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Classification Of Center Layout Problems

The reason for this books review is to explore the overall facility layout problem, the active facility design problem, the models which may have been used to signify the facility design problem and the algorithms that solve the models, .

Classification of Center Layout Problems

"Determining the most effective agreement of physical departments within the facility is defined as a facility layout problem (FLP)" (SMTF Ghomi et al, 2011). Over the time of several couple of generations, FLP have been analyzed by several experts to a significant extent for creating optimum and common solution to solve the condition and a large variety of solution types of procedures predicated on algorithms have been proposed.

Facility layout problems are categorized into two categories, static facility layout problem (SFLP) and powerful facility structure problem (DFLP).

Static facility structure problem (SFLP)

The static center structure problem (SFLP) is the perseverance of the very most efficient set up of departments inside a facility with opportunity of improvement only within the layout boundary. The service can be developing plant life, administrative office buildings, or service facilities, ( Alan R, jin et al 2005). The static center structure problem (SFLP) procedure generally assumes that move between machines, product demand, and levels of product combine are constant through the planning horizon.

Dynamic facility design problem (DFLP)

When material movement assumes varied avenue between departments through the planning horizon, the condition becomes the strong facility design problem (DFLP). Under a volatile environment, demand is not steady. It changes from one production period to some other. To operate proficiently under such surroundings, the facilities must be adaptive to changes of development requirements. From a design viewpoint, this example requires the perfect solution is of the dynamic design problem (DLP). (Adil, Turkay et al 2005)

Tree representation of the design problems

Essential feature of design problems are characterized in tree representation diagram as shown in fig.

Tree representation of the design problems (Amine, henri et al 2007)

Algorithms for Resolving facility design problems

There are two types of algorithms for fixing facility design problems. An example may be heuristic algorithm and another is maximum algorithm.

Heuristic algorithms

These algorithms provide a solution which possibly might not only be the best fit for the situation. An excellent heuristic procedure usually produces the best answer for almost all of the tiny problems. A heuristic algorithm works towards an optimal solution but ends its search when it discovers a 'good enough' solution. As computation boosts, these algorithms will approach the optimal solution. The purpose of the heuristic algorithm is not to find a very good or best solution but to find an acceptable solution within an acceptable amount of time using an acceptable amount of computer storage. Heuristic algorithms can be classified as structure algorithms and improvement algorithms.

Construction algorithms

In Development algorithms structure is constructed from the start and facilities are designated to a niche site, individually, until the complete layout is obtained. (andrew, et al 1987). The herb layout software utilizing a engineering type algorithm will first build a solution within an open floor area from fresh data. The algorithm essentially takes relationships between activity areas into consideration and generates a stop design. Their basic approach is to find a starting place or primary activity positioning and then add the rest of the activity areas matching to certain guidelines. In a few algorithms the rules act like Muther's vowel letter sequencing (A-E-I-O-U-X) for closeness interactions. Three well known examples of building algorithms are CORELAP, PLANET, and ALDEP.

CORELAP

Computerized Relationship Structure Planning (CORELAP) is a structure algorithm and was developed by Robert C. Lee. It's the oldest construction algorithm predicated on Richard Muther's manual process of converting the Relationship Chart into a design. The essential inputs required by CORELAP are the relationship graph and the area requirements of each department. CORELAP commences by calculating the "total closeness ranking" (TCR) for each and every section where TCR is the sum of the numerical beliefs allocated to the closeness romantic relationships (A=6, E=5, 1=4, etc. ).

A downside of CORELAP is that it includes problems when an attempt is made to fix departments in a certain location. CORELAP does not take into account the building and would depend on the structure arrangement. It really is useful for new plants where in fact the objective is to determine new building design rather than for buildings that already are around. ( Altaf et al 1995)

ALDEP

Automated Design Design Program (ALDEP) originated within IBM and was provided by Jerrold Seehof and Wayne Evans. It had been first published in 1967. ALDEP gets the same basic data input requirements as CORELAP. It varies from CORELAP in using the Total Closeness Score for placement of departments; ALDEP selects and places departments arbitrarily. CORELAP attempts to create the one best structure while ALDEP constructs many layouts and rates each design and therefore leaves the final decision of selecting the appropriate design to the facility designer. Benefits of using ALDEP include rectangular or rectangular layouts. It is also able to handle facilities with up to three flooring surfaces and provides the capability to fix departments in a certain location and also to include docks, elevators and stairwells. The disadvantage of ALDEP is that it arbitrarily picks departments for concern in the structure process. Hence, ALDEP should be carried out several times to make sure that the layouts generated will be the "best" designs. The "best" layout will eventually made will be offered to the center custom for selecting the most appropriate and feasible design.

PLANET

Plant Layout Analysis and Evaluation Technique (Globe) is another construction type algorithm. It uses the same type requirements as Art. PLANET is adaptable in that it'll accept material circulation data in three formats and that there are three different design construction stages available. The three phases that exist to generate a structure are as follows: The first stage includes the translation of the input data so that it pays to to the algorithm in PLANET. The second phase involves selecting the order where the departments should be considered in the design. The third phase involves the perseverance of the placements of the departments when they are believed for the design (placement main concern from the highest to the lowest is 1 to 9).

PLANET turns the materials circulation information from the from-to cost chart, a from-to chart or a penalty graph to a flow-between cost graph. That is done with the addition of the prices in both guidelines between departments and then entering the amount for the circulation in each course. The foundation for the PLANET selection algorithms will be the flow-between cost graph and placement priorities

The benefits of using Entire world are that it's very flexible in allowing inputs such as materials flow data to be came into in three formats and having three methods in making a layout. The down sides with Globe are that in its change of inputs to a flow-between cost chart, it considers the closeness romantic relationships between departments but conceals the direction of stream among departments. This might result in designs that have a considerable amount of backtracking on the list of departments.

Improvement algorithms

An improvement algorithm always commences with a short layout. The algorithm exchanges department locations until a structure is available that cannot be improved. The quality of the layout produced depends upon the initial layout and the ability of the algorithm to exchange multiple departments at the same time. The basic strategy of improvement algorithms is to reduce transportation cost or motion cost by lowering the distance on the most traveled routes. Popular examples of improvement type computer regimens are COFAD, Build and BLOCKPLAN.

CRAFT

Computerized Relative Allocation of Facilities Technique (Build) was the first improvement type algorithm used in computerized facilities design. CRAFT originated in 1964 by Armour and Buffa. Build begins with a short structure that is came into by the analyst. The layout is examined, and pair sensible exchanges of departments are made to try to improve the layout. Designs are assessed on the minimization of materials stream cost between departments. Set wise exchanges are only made between departments that are of equal size or have common boundaries. CRAFT are designed for up to 40 departments and is recommended by many over CORELAP and ALDEP because of its evaluation of layouts. CORELAP and ALDEP lessen the amount of stream between departments and maximize closeness scores, while CRAFT reduces the price tag on move between departments. The initial layout utilized by CRAFT restricts the restrictions of all designs produced from it. CRAFT can not work well with departments having unequal areas because it struggles to shift the structure to permit nonadjacent departments of unequal areas to be exchanged. (jin et al 1996) used CRAFT to resolve the failure-to-fit problem by changing the scale and/or form of the departments in a organized manner minus the help of humans

COFAD

Computerized Facilities Design (COFAD) is an adjustment of Build. COFAD's algorithm first attempts to enhance the initially inputted layout by an operation that Is comparable to CRAFT except that COFAD is with the capacity of considering straight range as well as rectilin2ar distances between departments being considered for interchange. This is ideal for materials controlling systems that use conveyors that do not have to follow aisles in a rectilinear fashion. COFAD then can determine the expense of executing each move using the feasible materials managing system alternatives available. This is dependent on the sort of material handling system chosen (ie. fixed way equipment such as conveyors or mobile equipment such as tote carts). COFAD's next function is to use the above move costs to find out a minor cost of materials managing system. The negatives of using COFAD are that the sensitivity evaluation within COFAD only considers variations in the full total flow volume level for a predefined product mix and does not assess changes in product mix. (Vic Kichodhan et al 1990)

BLOCKPLAN

BLOPLAN means Block Layout Analysis with Computerized Planning. Some type of computer routine which allows the use of random, development, and improvement type algorithms is BLOCPLAN. It was developed by Dr. Charles E. Donaghey, Chairman of the Industrial Engineering Team at the University of Houston. BLOCPLAN is an interactive program used to develop and improve both solitary and multi storey layout BLOCPLAN is a departmental location system that includes random, development and improvement type algorithms for growing layouts It is a straightforward program which generates good initial layouts because of its flexibility based on several imbedded options. It uses both quantitative and qualitative data to generate several block designs and their measure of fitness. ( Pinto, et al 2007). BLOCPLAN can screen a design graphically on the display.

The inputs that are essential are: the no of division (maximum 18) The Titles of the departments, their matching areas, and a romance chart. The graph relationship format is equivalent to recommended by Mather in his Organized Layout Planning techniques. Once the relationship chart has been joined, BLOCKPLAN then shows a relationship vector of "Code similar Score". The purpose of this is to allow the facility developer to indicate the value attached to the rating of the relationship chart, BLOCPLAN needs to use some quantifiable factor to rake decisions when it generates and scores designs. It uses the CES vector to assign a numeric value romance graph. The default CES vector prices are 10, 5, 3, 2, 1, 0, and -10. Which means that has "A" rating is worth 10, an "E" score will probably be worth 5 and so on. . An "X" score is worth -10. The center designer can also establish his/her own beliefs if desired. (Vic Kichodhan et al 1990)

The technique that BLOCPLAN uses to create layouts is that it first determines an Importance Ranking (IR) for each and every department in the layout. The score is the total of all relationship scores for every single team, using the CES vector values. Second, a menu for the center designer is viewed. The options are:

Random Structure.

Layout Algorithm.

Improvement Algorithm.

Adjust Romance Information.

Manually Place Departments.

Review Saved Layouts.

Stop.

Save Problem Data

Selecting option one, Random Layout, will cause structure to be developed without respect to the partnership chart. The Departments will be located arbitrarily in another of the eighteen zones that the program has generated. BLOCPLAM divides the building structure directly into three tiers, with three areas per tier. Each zone can be further split into its still left and right part supplying the possible eighteen zones.

BLOCPLAN randomly selects one of the eighteen locations for each office and assigns it to a particular location.

After all the departments have been given a location, the program proceeds to bring the structure. It talks about the departments that are found in Tier 1 up to six departments can be found in Tier 1. The full total required region of a tier is the amount of all areas for the departments positioned in that particular tier. Each team is drawn in percentage to its area and the departments are rectangular in form. If a division with a little area is the only person located in a tier, it'll be drawn as a long narrow department stretches across the complete layout. BLOCPLAN continues with this process for all the tiers.

The layout produced is obtained by the rating algorithm predicated on an adjacency criterion. The CES ratings for departments that talk about a standard boundary in the structure are summed and then divided by the amount of all the positive CES results from the relationship chart. A rating of just one 1. 0 indicates that all "good" connections in the relationship chart have been satisfied in the layout

Selecting option two, Layout Algorithm, may cause the software to make available to the service designer a layout algorithm. The algorithm places departments that contain high IR results in the heart of the layout and then surrounds them with departments with high romantic relationships. Departments with an "X" relationship are separated whenever you can. This method of seeking the departments produces layouts that are "better" than the arbitrary process.

Selecting option three, Improvement Algorithm, will cause the software to try to improve on a structure that has been saved in storage area. The improvement algorithm interchanges each pair of departments in the layout and then displays its score before moving to the next interchange when the facility designer hits the "Return Key". The amount of interchanges is the mixture of the amount of departments used two at a time. For example, for ten departments there will be forty five interchanges. The perfect layout can be obtained by using option two, Layout Algorithm, and then using this program, Improvement Algorithm, to improve on the previous saved structure.

Selecting option four, Adjust Marriage Info, allows the relationship information to be altered. The facility creator can change the relationship information and the CES ratings that were actually entered. This allows the effects of changes in the partnership graph to be evaluated

Selecting option five, Manually Insert Departments, allows the manual placement of departments in the layout. Each department can be personally placed in the required tier and zone. This is the same as fixing a office in a layout

The benefits of BLOCPLAN are that it's a useful tool to center designers for the reason that designs can be produced or evaluated, the consequences of changing the ideals in a romance chart can be analyzed, and it only requires a microcomputer instead of a mainframe to use. Although the processing time varies with the amount of departments that have to be located, the limitation of BLOCPLAN being able to only cope with eighteen departments limits the processing time for you to an acceptable amount. The drawbacks of BLOCPLAN are:

BLOCPLAN can only handle layouts with eighteen departments or less.

BLOCPLAN can only store twenty layouts in recollection.

All the layouts are viewed on the display screen in a rectangular drawing that has a horizontal amount of 6. 75 ins and a vertical height of 4. 75 inches whatever the number of departments in the layout or their placement in the layout.

Simulated Annealing Algorithms

Simulated Annealing (SA) is a way based on Monte Carlo simulation, which solves difficult combinatorial optimization problems. The name originates from the analogy to the patterns of physical systems by melting a compound and cutting down its temperature slowly but surely until it extends to freezing point (physical annealing). Simulated annealing was initially used for search engine optimization by Kirkpatrick et al. (1983). Within the numerical optimization platform, SA is an operation that has the capability to re-locate of locations near local minima. SA is dependant on random assessments of the target function, so that transitions out of an area least are possible. It generally does not promise, of course, to find the global minimum, but if the function has many good near-optimal solutions, it will find one (George D. et al 2002)

Simulated annealing was also found in General Facility Design Problems (GFLP) considering facilities areas, figures and orientations or in Machine Design problems (MLP) considering machine's pick-up and drop-off tips (Leonardo Chwif et al 1998).

SA was also used for powerful facility design problems for resolving the problems for arranging and rearranging (whenever there are changes between your moves of materials between departments) making facilities in a way that the sum of the materials handling and rearrangement costs is reduced (Alan R et al 2006).

Wang et al (2001) developed a model to resolve the facility structure problem in cellular manufacturing system. Within the model, they assumed that the demand rate varies over the product life cycle. The target function was to reduce the total material managing cost and solve both inter and intra cell service layout problems together.

Simulated annealing heuristic for the DFLP with budget constraint, and show the effectiveness of this heuristic on a couple of numerical experiments (Ramazan et al. , 2010).

Artificial Neural Networks

Neural networks are a powerful method of optimization which depends on producing systems that exhibits self corporation and adaptation in a similar, though basic, manner to the way in which natural systems work. A kind of artificial neural network model has been applied for computation to solve a wide variety of discrete combinatorial search engine optimization problems. A neural expert system is an interactive classification system with justification ability. This system begins with the data representatives from a couple of training instances, learns through staff, and then develops the capability to appropriately classify new cases based on discovered knowledge. This classification ability makes the proposed neural expert system create a conceptual engineering layout by means of the learned symbolic knowledge resonant to the type layout requirements.

ANN can be a system comprising N - N neurons predicated on an man-made two-dimensional maximum neural network for an N-facility design problem. ANN algorithm has given advanced solutions for many benchmark problems over the best existing algorithms (Kazuhiro Tsuchiya et al 1996).

The annealed neural network combines characteristics of the simulated annealing algorithm and the neural network for quick convergence of the neural network, while conserving the answer quality afforded by simulated annealing (Yeh, 2006). This also have found implementation in resolving the facility structure problem

Genetic Algorithms

GAs came to the fore in the 1960s, through the work of Holland for fixing many industrial and service sector issues that demonstrated extremely difficult to resolve with the available methods known in those days. The main contribution of GAs is handling search engine optimization and search problems by providing a solution which is not the perfect one but which is nevertheless a good approximation to the perfect one. Due to the enormous upsurge in the capacity of computer technology, applying GAs, lately has become increasingly more well-known, since the challenge of the price tag on using computer facilities which might have arisen, is in reality only a one (A. Gomez et al 2003).

With cyber technology getting impetus software predicated on GA have been developed for problem fixing. An improved cross genetic algorithm (IHGA) was developed to use a powerful local improvement process as well as an efficient restart device that is dependant on so-called 'switch mutations' and put on the well-known combinatorial marketing problem and quadratic project problem (QAP) (Alfonsas Misevicius et al 2004).

Extensive computational tests for solving quadratic task problems using various variants of a cross types hereditary algorithm were carried out (Zvi Drezner et al 2008). Simple tabu and modified sturdy tabu as improvement algorithms in a hybrid genetic algorithm are superior than other tabu queries (concentric tabu, engagement ring moves, all goes, sturdy tabu) (Jasmit singh kochher et al 1997) outline a GA structured algorithm for solving the sole floor facility layout problems for similar and unequal size team.

(Ming-Jaan Wang et al 2005) is give attention to the unequal areas team facilities layout problem, and implements examination of variance (ANOVA) of information to determine the best site size of layout by hereditary algorithm.

The dynamic seed design problem (DPLP deals with the look of multi-period structure plans Although an ideal solution method predicated on dynamic programming can be obtained, it isn't sensible for large DPLPs and heuristics predicated on genetic algorithms can solve large DPLPs. (Jaydeep Balakrishnan et al 2003) prolong and improve the use of hereditary algorithms by creating a hybrid genetic algorithm and a computational study is carried out to compare the suggested algorithm with the existing genetic algorithms and a recently available simulated annealing algorithm.

An important strategy in facility layout problems that may be used to gauge current and emerging fads in new design aims and methodologies that address combinatorial optimization aspects and presents a state-of-the-art review of the use of the Hereditary Algorithm (GA)(Kundu A et al 2010)

NP-hard issue of arranging a quantity of facilities on a line with bare minimum cost, known as the solo row facility layout problem (SRFLP) also to solve this kind of problems permutation-based hereditary algorithm (GA) is used. (Dilip Datta et al 2011)

Tabu Search Algorithm

TS approach is a meta-heuristic search that is used to solve the combinatorial marketing problems TS, is usually dominated by area solutions in looking for an optimum solution. Unlike the GA, it is highly dependent on the principles of the algorithm's control variables. TS is dependant on flexible memory buildings regarding the strategic constraints and aspiration levels as an approach for exploiting alternatives.

The search begins when the parameters are chosen and a possible solution to the problem is generated. The main parameters of TS technique are the area size, the size of tabu list, the aspiration criteria and stopping requirements. The operator that can be altered in order to generate area solutions is 'move'. This operator can place each element to move from its location to any other location in the answer. From 'move', a set of neighboring solutions is generated by way of a pre- identified change to the present solution. Then your best solution is chosen from the existing group of neighboring solutions which becomes the new current solution. Again, a fresh set of neighboring alternatives is made from the new current solution and the procedure repeats itself before stopping requirements are met. (Lou Y. Liang et al 2008).

There are two new effect approaches for the tabu search algorithm. The first strategy treats the tabu search algorithm as a goal system to be handled and runs on the control-theoretic approach to adjust the algorithm variables that affect search intensification. The second strategy is a versatile diversification strategy which can change the algorithm's guidelines based on the search history. These two strategies, combined with tabu search, form the Personal Controlling Tabu Search (SC-Tabu) algorithm. The algorithm is integrated and examined on the Quadratic Task Problem (QAP). The results show that the self-controlling top features of the algorithm make it possible to attain good performance on different kinds of QAP circumstances. (Nilgun Fescioglu-Unver et al 2011)

Two extensions were recommended and tested for concentric tabu seek out the quadratic assignment problem to add more permissible moves (Zvi Drezner et al 2005).

The ideal solution for special circumstance of Single Row Facility Structure Problem (SRFLP) was suggested by having a theorem by Hamed Samarghandi et al this year 2010. He suggested a new algorithm predicated on tabu search for the SRFLP and suggest computational results of the suggested algorithm on benchmark problems show the greater efficiency of the algorithm set alongside the other heuristics for solving the SRFLP.

Slicing tree structured tabu search heuristic for the rectangular, continual airplane facility structure problem (FLP) had been designed with treatment to assess the design corresponding to confirmed slicing tree on the basis of bounding curves (Daniel Scholz et al 2009). These designs are slicing structures which have the ability to contain empty spots to guarantee that stringent shape constraints of facilities are maintained. Due to these features this process is better suited for sensible use than up to now existing ones.

Graph Theory

Graph theory (Seppanen and Moore, 1970) can be utilized as a way to produce good layouts predicated on the move matrix. A relationship diagram can be attracted as a weighted graph with the nodes signifying the departments and the ends representing the circulation between the office pairs. The dual of this graph is a block diagram structure.

Graph theory strategy, relationships (or flows) among facilities can be symbolized by way of a (romantic relationship) graph in which vertices denote facilities and corners denote presence of flows or relationships between facilities. A requirement of existence of any block layout satisfying the relationships represented by a graph is that the graph be planar. A graph is planar if it could be used the planes and each border intersects no other corners and goes by through no other vertices. The relationship graph may well not be planar. A planar sub graph of any relationship graph is named a maximal planar graph (MPG) if no ends can be added without making the graph no planar. The dual of any (primal) planar graph can be created by placing a dual node in each face of the primal planar graph and by signing up for vertices corresponding to two faces (in the primal graph) that promote an advantage in their common boundary. (Here, faces are regions identified with a planar graph. ) The dual of any planar graph is planar as well. (J-Y KIM et al 1995)

Russell D. Meller et al 1996 says about creating a design in the graph-theoretic way requiring the following three steps:

(1) Expanding an adjacency graph from section human relationships (which departments are adjacent),

(2) Building the dual graph of the adjacency graph (represent departments as adjacent areas having specific boundaries),

(3) Changing the dual graph into a block layout (specifying departments with regular forms and specific areas)

Graph theoretic techniques were also used to take care of the unequal area stop plan. In these strategies a stop plan is constructed as the dual of an planar graph where nodes symbolize areas and links signify required adjacencies. While it is actually possible to construct a block plan from a planar graph which fulfills the given adjacency requirements between areas and between spaces and the outside area, the ensuing plan might not meet size and form requirements imposed on each space. Creating a stop plan that meets size and shape requirements is a nontrivial problem. (Robin S. Liggett et al 2000). Other commercial problems like furniture creation line developing were also solved using graph (Wilsten and Shayan 2007).

The main problem concerned with applying graph theory to facilities design is the conversion of the dual graph to a stop structure (S. A. IRVINE et al 2010) gives a new approach to creating a planar orthogonal layout or floor plan of a couple of facilities subject to adjacency and area constraints. It improves upon previous strategies by taking any maximal planar graph representing the adjacencies as suggestions. Simple selection conditions for choosing the next service to be inserted into the floor plan are used. Further, any smart orthogonal condition for the facilities in the causing floor plan can be made.

Optimal algorithm

During the 1960's extensive research was done in producing optimum algorithms. Optimal algorithms find a very good solution. However they are not practical due to restrictions on computer time and space. Some optimum algorithms are categorized as listed below.

Quadratic Assignment Model

The quadratic project model (Koopmans and Beckman 1957) presents the situation of locating numerous facilities that required materials move between them. The name QAP was presented with because the objective function is another level function of the factors and the constraints are linear functions of the parameters. The objective function maximizes the income gained by assigning the departments to a location, less the price tag on the material move between your departments. The numerical model of the quadratic task problem (QAP) is:

The integer varying, Xij is equal to 1 if division i is designated to location j, often the variable is add up to 0. The frequent aij is the region required for department i to location j and fik is the materials stream between departments i and k, and Cjl is the price of material movement between location j and l. The first constraint means that each location will be designated exactly one division and second constraint means that each division will be assigned to exactly one location.

Layouts produced using the quadratic project models are often used as an instrument in formulating your final design. The QAP will take into consideration the material circulation between departments, however, the model functions under the assumption that department areas are similar which in many cases is impractical to presume. Because of this, the layout generated by the quadratic project problem often acts as a starting place for developing a final layout. (Ekrem Duman et al 2007) used the quadratic assignment problem in the context of the printed out circuit board assemblage process. (A. S. Ramkumar et al 2008) specializes in multi-row machine design problems that can be accurately developed as quadratic task problems (QAPs). A hereditary algorithm-based local search methodology is proposed for resolving QAPs.

2. 4. 2. 2 Quadratic collection covering problem

The quadratic set covering problem needs the quadratic assignment problem a step further by including the area of every office in the model (Bazaraa 1975). A branch and certain approach is employed to optimize the issue. The quadratic collection covering problem models solitary or multi-story properties, and departments with abnormal shapes. It is able to generate initial layouts and add departments to the already existing structure.

The quadratic set covering problem has the advantage of varying areas, but it comes short of being sensible. The model assumes that the form of the departments is known and set. This restricts the problem's capability to find an improved solution by varying the figures of the departments to reduce the length between centroids.

2. 4. 2. 3 Linear Integer Coding Problem

The linear integer programming problem is a reformulation of the quadratic assignment problem. Lawler (1963) was the first ever to formulate the service structure problem as a linear integer development problem by determining Yijkl = Xij Xkl which reformulates the quadratic assignment problem into a linear integer encoding problem. The number of integer parameters required are n2+n4 the number of constraints required are nd+2n+l (n = quantity of departments). The amount of integer parameters and constraints required is much more than required for the quadratic task problem. These much larger problems are more challenging to solve optimally due to increased amount of computation and storage area required.

The facility layout problem (FLP) with unequal departmental areas which is came across in many processing, service facilities (Abdullah Konak et al 2006) and the challenge of getting a range of departments on the range (Andre R. S. 2006) were solved by the mixed-integer encoding formulation.

2. 4. 2. 4 Branch and Bound Algorithms

The first two branch and bound algorithms were produced by Gilmore (1962) and Lawler (1963) separately for handling the quadratic task problem. These algorithms implicitly examine all possible solutions to a problem, leading to an optimum solution. These algorithms require a sizable amount of recollection and computational time governed by the number of integer parameters required. In this particular algorithm, each facility is allocated to each location level by stage. The partial design is evaluated and in comparison to less bound. If the price of partial structure is less than the low bound, the partial layout is kept and used as a lesser bound in following iterations, usually it is discarded and the branch is fathomed.

The branch and bound algorithm are also used to solve the two-dimensional layout problem (Solimanpur and Jafari, 2008). A two-dimensional design is a organized arrangement, in which the developing facilities are laid in a planar area. A mathematical model was proposed for determining the optimum layout of machines in a two-dimensional area. The variables considered by the proposed model were (a) production capacity of machines, (b) multiple machines of each type (machine redundancy), (c) handling route of parts, (d) dimensions of machines. A new parallel Branch and Bound algorithm for the Quadratic Task Problem, which really is a Combinatorial Optimization problem known to be very hard to resolve exactly also are present in books (Bernard Mans et al 1995). The branch and bound algorithm are also used for continuous facility design problem (Xie and Sahinidis, 2008).

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