2 The Facility Layout Problem Given the activity relationship as well as the space of the department, how to construct plan the layout of the facilityThe basis of the layout planning is the closeness ratings or material flow intensitiesminimize the flow times distanceMaximize the closeness (adjacency)For most practical real world instances, the computational complexity has results in various heuristicsWhat is heuristic?Construction HeuristicImprovement Heuristic
3 Heuristic and Optimality Consider the knapsack problemZ = max 5 x1 + 7 x x x x5Subject to 2 x1 + 3 x x x x5 <= 10Heuristic: An intuitive problem solving method/procedureConstructiveHeuristic 1, Pick sequentially the ones with the best benefitHeuristic 2, Pick sequentially the ones with the best benefit per unitGreedy Improvement:Exchange two items in a solutionMeta-Heuristic: Simulated Annealing, Genetic AlgorithmOptimal SolutionMathematical Programming and OptimizationLinear Programming, Integer Programming, Nonlinear Programming
4 A Simple Facility Layout Problem Suppose we have 10 identical sized departments and the flow intensity between these 10 department is fijFind the best arrangement of the 10 department along an aisle so that the total travel (flow intensity times distance) is minimizedA Quadratic Assignment Model is necessary to Optimally solve the problem12345678910PositionDepartment
5 A Quadratic Assignment Model Decision VariablesX(i,j) : Department I will be located at position j0: OtherwiseConstraintsEach one position can hold exactly one departmentSUM( i in 1…10) x(i,j) = 1Each department has to be assigned exactly one positionSUM( j in 1…10) x(i,j) = 1ObjectiveSUM(I, j, m, n, all in ) x(i,j)* x(m,n)*f(i,m)*d(j,n)This is an integer quadratic assignment problem.
6 Pair-Wise Exchange Heuristic From\To1234--1015205Phase I: Construct Phase Initial Solution (1,2,3,4)Phase II: Improvement – Pair Wise Exchangea) Exchange two departmentsb) If results in better solution, accept; go to a)otherwise stop
11 Pair-Wise Exchange Heuristic LimitationsNo guarantee of optimality,The final solution depends on the initial layoutLeads to suboptimal solutionDoes not consider size and shape of departmentsAdditional work has to be re-arrange the department if shaper are not equal
12 Graph Based MethodGraph based method dates back to the later 1960s and early 1970s.The method starts with an adjacency relationship chartThen, we assign weight to the adjacency relationships between departmentsA graph, called adjacency graph is constructedNode: to represent departmentArc : to represent adjacency, weight on arc represents the adjacency scoreGoal: To find a graph with maximum sum of arc weightsHowever, not all the adjacency relations can be implemented in such a graph, that is the graph may not be planar.
13 A Planar GraphPlanar graph: A graph is a planar if it can be drawn so that each edge intersects no other edges and passes through no other verticesIntuitively, a planar graph is a graph where there is no intersection of arcs (flow of material)To find a maximum weight planar graph
14 Procedure to Find Maximum Weight Adjacent Planar Graph Step 1: Select a department pair with largest weightStep 2: Select a third department based on the sum of the weights with the two departments selected.Step 3: Select next unselected department to enter by evaluating the sum of weights and place the department on the face of the graph.Here, a face of a graph is a bounded region of a graphStep 4: Continuing the Step 3 until all departments are selectedStep 5: Construct a block layout from the planar graph
15 From Graph to Block Design Let us “blow” air into each node in the planar graphNodes explodeInterior faces becomes a dotThe edge in primal graph becomes the boundary between departmentsDual GraphNodes in dual faces in primalEdge in dual : if two faces connects in the primal graphThe faces in dual represents the departmentDraw Block Design
16 Limitation of Graph Based Method LimitationsThe adjacency score does not account for distance, nor does it account for distance other than adjacent departmentAlthough size is considered in this method, the specific dimension is not, the length between adjacent departments are also not considered.We are attempting to construct graphs, called planar graphs, whose arcs do not intersect.The final layout is very sensitive to the assignment of weights in the relationship chart.
23 Computer Relative Allocation of Facility Techniques (CRAFT) Discrete or Continuous RepresentationDiscrete RepresentationA two-dimension array with numbersEach cell represents a unit area & numbers represent the department occupied the cell
25 Valid Discrete Representation Valid RepresentationContiguous: If an activity is represented by more than one unit, every unit of the must share at least one edge with at least one other unitConnectedness: The perimeter of an activity must be a single closed loopNo Enclosed Void: No activity shape shall contain an enclosed void333
26 Computer Relative Allocation of Facility Techniques -- CRAFT (1963) Algorithm1) Any Incumbent LayoutDescribe a tentative layout in blocksDetermine centroids of each departmentCost= S distance (in the from-to matrix) X unit costDistance can be Euclidian or Rectilinear2) Improvement: make pair wise or three way exchangesequal area onlyadjacent (generally)3) If better solution exists; Choose the best, go to 1)Otherwise Stop
32 CRAFTIn the original design, exchange has to be departments of equal area or adjacent departments.
33 Shape Consideration Shape Consideration Shaper Ratio Rule: The ratio of a feasible shape should be with specified limitsCorner Counter: The number of corners for a feasible shaper may not exceed specified maximum
34 Excel Add-ins for facility Planning The Excel Add-InWritten by Prof. Paul Jensen (UT-Austin)Contains an implementation of CRAFT and can be downloaded atSequenceCreate a PlantDefine the FacilityOptimum SequenceCraft MethodFixed PointOptimize
35 Mixed Integer ProgramThe work begins latterly in the 1990s by MontreuilThe departments are assumed to be rectangular within a rectangular plant.PlantLength Bx, Width ByShape consideration:Area,The (minimum, maximum) width of a departmentThe (minimum, maximum) length of a departmentDecisions: Where to put the Departments (Centroid) and the shape (length,width) of the departmentObjective = flow_intensity* cost *distance
39 MIP model setup IIConstraint (6.13) ensures the upper corner of j is less than the lower corner of i if z_ij(x) =1 . i.e., to the east of i. Note if z_ij(x) = 0, (6.13) is redundant.Constraint (6.14) ensures to the north-south relationshipConstraint (6.15) ensures that no two departments overlap by forcing a separation at least in the east-west or north-south direction.
40 MIP Models Benefit of MIP Model Department shapes as well as their area can be modeled through individually specified lower and upper limits !!!It might be able to control length-width ratio as well(xi’’ – xi’ ) <= R (yi’’-yi’) or(yi’’ – yi’ ) <= R (xi’’-xi’)Heuristically, we can combine CRAFT with MIP.Get a initial layout using CRAFT, use MIP to find the best rectangular layout designSolving the problem exactly (optimal solution) is hard8~10 are the typical size solvable in a reasonable amount of time
41 Commercial Facility Layout Packages In the Instructor’s Opinion, there is no commercial package that will suit all the needs, partly due to the difficult of the problem, but more due to the fact that Facility Layout is a combination of Science and Art.There has been a trend to combine optimization techniques with interactive graphic procedures, especially people have an unique pattern reorganization capability than computers.We encourage the reader to use the web to keep abreast of new developments, resort to professional publications, which periodically publish survey of software packages for facilities planning, and new techniques
42 References Literature – Presentation topics General Survey Meller, R.D. and K. Gau, “The Facility Layout Problem: Recent and Emerging Trends and Perspective,” Journal of Manufacturing Systems, 15:5, ,1996Kusiak, A. and S. S. Heragu, “The Facility Layout Problem,” European Journal of Operational Research, v29, , 1987Mixed Integer ProgrammingMontreuil, B., “A Modeling Framework for Integrating Layout Design and Flow Network Design,” Proceedings of the Material Handling Research Colloquium, Hebron, KY, 1990Assignment Problem and the Location of Economic Activities, Econometrica,
43 Reference Reference (Continue) Graph Based Approach Hassan, M. M. D and G. L. Hogg, “On Constructing a Block Layout by Graph Theory,” International Journal of production Research, 29:6, , 1991Irvine, S. A. and I. R. Melchert, “A New Approach to the Block Layout Problem,” International Journal of Production Research, 35:8, , 1997Computerized Layout DesignBozer, Y.A., R.D. Meller and S.J. Erlebacher, “An Improvement Type Layout Algorithm for Single and Multiple Floor Facilities,” Management Science, 40:7,Tate, D.M. and A. E. Smith, “Unequal Area Facility Layout Using Genetic Search,” IIE Transactions, 27:4, , 1995Your Contribution In The Future !!
44 AssignmentsUsing Excel Add-ins as well as graph based method to solve the following problems6.8, 6.9, 6.10, , 6.15, 6.19, 6.20Compare the results and see if they make sense or not.Work in group, select one of the papers and present it in class at the end of the quarter.
50 MIP(Mixed Integer Program) Generally a construction type modelRequires some knowledge of linear and integer programmingSolutions to these types of problems are difficultWe will examine the general formulation
51 LOGIC Layout Optimization with Guillotine Induced Cuts Slice the area to partition the plant between departmentsSupersedes BLOCPLAN, because all BLOCPLANS are LOGIC plansImproved by pair wise exchange or simulated annealing