Design of robust layout for Dynamic Plant Layout Problems

Computers & Industrial Engineering 61 (2011) 813–823 Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie Design of robust layout for Dynamic Plant Layout Problems V. Madhusudanan Pillai a,⇑, Irappa Basappa Hunagund a,1, Krishna K. Krishnan b,2 a b Department of Mechanical Engineering, National Institute of Technology Calicut, Calicut 673 601, Kerala, India Department of Industrial and Manufacturing Engineering, Wichita State University, Wichita, KS 67260-0035, USA a r t i c l e i n f o Article history: Received 27 September 2010 Received in revised form 22 May 2011 Accepted 24 May 2011 Available online 30 May 2011 Keywords: Cellular layout Robust and adaptive designs Facility layout Quadratic Assignment Problem Simulated Annealing a b s t r a c t In this paper, a design for robust facility layout is proposed under the dynamic demand environment. The general strategy for a multi period layout planning problem is adaptive approach. This approach for Dynamic Plant Layout Problem (DPLP) assumes that a layout will accommodate changes from time to time with low rearrangement and production interruption costs, and that the machines can be easily relocated. On the other hand the robust layout approach, assumes that rearrangement and production interruption costs are too high and hence, tries to minimize the total material handling costs in all periods using a single layout. Robust approach suggests a single layout for multiple scenarios as well as for multiple periods. As a solution procedure for the proposed model, a Simulated Annealing (SA) algorithm is suggested, which perform well for the problems from literature and QAPLIB website. The application of suggested model for robust layout to cellular layouts has given better results compared to the robust cellular layout model of literature. For the standard DPLP of the literature, the solution values of the suggested model are very near to the results of adaptive approach. The Total Penalty Cost (TPC) is used to test the suitability of the suggested layout to be a robust layout for the given data set. TPC values indicate that the suggested layout is suitable as robust layout for the given data sets. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction To operate production and service systems efficiently, systems should not only have to be operated with optimal planning and operational policies, but also have a facility layout that is well designed. Optimal design of physical layout is an important issue in the early stage of system design and has a big influence on the long-term viability of the manufacturing system. A poorly designed layout will results in reduced productivity, increased work-in-process, increased manufacturing lead time, disordered material handling and so on. In general, the objective function for the facility layout problem is focused on reducing the Material Handling Cost (MHC). According to Chan, Chan, and Kwong (2004) the MHC assumes about 20–50% of the total operating cost of the facility layout. MHC is a non-value added cost. Efficient facilities planning can reduce these costs by at least 10–30% and thus increase the productivity. The facility layout problem is a long term, costly proposition, and any modifications or rearrangements of the existing layout represent a large expense and cannot be easily accom- ⇑ Corresponding author. Tel.: +91 9895367804; fax: +91 495 2287250. E-mail addresses: [email protected] (V. Madhusudanan Pillai), iranna346@yahoo. com (I.B. Hunagund), [email protected] (K.K. Krishnan). 1 Tel.: +91 9448641567; fax: +91 495 2287250. 2 Tel.: +1 316 978 5903; fax: +1 316 978 3175. 0360-8352/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2011.05.014 plished. Hence, an efficiently planned facility can reduce these costs, and thus increase productivity. A facility layout is concerned with the location and arrangement of departments, cells or machines within the cells. The facility layout problem is often formulated as a Quadratic Assignment Problem (QAP), which assigns m departments to m locations while minimizing the MHC. However, QAP is known to be NP-complete, and optimization methods are not capable of solving problems with 15 or more facilities in a reasonable amount of time. Therefore, there is a need for heuristic methods that provide good suboptimal solutions. In a competitive environment, markets are heterogeneous and volatile in nature. In order for a manufacturing firm to sustain its productivity under volatile demand conditions, its production process has to be configured suitably. The ability to design and operate manufacturing facilities that can quickly and effectively adapt to changing technological and marketing requirements is becoming increasingly important to the success of any manufacturing organization. Hence, manufacturing facilities must be able to exhibit high levels of flexibility and robustness in order to deal with significant changes in their operating requirement. When the demand is more or less constant with time, Static Plant Layout Problem (SPLP) approach is a suitable method for obtaining a good facility layout. But when demand is varying frequently with time, static layout generation approaches may not be efficient in various periods of the planning horizon. Fluctuations 814 V. Madhusudanan Pillai et al. / Computers & Industrial Engineering 61 (2011) 813–823 Nomenclature total traveling score for average demand scenario or this represents the objective function for robust design ki;jk part handling factor of part i when transported from facility j to facility k x cost for unit traveling score m number of facilities (departments) drs rectilinear distance between locations r and s {xr, yr}, {xs, ys} co-ordinates of measuring points of location ‘r’ and ‘s’ in location grid fjk part flow weight from facility j to facility k part flow weight from facility j to facility k in period p fp,jk 1 if facility j is assigned to location r, 0 otherwise xjr 1 if facility j is assigned to location r in period p, 0 otherxp,jr wise Ap,j,rs relocation cost of facility j in period p if it is shifted from location r to s Bi,jk number of parts i per transportation when transported from facility j to facility k demand of part i, where i = 1, 2, . . . , N Di fE in product demand, changes in product mix, introduction of new products, and discontinuation of existing products are all factors that render the current facility layout inefficient and can increase MHC, which might necessitate a change in the layout (Afentakis, Millen, & Solomon, 1990). Maintaining a good facility layout requires a continuous assessment of the variations in product demands and flow between departments, and the need for Dynamic Plant Layout Problems (DPLP) approaches for the development of layouts. The approaches that have been followed to solve the dynamic facility layout fall into two major categories.  Adaptive or flexible or agile approach.  Robust approach. The first approach assumes that layout will accommodate changes from time to time with low rearrangement costs and that the machines can be easily relocated. On the other hand, a robust layout approach assumes that rearrangement costs are too high and hence tries to minimize the total material handling costs in all periods using a single layout. Robust layout approach is one of the methods used for developing layouts for multiple production scenarios of a single period problem and for multi-period problems. Robust approach suggests a single layout for multiple scenarios as well as for multiple periods. Rosenblatt (1986) made the first attempt to model the DPLP. A dynamic programming approach was used for solving the model for multiple periods. At each phase, the designer solves a static layout design problem for a specific number of alternatives. Lacksonen (1997) proposed a model to handle both rearrangement and unequal area constraints for the DPLP. This model employed a pre-processing strategy that generated solutions for large-sized problems and uses an improved branchand-bound algorithm to yield feasible layouts. Yang and Peters (1998) proposed a flexible machine layout model which includes both material handling and machine rearrangement costs. This layout design uses a rolling horizon planning time window. The present paper suggests a robust model for DPLP. Simulated Annealing (SA) method of layout formation is used as a solution procedure for the suggested robust model. The proposed SA method is tested for problems in literature and it is performing well in all cases of problems except in one case where the result is inferior by 0.07%. The robust model is applied to the problems of literature. demand of part i in period p, where p = 1, 2, . . . P average demand of part i, where i = 1, 2, . . . N material handling cost when the given layout is used in period p N total number of parts P number of periods in planning horizon rectilinear distance between facilities j and k in a given Rjk layout TMHC total material handling cost in the planning horizon {Xj, Yj}, {Xk, Yk} co-ordinates of measuring points of facilities j and k in the layout average part flow weight from facility j to facility k WEjk 1 if facility j is shifted from location r to s in period p, 0 Yp,j,rs otherwise ZA objective function for dynamic layout problems or it represents the traveling score plus the relocation cost under dynamic demand situations objective function for static layout problems or it repreZs sents the traveling score for static layout problems Dp,i DEi MHCp Then, robust layout solution is compared with the adaptive layout solution of the problems tested. Robust layout strategy provides solution quality almost equal to the solution quality of adaptive layout strategy without production interruption and relocation. We define robust layouts as those that can effectively cope with product demand variability, over various periods of planning horizon. 2. Literature review Approaches to the generation of layouts can be classified into two: (i) qualitative and (ii) quantitative. Qualitative approaches provide a layout based on the closeness rating between the departments. On the other hand, quantitative approaches typically involve the minimization of the total MHC between the departments. For a comprehensive review of the existing methods for the facility layout problem, see Kusiak and Heragu (1987), Yaman, Gethin, and Clarke (1993), Singh and Sharma (2006), and Drira, Pierreval, and Hajri-Gabouj (2007). Considerable amount of research has been made to static layout problems with exact, heuristic, meta-heuristic and hybrid solution approaches. Some researchers (Bock & Hoberg, 2007; Chan, Chan, & Ip, 2002; Foulds, Hamacher, & Wilson, 1998; Tam & Li, 1991; Tang & Abdel-Malek, 1996) suggest heuristic methods to solve SPLP. They considered various constraints like forbidden areas, equal and unequal areas, aisles and barriers within the layout like existing walls or columns. Some researchers (El-Baz, 2004; Hu & Wang, 2004; Mak, Wong, & Chan, 1998; Wilhelm & Ward, 1987) have attempted the SPLP with the meta-heuristics like simulated annealing, genetic algorithm to solve the large size layout problems. The SPLPs are also formulated with multiple objectives like combining both quantitative and qualitative factors and they are solved with heuristic, metaheuristics, or hybrid approaches. Various literatures in this category are Rosenblatt (1979), Dutta and Sahu (1982), Malakooti and D’Souzas (1987), Soundar, Kashyap, and Moodie (1988), Heragu and Kusiak (1990), Catherine and Tothero (1992), Raoot and Rakshit (1993), Meller and Gau (1996), Islier (1998), Sha and Chen (2001), Tuzkaya, Ertay, and Ruan (2005), Ertay, Ruan, and Tuzkaya (2006), and Khilwani, Shankar, and Tiwari (2008). Enormous amount of research has been done into SPLP from mid fifties to mid nineties. However, in recent years the researchers are making efforts to address the DPLP. Various researchers proposed 815 V. Madhusudanan Pillai et al. / Computers & Industrial Engineering 61 (2011) 813–823 new and improved models and algorithms to solve DPLP. Rosenblatt (1986) first developed a model and solution procedure to DPLP with adaptive approach for small size problems. A review of research on the dynamic layout problem is available in Balakrishnan and Cheng (1998), Ulutas and Islier (2005), and Konak (2007). These papers categorize different algorithms for equal and unequal sized departments, and deterministic and stochastic material flow. Many researchers (Balakrishnan & Cheng, 2000, 2009; Balakrishnan, Cheng, & Conway, 2000; Balakrishnan, Cheng, Conway, & Lau, 2003; Balakrishnan, Jacobs, & Venkataramanan, 1992; Baykasoglu, Dereli, & Sabuncu, 2006; Baykasoglu & Gindy, 2001; Corry & Kozan, 2004; Dunker, Radons, & Westkamper, 2005; Kochhar & Heragu, 1999; Lacksonen, 1997; Lacksonen & Enscore, 1993; Lahmar & Benjaafar, 2005; McKendall & Shang, 2006; McKendall, Shang, & Kuppusamy, 2006; Norman & Smith, 2006; Palekar, Batta, Bosch, & Elhence, 1992; Rosenblatt, 1986; Sugiyama & Ueda, 2002) have developed the adaptive or flexible or agile layouts that can be easily rearranged to meet the changes in production requirements. Researchers (Balakrishnan & Cheng, 2009; Balakrishnan, Jacobs, et al., 1992; Balakrishnan, Cheng, et al., 2000; Lacksonen, 1997; Lacksonen & Enscore, 1993; Lahmar & Benjaafar, 2005; Palekar et al., 1992; Rosenblatt, 1986) use exact and heuristic methods to solve the DPLPs. Researchers (Balakrishnan & Cheng, 2000; Baykasoglu & Gindy, 2001; Baykasoglu et al., 2006; Corry & Kozan, 2004; Kochhar & Heragu, 1999; Norman & Smith, 2006) made use of meta-heuristics like simulated annealing, genetic algorithm and ant colony optimization techniques to DPLPs. Recently the hybrid approaches are also attempted in Balakrishnan and Cheng (2000), Sugiyama and Ueda (2002), Balakrishnan et al. (2003), Dunker et al. (2005), McKendall and Shang (2006) and McKendall et al. (2006). Balakrishnan et al. (1992) and Baykasoglu et al. (2006) have modeled with budget constraint on rearrangement costs. Balakrishnan and Cheng (2009) investigated the performance of various algorithms under fixed and rolling horizons, under different shifting costs and flow variability, and under forecast uncertainty as compared with most DPLP that assumed the fixed planning horizon and no forecast error. Lahmar and Benjaafar (2005) presented the procedure for design of distributed layout (multiple copies of the same department type) in multi period. Kochhar and Heragu (1999) explored the design of a multiple-floor dynamic facility that is able to respond to frequent production demand and mix changes. Some researchers (Aiello & Enea, 2001; Benjaafar & Sheikhzadeh, 2000; Kouvelis, Kuawarwala, & Gutierrez, 1992; Yang & Peters, 1998) have developed robust layouts for multiple production scenarios in a single period and for multi period. Kouvelis et al. (1992) mentioned the importance of robustness for dynamic layout problems and developed an algorithm to generate the robust layouts for the manufacturing systems. Pillai and Subbarao (2008) presented a robust approach for forming part families and machine cells, which can handle all the changes in demands and product mixes without any relocations. A genetic algorithm based solution procedure is adopted to solve the problem. Different criteria are used to measure the layout effectiveness and strategies to be followed to go for robust or adaptive approaches to layout formation in dynamic environment. Braglia, Simone, and Zavanella (2003) proposed the adoption of indices that will help in identifying the strategy to be preferred for the identification of either a robust or an agile layout. Pillai (2005, chap 6) explained about the general measures of effectiveness used to evaluate the performance of the layout under various conditions. The measures mentioned are (i) average percentage of cost difference, (ii) percentage of situations for which a layout is optimum, (iii) maximum percentage of cost difference and (iv) robustness indicator. Robustness indicator is the percentage of cost difference that is less than or equal to a fixed percentage. It measures the flexibility of layout to adapt to demand changes. Raman, Nagalingam, Gurd, and Lin (2007) developed a model to measure the effectiveness of the layouts with respect to layout flexibility, area utilization and closeness gap. The closeness gap refers to bringing of highly interactive facilities/departments closer considering the empty travel of material handling equipment, information flow, and personnel flow. They contended that measuring these parameters for checking the effectiveness of the layout help in productivity improvement. The concept of robust design for the cellular manufacturing system under dynamic demand proposed in the research work of Pillai and Subbarao (2008) is used for the development of robust layout model for DPLP. They suggested a cellular manufacturing system design for an average scenario to use in all periods of the multi period planning horizon and the same concept is applied to the multi period layout design to develop a robust design. The layout effectiveness measures from Braglia et al. (2003) are used for evaluating suitability of solution obtained using suggested robust model for DPLP problems. 3. Problem description and formulation for SPLP In a static environment, the plant layout problem is solved for a single period, when the interdepartmental flow is nearly constant from period to period. In such cases, layout design problem is concerned with the assignment of ‘m’ facilities to ‘m’ discrete locations with the objective of minimizing the assignment cost. The assignment cost is the sum of the product of flow of materials between the facilities, the distances between their locations and the cost of installation. Part handling factor as suggested in Chan et al. (2004) is also taken into account. That is, the attributes of a part will change from process to process. For example, in an assembly cell, a part can change in size, weight, shape and so on. In some cases, an initial 1 kg part is increased to 5 kg after some assembly operations or, the opposite, a finished part may reduce in weight in comparison with its initial state if there is a material removal action involved. As a result, even though the quantitative demand of a part remains unchanged, the best possible layout can be different if the part-handling factor is taken into account. Inputs to the problem are number of parts to be manufactured, demand of parts, machine sequence or route sheet of parts, part handling factor and location layout grid. The inputs to QAP are flow between facilities and distance between locations. Flow between facilities is computed from demand of parts and machine sequence or route sheet of parts. Eq. (5) in the formulation is used to calculate the flow between facilities based on the part handling factor and number of units of part moving per batch in between the facilities. The flow between facilities is represented in a from-to chart matrix. In the from-to chart matrix, element indices are facilities, and the value of the element is the volume of material flow. 3.1. Model The mathematical model of static layout problem Minimize Z s ¼ m X m X m X m X fjk drs xjr xks ð1Þ j¼1 r¼1 k¼1 s¼1 Subjected to m X xjr ¼ 1 8 j ¼ 1; 2; . . . ; m ð2Þ xjr ¼ 1 8 r ¼ 1; 2; . . . ; m ð3Þ r¼1 m X j¼1 816 V. Madhusudanan Pillai et al. / Computers & Industrial Engineering 61 (2011) 813–823 where drs ¼ jxr  xs j þ jyr  ys j fjk ¼ N X Di ki;jk B i;jk i¼1 xjr 2 f0; 1g; 8 j ¼ 1; 2; . . . :m and 8 k ¼ 1; 2; . . . :m 8 r ¼ 1; 2; . . . ; m and 8 j ¼ 1; 2; . . . ; m ð4Þ ð5Þ ð6Þ 4. Proposed Simulated Annealing (SA) algorithm for layout formation SA is a technique which is suitable for solving large combinatorial optimization problems. This technique is based on probabilistic methods that avoid being stuck at local (non-global) minima and has proven to be a simple but effective method for large-scale combinatorial optimization. The concept is based on the manner in which metals recrystallize in the process of annealing. If the heating temperature is sufficiently high to ensure random state and the cooling process is slow enough to ensure thermal equilibrium, then the atoms will place themselves in a pattern that corresponds to the global energy minimum of a perfect crystal. As the cooling proceeds, the system becomes more ordered and approaches a ‘‘frozen’’ ground state at T = 0. If the initial temperature of the system is too low or cooling is done quickly, the system may become quenched forming defects or freezing out in metastable states (that is, trapped in a local minimum energy state). In the SA method, the goal is to bring the system, from an arbitrary initial state, to a state with the minimum possible energy. At each step, the SA heuristic considers some neighbor s0 of the current state s, and if the movement to s0 is not economical, probabilistically decides in moving the system to state s0 . The parameters are chosen so that the system ultimately tends to move to states of lower energy. The SA parameters used for the solving the various size problems are given below. 1. Configuration changes: Configuration changes are obtained by swapping operation. 2. Initial temperature: Initial temperature is set in such way that 90% of the configuration changes are accepted at starting stage. 3. Cooling ratio: It is taken as 0.98. A study on the cooling rate was conducted. The results were analyzed for different values of cooling rate from 0.99, 0.98, 0.97. . . 0.91. For the set of problems considered in this paper, the quality of the solutions and its repeatability were found to be the best when the cooling rate was set at 0.98. However, for different set of problems, the same testing for the best cooling rate has to be completed. 4. Number of samples in each temperature level: The number of configuration changes attempted at each temperature level was determined by the expression, L = a m2, where ‘a’ is a constant having value between 0.8 to 1 and ‘m’ is the number of facilities. 5. Termination condition or final temperature: The termination condition or final temperature is set to 3, which is established by experimentation on 288 problems. It means that the solution quality is not improving below final temperature 3. 6. Configuration change acceptance criteria: The Metropolis criterion was selected to govern the acceptance or rejection of configuration changes. It involves the following cases: 1. If the configuration change results in a net reduction of the objective function, then it is accepted. 2. If the configuration change increases the objective function, then it is accepted with a probability of R[0 1] < exp (DE/Ti) where ‘DE’ represents the change in the value of the objective function and Ti is the temperature of the system at the corresponding stage of the procedure. In the present application, the configuration changes are accepted if a random number between 0 and 1 is less than the value of expression exp (DE/Ti). 4.1. SA pseudo code  initialize: temperature, ntemp, final temperature, initial layout, number of samples in each temperature (L), cooling ratio  for i:= 1. . . ntemp do  for j:= 1. . .L do  Try a random swap between two facilities of the layout  DE:= current_cost  trial_cost  if DE<0 then  make the swap permanent  increment good_swaps  else R:= random number in range [0. . .1]  m:= exp(DE/temperature)  if R<m then // Metropolis criterion  make the swap permanent  increment good_swaps  end if  end if  end for  temperature:= cooling ratio  temperature  exit when temperature > final temperature  end for The layout model defined above is solved using the Simulated Annealing (SA) algorithm coded in MATLAB. The performance of this solution procedure is tested by solving cellular layout cases from Yaman et al. (1993), and cases given in Nugent et al., and Wilhelm and Ward. The cases of Nugent et al., and Wilhelm and Ward are obtained from the QAPLIB website (2007). The numerical illustration of the cases and analysis of results are provided in Section 6. 5. Problem description and formulations for DPLP The DPLP assumes different flow matrices in the different periods of planning horizon and arrives at best layouts for the entire planning horizon. Several researchers solved the DPLP by adaptive approach, which considers rearrangement of facilities with some relocation costs. The shifting of departments from one period to the next period is done to offset the increase in MHC. Therefore, the objective of the adaptive DPLP model is to minimize the sum of MHC and relocation costs over all periods in the planning horizon. It consists of quadratic assignment model of the layout under dynamic situation which involves assigning ‘m’- facilities to ‘m’-potential candidate locations in the layout grid in the various periods of planning horizon by considering the rearrangement cost. Hence, a typical mathematical formulation of the adaptive approach is as given below. Minimize Z A ¼ P X m X m X m X m X fp;jk drs xp;jr xp;ks p¼1 j¼1 r¼1 k¼1 s¼1 þ P X m X m X m X p¼2 j¼1 r¼1 s¼1 Ap;j;rs Y p;j;rs ð7Þ 817 V. Madhusudanan Pillai et al. / Computers & Industrial Engineering 61 (2011) 813–823 Subjected to m X xp;jr ¼ 1 Subjected to 8 j ¼ 1; 2; . . . ; m 8 p ¼ 1; . . . ; P ð8Þ r¼1 m X m X xjr ¼ 1 8 j ¼ 1; 2; . . . ; m ð13Þ xjr ¼ 1 8 r ¼ 1; 2; . . . ; m ð14Þ 8 r ¼ 1; 2; . . . ; m and 8 j ¼ 1; 2; . . . ; m ð15Þ r¼1 xp;jr ¼ 1 8 r ¼ 1; 2; . . . ; m 8 p ¼ 1; . . . ; P ð9Þ j¼1 xp;jr ¼ f0; 1g; m X j¼1 r; j ¼ 1; 2; . . . ; m p ¼ 1; . . . ; P Y p;j;rs ¼ xðp1Þ;jr xp;js j; r; s ¼ 1; 2; . . . ; m p ¼ 2; 3; . . . ; P ð10Þ xjr 2 f0; 1g; ð11Þ DEi ¼ drs ¼ jxr  xs j þ jyr  ys j The robust approach to dynamic layout problem involves development of a layout for the expected flow between facilities or expected demand scenario of the various periods. The layout of expected flow or expected demand scenario is applied in all the periods. Thus, the entire planning horizon uses a single layout even though the demand or flow between facilities is different in different periods of the planning horizon. Quadratic assignment model of robust approach is developed and the Eqs. (12)-(21) represent this model. In this model, a layout is developed for an average scenario and this layout is used in every period without relocation of facilities in any period of planning horizon. In this model the computational effort required to solve the dynamic layout problem is same as that of the static layout problem. That is, the adaptive approach for the dynamic layout problems requires (m!)P computational effort, where as the proposed robust approach requires only m! computational effort. The proposed layout for dynamic environment is most effective when the facilities are difficult to relocate, rearrangement costs are too high and the chances of operational disruption are high due to rearrangement. Inputs to this model are the number of parts to be manufactured, demand of parts in various periods, machine sequence or route sheet of parts, part-handling factor and distance between locations. Eq. (16) gives the expected demand of parts for the planning horizon and Eq. (17) in the formulation is used to calculate the expected flow between facilities, when demand of parts with part handling factor and number of parts moving per batch from one facility to other facility are given. MATLAB code is written for above computation. If the flows between facilities in the various periods of the planning horizon are available, then the expected flow matrix can be derived by arithmetic averaging of all flows between the facilities of the various periods. The total MHC of the planning horizon is determined by applying the layout of the expected scenario to every period of the planning horizon. Eq. (18) gives the actual flow between the facilities in the various periods of planning horizon. Eqs. (19) and (20) are used to calculate the MHC of each period and the entire planning horizon, respectively by applying the robust layout. Eq. (21) is used to find the rectilinear distances between facilities in the robust layout. 5.1.1. Mathematical model for robust approach m X m X m X m X j¼1 r¼1 k¼1 s¼1 WEjk drs xjr xks p¼1 Dp;i WEjk ¼ 5.1. Proposed robust approach to DPLP Minimize fE ¼ PP ð12Þ ð16Þ P N X DEi ki;jk B i;jk i¼1 8 j ¼ 1; 2; . . . ; m; and 8 k ¼ 1; 2; . . . :; m ð17Þ 5.1.2. Robust layout MHC calculation fp;jk ¼ N X Dp;i ki;jk B i;jk i¼1 8 j ¼ 1; 2; . . . ; m; 8k ¼ 1; 2; . . . :; m 8 p ¼ 1; . . . ; P MHC p ¼ x  m X m X Rjk fp;jk j¼1 k¼1 TMHC ¼ P X ! MHC p ð18Þ ð19Þ ð20Þ p¼1 Rjk ¼ jX j  X k j þ jY j  Y k j ð21Þ Performance of this model is demonstrated in Section 6.2. 6. Numerical demonstrations and analysis of results Data set used for evaluating the performance of the layout formation method consists of data from case studies from Yaman et al. (1993), Chan et al. (2004) and QAPLIB website (2007). Data used from QAPLIB website (2007) consists of problems of Nugent et al. and Wilhelm and Ward (1987). Data from Yaman et al. (1993), Chan et al. (2004), and the data obtained from Balakrishnan and Cheng are used to demonstrate the performance of robust layout model. The data set of Yaman et al. (1993) consists of five periods in the planning horizon and five parts to process with nine machines. This data set uses location grid of 3  3 for locating machines. For each part in the family, the operational sequence and demand in a five-period planning horizon are provided in Tables 1 and 2, respectively. Chan et al. (2004) proposed a part-handling factor for Yaman et al. (1993) case and are given in Table 3; parts 1 and 3 gradually decrease in part-handling factors, while that of others increased. The cost per unit part movement is taken as 10 (x = Rs.10/unit traveling score), the part transportation quantity Table 1 Machine operational sequence of Yaman et al. (1993) case. Part Machine operational sequence 1 2 3 4 5 01 ? 03 ? 05 ? 07 ? 02 ? 07 ? 09 01 ? 04 ? 02 ? 05 ? 06 ? 08 ? 09 01 ? 05 ? 07 ? 08 ? 05 ? 06 ? 02 ? 09 01 ? 02 ? 04 ? 06 ? 07 ? 08 ? 02 ? 03 ? 09 01 ? 07 ? 06 ? 04 ? 02 ? 08 ? 03 ? 05 ? 06 ? 09 818 V. Madhusudanan Pillai et al. / Computers & Industrial Engineering 61 (2011) 813–823 Table 2 Demand profiles in Yaman et al. (1993) case. Part 1 2 3 4 5 Table 5 Comparison of heuristic solution and SA solution values for QAPLIB problems. Period p=1 p=2 p=3 p=4 p=5 10 30 45 70 85 35 50 15 80 60 90 25 40 55 70 40 65 70 90 20 55 20 15 85 30 Problem size 9 12 15 20 25 30 50 Table 3 Proposed part-handling factors by Chan et al. (2004) for Yaman et al. (1993) case. * Part Part-handling factor 1 2 3 4 5 6?5?4?3?2?1 1?2?3?4?5?6 6?5?4?4?3?2?1 1?1?2?2?2?3?3?4 1?2?3?4?5?5?5?5?6 Table 4 Comparison of MHC for Yaman et al. (1993) case by various methods. Period Chan et al. (2002) heuristic Yaman et al. (1993) Spiral-1 Yaman et al. (1993) Spiral-2 Tang and Abdel-Malek (1996) approach SA method 1 2 3 4 5 28,500 27,900 31,600 31,650 22,750 36,300 31,800 36,900 39,750 30,450 34,700 33,500 35,700 40,650 29,750 28,200 29,800 32,000 31,000 23,550 27,800 26,400 29,500 30,200 22,000 per move is taken as 1 (Bi,j?k = 1). Chan et al. (2004) uses the basic data of Yaman et al. (1993). Data obtained from Balakrishnan and Cheng and QAPLIB website are with authors. 6.1. Analysis of results of SPLP using SA method of layout formation The performance of suggested layout formation method is to be established. Standard problem instances from Yaman et al. (1993) and QAPLIB website (2007) are used for this purpose. Instances from Yaman et al. (1993) mainly concerned with cellular layout problems. 6.1.1. Results for Yaman et al. (1993) case In the static environment, the performance of SA method of layout formation is compared with the Yaman et al. (1993) case. The five-period dynamic demand situation is considered as five static equivalent problems. All the part transportation quantity per move and the part handling factors are set to ‘1s’, with the purpose of comparisons to see the performance of proposed method with the established research work. Table 4 shows comparison of MHCs of SA method of layout formation with the various methods in literature, when the five-period dynamic demand situation is considered as five static equivalent problems. SA method is giving optimal solution values for this size problem (9-size). 6.1.2. Results for QAPLIB problems The problem instances obtained from QAPLIB website (2007) are mainly used for evaluation. These standard problem instances (Nugent et al.; Wilhelm & Ward, 1987) as given in the QAPLIB with problem sizes 12, 15, 20, 30, 50 and also 9-size first period problem of Yaman et al. (1993) are solved with SA. The SA is initialized with random solution value and each problem is run for 20-times. Table 5 shows the comparison of solution values and program run Optimal solution value 2780 289 575 1285 1872 3062 24,408* SA solution values % Deviation of SA best value from optimal Best Worst Mean Time (s) 2780 289 575 1285 1872 3064 24,408 2780 296 587 1312 1894 3129 24,479 2780 291.75 579.05 1299 1879.2 3090.15 24,444.8 3.15 3.63 7.72 21.82 44.37 89.93 582.74 0.00 0.00 0.00 0.00 0.00 0.07 0.00 Best value. timings of SA for various size problems. The SA run time given in Table 5 is the average run time based on the 20 run of a problem. The solution values when SA used for layout problems of Nugent et al. given in QAPLIB website up to 25-size are not differing from published (optimum) values in QAPLIB. But, for 30 size problem the solution value is slightly differ by 0.07% and for 50-size problem of QAPLIB, the solution value of SA is same as the solution of Wilhelm & Ward, 1987. In general, we can say that the suggested SA method of layout formation is able to provide good result for the layout problems. Program coded in MATLAB for SA algorithm were run on the Pentium – 4, 2.60 GHZ, 248 MB RAM processor and program run CPU timings were saved for various size problems and it is shown in Table 5. 6.2. Analysis of results of Dynamic Plant Layout Problem This section describes the performance of robust layout model. Problem instances from different situations are considered here and the results for these problems are given below. 6.2.1. Results of Yaman et al. (1993) case and Chan et al. (2004) [S1] and [S2] cases In dynamic environment, expected demand of each part in the planning horizon is calculated using Eq. (16) for the Yaman et al. (1993) case. Table 6 shows expected demand profile of the parts for the above problem. Using expected demand of parts and machine operational sequence of parts, a layout is formed by applying proposed SA method of layout formation. Table 7 shows the robust layout and its total traveling score (fE) for expected demand scenario. This robust layout is then applied to different demand profile Table 6 Expected demand profile of the parts. Part 1 2 3 4 5 Average demand 46 38 37 76 53 Table 7 Robust layout with SA method and its total traveling score for expected demand profile when ki;jk ¼ 1 and Bi,jk = 1. (a) Layout 01 02 04 05 07 06 (b) Traveling score (MHC) for average demand scenario fE = 2740 03 08 09 819 V. Madhusudanan Pillai et al. / Computers & Industrial Engineering 61 (2011) 813–823 Table 8 Robust layout material handling cost of each period (MHCp) and TMHC of planning horizon when ki;jk ¼ 1 and Bi,jk = 1. Period p=1 p=2 p=3 p=4 p=5 Planning horizon Material handling cost 27,800 26,400 29,500 31,300 22,000 137,000 Table 9 Robust layout with SA method and its total traveling score for expected demand profile when ki;jk is varying and Bi,jk = 1. (a) Layout 03 05 01 08 07 02 09 06 04 (b) Traveling score (MHC) for average demand scenario fE = 8759 of various periods in the planning horizon and material handling cost of each period is aggregated to get the TMHC of cell layout over the entire planning horizon. Table 8 shows the MHC of each period and TMHC of planning horizon for the above problem. The above problem is also solved by including part handling factor; that is, varying part handling factor (ki;jk ) as given in Table 3 and the corresponding results are provided in Tables 9 and 10. Problems of [S1] and [S2] cases of Chan et al. (2004) are also solved with and without considering part handling factor using robust approach and the corresponding results are provided in Table 11. When we compare these results with those available in literature, the robust layout procedure suggested in this paper provides less total MHC under dynamic demand. This shows the better performance of the proposed layout model (robust) and relocation of machines from one period to the next is also not necessary. That is, this layout model performs well in the dynamic demand situation even though the demand is varying from period to period. Also, it can be seen that the proposed method is providing good result under varying part handling factor. For better illustration and comparison, the results of all these cases and the results of Chan et al. (2004) are provided in Table 11. 6.2.2. Results of case study from Balakrishnan and Cheng This section describes the performance of robust layout model using the data obtained from Balakrishnan and Cheng. These data set consists of eight problems in each of the six situations (6 – departments 5 and 10 periods; 15 – departments 5 and 10 periods; and 30 – departments 5 and 10 periods) and thus a total of 48 problems which are solved using proposed robust model for DPLP. The SA is used as solution procedure to the robust layout model. The SA is run for 20 replications and the details of the results are shown in Table 12. A best layout is one which minimizes cost over the planning horizon. Adaptive approach layout results available for the data set from Balakrishnan and Cheng are compared with the results of the robust strategy in the present paper. The results of the robust approach solution values are not significantly different compared to the values obtained by Balakrishnan and Cheng, although there is no relocation of facilities and no operational disruptions in any periods of planning horizon in the robust method. The following research papers on adaptive approach are used for comparison of results of robust approach for Balakrishnan and Cheng’s data set. 1. Conway and Venkataramanan (1994) – Conway and Venkataramanan Genetic Algorithm (CVGA). 2. Balakrishnan and Cheng (2000) – Nested Loop Genetic Algorithm (NLGA). 3. Baykasoglu and Gindy (2001) – Simulated Annealing (SA). 4. Balakrishnan et al. (2003) – Genetic Algorithm with Dynamic Programming (GADP). Some comparison below involve parent pool generated randomly which is represented as GADP(R) and generated with the Urban’s method as GADP(U). 5. Baykasoglu et al. (2006) – Ant colony. 6. McKendall et al. (2006) – Modified SA-I and SA-II. For the robust approach, the solution values show (for 8 sets of data in each size of problem) 0.32–1% deviation among periods of planning horizon and 2.1–9.89% deviation for the entire planning horizon from the best results of adaptive approach. The results also show that the deviation is less for 5-period problems compared to 10-period problems of various sizes. These results are shown in the Tables 13–18. The Total Penalty Cost (TPC) suggested by Braglia et al. (2003) is used to test the suitability of the suggested layout to be a robust layout for the given data set. They suggested that TPC should be less than 15% to go for robust strategy. The TPC is defined as the minimum re-layout cost acceptable to support an agile strategy. The TPC equation is Table 10 Robust layout material handling cost of each period (MHCp) and TMHC of planning horizon when ki;jk is varying and Bi,jk = 1. Period p=1 p=2 p=3 p=4 p=5 Planning horizon Material handling cost 90,850 85,200 98,050 95,800 68,050 437,950 Table 11 Comparison of results of Chan et al. (2004) and the results of robust approach for various problem cases. Problem case Chan et al. (2004) MAIN algorithm Robust layout cost with SA % Change in TMHC Yaman et al. (1993) with ki;jk ¼ 1, Bi,jk = 1. Yaman et al. (1993) with varying ki;jk and Bi,jk = 1. Chan et al. (2004) [S1] with ki;jk ¼ 1, Bi,jk = 1. Chan et al. (2004) [S1] with varying ki;jk and Bi,jk = 1 Chan et al. (2004) [S2] with ki;jk ¼ 1, Bi,jk = 1. Chan et al. (2004) [S2] with varying ki;jk and Bi,jk = 1. 145,700 456,550 289,900 6,052,000 3,243,600 6,983,514 137,000 436,450 289,900 5,983,600 3,243,600 6,937,900 5.97 4.40 0 1.13 0 0.65 820 V. Madhusudanan Pillai et al. / Computers & Industrial Engineering 61 (2011) 813–823 Table 12 Robust approach results for the data of Balakrishnan and Cheng using proposed method. Description Data 1 Best Data 2 Worst Average Best Data 3 Worst Average Best Data 4 Worst Average Best Worst Average Results for data sets 1–4 6-departments 5106,419 106,419 106,419 105,731 105,731 105,731 107,650 107,650 107,650 108,260 108,260 108,260 periods 6-departments 10220,776 220,776 220,776 217,412 217,412 217,412 219,024 219,024 219,024 217,350 217,350 217,350 periods 15-departments 5506,847 507,248 506,967.3 500,284 501,893 500,879.3 508,011 510,007 508,110.8 503,699 504,506 503,974.3 periods 15-departments 1,059,100 1,060,304 1,059,732 1,022,447 1,023,017 1,022,561 1,068,402 1,069,885 1,069,517 1,054,997 1,056,157 1,055,352 10-periods 30-departments 5579,704 580,820 580,072.2 576,350 577,370 576,844 586,831 588,554 587,613.5 584,318 585,359 584,752.8 periods 30-departments 1,172,691 1,174,563 1,173,088 1,182,286 1,184,403 1,182,954 1,188,620 1,190,966 1,189,691 1,198,487 1,199,788 1,199,190 10-periods Results of data sets 5–8 Description Data 5 Data 6 Data 7 Data 8 6-departments 5108,188 108,188 108,188 107,765 107,765 107,765 108,114 108,114 108,114 107,248 107,248 107,248 periods 6-departments 10217,142 217,142 217,142 217,397 217,397 217,397 219,788 219,788 219,788 220,144 220,144 220,144 periods 15-departments 5502,622 502,913 502,796.6 499,891 500,325 499,912.7 502,919 504,474 503,217.6 507,970 507,970 507,970 periods 15-departments 1,051,395 1,053,081 1,051,651 1,057,543 1,060,375 1,058,380 1,037,066 1,038,925 1,037,227 1,040,450 1,040,450 1,040,450 10-periods 30-departments 5570,492 571,736 570,958.9 572,782 574,638 573,285.2 571,703 573,072 572,355 596,835 598,280 597,134.2 periods 30-departments 1,198,674 1,201,556 1,199,215 1,202,033 1,203,802 1,202,968 1,210,573 1,212,569 1,211,127 1,209,088 1,211,241 1,210,039 10-periods Table 13 Comparison of adaptive and robust approach results for the 8-data set of 6 Department and 5 Period problems. Description Data1 Data 2 Data 3 Data 4 Data 5 Data 6 Data 7 Data 8 CVGA (Adaptive) NLGA (Adaptive) GADP (Adaptive) SA (Adaptive) Ant colony (Adaptive) Modified SA (Adaptive) SA (Robust) Best cost % Deviation of SA (Robust) value from best 108,976 106,419 106,419 107,249 106,419 106,419 106,419 106,419 0.00 105,170 104,834 104,834 105,170 104,834 104,834 105,731 104,834 0.86 104,520 104,320 104,529 104,800 104,320 104,320 107,650 104,320 3.19 106,719 106,515 106,583 106,515 106,509 106,399 108,260 106,399 1.75 105,628 105,628 105,628 106,282 105,628 105,628 108,188 105,628 2.42 105,606 104,053 104,315 103,985 104,053 103,985 107,765 103,985 3.64 106,439 106,978 106,447 106,447 106,439 106,439 108,114 106,439 1.57 104,485 103,771 103,771 103,771 103,771 103,771 107,248 103,771 3.35 Table 14 Comparison of adaptive and robust approach results for the 8-data set of 6 department and 10 period problems. Description Data1 Data 2 Data 3 Data 4 Data 5 Data 6 Data 7 Data 8 CVGA (Adaptive) NLGA (Adaptive) GADP (Adaptive) SA (Adaptive) Ant colony (Adaptive) Modified SA (Adaptive) SA (Robust) Best cost % Deviation of SA (Robust) value from best cost 218,407 214,397 214,313 215,200 217,251 214,313 220,776 214,313 3.02 215,623 212,138 212,134 214,713 216,055 212,134 217,412 212,134 2.49 211,028 208,453 207,987 208,351 208,185 207,987 219,024 207,987 5.31 217,493 212,953 212,741 213,331 212,951 212,530 217,350 212,530 2.27 215,363 211,575 210,944 213,812 211,076 210,906 217,142 210,906 2.96 215,564 210,801 210,000 211,213 210,277 209,932 217,397 209,932 3.56 220,529 215,685 215,452 215,630 215,504 214,252 219,788 214,252 2.58 216,291 214,657 212,588 214,513 214,621 212,588 220,144 212,588 3.55 TPC ¼ PP P robust  Pp¼1 C optimum p p¼1 C p PP robust C p¼1 p  100 ð22Þ where C robust is the material handling cost when robust layout is p applied to period p, C optimum is Material handling cost of optimum p layout of the period p. They established that the maximum acceptable limit of TPC for a layout to be robust is 15%. Using Eq. (22) the TPC is calculated for all the data set and, it is within the specified limit and hence we can infer that the robust layout strategy is suitable for the given data set. This indicates that the suggested layout is suitable as robust layout for the given data sets. A poor TPC (high value) of a layout may be interpreted as the need for an agile plant, suitable 821 V. Madhusudanan Pillai et al. / Computers & Industrial Engineering 61 (2011) 813–823 Table 15 Comparison of adaptive and robust approach results for the 8-data set of 15 department and 5 period problems. Description Data1 Data 2 Data 3 Data 4 Data 5 Data 6 Data 7 Data 8 CVGA (Adaptive) NLGA (Adaptive) GADP (R) (Adaptive) GADP (U) (Adaptive) SA (Adaptive) Ant colony (Adaptive) Modified SA-I & SA-II (Adaptive) SA (Robust) Best cost % Deviation of SA (Robust) value from best cost 504,759 511,854 493,707 484,090 484,695 501,447 480,453 506,847 480,453 5.49 514,718 507,694 494,476 485,352 486,141 506,236 484,761 500,284 484,761 3.20 516,063 518,461 506,684 489,898 496,617 512,886 488,748 508,011 488,748 3.94 508,532 514,242 500,826 484,625 490,869 504,956 484,405 503,699 484,405 3.98 515,599 512,834 502,409 489,885 491,501 509,636 487,882 502,622 487,882 3.02 509,384 513,763 497,382 488,640 491,098 508,215 487,147 499,891 487,147 2.62 512,508 512,722 494,316 489,378 491,350 508,848 486,779 502,919 486,779 3.32 514,839 521,116 500,779 500,779 496,465 512,320 490,812 507,970 490,812 3.50 Table 16 Comparison of adaptive and robust approach results for the 8-data set of 15 department and10 period problems. Description Data1 Data 2 Data 3 Data 4 Data 5 Data 6 Data 7 Data 8 CVGA (Adaptive) NLGA (Adaptive) GADP (R) (Adaptive) GADP (U) (Adaptive) SA (Adaptive) Ant colony (Adaptive) Modified SA-I & SA-II (Adaptive) SA (Robust) Best cost % Deviation of SA (Robust) value from the best cost 1,055,536 1,047,596 1,004,806 987,887 950,910 1,017,741 979,468 1,059,100 950,910 11.38 1,061,940 1,037,580 1,006,790 980,638 947,673 1,016,567 978,065 1,022,447 947,673 7.89 1,073,603 1,056,185 1,012,482 985,886 968,027 1,021,075 982,396 1,068,402 968,027 10.37 1,060,034 1,026,789 1,001,795 976,025 950,701 1,007,713 972,797 1,054,997 950,701 10.97 1,064,692 1,033,591 1,005,988 982,778 948,470 1,010,822 977,188 1,051,395 948,470 10.85 1,066,370 1,028,606 1,002,871 973,912 948,630 1,007,210 967,617 1,057,543 948,630 11.48 1,066,617 1,043,823 1,019,645 982,872 965,844 1,013,315 979,114 1,037,066 965,844 7.37 1,068,216 1,048,853 1,010,772 987,789 956,170 1,019,092 983,672 1,040,450 956,170 8.81 Table 17 Comparison of adaptive and robust approach results for the 8-data set of 30 departments and 5 periods problems. Description Data1 Data 2 Data 3 Data 4 Data 5 Data 6 Data 7 Data 8 CVGA (Adaptive) NLGA (Adaptive) GADP (R) (Adaptive) GADP (U) (Adaptive) SA (Adaptive) Ant colony (Adaptive) Modified SA-I & SA-II (Adaptive) SA (Robust) Best cost % Deviation of SA (Robust) value from best cost 632,737 611,794 603,339 578,689 562,405 604,408 576,039 579,704 562,405 3.08 647,585 611,873 589,834 572,232 569,251 604,370 568,095 576,350 568,095 1.45 642,295 611,664 592,475 578,527 564,464 603,867 573,739 586,831 564,464 3.96 634,626 611,766 586,064 572,057 552,684 596,901 566,248 584,318 552,684 5.72 639,693 604,564 580,624 559,777 559,596 591,988 558,460 570,492 558,460 2.15 637,620 606,010 587,797 566,792 592,515 599,862 566,077 572,782 566,077 1.18 640,482 607,134 588,347 567,873 582,409 600,670 567,131 571,703 567,131 0.81 635,776 620,183 590,451 575,720 578,549 610,474 573,755 596,835 573,755 4.02 Table 18 Comparison of adaptive and robust approach results for the 8-data set of 30 departments and10 periods problems. Description Data1 Data 2 Data 3 Data 4 Data 5 Data 6 Data 7 Data 8 CVGA (Adaptive) NLGA (Adaptive) GADP (R) (Adaptive) GADP (U) (Adaptive) SA (Adaptive) Ant colony (Adaptive) Modified SA-I & SA-II (Adaptive) SA (Robust) Best cost % Deviation of SA (Robust) value from best cost 1,362,513 1,228,411 1,194,084 1,169,474 1,122,154 1,223,124 1,163,222 1,172,691 1,122,154 4.50 1,379,640 1,231,978 1,199,001 1,168,878 1,120,182 1,231,151 1,161,521 1,182,286 1,120,182 5.54 1,365,024 1,231,829 1,197,253 1,166,366 1,125,346 1,230,520 1,156,918 1,188,620 1,125,346 5.62 1,367,130 1,227,413 1,184,422 1,154,192 1,120,217 1,200,613 1,145,918 1,198,487 1,120,217 6.99 1,356,860 1,215,256 1,179,673 1,133,561 1,158,323 1,210,892 1,126,432 1,198,674 1,126,432 6.41 1,372,513 1,221,356 1,178,091 1,145,000 1,111,344 1,239,255 1,145,146 1,202,033 1,111,344 8.16 1,382,799 1,212,273 1,186,145 1,145,927 1,128,744 1,248,309 1,140,744 1,210,573 1,128,744 7.25 1,383,610 1,245,423 1,208,436 1,168,657 1,136,157 1,231,408 1,161,437 1,209,088 1,136,157 6.42 for frequent relocation of the facility. The TPC for all data sets is shown in Table 19. 7. Conclusions In this research paper a SA based meta-heuristic is developed for solving layout formation problems. The developed approach has given optimal values to case studies from Yaman et al. (1993) and for the problem instances obtained from QAPLIB website. In addition to the SA approach, a robust layout procedure is developed for dynamic environment, which generate a layout for an expected demand scenario or expected flow matrix. The robust layout does not change from period to period of the planning horizon. Even though, this layout may not be optimal for any period in the planning horizon, the performance of this layout over the entire period of the planning horizon is better. The robust approach has been applied to the problems from Chan et al. (2004). The results show that the developed robust method provides better performance compared to Chan et al. (2004) for cell layout problems. For standard DPLP problems from Balakrishnan and Cheng, the 822 V. Madhusudanan Pillai et al. / Computers & Industrial Engineering 61 (2011) 813–823 Table 19 Total Penalty Cost (in percentage) for the proposed robust layout. Problem size and the number of periods Description Data 1 Data 2 Data 3 Data 4 Data 5 Data 6 Data 7 Data 8 6-Departments 5-Periods a b c 106,419 98,229 7.70 105,731 98,749 6.60 107,650 98,229 8.75 108,260 100,032 7.60 108,188 98,996 8.50 107,765 98,326 8.76 108,114 99,183 8.26 107,248 98,792 7.88 6-Departments 10-Periods a b c 220,776 202,561 8.25 217,412 199,794 8.10 219,024 194,900 11.01 217,350 199,351 8.28 217,142 199,125 8.30 217,397 197,230 9.28 219,788 198,439 9.71 220,144 199,189 9.52 15-Departments 5-Periods a b c 506,847 454,741 10.28 500,284 463,603 7.33 508,011 463,497 8.76 503,699 458,070 9.06 502,622 462,155 8.05 499,891 461,424 7.70 502,919 462,000 8.14 507,970 465,704 8.32 15-Departments 10-Periods a b c 1,059,100 921,566 12.99 1,022,447 926,196 9.41 1,068,402 922,129 13.69 1,054,997 916,811 13.10 1,051,395 916,995 12.78 1,057,543 912,386 13.73 1,037,066 921,472 11.15 1,040,450 926,832 10.92 30-Departments 5-Periods a b c 579,704 528,688 8.80 576,350 527,323 8.51 586,831 530,222 9.65 584,318 522,299 10.61 570,492 521,093 8.66 572,782 527,647 7.88 571,703 529,269 7.42 596,835 533,449 10.62 30-Departments 10-Periods a b c 1,172,691 1,054,922 10.04 1,182,286 1,055,902 10.69 1,188,620 1,056,650 11.10 1,198,487 1,045,993 12.72 1,198,674 1,044,651 12.85 1,202,033 1,056,419 12.11 1,210,573 1,063,984 12.11 1,209,088 1,060,993 12.25 a – Robust with SA, b – Agile with zero relocation cost (layout of each period is developed independently), c – Total Penalty Cost in%. MHC for the layouts from the robust method are not significantly different from the best results for the adaptive approach. The robust approach has the advantage of no relocation of facilities in the periods of planning horizon and hence no disruptions of the operations. Also, robust model is computationally efficient compared to the adaptive model. The robustness of the layouts is measured with the Total Penalty Cost (TPC) as suggested in Braglia et al. (2003). TPC is calculated for 48-data set from Balakrishnan and Cheng and it is within the percentage mentioned in Braglia et al. (2003). This indicates that the suggested layout is suitable as robust layout for the given data sets. That is, the suggested robust layout procedure provides good robust layout for the given problem situations. Some directions for future studies are: (i) The present algorithms for DPLP consider only single objectives. Multiple objective cases of DPLP can also be modeled and solved. 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