Layout of process plants: A novel approach

Computers chem. Engng, Vol. 21, Suppl., pp. $337-$342, 1997 © 1997 Elsevier Science Ltd All rights reserved Printed in Great Britain

Pergamon PII:S0098-1354(97)00071-9

0098-1354/97 SlT.O0+O.O0

Layout of Process Plants: A Novel Approach M.C. Georgiadis and S. Macchietto* Centre for Process Systems Engineering Imperial College of Science, Technology and Medicine London SW7 2BY, U.K.

Abstract. Plant layout is concerned with the spatial arrangement of processing equipment, storage vessels and their interconnecting pipework. This is an important aspect in the design of chemical and process plants since a good layout will ensure that the plant functions correctly and will provide an economically acceptable balance between the many, often conflicting, design constraints. These constraints are derived from safety, environmental, construction, maintenance and operational considerations. Process relationships, for example the use of gravity flow, and issues such as the provision of space for future expansion must also be taken into account. Traditional methods for locating equipment within chemical plants are based on mixtures of process heuristic rules and exact-to-the-inch distance information. Such techniques are unsystematic and they do not make use of all the relevant and appropriate data. In this paper an optimization based approach is used to determine a good preliminary plant layout, subject to all of the above constraints. A novel mathematical formulation is presented which addresses the problem of locating items of equipment within a given two or three dimensional space. The objective function to be minimized is the sum of the relevant operation, connection and floor construction costs. Detailed cost factors are used to account for the flow direction between two connected units. The problem is formulated as a mixed integer linear programming problem. Specific attention is paid to constructing a formulation which is suitable for the solution of large scale problems. The method presents the rigorous solution of problems with around 30 process equipment and of essentially unlimited size problems when combined with single heuristic rules. The approach is demonstrated with several practical scale problems, including an industrial multi-purpose plant. INTRODUCTION Process plant layout is an important part of plant design. Access to the plant and the supply of maintenance, construction and emergency services are all affected by the plant layout (Mecklenburgh, 1985). The traditional method of locating equipment within plants has been to distribute it over a large area, with land usage as a secondary concern. However, the current trend in plant design is directed more towards compact plots and enclosed structures. Layout is a particularly important consideration during retrofit design since here, typically, there is little opportunity for changing the plant boundaries. This may pose significant limitations on the type, size and location of new equipment. Plant layout in its own right has only recently become the subject of study. However, the combined problem of facility layout and location has been extensively studied in the area of industrial engineering, with applications to work shops and manufacturing units (Francis and White 1974). The most widely studied approach to the facility layout problem is known as the "Quadratic Assignment Problem". This problem is aimed at the allocation of a number of production facilities to an equal number of locations. The cost objective function depends on * Author to whom all correspondence should be addressed, Emaih s.maeehietto©ic.ae.uk Fax: (44)-1715946606

the flow between the facilities and their respective positions. Several formulations and algorithms based on heuristics have been proposed for the solution of these problems (Fortenberry and Cox, 1985), (Bozer et al., 1994). Combined exact and heuristic methods have also been put forward by Bazaraa and Sherali, (1982), Bazaraa and Kirka (1983) and Adams and Sherali, (1986). The general layout problem for chemical plants has been considered by other researchers (Gunn and A1-Asadi, 1987; Suzuki et al., 1991; Amorese et al., 1991). All these approaches incorporate a large number of heuristic rules which may lead to the development of sub-optimal solutions. Recently, Penteado and Ciric (1996) presented an optimization formulation that can generate safe and economical layouts. The problem was solved as a relaxed MINLP and illustrated with a small case study for a two dimensional layout. In the batch plant area Jayakumar and Reklaitis (1994) proposed a graph theory approach to single floor layout problems, establishing an analogy between distributing units on a single floor and the traditional graph partitioning problem. In a subsequent publication, the same authors considered the multilevel or multi-floor layout of batch plants (Jayakumar and Reklaitis, 1996). In fact, the objective of their work was not to find the optimal plant layout but to determine the optimal partitioning of units

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among floors - a sub-problem of the general layout problem. A graphical heuristic approach was presented which provides an upper bound to the true optimal value, and a mathematical programming approach was also proposed to provide a lower bound. These two bounds together "bracket" the optimal value. This approach includes a very tight linearization scheme for the initial MINLP (compared with the well known Glover transformations). The continuous relaxation of the resulting MILP was solved using the Lagrangean relaxation technique together with sub-gradient optimization for the calculation of the required Lagrange multipliers. Large scale problems were solved using these approaches. However, the solution of the relaxed LP may lead to in noninteger results. In fact, for the small illustrative example these authors presented, the solution in one of the cases examined was fractional. In general, for large problems, non-integer solutions cannot be easily converted into integer solutions, even if the continuous optimal point is very close to the integer one. In this paper a mathematical programming approach for process plant layout is proposed and trade-offs between the capital and operating costs are illustrated with a large scale case study concerning an industrial multi-purpose plant. MATHEMATICAL MODEL AND RIGOROUS SOLUTION The plant layout problem is formulated as an optimization problem with minimum cost as the objective. This cost is divided into four elements: the upward pumping cost, the horizontal pumping cost, the connection cost and the floor construction cost. The pumping costs represent the fluid transport costs. The allocation of units to different floors requires the use of cost factors which differ from those in the 2-D case. The piping cost for the movement of materials to higher floors cannot be neglected and must be included in the objective function. In the case of a downward flow there is no pumping cost due to gravity, but the connection cost (S/m) still exists. Finally, where two connected units are allocated to the same floor, a (low) pumping and connection cost are taken into account. This is a much more realistic objective function than in other works, where arbitrary cost values are typically used (especially where distances cannot be calculated explicitly). Three different annualised cost factors are proposed: • Upward cost, CUP. There is a high cost associated with pumping materials to higher floors. This cost is a function of the flowrate between the connected process units and the height differences. This function is given approximately by Coulson and Richardson (1985):

CUP = unit cost. Flow. g

$/m. yr

The above cost must be multiplied by suitable variables which represent the height difference.

The gravity constant g is taken as 9.8Ira~see 2, the flow must be given in kg/s and the unit transport cost in S/kwh (energy cost). The above expression is multiplied by 1.2 to account for other factors such as the bends in the pipes, the roughness of the pipe surface etc. The final expression for the CUP is:

CUP = 1515. Flow $/m. yr To this, the connection cost must be added. • Downward flow. In this case there is no pumping cost since the flow is gravity driven. However, there is a connection cost which is a function of the vertical distance between the connected units. • Horizontal cost, CHOR. This case concerns two connected units on the same floor. A low pumping cost together a connection cost is taken into account. The pumping cost is a function of flowrate and distance. It is estimated as 10 % of the upward pumping cost and is given as follows (Coulson and Richardson, 1985):

CHOR = 15.15. Flow

$/m. yr

This cost coefficient must be multiplied by the horizontal distance. The direction matrix dq between two units is determined as in Jayakumar and Reklaltis (1996):

dq =

1, 0,

if flow is from unit i to unit j otherwise

Now suppose that n is the total number of equipment units and K is the number of available locations (3-D grids) which are represented by unique x, y and z coordinates. Then, the proposed 3-D layout mathematical model is presented below: m i n O F = E [dij(BELij. CUP) + ij,i
dji( ABq • CUP] + E

ICHOR. (Rij + Lq + Aij + B,j)] +

i,j,i
E

[CCOq. C C F . (Rq + Lij +

i,j,i
Aij + Bij + ABij + BELij)] + (CCF.FCC.FA.NF) [$/yr] [P1] subject to 1 g

Rij - Lq = ~ E ( D k + Dlk). (Yik - Yjk),

(1)

k=l 1 g

Aij - Bq = ~ E ( s k - D l k ) . (Yik - Yjk),

(2)

k=l

1

ABij - B E L i j = -~ ~-~(Sk -- Ok)" (Yik -- Yjk), k=l

(3)

PSE '97-ESCAPE-7 Joint Conference K Z yik = 1, k=l n

Z

Yik --< 1,

(4)

-

yik = 0 or 1

-

k = 1,.--,K

i = 1,...,n

(8)

The integer variables Yik are defined as Yik =

1, 0,

if unit i is allocated to site area k otherwise

The coordinates for each unit can be easily calculated if required, by adding the following set of constraints: K

SkYik

(9)

Dkyik

(10)

DlkYik

(11)

Xi -t- Yi q- Zi = ~ k=l K

xi + Yi - zi = ~

problem, as a sub-problem, using the same number of integer variables.

(5)

i=1 Rij >_ 0 Lij >_0 Aij >_0 (6) Bij > 0 ABij > 0 B E L i j > 0 (7)

k=l

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HANDLING

DEGENERACIES

A problem with the above MILP layout formulation is the presence of degenerate solutions, i.e. there may exist permutations of feasible layouts with exactly the same value of the objective function. Each permutation may be obtained from the previous one by shifting the allocation by one site area. In fact, degeneracies always arise when there exists any symmetric layout geometry. As shown in Figure 1, the two solutions (denoted by circles and crosses) are equivalent. It is difficult to develop general methods to avoid degeneracies since they depend not only on the layout geometry but also on the number of units. Here we propose one way to handle degeneracies for the most symmetric 2-D case (grids of equal size in the x and/or y directions). If there are no constraints imposing minimum distance (e.g. due to safety considerations), then the following set of constraints is added to the formulation:

K

xi - Yi + zi = ~ k=l

It is worth noting that the variables ABij and B E L i j automatically determine the floor sequence. Constraints (1), (2) and (3) define the relative positions of two connected units and serve to calculate the total distance between them. Here, this distance is taken to be rectangular, which is more appropriate in industrial problems than a straight-line distance. Constraint (4) specifies that unit i must be allocated in one of the given n locations, while constraint (5) specifies that each location may be occupied by at most one unit. This constraint provides the flexibility to have more grids than units in cases where there is sufficient space. It must be emphasised that the variables Rij, Lij, Aij, Bij, ABij, B E L i j are not integer but continuous variables. For example, when unit 2 is allocated to the right of unit 3, then R23 takes a positive value while L32 is zero. The sum (Rij + Lij + Aij + Bij + ABij + B E L i j ) represents the total distance between units i and j. FCC represents the floor construction cost ($/m2), FA is the area (m 2) per floor, NF is the number of floors and CCF is a capital charge factor. Here, FA and NF are given and fixed. The main advantage of the above formulation is the fact that it involves n 2 binary variables. Thus, even reasonably large scale problems (number of units greater than 15) can be solved efficiently using a commercial mixed integer linear code (e.g. CPLEX). Furthermore, it can easily accommodate locational constraints. For example, if it is necessary to fix the location of unit i to area k we can force the corresponding integer variable Yik to take the value of 1. Similar constraints for a minimum distance (e.g. for safety) between two units can be easily included. Clearly, the proposed formulation can also be applied to the two dimensional layout

~

~

i keLp(k)

Yik <_ ~_~

~_,

yik "MNU

(12)

i keL~-l(k)

where Lm (k) and L m - 1(k) are two neighbouring subsets of grids (rows or columns in the 2-D layout area for which the side difference in the x and/or y directions is zero). MNU is the maximum number of units which can be allocated in each subset of areas considered. For example, according to Figure 1 the above constraints must be applied for L2 (grids 5, 6, 7 and 8) and L1 (grids 1, 2, 3 and 4) since they have the same size (in the y direction). Here MNU is 4. Similarly, the same constraints will be also applied for L5 (grids 2, 6 and 10) and L4 (grids 1, 5 and 9) (same size in the x direction). In this case MNU is 3. The above constraints enforce the generation of solutions preferentially occupying the bottom left-hand corner, i.e. the solution denoted by circles instead of crosses, for the simple example considered in Figure 1. Constraints can be imposed for the 3-D problem taking into account the z direction as well. However, it should be emphasised that for the case of unequal segments in the x and y directions (i.e. a 2D rectangular area) the constraints proposed above can generate sub-optimal solutions. A HEURISTIC METHOD FOR VERY L A R G E PROBLEMS The above rigorous mathematical approach for the plant layout problem can lead to good solutions in a reasonable computational time for medium to large size problems (up to 30 units). However, for some problems of an industrial nature, with a number of equipment units greater than 30, the computational cost can increase considerably. Two solutions are proposed for this case only. First, a heuristic method is proposed based on the pre-allocation of specific units. Although in this way,

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L7

sub-optimal solutions may be obtained, several tests have shown that these solutions are very close to the optimal. The steps of the proposed method are described as follows:

9

0

7

×

I

8

2

~L ! 3

4

10m ~

2

! 10m

©

3. Allocate this unit to an area such that the connection cost between itself and the two fixed units is minimized. 4. The fourth unit is chosen and located in a similar way by treating the first three units as fixed.

6

12

00x

1

2. Find a third unit which has the highest degree of connectivity to the two already fixed units.

I1

×

$

1. Find the two units with the largest sum of flowrates (highest degree of connectivity) and locate them at two central site areas.

30m

10

Li

20m

30m

40m

Degeneratesolutions

Figure 1: Degeneracies in the optimal plant layout problem

5. Continue in the same way until the number of unallocated units can be efficiently handled by the optimizer. This number is about 25-30 units. In practice, the allocation of units to some of the site areas may be prohibited due to safety or operational considerations. By imposing such constraints the computational cost can be reduced without the need to use the above algorithm. An alternative way is to use aggregate equipment modules. A module would include a group of units which have similar operating characteristics or a high degree of interconnection. Once the choice of modules has been made, and equipment in each module has been specified (including relative position), the plant layout problem takes the form of deciding the relative positions of the plant modules. EXAMPLE The applicability of the proposed formulation for the determination of optimal layout is illustrated with a case study taken from Barbosa-Povoa (1994)(Figure 2) which is solved rigorously (no heuristic). This is an industrial multi-purpose batch plant in which the number of equipment units is 18, with a high degree of connectivity. It is assumed that the connections have different nominal diameters. The purchase cost in $/m is given in Table 2. The floor construction cost, for a typical floor height of 4 m, in approximate current values is given in Table 1 (The Institution of Chem. Eng., 1988). The optimal multi-floor layout problem is considered for the cases of two, three, four and five floors. The layout geometry for each problem is discretised into a number of 3-D grids. For example, for the two floor case, the sets which define the grid coordinates are given in Table 3. Note that these coordinates represent the centre of the corresponding grid. Here, each floor is discretised into 9 grids. Full flexibility is allowed for the allocation of units to areas. The total area per floor is 504 m 2 and the corresponding floor construction cost is 500 $/m 2 (Table 1). The optimal plant layout on two, three, four and five floors is depicted in Figures 3, 4, 5, and 6 respectively. In all cases, units are allocated in such a

~2 i

vv$

'~

1

Figure 2: Flowsheet of the case study

way that the high upward cost is minimised. Substantial reduction of the pumping cost is achieved as the number of floors increases (see Table 4). The units become stacked one above another in order that as many flows as possible are aided by gravity. However, there is a trend which should be noted - the floor construction cost increases ~s the number of floors increases, but the pumping cost decreases. These trade-offs result in the optimal number of floors together with the optimal equipment layout. For the example considered here, the optimal number of floors is three (Table 4) and the corresponding total annual cost is 240000 S/yr.

1/1

I/2

~,'V3

RI

!~

~'S

~

~,4rF.l~

Wl

Figure 3: Optimal plant layout in two floors

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'EArES

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V2

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t

W3

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l

/

i

tntlnt

R4/ t V4

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I,'/7nt

lint

Figure 4: Optimal plant layout in three floors

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VV3

Wl

1%'5

~s

¢

i / ,m /

I

I+4 J V6 b V8

/ V9

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Connection number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

7m

23 24

8m

Figure 5: Optimal plant layout in four floors

Cost, $/m 200 240 230 240 230 160 240 160 230 170 270 270 280 170 170 3OO 25O 25O 25O 175 170 175 140 23O

Table 2: Cost of connections for the case sudy

lm

VI

V2

VV3

Wl lm

R4

/

W2

I

RI

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114

i

R2

i

Grid number

EAr~S

4m VV5

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Z lnt

R3

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V5

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VV7

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f

Still

i V8

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4m

1"9

i

Im

8m

Figure 6: Optimal plant layout in five floors

Cost, $/m 2

Area, m 2

500

400-500

520 560

300-400 250-300

610

200-250

Table 1: Floor construction cost data

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

S, m 13.5 21.5 29.5 37.5 45.5 53.5 61.5 69.5 77.5 9.5 17.5 26.5 33.5 41.5 49.5 57.5 65.5 73.5

D, m 6.5 14.5 22.5 30.5 38.5 46.5 54.5 62.5 70.5 2.5 10.5 18.5 26.5 34.5 42.5 50.5 58.5 66.5

D1, m 1.5 9.5 17.5 25.5 33.5 41.5 49.5 57.5 65.5 5.5 13.5 21.5 29.5 37.5 45.5 53.5 61.5 69.5

Table 3: Sets for the definition of grid coordinates for the two floor layout

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Number of Floors

Pumping and connection cost, $/yr 102000 66000 45000 36000

Total Cost, $/yr 270000 240000 254000 267000

Table 4: Optimization Results

Dlk = set which defines the difference of coordinates of location k, value of the first coordinate minus value of the second coordinate plus value of the third coordinate Lij = horizontal distance between unit i and unit j if unit i is to the left of unit j

Rij = horizontal distance between unit i and unit j if unit i is to the right of unit j

Sk = set which defines the summation of coordinates of location k CONCLUSIONS A general optimization formulation has been presented for the 2-D and 3-D process plant layout problem including a number of cost factors and constraints. The important decision in the proposed approach is the discretisation of the layout area into a number of locations characterized by unique coordinates. Although this may result in some overestimation of the required area (for example if a small unit is allocated to a large area), the presented approach provides the rigorous solution of problems with up to 30 process units. The technique for generating the 3-D optimal plant layout results in the optimal allocation of processing units (coordinates and distances between them) using a detailed objective function. Trade-offs between capital and operating costs are captured so that the optimal number of required floors may be determined. The main characterisitic of the proposed formulation is the low number of integer variables, which makes it attractive for the solution of large scale problems. Different types of constraints are easily imposed. A way to handle degenerate solutions, for a specific layout geometry, in order to obtain compact layouts, was put forward. Since the problem is formulated in a linear form it can be incorporated in a simultaneous approach with the retrofit design/scheduling problem for multipurpose batch plants as considered by BarbosaPovoa (1994). NOTATION Aij -- vertical distance between unit i and unit j if unit i is above unit j ABij = vertical distance between unit i and unit j if unit i is at a higher floor than unit

J Bij = vertical distance between unit i and unit j if unit i is below unit j

BELij -- vertical distance between unit i and unit j if unit i is at a lower floor than unit

J CCF = capital charge factor CCO~j -- connection cost between unit i and unit j in $/m Dk = set which defines the difference of coordinates of location k, value of the first coordinate plus value of the second coordinate minus value of the third coordinate

REFERENCES Adams, W.P. and H.D. Sherali, 1986. A tight linearization and an algorithm for zero-one quadratic programming problems. Management Science 32, 1274-1290 Amorese, L., V. Gena and C. Mustacchi, 1991. A heuristic for the compact location of process components. Chemical Eng. Science 32, 119-124 Bazaraa, M.S. and H.D. Sherali, 1982. On the use of exact and heuristic cutting plane methods for the Quadratic Assigment Problem. Op. Res. Soc., 33, 991-1003 Bazaraa, M.S. and 0. Kirca, 1983. A branch and bound based heuristic for solving the Quadratic Assigment Problem. Naval Res. Logist., 30, 287-304 Barbosa-Povoa, A., 1994. Detailed Design and Retrofit of Multipurpose Batch Plants. PhD Thesis. University of London. Bozer, Y., R. Meller and S. Erlebacker, 1994. An improved type layout algorithm for single and multiplefloor facilities. Management Science 40, 918-932 Coulson, J.M. and J.F. Richardson, 1985. Chemical Engineering. 3rd edition. Pergamon Press. Gunn, D.J. and H.D. A1-Asadi, 1987. Computer aided layout of Chemical plants: A computational study method and case study. Computer aided design 19, 131-140 Fortenberry, B.J.C., and J.F Cox, 1985. Multiple Criteria approach to the facilities layout problem. Int. J. Prod. Res. 23, 773-782 Francis, R.L. and J.A. White, 1974, Facility Layout and Locations. 1st edition. Prentice-Hall International Series Jayakumar, S. and G.V. Reklaitis, 1994. Chemical plant layout via graph partitioning-I. Single level. Comput. Chem. Engng., 14, 441-458 Jayakumar, S. and G.V. Reklaitis, 1996. Chemical plant layout via graph partitioning-II. Multiple levels. Comput. Chem. Engng., 20, 563-578 Mecklenburgh, J.C., 1985. Process Plant layout. 1st edition. Institution of Chemical Engineers Penteado F.D. and A.R. Ciric, 1996. An MINLP approach for safe process plant layout. Ind. Eng. Chem. Res. 35, 1354-1361 Suzuki, A., T. Fuchino and M. Murald, 1991. An evolutionary method of arranging the plot plan for process plant layout. Journal of Chemical Eng. of Japan 42, 226-231. The Institution of Chemical Engineers, UK, 1988. A guide to capital cost estimating. 3rd edition