Proceedings of the International Conference on Advances in Production and Industrial Engineering 2015
Facility Layout Optimization using Linear and Non-Linear Mixed Integer Programming Manish Patil1, S.Kumanan2* Department of production engineering, NIT Tiruchirappalli, 620015 E-mail:
[email protected] *E-mail:
[email protected] Abstract The block layout problem of a multi-objective facility layout is an important sub class of the facility layout problem with practical applications when the price of land is high or when a compact building allows for more efficient control. Every department and shop floor in tested problem were fixed dimension of length and width. Proposed model considered a facility layout problem where size and location of each department, overall length and width of facility are all modeled as decision variables. Formulation is large mixed integer programming (MIP) problems. Model consists of two objective functions, minimizing material handling cost and facility building cost and proposed a lexicographic technique to handle multiple objectives. The results of the numerical experiments are showed by solving these problems with MIP solver for the base formulation and acceleration techniques such as symmetry breaking constraints to reduce the computation time and approximation to linearize constraints of area of each department. Keywords: Facility layout, Mixed integer programming, Approximation to linearize area constraints, Symmetry breaking constraints
1. Introduction Building a new facility involves investing a significant amount of resources in a project. Once construction of the facility is complete, it is even harder and costlier to redesign or correct flaws and other issues. Therefore design phase of such a project is of critical importance. Industrial engineering focuses on the design of conceptual block diagram of the facility. The block diagram shows the location and dimensions of the building and departments, without details related to utilization of networks. Operations executed in the facility are represented by good block diagram. Typically, the total distance travelled by the ‘materials’ is used as a cost of the facility layout and as indicator of the quality of the facility design. But depending upon the resource available for the project our objectives and constraints may change. For example in the case of scarce land, the department must fit into an area of given width and length. In other cases, however the length and width of the facility might be part of the set of decision variables and one might need to determine the size i.e. length and width of the facility so that it is large enough to efficiently place all the departments and not waste any resources. The focus of this paper is on minimizing the material handling cost as well as facility building costs. In this paper we concentrate on solving a mixed integer programming model for multi objective facility layout
problem by the use of the symmetry breaking constraints to reduce the computation time and by linearizing the area constraints.
2. Literature Review Machine layout problems have been treated as typical facility layout problems. However, as research on layout problems progressed, researchers began to focus on more specific problems, Such as minimizing material handling cost and maximizing closeness rating score. Love and Wong (1976b) presented a linear mixedinteger program for the single-row facility layout problem. The single row facility layout problem is a special case where facilities of equal or unequal dimensions are arranged on a line. Picard and Oueyranne (1981) extended the algorithm developed by Karp and Held (1967) For the general one-dimensional space allocation problem. A number of models and algorithms have been proposed for the single row facility layout which is known to be an complete NP-complete problem (Beghin-Picavet and Hansen,1982). Karp and Held (1967) developed a dynamic programming algorithm for the solution of the module placement problem which is equivalent to the optimal linear ordering problem. The facility layout problem (Meller and Gau 1996, Singh and Sharma 2006) is concerned with solving the
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physical organizational puzzle within a production system to minimize the material handling flow costs between departments. Solutions arrange a discrete number of departments within the bounds of a facility. With any set of objectives, the layout must also satisfy numerous constraints. Constraints might include departmental area requirements (Lacksonen 1997), departmental location restrictions, aisle network requirements (Wu and Appleton 2002), or department overlapping restrictions. The facility layout problem formulation can also be applied in many other applications, e.g. the arrangement of electronic components on a circuit board (Duman and Or 2007). In developing solutions to the facility layout problem, researchers have utilized various modeling approaches, formulations, and solution algorithms. Three of the most popular modeling approaches for the facility layout problem are quadratic assignment (Lawler 1963), mixed-integer programming (Montreuil 1990), and the graph-theoretic approach (Hassan and Hogg 1987). Montreuil (1990) extended the QAP for the continuous layout representation and proposed a mixedinteger programming (MIP) formulation. This MIP formulation has the same objective as the QAP approach: to minimize the material handling cost within a facility. However, unlike the QAP, which assigns a discrete number of equal-sized departments to a set of discrete locations, the MIP approach attempts to position a discrete number of unequal-sized departments into a continuous solution space (constrained by the overall facility size). (Konak et al. 2006). In addition, the incorporation of practical considerations, such as establishing an aisle network or locations for pick-up and drop-off, severely increase the complexity of the model. Tractability becomes an issue as the number of departments increases, therefore several heuristics have been proposed to solve the facility layout problem using the graph-theoretic approach (Leung (1992)). In addition, there have been numerous studies that use heuristic approaches such as genetic algorithms, simulated annealing, Particle swarm optimization, Ant colony algorithm. The most common objective is to minimize the material handling cost – the sum of the distances between each department multiplied by the flow or weight between those departments, But there are many other important objectives such as maximizing a closeness rating measure between departments (Rosenblatt 1979) and minimizing land and facility construction costs (Georgiadis et al.1999). Patsiatzis and Papageorgiou (2002) propose a multi-floor process plant model with a comprehensive objective function to minimize the total plant layout cost by determining the number of floors, land area,
equipment–floor allocation and detailed layout of each floor. Practical applications of considering building cost occurs when the land costs are high and compact building configuration allows for easier or more efficient environmental control. (Goetschalckx and Irohara, 2007) We concentrated on solving a mixed-integer programming model for the multi-objective facility layout problem by linearizing the area constraints and showing the effect of using the symmetry breaking constraints on computation time as our model is large mixed integer programming formulations with extensive computation run times. We are considering the facility cost as well as cost of the building as an objective function as well as use of lexicographic technique to handle multiple objectives.
3. Formulation of facility Layout Model Among the many possible objectives, we are considering (1) minimizing the material handling cost and (2) minimizing the building cost. Finding an optimal facility layout so as to minimize these costs requires the following decisions, the land area required by the facility, minimum side length and area of each department, material flow between departments and building facility cost per unit length. We made several simplifying but relatively mild assumptions to improve the tractability of the formulation. 1. Facility floors have a rectangular shape. 2. All departments have a rectangular shape and their height is equal to floor height. 3. All department area lower bound are specified as parameters. 4. Movement between departments is modeled as rectilinear centroid- to-centroid movement. 5. Movement is not explicitly modeled through an aisles network and the movement aisles do not consume any area. 6. Material flow volumes between departments are given. We first define the notation below, which is followed by the multi-objective MIP for the facility layout problem.
4. Notations i &j
–
To represent department.
Parameters N = Number of departments. L, W = maximum length and width of the facility.
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= flow between departments i and j respectively. = horizontal cost per unit flow per unit distance between departments i and j respectively. = lower bound of the area for department i. = Minimum length of any side for department i. = cost of making facility in making one unit greater in length and width respectively.
side length for that department and the maximum length and width of the facility. Constraints (5) require that each department’s area be greater than the minimum department area specified for each department. This constraint is in non-linear form. We use approximation to linearize these area constraints. Non overlapping constraints
Binary variables one if departments i is located to the left (L), right(R), above(A), below(B) of department j, zero otherwise.
(
Continuous variables
= distance between departments in x and y directions.
= horizontal rectilinear distance between the centroids of departments i and j
5. Mathematical Model Objective function- Minimize (
)
Objective function (1) is to minimize the material handling cost, which is equal to the sum of the flowweighted transportation costs between each pair of departments. The transportation cost between each set of departments is equal to the sum of the horizontal distance multiplied by horizontal transportation cost and objective function (2) represents total facility building cost. The first and second term represents the cost of making the building greater in length and width respectively. Department dimension constraints
Constraints (3) and (4) require each department’s length and width to be between the specified minimum
(
)
(
(
= x and y co-ordinates of the centroid of department i, respectively = length and width of facility along x- and yaxes respectively = length and width of department i along x- and y-axes respectively.
∑ ∑
) ) ) (
)
( (
)
) (
)
Constraint (6) forces any one of four constraints to remain active. Constraints (7) and (8) prevent departments from overlapping on the x-axis, while constraints (9) and (10) prevent departments from overlapping on the y-axis. Distance constraints
(
)
(
)
(12)
Constraints (11), (12) and (13), (14), define the rectilinear distance between centroids of two departments on x and y axis respectively. Constraint (15) computes the total rectilinear distance between two departments. Facility bounding constraints
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Proceedings of the International Conference on Advances in Production and Industrial Engineering 2015
When solving the model, we substitute the linear approximation constraints (23) and (24) instead of nonlinear department area constraints (4).
Constraints (16) and (17) require each department to be located within the lower bounds of the facility. Constraint (18) and (19) requires each department to be located within the upper bounds of the facility. Constraints (20) constrain the facility to the nonnegative region less than the upper bounds specified as parameters. Additional constraints
Figure 1a: Feasible region for li & Wi Constraints (21) and (22) are non-negativity constraints.
5.1 Approximation to linearize the area constraints Constraint (5) requires the area of each department to be at least as great as the minimum area parameter specified for that department; however, the area calculation is not linear. The feasible region for a department’s width and length subject to the minimum area requirement is shown Figure 1a. The horizontal axis of this graph represents the length of department i (li) and the vertical axis represents its width (wi). The region shaded red in the figure represents all feasible length–width combinations. As can be seen from the figure, the boundary of the feasible region (red curve) is Non-linear due to constraint (5). The length–width combinations below the red curve are infeasible because the resulting area will be less than the minimum area required. To linearize these constraints, we approximate the curve with a piecewise linear functions using the two lines shown in Figure 1b. √
√
√ √
√ √
(
√ )
(
√ )
Figure 1b: Linearization of the boundary of feasible region
5.2 Symmetry breaking constraints 1) Position q method for department q.
To reduce the solution effort, Sherali et al proposed symmetry breaking constraints for the single floor facility layout problem and demonstrated that these constraints can reduce the computation time effectively by eliminating the duplicate feasible solution. Constraints (25) and (26) are used to model symmetry
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Proceedings of the International Conference on Advances in Production and Industrial Engineering 2015
breaking constraints for department which has the largest sum of flows with other departments. These constraints restrict department q’s centroid to the left bottom corner of the facility. This acceleration technique is referred as position q method. 2) Position pq method for departments p and q.
Constraints (27) and (28) are used to model the symmetry-breaking constraints for the pair of departments has the largest flow between them.
Dep. (Si) meter (Ai)× Dep. 1 2 3 4 5 6 7 8
1 0 0 0 0 0 0 0 0
1 2 3 4 5 6 7 100 80 60 100 80 60 60 12 8 6 12 8 12 12 Table 2: Flow of material Flow frequency 2 3 4 5 6 7 500 0 0 700 0 0 0 200 0 800 0 0 0 0 500 0 0 0 0 0 0 900 0 0 0 0 0 0 600 0 0 0 0 0 0 800 0 0 0 0 0 0 0 0 0 0 0 0
574
8 40 2 8 0 0 0 0 0 0 0 0
5.3 Solving multiple objectives We considered two objectives function (1) minimizing the material handling cost and (2) facility building cost. The terms for the facility building costs could be in a monetary unit and facilities are for a longer period. But at the same time converting the material handling cost into monetary for this much and comparing those two objectives by balancing the weight is too difficult. If we place too much weight on the material handling cost results in an efficient layout but causes inefficiency in utilizing the facilities space. For using weighted sum approach, first we have to define weights which may not be meaningful or easy to interpret. In the case of two different objectives, we employed a lexicographic ordering technique in which LO solves a sub problem that considers only one objective. The solution is then provided to the final problem in the form of a constraint. For ex. Suppose facility is taking ₹1 crore to build the set up. This facility cost can be used as a constraint for the next objective of material handling cost by giving some slack say ₹1.1 crore.
Figure 2a: First sub-problem
6. Numerical results The unit of length is in meters and the units related to building cost are in ₹. 60000 per unit length. The maximum length and width of the facility were determined by the sum of each department’s minimum area divided by that department’s minimum side length. This is an upper bound for the possible length of the facility. We used GAMS to interact with CPLEX solver to solve our model. All computations are executed on the Windows-8 32 bit operating system on i3 intel processor with 2.4 GHz and 2 GB of RAM. Table 1: Values for department sizes
Figure 2b: Final problem with 11% slack. The facility cost of this layout is ₹ 3,26,68,222 while the material handling cost is ₹. 6,15,867 (Recall that material handling cost is not a monetary unit.).
Proceedings of the International Conference on Advances in Production and Industrial Engineering 2015
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After solving the second optimization model of material handling cost with different slacks i.e. increase in
percentage of the facility building cost as a constraint. The building facility cost increases to ₹3,62,40,000 and
material handling cost reduces to ₹ 4,77,000. The figure 2a and 2b shows the optimized layout with building facility cost and 11% slack area.
the considered objective functions are minimizing the material handling cost and building facility cost. We used an approximation to linearize the minimum area requirement constraint for each department and lexicographic ordering technique to prioritize the objectives and to solve sub-problems in sequence. Proposed model minimizes the building facility cost to the extent that it can accommodate objective of material handling cost minimization. Future research will focus on the decomposition techniques for the solution algorithms so that realistic problem sizes can be solved to optimality.
Figure 3: Flow cost for facility as area slack is increased
References Bukchin, Y., & Tzur, M. (2014). A new MILP approach for the facility process-layout design problem with rectangular and L/T shape departments. International Journal of Production Research, (aheadof-print), 1-21. Deechongkit, S., & Srinon, R. (2014). New Mixed Integer Programming for Facility Layout Design without Loss of Department Area. Industrial Engineering Letters, 4(8), 21-27.
Table 3: Numerical results
Nonlinear
Linear
Linear + q method Linear + pq method
Dep. 8
Objective function value
Computation time(Sec.)
1st Obj.
3,23,73,160
1004.13
2nd Obj.
4,91,900
1000.2
Total
-
2004.33
1st Obj.
3,26,74,372
129.00
2nd Obj.
4,77,000
14.62
Total
-
143.62
1st Obj.
3,26,68,222
66.613
2nd Obj.
4,77,000
16.422
Total
-
83.035
st
3,26,68,222
65.2
nd
2 Obj.
4,77,000
4.6
Total
-
69.8
1 Obj.
st
1 Objective – Minimize Building facility cost 2nd Objective – Minimize Material handling cost
7. Conclusion
Formulation for facility layout optimization was presented in this paper that incorporate length and width of facility as a decision variables. In this paper
Drira, A., Pierreval, H., & Hajri-Gabouj, S. (2007). Facility layout problems: A survey. Annual Reviews in Control, 31(2), 255-267. Goetschalckx, M., & Irohara, T. (2007). Efficient formulations for the multi-floor facility layout problem with elevators. Optimization Online, 1-23. Hathhorn, J., Sisikoglu, E., & Sir, M. Y. (2013). A multi-objective mixed-integer programming model for a multi-floor facility layout. International Journal of Production Research, 51(14), 4223-4239. Jannat, S., Khaled, A. A., & Paul, S. K. (2010). Optimal solution for multi-objective facility layout problem using genetic algorithm. In Proceedings of the 2010 International Conference on Industrial Engineering and Operations Management. Dhaka, Bangladesh. Rasmussen, R. (2007). QAP—not spreadsheets. Omega, 35(5), 541-552.
so
hard
in
Sherali, H. D., Fraticelli, B. M., & Meller, R. D. (2003). Enhanced model formulations for optimal facility layout. Operations Research, 51(4),629-644.
