scores between departments based on a closeness function.

Disadvantages

•Qualitative approach is based on pre-assigned numerical values for different closeness ratings but does not consider the materials handling cost due to work flow between the departments.

• The quantitative approach does not consider the qualitative objectives. The layout designer may have to keep certain departments closer while other departments are to be kept further apart.

Solution Approach

•In order to do a more effective layout, both objectives should be considered can be done by adopting a bi-objective approach for facility layout problem.

•SA algorithm is presented minimizing the objective function that combine the total materials handling cost and the closeness rating score.

Literature reviewFacility layout problem is formulated as quadratic assignment problem (QAP). Consider the problem of allocating N facilities to N locations, with the cost being a function of the distance and flow between the facilities.

QAP is one of the hardest optimization problems and no exact algorithm can solve problems of size N > 30 in reasonable computational time.

For multi objective facility layout problem (MOFLP), the quadratic assignment formulation is shown in Eqs. (1)–(4) in Sha and Chen (2001)

Where,

Xij = 1 if facility i is assigned to location j;

• 0 otherwise.Aijkl the cost of locating facility i at location j and facility k at location l.

Eqn (1) It represents the combination of the total materials handling cost and the closeness rating score in MOFLP formulation.

Eqs. (2) and (3) ensure that each location is assigned only one facility and each facility is assigned to only one physical location, respectively.

Numerous methods have been suggested in the literature up to now in order to solve the MOFLP.

These studies can be assigned into four different groups according to their ways of combining the objectives;

Rosenblatt (1979)

defines the parameter Aijkl as Aijkl = w2Cijkl -w1Rijkl. Here, Cijkl representsthe total material handling cost, and Rijkl represents the total closeness rating score. w1 and w2 are the weight used for the objectives.

Fortenberry and Cox (1985)

sets Aijkl = fikdjlrik. It is named as multiplicity model. In this expression fik shows the handling of material between i and k facilities, rik shows the closeness rating score between i and k facilities, and djl shows the distancebetween locations j and l.

Urban (1987)

Aijkl = djl(fik + crik) is used where parameters have the same meaning as those used in Fortenberry and Cox (1985). The additional parameter c is a constant which states the importance of closeness rating score due to handling of material. C is taken as equal to the highest level of material handling between facilities.

Khare, Khare, and Neema (1988)

Defined Aijkl term as Aijkl = w1djlfik + w2rikdjl. The definition of the parameters are same as given before, and w1 and w2 are the weightsused to unify the two objectives, respectively.

Deficiencies

All factors may not be represented on the same scale: for example, values for work flow may range from zero to a tremendous amount, while closeness rating values may range from 1 to 4. As a result, the closeness ratings would be dominated by work flow and have little impact on the final layout.

•Measurement units used for the objectives may be incomparable:The total closeness rating score is only an ordinal value; on the other hand, the material flow handling is measured according to cost. Aggregation of these two values with different measurement units in an algebraic operation is unsuitable.

Harmonosky and Tothero (1992) suggests an approach that normalizes all objectives, before combining them.To normalize an objective, each relationship value is divided by the sum of all relationship values for that objective. Eq. (5) is used to do this

where Sikm represents the relationship value between i and k for objective m and Tikm represents the normalized relationship value between i and k for objective m.

Next, all values are multiplied by the weights representing the relative importance of each objective m (wm). Then, the total of all values for each pair of departments is calculated.

Where wm is the weight for objective m, T is the number of objectives, N is the number of departments, dij is the distance between locations j and l.

Xij = 1 if department i is assigned to location j or, 0 otherwise.

i and k are indices for the department numbers and j and l are indices for the location numbers.

Simulated annealing for the MOFLP

• SA is a stochastic search method, which imitates the physical annealing of solid, for finding solution to combinatorial optimization problems.

• SA has an advantage over the other meta-heuristic algorithms in terms of the ease of implementation and gives reasonably good solutions for many combinatorial problems.

step procedureStep 1 Read the input data (normalized flow matrix, size of the problem,

normalized relationship matrix, the weights for two objectives) and the parameters of simulated annealing (Tin = initial temperature, a = cooling rate, NIET = the number of trials to be performed with thesame temperature value, iteration number).

Step 2 Start temperature counter: el = 0

Step 3 Create a random initial solution (S0) and calculate the weighted cost of initial solution (E0) using the Eq. (6).Sbest = Sc = S0; Ebest = Ec = E0

Step 4 Make the iteration counter 0 at each temperature level: il = 0

Annealing schedule

• In general, SA algorithms start with a randomly generated initial solution or with a solution produced using a heuristic. In this work, we use a randomly generated initial solution.

• In the proposed SA, the search moves from the current solution to a neighbouring solution by swapping two departments. Two departments are selected randomly and these departments are swapped.

• The probability of acceptance is defined as the probability of accepting a non-improving solution. This is determined based on the following probability: P(E)= exp(-E/T);

where T is the temperature and DE represents the change in the cost of the neighbouring solution and the cost of the current solution.

• If x is a randomly generated number between 0 and 1, and x < P(E), then accept the non-improving neighbouring solution as the current solution. Otherwise, reject the non-improving neighbouring solution, and keep the current solution.

• The value of the temperature at the beginning of the schedule should be large enough so that most of the initial movements are accepted. However, as the temperature is reduced, the probability of accepting a non improving solution reduces.

• For a good solution the decrease in temperature must be as fast as possible to be computationally efficient, but slow enough to maintain equilibrium within the system. The most common cooling function uses a geometric decrement function (Tk = Tk-1). Here a is cooling rate and its value is between 0 and 1 but it is close to 1.

• To finalize the algorithm various methods are used. For example some stops the iteration s when the total iteration number reaches, a specified value while other stop when there is no accepted move in a given number of trials.

• In the concerned experiment the search is ended when we reach the specified maximum number of iterations (elmax).

ResultAlgorithm is tested on two sets of test problems.

• Set 1: contains two problems given by Harmonosky and Tot- hero (1992) which consist of 8 and 12 departments

• Set 2: contains four problems given by Chen and Sha (1999) which consist of 8, 12, 15, and 20 departments.

Two comparisons are made for the mentioned two sets.

1. The first data set results obtained by the proposed SA algorithm are compared with H & T (1992) and C & S (1999) results.

2. The second data set results are compared with C & S (1999) results.

SET 1

• For 8 weight combination: the proposed SA algorithm and Chen and Sha’s procedure produced same results while

• For 12 weight combinations: the proposed SA algorithm clearly outperformed the other two procedures.

SET 2

• For 8 weight combination: the proposed SA algorithm provide better results than C & S procedure for six weight combinations. Reduction in the cost are quiet significant for weights 0.5, 0.6, and 0.7.

• For 12 weight combinations: the proposed SA algorithm procedure produced same results for weights 0.0, 0.1, and 0.9 and for the rest, the proposed SA algorithm is slightly better than C & S procedure .

• For 15 weight combination: for the seven weight combination the proposed SA algorithm obtained slightly better results than C & S procedure and for the rest the results are same.

• For 18 weight combinations: the proposed SA algorithm obtained better results for all weight combinations

Conclusion• The proposed SA algorithm for BOFLP has obtained the best results for the

most weight combination for each problem.

• For the rest of the weight combination, the proposed SA algorithm has obtained the same results with Harmonosky and Tothero’s (1992) results and Chen and Sha’s (1999) results.

• Above results proved that the proposed SA algorithm performed better than two other heuristics for all test problems according to solution quality.

Future Work• For future studies, using other meta-heuristic methods (genetic algorithm,

tabu search, ant colony algorithm, etc.) or employing hybrid methods can be thought.

• Also, it will be suitable to search solutions to problems where the data is dynamic, fuzzy or stochastic

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