DESIGN OF ROBUST LAYOUT FOR DYNAMIC

PLANT LAYOUT PROBLEM

PRESENTED BY:

SREEPATHY R NAIK

IEM 2014-16

NIT CALICUT

INTRODUCTION

• 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.

• Poorly designed: reduced productivity, increased work-in-process, increased manufacturing lead time, disordered material handling

• Facility layout problem is focused on reducing the Material Handling Cost (MHC)

• MHC assumes about 20–50% of the total operating cost of the facility layout.

• Efficient facilities planning can reduce these costs by at least 10–30% and thus increase the productivity

INTRODUCTION (contd..)

• 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

• Markets are heterogeneous and volatile in nature hence, manufacturing facilities must be able to exhibit high levels of flexibility and robustness

• demand is constant with time: Static Plant Layout Problem (SPLP)

• Fluctuations in product demand, changes in product mix, introduction of new products, and discontinuation of existing products: DPLP

Approaches followed to solve DPLP

• Adaptive/flexible approach: Layout will accommodate changes from time to time with low rearrangement cost; Machines can be easily relocated

• Robust approach: Rearrangement costs are too high, hence minimize MHC in all periods using single layout. Multiple production scenarios of a single period problem and for multi-period problems

Literature Review

Author Year Problem statement content

Rosenblatt 1986 Solution procedure to DPLP with adaptive approach

Small size problems

Kouvelis et al. 1992 Algorithms for robust single and multiple period layout planning for manufacturing systems

Developed algorithms to generate robust layout

Lackson&Enscore 1993 Quadratic assignment algorithms for the dynamic layout problem

Exact and heuristic methods

Balakrishnan and Cheng 1998 Dynamic layout algorithms: a state of the art survey

categorize different algorithms for equal andunequal sized departments

Literature Review (contd.)

Author Year Problem statement content

Braglia et al. 2003 Layout design in dynamicenvironments: Stratégies and quantitative indices

Layout effectiveness measures

Baykasoglu et al. 2006 An ant colony algorithm for solving budget constrained and unconstrained dynamic facility layout problems

Solved DPLP using ant colony optimization

Pillai and Subbarao 2008 robust cellular manufacturing system design for dynamic part population using a genetic algorithm

Forming machine cells, which can handle all the changes in demands and product mixes withoutany relocations.

Nomenclature

• 𝜁𝐸 total traveling score for average demand scenario or this represents the

objective function for robust design

• λ𝑖,𝑗𝑘 part handling factor of part i when transported from facility j to facility k

• ω 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

• 𝑓𝑝,𝑗𝑘 part flow weight from facility j to facility k in period p

• 𝑥𝑗𝑟 1 if facility j is assigned to location r, 0 otherwise

• 𝑥𝑝𝑗𝑟 1 if facility j is assigned to location r in period p, 0 otherwise

• 𝐴𝑝𝑗,𝑟𝑠 relocation cost of facility j in period p if it is shifted from location r to s

• 𝐵𝑖𝑗𝑘 number of parts i per transportation when transported from facility j to facility k

• 𝐷𝑖 demand of part i, where i = 1, 2, . . . , N

• 𝐷𝑝,𝑖 demand of part i in period p, where p = 1, 2, . . . P

• 𝐷𝐸𝑖 average demand of part i, where i = 1, 2, . . . N

• 𝑀𝐻𝐶𝑝 material handling cost when the given layout is used in period p

• N total number of parts

• P number of periods in planning horizon

• 𝑅𝑗𝑘 rectilinear distance between facilities j and k in a given 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

• 𝑊𝐸𝑗𝑘 average part flow weight from facility j to facility k

• 𝑌𝑝𝑗,𝑟𝑠 1 if facility j is shifted from location r to s in period p, 0 otherwise

• 𝑍𝐴 objective function for dynamic layout problems or it represents the traveling score plus the relocation cost under dynamic demand situations

• 𝑍𝑆 objective function for static layout problems or it represents the traveling score for static layout problems

Simulated Annealing Algorithm

• Used for large combinatorial optimization problem

• Probabilistic method that avoid being stuck at local minima

• Goal: bring the system from arbitrary initial state to a state with minimum possible energy

• SA heuristic considers some neighbor s’ of current state s in each step

• If movement to s’ is not economical, then probabilistically decides to move the system to state s’

“

”

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 global energy

minimum of a perfect crystal

Annealing process

SA Parameters

• Configuration changes: obtained by swapping operation

• Initial temperature: set such that 90% configuration changes are accepted in starting stage

• Cooling ratio: the quality of solution and repeatability were found to be best at 0.98

• No of samples in each temperature level: L=a𝑚2 ,a is the constant having value 0.8 to 1 ,m is no of facilities

• Termination condition or final temperature: it is set to 3 , that is solution is not improving below temperature 3

• Configuration change acceptance criteria: Metropolis criterion ;it involves following cases

• If configuration change result in a net reduction of the objective function ,then it is accepted

• If configuration change increases the objective function, then it is accepted with probability R[0,1]< exp(-DE/Ti)

Parameters are chosen so that the system ultimately tends to move to a

state of lowest energy

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

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.. 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.

• 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.

Model for Adaptive Approach

• Minimize 𝑍𝐴 = 𝑝=1𝑝 𝑗=1

𝑚 𝑟=1𝑚 𝑘=1

𝑚 𝑠=1𝑚 𝑓𝑝,𝑗𝑘𝑑𝑟𝑠𝑥𝑝,𝑗𝑟𝑥𝑝,𝑘𝑠 +

𝑝=2𝑝 𝑗=1

𝑚 𝑟=1𝑚 𝑠=1

𝑚 𝐴𝑝,𝑗𝑟𝑠𝑌𝑝,𝑗𝑟𝑠

S.T.

• 𝑟=1𝑚 𝑥𝑝,𝑗𝑟 = 1 ∀ j=1,2,…..m ∀ p=1….p

• 𝑗=1𝑚 𝑥𝑝,𝑗𝑟 = 1 ∀ r=1,2,…..m ∀ p=1….p

• 𝑥𝑝,𝑗𝑟 𝜖 {0,1} ∀ j,r=1,2,…..m & ∀ p=1,2,…..m

• 𝑌𝑝,𝑗𝑟𝑠 = 𝑥(𝑝−1),𝑗𝑟𝑥𝑝,𝑗𝑠 ∀ j,r,s=1,2,…..m & ∀ p=2,…..p

• 𝑑𝑟𝑠 = 𝑥𝑟 − 𝑥𝑠 + 𝑦𝑟 − 𝑦𝑠

Proposed Robust approach to DPLP

• 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

• 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

• Adaptive approach for the dynamic layout problems requires (m!)P computational effort, where as the proposed robust approach requires only m! (same as static layout)

• when the facilities are difficult to relocate, rearrangement costs are too high and the chances of operational disruption are high due to rearrangement.

• Input

• number of parts to be manufactured,

• demand of parts in various periods

• machine sequence or route sheet of parts

• part-handling factor

• distance between locations

• 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

Model for ROBUST Approach

• Minimize 𝜁𝐸 = 𝑗=1𝑚 𝑟=1

𝑚 𝑘=1𝑚 𝑠=1

𝑚 𝑊𝐸𝑗𝑘𝑑𝑟𝑠𝑥𝑗𝑟𝑥𝑘𝑠 [1]

S.T.

• 𝑟=1𝑚 𝑥𝑗𝑟 = 1 ∀ j=1,2,…..m [2]

• 𝑗=1𝑚 𝑥𝑗𝑟 = 1 ∀ r=1,2,…..m [3]

• 𝑥𝑗𝑟 𝜖 {0,1} ∀ j=1,2,…..m & ∀ r=1,2,…..m [4]

• 𝐷𝐸𝑖 = 𝑝=1𝑝

𝐷𝑝,𝑖

𝑃[5]

• 𝑊𝐸𝑗𝑘 = 𝑖=1𝑛 𝐷𝐸𝑖

𝐵𝑖𝑗,𝑘λ𝑖,𝑗𝑘 ∀ j=1,2,…..m & ∀ k=1,2,…..m [6]

ROBUST LAYOUT MHC Calculation

• 𝑓𝑝,𝑗𝑘 = 𝑖=1𝑁 𝐷𝑝,𝑖

𝐵𝑖𝑗,𝑘λ𝑖,𝑗𝑘 ∀ j=1,2,…..m , ∀ k=1,2,…..m & ∀ p=1,…..p [7]

• 𝑀𝐻𝐶𝑃 = 𝜔 x 𝑗=1𝑚 𝑘=1

𝑚 𝑅𝑗𝑘𝑓𝑝,𝑗𝑘 [8]

• 𝑇𝑀𝐻𝐶 = 𝑃=1𝑃 𝑀𝐻𝐶𝑃 [9]

• 𝑅𝑗𝑘 = 𝑋𝑗 − 𝑋𝑘 + 𝑌𝑗 − 𝑌𝑘 [10]

Solution methodology

• 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).

REFERENCES

• Aiello, G., & Enea, M. (2001). Fuzzy approach to the robust facility layout in uncertain production environments. International Journal of Production Research,39(18), 4089–4101.

• Balakrishnan, J., & Cheng, C. H. (1998). Dynamic layout algorithms: A state-of-theart survey. Omega The International Journal of Management Science, 26(4),507–521

• Balakrishnan, J., & Cheng, C. H. (2000). Genetic search and the dynamic layout problem. Computers and Operations Research, 27(6), 587–593

• Baykasoglu, A., Dereli, T., & Sabuncu, I. (2006). An ant colony algorithm for solving budget constrained and unconstrained dynamic facility layout problems. Omega The International Journal of Management Science, 34, 385–396

• Braglia, M., Simone, Z., & Zavanella, L. (2003). Layout design in dynamic environments: Strategies and quantitative indices. International Journal of Production Research, 41(5), 995–1016.

• Braglia, M., Simone, Z., & Zavanella, L. (2003). Layout design in dynamic environments: Strategies and quantitative indices. International Journal of Production Research, 41(5), 995–1016

• Pillai, V. M., & Subbarao, K. (2008). A robust cellular manufacturing system design for dynamic part population using a genetic algorithm. International Journal of Production Research, 46(18), 5191–5210

• Rosenblatt, M. J. (1986). The dynamics of plant layout. Management Science, 32(1),76–86

• Wilhelm, M. R., & Ward, T. L. (1987). Solving quadratic assignment problems by simulated annealing. IIE Transactions, 19(1), 107–117.

• Yaman, A., Gethin, D. T., & Clarke, M. J. (1993). An effective sorting method for facility layout construction. International Journal of Production Research, 31(2),413–427.