an approach for finding the optimal location of elevator in multi- floor ...

Proceedings of the 37th International Conference on Computers and Industrial Engineering, October 20-23, 2007, Alexandria, Egypt, edited by M. H. Elwany, A. B. Eltawil

AN APPROACH FOR FINDING THE OPTIMAL LOCATION OF ELEVATOR IN MULTI- FLOOR LAYOUT PROBLEMS Ehab Abdelhafiez Mechanical Design and Production Department Cairo University Egypt [email protected]

M. Adel Shalaby Tarek Nayer Mechanical Design and Production Department Cairo University Egypt [email protected]

ABSTRACT The Multi-floor layout problem is defined as the problem of finding a non-overlapping location for each department on the floors plan and a unified location for the elevator in all floors such that the overall material handling cost is minimized. This paper addresses the problem of finding the best position of a single elevator or several elevators grouped together that serve the vertical flow for a given multi-floor layout. The proposed technique presented in this paper simulates the single-location problem known in literature after projecting the centroids of all departments on one plan. It solves the problem mathematically in either the rectilinear or the Euclidean distance situations giving a unique optimal solution to the problem. The proposed technique gave better results to benchmarked problems when compared with most known techniques.

KEYWORDS Layout, Facilities Planning, Multi-floor Layout, Elevator Location, SFC, CRAFT

1. INTRODUCTION The multi-floor layout problem involves the existence of many single floor layouts, one for each floor with the use of; stairs, elevators, or vertical conveyors to move material flows vertically across the building of the facility. The positions of these stairs, elevators, or vertical conveyors affect the material handling cost. The problem of finding the optimum location of elevator(s) concerns minimizing the distances traveled by material flows between departments in

different floors, to/from/through the elevator position. In most of the surveyed approaches, the elevators' positions have been assigned in advance, and then the procedure of finding the layout is processed regardless the number of elevators used. In MULTIHOPE "Kochhar J.S., and Heragu S.S. (1998)" and MSLP "Kaku B.K., et al (1988)", a single central position for one elevator or a group of elevators is used, while in SPACECRAFT "Johnson R.V. (1982)"; MULTIPLE "Bozer Y.A., et al (1994)"; SABLE "Meller R.D., and Bozer Y.A. (1996)"; and STAGES "Meller R.D., and Bozer Y.A. (1997)", a position for each elevator is specified in the initial layout. In all of the algorithms that use more than one elevator for vertical transportation, a decision of determining the shortest path through the existing elevators is made, to route each material flow between two departments on different floors. MUSE "Matsuzaki K., et al (1999)", up to the authors knowledge, is the only approach that was found to be used to solve the layout problem taking into account the search for the number and locations of the elevators to be used. The layout problem is solved using simulated annealing through a slicing tree algorithm, then the number and positions of the elevators are reconsidered using Genetic Algorithms. The two steps of finding the layout and then reconsidering the number and positions of the elevators are repeated until no further improvement is achieved, or a certain number of iterations are reached. In this paper, a solution technique is proposed to be used in finding the optimal location of the elevator in a given multi-floor layout. This means that the departments' centroids are already estimated, and distances in-between are known.

1485

Proceedings of the 37th International Conference on Computers and Industrial Engineering, October 20-23, 2007, Alexandria, Egypt, edited by M. H. Elwany, A. B. Eltawil

This solution technique considers the material flow is to be calculated from the geometric centroid of each department, so, it solves the problem considering both rectilinear and Euclidean distance functions. In addition, this solution does not allow splitting of departmental areas over floors.

c21: handling cost of one item per unit distance out/to department (a21), h21: horizontal distance between department (a11) and the elevator location. Then, the above formula can be represented as: i=2

2. PROBLEM FORMULATION Figure 1, shows the case of transferring materials "f11,21" from department "1" at floor number "1", denoted as "a11", to department "1" at floor number "2", denoted as "a21", through the elevator.

a11

f11,21

a21

a31

Elevator

Elevator

Elevator

Z Y

X

Figure 1 material handling between different floors using elevator The cost of transferring (handling) "f11,21" can be estimated as the summation of handling cost through floor-1, handling cost through elevator, and handling cost through floor-2. Handling cost = (c11*f11,21*h11)+(s12*t12)+(c21* f11,21*h21) Where: c11: handling cost of one item per unit distance out/to department (a11), f11, 21: number of items to be handled between departments (a11) and (a21), h11: horizontal distance between department (a11) and the elevator location, s12: cost of one travel of the elevator from floor-1 to floor-2, t12: number of elevator travels required to transfer "f11,21" from floor-1 to floor-2,

Handling cost = ∑ (ci1 * f11,21* hi1) + (s12 * t12) i=1

Now, for calculating the total handling cost considering all departments and all floors: j=m i=n

l=m k=m

j=1 i=1

l=1 k=1

Handling cost = ∑∑ (cij * fij* hij) + ∑∑ (skl * tkl) Where: m: number of floors, n: number of departments in floor j, fij: number of units transferred between department-i and elevator at floor-j, skl: cost of one vertical travel of the elevator from floor-k to floor-l, tkl: number of vertical travels of the elevator between floor-k and floor-l, The vertical cost of traveling does not contribute in the problem of finding the elevator position. The vertical projection of an elevator is the only view that involves the (x, y) position of that elevator, and so, that position is only influenced by horizontal distances of material flows that pass through the elevator, and come out and to departments in different floors. Any (x, y) position of an elevator will result in the same vertical traveling cost for a certain layout.

3. SOLUTION ALGORITHM The effect of elevator position is relevant to horizontal distances traveled by material flows from and to the elevator position which acts as a contact point between any two departments on two different floors and exchange material flows in-between and hence, there is a similarity between finding an elevator position and the well-known single-facility location problem where the objective is to find where to place a new facility among some existing facilities, such that some cost function relating to transportation or handling between the existing facilities and the new facility is minimized. However, some differences exist where; in the location problem all the existing facilities are on the same ground without any interrelationships between each other. In

1486

Proceedings of the 37th International Conference on Computers and Industrial Engineering, October 20-23, 2007, Alexandria, Egypt, edited by M. H. Elwany, A. B. Eltawil

addition, the existing location model for solving the problem deals only with Euclidean distances, and the proof of its optimality was built on that basis. In solving the elevator problem, the following adaptations are required to convert the elevator problem to a location one. These steps are as follow: 1. Discard all relations between departments on the same floor, as those relations don’t affect the elevator location. 2. For each department, all of its flows to (or from) departments on other floors are pooled together as one flow between that department and the elevator. 3. Ignore vertical (Z) Coordinate of all of departments, and take only their x, y centroids as their locations. This step may be looked at as a plan view projection process for all the departmental centroids. Through the previous steps, a single facility location problem can be solved with all departments' centroids projected on the x-y plane, and having quantitative flows with the new facility, which is the elevator. The problem previously stated shall be presented next and solved applying both Euclidean and Rectilinear distance assumptions. 3.1 Elevator Location in Euclidean coordination system

C

= ∑ C i ( x − xi ) + ( y − y i )

= 0, and

∂x

2

(1)

i

It will be recalled from calculus that a necessary condition for a function of two variables to have a minimum is that first-partial derivatives of the function with respect to each of the variables be zero. Therefore, it is required that:

∂C ∂y

=0

This yield:

∂C

Ci (x − xi )

n

=∑

∂x

(x − xi )2 + ( y − yi )2

i =1

∂C

Ci ( y − yi )

n

=∑

∂y

(x − xi )2 + ( y − yi )2

i =1

=0

(2)

=0

(3)

Equations (2) and (3) can not be solved explicitly for (x, y). However, an iterative numerical scheme solving (2), and (3), can be derived as follows. For simplicity let’s denote that:

d i = ( x − xi ) 2 + ( y − y i ) 2 So equations 2), and 3) can be written as:

∂C

n

=∑

∂x

i =1

∂C

n

=∑

∂x

The elevator problem is analogous to the well-known “Single Source Continuous Location Problem, Sule D.R. (1944)", which seeks the determination of a source (elevator) position (x, y) such that the total traveling cost from and to the source is minimized. In order to develop the model, the following is defined: ƒ xi , y i : are coordinates of departmental centroids, i =1 … n : are coordinate of the elevator to be ƒ x, y determined ƒ Ci : is summation of handling cost of the flows passing through the elevator, to and from department i per unit distance (cij * fij). Since it is required to minimize the total horizontal traveling cost from and to the elevator, the expression of that cost, is noted as: 2

∂C

i =1

C i ( x − xi ) =0 di

(4)

Ci ( y − yi ) =0 di

(5)

Then equations (4) and (5) can be written as: n

x∑

Ci

y∑

Ci

i =1 n

i =1

n

di

=∑

C i xi

di

=∑

Ci yi

i =1 n

i =1

(6)

di

(7)

di

Formally, x, and y can be solved in equations (6), and (7) to yield: n

∑C x i

x=

i =1 n

∑

Ci

i =1 n

i

i

y=

∑ i =1

(8)

di

∑C y i =1 n

di

Ci

i

di

(9)

di

Equations (8) and (9) are nonlinear complex equations as x is a function of x and y and y is a function of x and y too ( d i is function of x and y). So the solution of these equations has not been obtained analytically, and the numerical methods are applied instead.

1487

Proceedings of the 37th International Conference on Computers and Industrial Engineering, October 20-23, 2007, Alexandria, Egypt, edited by M. H. Elwany, A. B. Eltawil

The following iterative algorithm is the one used in literature to obtain the values of x, and y of the elevator position and has proven to give a converging solution "Sule D.R. (1944)". 1. Initiate values for x, and y position of the elevator, let ( x o , y o ) be the centroids of a floor,

( x , y ) = (floor length/2, floor width/2) 2. Let ( x i , y i ) = ( x o , y o ) 3. Update the values of ( x i , y i ) by substituting ( x i , y i ) in equation (8), and equation (9), to obtain ( x i +1 , y i +1 ) 4. If ( x i +1 , y i +1 ) = ( x i , y i ) (in the order of 2 digits o

And, to minimize: n

C = ∑C [ y − y y

i

i =1

(12)

i

Where

C = C +C x

y

o

after the decimal point) then stop, a solution is obtained, otherwise go to step 2 The previous algorithm was proven to be convergent to an optimal solution for the single facility location problem, and under the stated assumptions, it gives the optimal location of the elevator too. Better starting points (near to the solution) leads to a faster obtained solution in a less number of iterations. 3.2 Elevator Location in Rectilinear coordination system The nature of this problem is different than the previous one in the essence of cost function, because of the difference in handling the distance here. Same as before, centroids of the departments are regarded ignoring Z-coordinate of all, and total interaction between each department and the elevator is calculated as the summation of flows in and out of each department going to a different floor. Since it is required to minimize the horizontal traveling cost from and to the elevator, the expression of the cost can be noted as: i

= ∑ C i [ x − xi + y − y i ]

C

(10)

i =1

As seen the nature of the function states that it can not be minimized by the simple differentiation method as before, but some different handling is required. The problem presented is well-known in literature [9], and can be handled into two separate problems, one for x-coordinates, and the other for ycoordinates. To minimize: n

C = ∑C [ x − x x

i =1

i

i

(11)

Equations (2) and (3) can be expressed in a different way:

C

x

C

y

=

∑ C (x − x ) + ∑ C (x

x > xi

=

i

i

x ≤ xi

i

∑C (y − y ) + ∑C (y

y > yi

i

i

y ≤ yi

i

− x)

i

i

(13)

− y ) (14)

Applying the minimization condition, it gives:

∂Cx ∂x

∂C y ∂y

=

∑C − ∑C

x > xi

=

i

x ≤ xi

i

∑C − ∑C

y > yi

i

y ≤ yi

i

=0

(15)

=0

(16)

The optimal solution is obtained at a location x where the summation of all costs to the right of this location is equal (closest) to the summation of all costs to the left of this location. The same can be done for the y location. The following algorithm finds the optimal location (x, y) under the stated assumptions. 1. Order set { xi } which contains x-coordinates of departments in an ascending order, Then construct a related set {C i } that contains the values of flows in and out of these departments, and in the same order of { xi }. 2. Construct two arrays, the first {Ri } contains the

cumulative ascending values of set {C i } , and the

other {Li } contains the cumulative descending values of set {C i } .

3. Subtract the elements of set {Li } from its equivalent elements in set {Ri } , then the resulted values are placed in a new set {results}. 4. Search the new set {results} for the smallest nonnegative value, and record its order in the set, then the xi coordinate that has the equivalent order in set { xi } is the optimum solution. Same steps are performed to find the optimum ycoordinate.

1488

Proceedings of the 37th International Conference on Computers and Industrial Engineering, October 20-23, 2007, Alexandria, Egypt, edited by M. H. Elwany, A. B. Eltawil

4. APPLICATION As shown in this paper, that the proposed solution technique can be applied for any given layout problem and under all conditions with no requirements other than the pre-estimates of the centroids of all departments. Then, by projecting the centroids of all departments on one plane, one can solve the problem as a single-location problem after discarding all the relations between departments on the same floor, and pooling together all flows to/from each department coming from/going to other departments on other floors as one flow between that department and the elevator. Nayer T, et al (2004), has proposed a two-stage solution algorithm for solving multi- floor plant layout problem assuming the handling costs per unit distance per unit flow are to be one unit and are the same for all horizontal transports.. The first stage is a hierarchy of five main steps that are: • Determination of Seed Departments • Assignment of the Rest of Departments to Floors • Graphical Area Representation and Departmental Centroids Evaluation • Finding Elevator Position • Cost Evaluation and Layouts Ranking Where, the second-stage is an improvement stage that performs a series of departmental exchanges infloor, and deals with each floor as a separate single layout problem. The first-stage starts with selecting seed departments. Those departments will act as the base units for each floor. Then, other departments will be assigned sequentially to the different floors, first according to their relation with the seed departments, then according to their relation with the growing set of the

assigned departments at each floor. As a result, the given departments are grouped into a number of sets equal to the number of floors, each set represents a floor such that the interrelations between its inside departments are maximized while the extra-relations with departments in other sets are minimized. Following this is the use of the developed graphical tool set “SFC” to build the graphical layout of each floor. By the end of this step, all departments' centroids were to be estimated. Hence, the technique proposed in this paper for finding the optimal location of the elevator can be applied. And, at the end of the first-stage, the total material handling cost is to be estimated. The flowchart of the Nayer solution approach including the technique of finding the optimal location of elevator as presented in this paper is shown in figure 2. According to Nayer approach, the first stage generates solutions corresponding to different seed departments' combinations, and selects the best K of such solutions to proceeds to the second stage. In the second stage, each recorded valid layout will have a sort of improvement by a series of re-layout action for each single floor, and the elevator is repositioned to its new location that may result in even better solutions. Nayer used a selected set of Benchmark problems that are applied by several researchers and that their full data and results documentation are available. This set is first reported in MULTIPLE [4], and are published in details through the internet (web-site of Meller). Those problems have been used extensively by other researchers for comparison. Table 1, shows the source data of Benchmark Problem 11x2 v.1 as an example.

1489

Proceedings of the 37th International Conference on Computers and Industrial Engineering, October 20-23, 2007, Alexandria, Egypt, edited by M. H. Elwany, A. B. Eltawil

Start No. of floors, No. of departments, Flow matrix, Arearequirements, Vertical to horizontal cost ratio, Floor shape, Fixed departments Determine different combinations of seed departments

Do until end of seed-combinations

Assign one seed department to each floor

Assign other departments to floors using the proposed assignment algorithm

Apply SFC to generate the layout of each floor

The Elevator- Finding Module can be inserted here

Find the optimal location of elevator using the proposed algorithm

Estimate the corresponding handling cost for the developed solution

Keep solution

Rank the developed solutions

Issue the list of the best k solutions

Improve each of the k solutions using the proposed improving algorithm

Provide improved solutions as final solutions

End

Figure 2 Flowchart of the Nayer solution approach

1490

Proceedings of the 37th International Conference on Computers and Industrial Engineering, October 20-23, 2007, Alexandria, Egypt, edited by M. H. Elwany, A. B. Eltawil

Table 1 Source Data of Benchmark Problem 11x2v.1 Problem Code: 11X2 V.1 Floor width: 7.5 m Department A B C Area 18.75 12.5 25 From:To A-B A-E A-F Flow 10 140 90 From:To G-H H-K I-J Flow 10 10 20

#Departments: 11 Floor length: 15 m D E F G 31.25 12.5 18.75 25 A-G A-I B-C C-D 20 40 10 10 J-K K-A 20 146

H 31.25 D-K 4

I 6.25 E-B 10

#Floors: 2 Floor area: 112.5 m 2 J K 6.25 37.5 E-G E-J F-C F-I 40 20 10 20

Other assumptions: Department K is fixed As mentioned before that MULTIPLE, SABLE, STAGES and MULTI-HOPE, which are the best known solution approaches for the multi-floor layout problem, use pre-assigned locations for elevators. They use two elevators in solving the benchmark problems. On the other hand, MUSE optimizes the number and locations of the used elevators. Table 2, shows the total handling cost associated with these

solution approaches when applied to solve the benchmark problems and the corresponding handling cost when replacing the elevators located by those solution-approaches by the one that can be located when applying the proposed solution algorithm presented in this paper.

Table 2 Solutions of the benchmark problems: original number of elevators vs. single optimum elevator Literature results Problem Code

Handling Cost (single optimum elevator)

Algorithm name

Solution cost

No. of originally used elevators

11X2 v.1

SABLE, STAGES

8477.3

2

IFBEAJK-CDHG

8365

11X2 v.2

SABLE

2493.9

2

IGCK-DHAJFEB

2363.3

12X3

SABLE, STAGES

1513.2

2

GFJAL-DEHB-KIC

1481.3

21X4 v.1

SABLE

14505.5

2

TSVU-AKEHIMQBDCNPOR-JLGF

13623.5

21X4 v.2

SABLE

11315

2

RLQTU-BPNFIKCESGO-JAHMD

10817

21X4 v.3

SABLE

10678.2

2

KQADU-SRLNJMFPTIOG-CEBH

11827.2

21X4 v.4

SABLE

8556.3

2

JEU-QMPTSNKRLDFHC-IBOAG

9101.3

The results support the idea of placing the elevator in its optimum position through the solution process, and that the single optimum elevator position gives

Layout arrangement

better results than pre-assigning many elevators at arbitrary positions.

1491

Proceedings of the 37th International Conference on Computers and Industrial Engineering, October 20-23, 2007, Alexandria, Egypt, edited by M. H. Elwany, A. B. Eltawil

5. CONCLUSIONS In searching elevator location in a multi-floor layout, one can discard all relations between departments on the same floor, as those relations don’t use the elevator, and have no effect on elevator position. In addition, all flows to/from a department from/to other departments on other floors can be pooled together as one flow between that department and the elevator. And given that the departments' centroids are already estimated, and distances in-between are calculated then, by projecting the centroids of all departments on one plane makes the problem very similar to the single-location problem known in literature. Considering this similarity, this paper introduced a foundation of an efficient technique that gives the optimal location of a single elevator to any given multi-floor layout. Also, in addition to solving the problem considering the rectilinear distance function, this paper solved the problem considering the Euclidean distance function as well which can be considered a new foundation for solving the singlelocation problem known in literature considering the Euclidean distance function. Moreover, the proposed solution technique does not require all floors to be identical, or handling costs per unit distance per unit flow to be the same for all horizontal transports.

8. Meller R.D., and Bozer Y.A., "Alternative Approaches to solve the Multi-Floor Facility Layout Problem", Journal of Manufacturing Systems, Vol.16, No.3, PP.192-203, 1997. 9. Nayer T., Abdelhafiez E., Shalaby M. A., "A New Algorithm for Solving Multi- Floor Plant Layout Problem", Proceedings of 8th International Conference on Production Engineering Design and Control, PEDAC’2004, Alexandria, Egypt. 10. Sule D.R., "Manufacturing Facilities", 2nd edition, pp659-660, 1944. 11. Tompkins J., et al., "Facilities Planning", John Wiley & Sons. Inc., Second Edition, PP.344-345, 1996.

REFERENCES 1. Bozer Y.A., et al., "An Improvement Type Layout Algorithm for Single and Multiple Floor Facilities", Management Science, Vol.40, No.7, PP.918-932, July, 1994. 2. Hyper Link ( http://ise.vt.edu/meller/), The web Site of Meller,R.D. 3. Johnson R.V., "SPACE-CRAFT for Multi-Floor Layout Planning", Management Science, Vol.28, No.4, PP.407-417, 1982. 4. Kaku B.K., et al., "A Heuristic Model for the MultiStory Layout Problem", European Journal of Operational Research, Vol.37, PP.384-397, 1988. 5. Kochhar J.S., and Heragu S.S., "MULTI-HOPE: A tool for Multiple Floor Layout Problems", International Journal of Production Research, Vol.36, No.12, PP.3421-3435, 1998. 6. Matsuzaki K., et al., "Heuristic Algorithm to solve the Multi-Floor Layout Problem with the Consideration of Elevator Utilization", Computers and Industrial Engineering, Vol.36, PP.487-502, 1999. 7. Meller R.D., and Bozer Y.A., "A New Simulated Annealing Algorithm for the Facility Layout Problem", International Journal of Production Research, Vol.34, No.6, PP.1675-1692, 1996.

1492