An efficient tabu algorithm for the single row facility layout problem

An efficient tabu algorithm for the single row facility layout problemByRajeev. AIntroductionGeneral goal of the facility layout problem is to arrange a given number of facilities to minimize the total cost associated with the known or projected interactions between them (service centre layout, hospital layout etc)Single Row Facility Layout Problem (SRFLP) consists of finding an optimal linear placement of rectangular facilities with varying dimensions on a straight lineSRFLP is proven to be a NP-complete problem (Suresh and Sahu, 1993) defined as follows: n rectangular departments are to be arranged on a straight line in a given direction. The length li of each department i, and a n * n matrix C = [cij] where cij is the flow between departments i and j, are also givenThe distance between two departments is calculated as the distance between their centroidsThe objective of the problem is to arrange the departments so as to minimize the weighted sum of the distances between all department pairsreviewformer attempts drawbacks:(a) The dimensions of the machines either are not considered or are assumed to be equal (Braglia, 1996)(b) The locations of facilities are predetermined (Kumar et al., 1995; Braglia, 1996)(c) The size of the machines is only considered in the physical layout of the machines rather than in the computation process(d) The method requires too much time to construct a layout, especially when applied to large instances of the SRFLP (Anjos et al., 2005)special case of the SRFLP makes it possible for the proposed algorithm to start its initial search from high-quality solutions in feasible spaceSo the proposed algorithm takes less time to reach better solutions in the feasible space compared to all other algorithmsABSMODEL

special caseTheorem 1. Suppose that for all i&j : cij = c; Sij = 0. Now, sort the facilities in non-descending order such that the shortest facility is denoted by 1 and the longest facility by n. Then, Fig. 2 shows the optimum solution when n is an odd number, and Fig. 3 shows the optimum solution when n is an even number

Tabu searchInitial solutionsthe number of required initial solutions is a parameter defined by the user in the proposed algorithm. layouts of Figs. 2 and 3 are still optimum by exchanging the locations of facilities I and j, provided that the same number of facilities are at the left and right sides of i and j, respectivelyIf this parameter is set to a number greater than 2, it is still possible to use randomly generated solutions to initiate the algorithmNeighbourhood structurealgorithm is based on binary movements. algorithm randomly chooses two facilities and exchanges their locationsIf new solution improves objective function then the algorithm inserts the solution into the AM and adds the indices of the exchanged facilities to the tabu list for the next h iterations unless either the exchange is the only one that improves the objective function of ABSMODEL, or if the new solution generated by the exchange has the best objective function obtained so farCont..After a futile efforts, the algorithm gives up probing in the neighbourhood of the current solution and selects a new solution from AM to advance. a is a parameter defined by the userIntensification, long-term memory, and termination criterionAdaptive Memory (AM) is a list capable of storing best L layouts of the search procedure with their objective functions as labelsfollowing iterations, algorithm constructs new layouts, stores them into the AM, and triggers the sorting process againSelecting a layout from the AMselection process is done through a probabilistic procedure, tending to select better solutions with more frequencyprobability function assigns selection probability to the ith worst solution stored in the AM as

Cont..Final intensificationWhen the TS stops, the algorithm will apply the final intensification procedure to the best solution stored in the AM by applying the binary exchange to adjacent facilitiesStepwise procedure1: Set the values of the control parameters: L (length of the AM list), a (number of attempts to find a better solution in the neighbourhood of the current solution), h (tabu list length) and k (number of iterations of the algorithm).2: Find L initial solutions based on the procedure described, store and sort them in the AM according to their objective functions in non-descending order3: Use a probability function to choose one of the solutions stored in the AM for further search in its neighbourhood.4: Randomly choose two facilities such as i and j from the solution selected in Step 3 and exchange their locations in the layout. If this exchange reduces the objective function, accept it and add (i, j) to the tabu list. Exchanging I and j will be prevented in the next h iterations unless it constructs a solution better than all the current solutions stored in AM, or if this exchange is the only exchange that improves the objective function. Then proceed to Step 5.If the exchange does not improve the current solution, repeat this step for a times. Go to Step 3 if no successful exchange is found.5: Store the improved layout into the AM, and sort the AM again. Remove the worst layout stored in the AM to limit its length to L.6: Select the best solution stored in the AM and proceed to Step 7 provided that the above procedure has been repeated for k times. Otherwise, go to Step 37: Apply the final intensification procedure to the solution generated by Step 6.experimentally derived values are proposed to be set as the parameters

Computational resultsThe set of problems with an optimal solution includes 30 problems and are shown in Table 1.The reference where the problem has been introduced for the first time is in the second column. The third column specifies the number of facilities of the corresponding problem, and the next column has information about the best non-exact solution found so far. The fifth column shows the results obtained by the proposed algorithm. As mentioned before, the proposed algorithm follows a probabilistic approach. Thus, we have run it five times for each problem. The information shown in the fifth column reflects the best objective function value found by the five independent runs and its relative CPU time

Result

Iterated tabu search for the circular open dimension problemIntroductionThis paper investigates the circular open dimension problem (CODP), which consists of packing a set of circles of known radii into a strip of fixed width and unlimited length without overlapping. The objective is to minimize the length of the strip. ITS starts from a randomly generated solution and attempts to gain improvements by a tabu search procedure. After that, if the obtained solution is not feasible, a perturbation operator is subsequently employed to reconstruct the incumbent solution and an acceptance criterion is implemented to determine whether or not accept the perturbed solution.The search terminates and returns the best found solution after the allowed computation time has been elapsed. Computational experiments based on numerous well-known benchmark instances show that ITS produces quite competitive results, with respect to the best known results, while the computational time remains reasonable for each instance.According to the size of the circles to pack, CPP can be classified into two categories (Castillo et al., 2008): the arbitrary sized circle packing problem (denoted by ACP) and the uniform sized circle packing problem (denoted by UCP)Given a strip of fixed width W and unlimited length L, as well as N arbitrary sized circles Ci of known radii ri(i = 1, 2, . . . , N). Take the origin of two-dimensional Cartesian coordinate system at the midpoint of the container, and denote the coordinates of the midpoint of Ci by (xi,yi). The objective of CODP is to obtain a solution (X,L), where X is a configuration denoted by (x1, y1, . . . , xi, yi, . . . , xN, yN), such that.Minimize L, subject to:

first two constraints state that each circle should not extend outside the container. The third constraint requires that any pair wise circles cannot overlap with each otherThere may exist two kinds of overlaps: overlaps between two circles and overlaps between a circle and a border of the stripin order to measure the feasibility of a given solution (X,L),

Constructive approaches attempt to pack the circles one by one in sequence into the container according to some constructive rules, until all the circles have been packed feasibly. In contrast, perturbation-based approaches start from one or several initial configuration(s) which contain(s) all the circles (generally with overlapping) and then attempt to gain further improvements by continuous optimization and solution perturbations, until a feasible solution has been obtainedprocessEstimate an upper bound Lub of the strip length L, which should be large enough for ITS to easily obtain a feasible solution. In this paper, Lub is uniformly set to equalPre-set L to Lub, and then independently launch ITS once (as detailed in Algorithm 2) to obtain a solution (X,L) with as ow penalty as possible. Note that L keeps unchanged while searching by ITSIf the obtained solution (X,L) is feasible, i.e., E(X,L) = 0, then employ some post-processing techniques to further improve (X,L) to solution (Xbest, Lbest), without destroying the feasibility (it means that E(Xbest, Lbest) = 0 and Lbest