Computational Facility Layout CHAPTER SIX Computational Facility Layout

The Facility Layout Problem Given the activity relationship as well as the space of the department, how to construct plan the layout of the facility The basis of the layout planning is the closeness ratings or material flow intensities minimize the flow times distance Maximize the closeness (adjacency) For most practical real world instances, the computational complexity has results in various heuristics What is heuristic? Construction Heuristic Improvement Heuristic

Heuristic and Optimality Consider the knapsack problem Z = max 5 x1 + 7 x2 + 11 x3 + 12 x4 + 17 x5 Subject to 2 x1 + 3 x2 + 4 x3 + 5 x4 + 7 x5 <= 10 Heuristic: An intuitive problem solving method/procedure Constructive Heuristic 1, Pick sequentially the ones with the best benefit Heuristic 2, Pick sequentially the ones with the best benefit per unit Greedy Improvement: Exchange two items in a solution Meta-Heuristic: Simulated Annealing, Genetic Algorithm Optimal Solution Mathematical Programming and Optimization Linear Programming, Integer Programming, Nonlinear Programming

A Simple Facility Layout Problem Suppose we have 10 identical sized departments and the flow intensity between these 10 department is fij Find the best arrangement of the 10 department along an aisle so that the total travel (flow intensity times distance) is minimized A Quadratic Assignment Model is necessary to Optimally solve the problem 1 2 3 4 5 6 7 8 9 10 Position Department

A Quadratic Assignment Model Decision Variables X(i,j) -- 1: Department I will be located at position j 0: Otherwise Constraints Each one position can hold exactly one department SUM( i in 1…10) x(i,j) = 1 Each department has to be assigned exactly one position SUM( j in 1…10) x(i,j) = 1 Objective SUM(I, j, m, n, all in 1..10 ) x(i,j)* x(m,n)*f(i,m)*d(j,n) This is an integer quadratic assignment problem.

Pair-Wise Exchange Heuristic From\To 1 2 3 4 -- 10 15 20 5 Phase I: Construct Phase Initial Solution (1,2,3,4) Phase II: Improvement – Pair Wise Exchange a) Exchange two departments b) If results in better solution, accept; go to a) otherwise stop

Pair Wise Exchange Heuristic

Pair Wise Exchange Heuristic

Pair Wise Exchange Heuristic

Pair Wise Exchange Heuristic

Pair-Wise Exchange Heuristic Limitations No guarantee of optimality, The final solution depends on the initial layout Leads to suboptimal solution Does not consider size and shape of departments Additional work has to be re-arrange the department if shaper are not equal

Graph Based Method Graph based method dates back to the later 1960s and early 1970s. The method starts with an adjacency relationship chart Then, we assign weight to the adjacency relationships between departments A graph, called adjacency graph is constructed Node: to represent department Arc : to represent adjacency, weight on arc represents the adjacency score Goal: To find a graph with maximum sum of arc weights However, not all the adjacency relations can be implemented in such a graph, that is the graph may not be planar.

A Planar Graph Planar graph: A graph is a planar if it can be drawn so that each edge intersects no other edges and passes through no other vertices Intuitively, a planar graph is a graph where there is no intersection of arcs (flow of material) To find a maximum weight planar graph

Procedure to Find Maximum Weight Adjacent Planar Graph Step 1: Select a department pair with largest weight Step 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 graph Step 4: Continuing the Step 3 until all departments are selected Step 5: Construct a block layout from the planar graph

From Graph to Block Design Let us “blow” air into each node in the planar graph Nodes explode Interior faces becomes a dot The edge in primal graph becomes the boundary between departments Dual Graph Nodes in dual  faces in primal Edge in dual : if two faces connects in the primal graph The faces in dual represents the department Draw Block Design

Limitation of Graph Based Method Limitations The adjacency score does not account for distance, nor does it account for distance other than adjacent department Although 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.

Graph-based Method

Graph-based Method

Graph-based Method

Graph-based Method

Graph-based Method

Graph-based Method

Computer Relative Allocation of Facility Techniques (CRAFT) Discrete or Continuous Representation Discrete Representation A two-dimension array with numbers Each cell represents a unit area & numbers represent the department occupied the cell

A Sample Problem

Valid Discrete Representation Valid Representation Contiguous: 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 unit Connectedness: The perimeter of an activity must be a single closed loop No Enclosed Void: No activity shape shall contain an enclosed void 3 3 3

Computer Relative Allocation of Facility Techniques -- CRAFT (1963) Algorithm 1) Any Incumbent Layout Describe a tentative layout in blocks Determine centroids of each department Cost= S distance (in the from-to matrix) X unit cost Distance can be Euclidian or Rectilinear 2) Improvement: make pair wise or three way exchanges equal area only adjacent (generally) 3) If better solution exists; Choose the best, go to 1) Otherwise Stop

CRAFT

CRAFT

CRAFT

CRAFT

CRAFT

CRAFT In the original design, exchange has to be departments of equal area or adjacent departments.

Shape Consideration Shape Consideration Shaper Ratio Rule: The ratio of a feasible shape should be with specified limits Corner Counter: The number of corners for a feasible shaper may not exceed specified maximum

Excel Add-ins for facility Planning The Excel Add-In Written by Prof. Paul Jensen (UT-Austin) Contains an implementation of CRAFT and can be downloaded at http://www.me.utexas.edu/~jensen/ORMM/frontpage/jensen.lib/index_omie.html#ormm Sequence Create a Plant Define the Facility Optimum Sequence Craft Method Fixed Point Optimize

Mixed Integer Program The work begins latterly in the 1990s by Montreuil The departments are assumed to be rectangular within a rectangular plant. Plant Length Bx, Width By Shape consideration: Area, The (minimum, maximum) width of a department The (minimum, maximum) length of a department Decisions: Where to put the Departments (Centroid) and the shape (length,width) of the department Objective = flow_intensity* cost *distance

MIP(Mixed Integer Program) Parameters

MIP(Mixed Integer Program ) Decision variables

MIP model setup

MIP model setup II Constraint (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 relationship Constraint (6.15) ensures that no two departments overlap by forcing a separation at least in the east-west or north-south direction.

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 design Solving the problem exactly (optimal solution) is hard 8~10 are the typical size solvable in a reasonable amount of time

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

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, 351-366,1996 Kusiak, A. and S. S. Heragu, “The Facility Layout Problem,” European Journal of Operational Research, v29, 229-251, 1987 Mixed Integer Programming Montreuil, B., “A Modeling Framework for Integrating Layout Design and Flow Network Design,” Proceedings of the Material Handling Research Colloquium, Hebron, KY, 1990 Assignment Problem and the Location of Economic Activities, Econometrica,

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, 1263-1278, 1991 Irvine, S. A. and I. R. Melchert, “A New Approach to the Block Layout Problem,” International Journal of Production Research, 35:8, 2359-2376, 1997 Computerized Layout Design Bozer, Y.A., R.D. Meller and S.J. Erlebacher, “An Improvement Type Layout Algorithm for Single and Multiple Floor Facilities,” Management Science, 40:7, 451-467 1994 Tate, D.M. and A. E. Smith, “Unequal Area Facility Layout Using Genetic Search,” IIE Transactions, 27:4, 465-472, 1995 Your Contribution In The Future !!

Assignments Using Excel Add-ins as well as graph based method to solve the following problems 6.8, 6.9, 6.10, 6.11 6.14, 6.15, 6.19, 6.20 Compare 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.

Thanks

BLOCPLAN Set up all departments in bands (2or3) Continuous areas not blocks Use From to or a relationship chart Uses two way exchanges

BLOCPLAN

BLOCPLAN

BLOCPLAN

MIP(Mixed Integer Program) Generally a construction type model Requires some knowledge of linear and integer programming Solutions to these types of problems are difficult We will examine the general formulation

LOGIC Layout Optimization with Guillotine Induced Cuts Slice the area to partition the plant between departments Supersedes BLOCPLAN, because all BLOCPLANS are LOGIC plans Improved by pair wise exchange or simulated annealing

LOGIC

LOGIC

LOGIC

LOGIC

LOGIC

LOGIC

LOGIC