Optimal multi-floor process plant layout

European Symposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.

475

Optimal Multi-floor Process Plant Layout Dimitrios I. Patsiatzis and Lazaros G. Papageorgiou* Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, United Kingdom. This paper presents a general mathematical programming formulation for the multi-floor process plant layout problem, which considers a number of cost and management/engineering drivers within the same framework thus resolving various trade-offs at an optimal manner. The proposed model determines simultaneously the number of floors, land area, floor allocation of each equipment item and detailed layout of each floor. The overall problem is formulated as a mixed integer linear programming (MILP) model based on a continuous domain representation. The applicability of the model is then demonstrated by an illustrative example. 1. INTRODUCTION The process plant layout problem involves decisions concerning the spatial allocation of equipment items and the required connections among them [1]. Usually, these plant layout decisions are ignored or do not receive appropriate attention during the design or retrofit of chemical plants. However, increased competition leads contractors and chemical companies to look for potential savings at every stage of the design process. In general, the process plant layout problem may be characterised by a number of cost or management/engineering drivers such as: (a) connectivity cost by involving cost of piping and other required connections between equipment items or any other network related operating costs (e.g. pumping); (b) construction cost thus leading to the design of more compact plants (e.g. off-shore plants); (c) retrofit by fitting new equipment items within an existing plant and (d) safety aspects by introducing, for example, constraints with respect to the minimum allowable distance between specific equipment items. Here, we concentrate on the process plant layout problem, which has recently received attention from the research community. A number of different approaches have been presented for the single-floor case. The allocation of units to sections created by aisles or corridors was formulated as a graph partitioning problem in [2]. A mixed integer non-linear programming (MINLP) model was proposed by [3], integrating safety and economic aspects. The use of genetic algorithms [4] has also proved to be effective in obtaining good and practical solutions for the layout problem. Continuous domain MILP mathematical models were presented in [5], determining simultaneously orientation and allocation of equipment items. An alternative continuous MILP formulation was also suggested in [6] for equipment allocation, utilising a piecewise-linear function representation for absolute

*Author to whom correspondence should be addressed: Fax:+44 20 7383 2348; Phone +44 20 7679 2563; emaih 1. p a p a g e o r g i o u @ u c l , ac. uk

476 value functionals (distances between equipment items). The assignment of equipment items to different floors, has been considered in [7] by satisfying a number of equipment arrangement preferences and also taking into account vertical pumping and land costs. The partitioning of units in different floors was studied in [8], combining a graph theory approach and a mathematical programming solution procedure. Grid-based MILP mathematical models have been described in [9, 10], considering equipment of different sizes and geometries, based on rectangular shapes. In this work, a general mathematical programming formulation for the multi-floor process plant layout problem is presented. This work extends the single-floor work of Papageorgiou and Rotstein [5], which is based on a continuous domain representation.

2. PROBLEM STATEMENT In the formulation presented here, rectangular shapes are assumed for equipment items following current industrial practices. Rectilinear distances between the equipment items are used for a more realistic estimate of piping costs [3, 5, 10]. Equipment items, which are allowed to rotate 90 ~ are assumed to be connected through their geometrical centres. Overall, the multi-floor plant layout problem can be stated as follows: Given:

* A set of equipment items (i or j = 1..N) and their dimensions (ai, fli) 9 A set of potential floors; k = 1..K 9 Connectivity network 9 Cost data (connection, pumping, land and construction) 9 Floor height, Fh 9 Space and equipment allocation limitations 9 Minimum safety distances between equipment items Determine:

The number of floors, land area, equipment-floor allocation and detailed layout (orientation, coordinates) of each floor So as to minimise the total plant layout cost. 3. M A T H E M A T I C A L F O R M U L A T I O N 3.1. Floor Constraints Each equipment item should be assigned to one floor: Z v~k - 1

v i

(1)

k

where Vik = 1 if item i is assigned to floor k; 0 otherwise. We introduce a new set of binary variables, zij, which have the value of I if equipment items i and j are allocated to the same floor; 0 otherwise. Their value can be obtained by: zij >__ vik + vjk - 1

V i = 1 . . N - 1, j = i + 1..N, k = 1..K

(2)

zij <_ 1 - - vik + vjk

Vi=I..N-I,j=i+I..N,k=I..K

(3)

zij < 1 + Vik -- Vjk

V i = 1 . . N - 1, j = i + 1..N, k = 1..K

(4)

477

The number of floors, N F , will be determined by: NF

>_ ~

kvik

(5)

V i

k

3.2. Distance Constraints The single floor distance constraints presented in [5] are here extended for the multi-floor case: R i j - Lij - xi - x j

V i - 1..N-

1, j - i + 1 . . N " fij - 1

(6)

A i j - B i j - y~ - yj

V i - 1..N-

1, j - i + 1 . . N " f~j - 1

(7)

U~} - Di~ - Fh ~

k(vik - Vjk)

V i - 1 . . X - 1, j - i + 1 . . N " fij - 1

(8)

k

where the relative distance in x coordinates between items i and j, is R i j if i is to the right of j or Lij if i is to the left of j. The relative distance in y coordinates between items i and j, is A i j if i is above j or B i j if i is below j. The relative distance in z coordinates between items i and j is U/~ if i is higher than j or D~j if i is lower than j. The coordinates of geometrical centre of item i are denoted by xi, yi. Parameter fij is equal to 1 if units i and j are connected ; 0 otherwise. Thus, the total rectilinear distance, Dij, between items i and j is given by: D i j -- R i j nt- Lij + A i j + B i j + U~. + Di).

V i - 1..N-

1 , j - i + 1 . . N " f~j - 1

(9)

3.3. Equipment Orientation Constraints The length, li, and the debth, di, of equipment item i can be determined by: li -- o~ioi + ~i(1 - oi) di - c~i + ~i - li

(10)

V i

(11)

V i

where oi is equal to 1 if li=oq; 0 otherwise (i.e. li=fli).

3.4. Non-overlapping Constraints In order to avoid situations where two equipment items i and j occupy the same physical location, when allocated to the same floor (i.e zij = 1), constraints are included in the model that prohibit overlapping of their equipment footprint projections, either in x or y direction: xi -- x j + M ( 1 - zij + E l i j + E 2 i j ) >

xj - xi + M ( 2 - zij - E l i j + E 2 i j ) >_

yi - yj + M ( 2 -

zij + E l i j - E2ij) >_

li + lj

l~ + lj

di + dj

V i - 1..N-

1 , j = i + 1..N

(12)

Y i - 1..N-

1 , j - i + 1..N

(13)

V i-

1..N-

1,j-

i + 1..N

(14)

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yj - yi + M(3 - zij - E l i j - E2ij) >_

di + dj

V i = 1 . . N - 1,j = i + 1..N

(15)

where M is an appropriate upper bound and Elij, E2~j are binary variables as used in [5]. Note that the above constraints are inactive for equipment allocated to different floors (i.e z~j = 0).

3.5. Additional Layout Design Constraints Intersection of items with the origin of axes should be avoided: xi > li -2

Yi

(16)

yi > _di -2

V i

(17)

A rectangular shape of land area is assumed to be used and its dimensions (x max, yma~) are determined by: li

xi + -~ 5 x max

g i

(18)

di _ yma~ Yi + ~ <

g i

(19)

These dimensions can then be used for the land area, FA, calculations: F A = x 'nax . ymaz

(20)

3.6. Objective Function The overall objective function used for the plant layout problem is as follows: rain

~

~[C~

. Dij + C~ " Di~ + C h " (Rij + Lij + Aij + Bij)]

i ir

+FC1. NF + FC2. NF.

FA + LC. FA

where the first term represents the total connection cost (C~j is the unit connection cost between i and j), and the second and third terms represent vertical (C~ is the unit vertical pumping cost if i is below j) and horizontal (C h is the unit horizontal pumping cost) pumping costs, respectively. The construction cost incurred is described by the fifth and sixth terms ( F C 1 and F C 2 are the floor construction cost parameters), while the last term is associated with the land cost ( L C is the land cost parameter). The above problem is an MINLP model because of the non-linearities involved in the last two terms of the objective function. However, the x max, ymaz variables required for the F A calculations can be discretised similarly to the work presented in [ 10]. Consequently, the nonlinear terms can easily be linearised. Due to space limitations, these linearisation constraints are not presented here. The linearised problem corresponds to an MILP model which can then be solved using standard branch-and-bound techniques. Next an illustrative example demonstrates the applicability of the MILP model.

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Figure 1" Flowsheet for Ethylene Oxide Plant ,_. . . . . . . . . . . . . . . . . . . . . . F. .L. .O . .O . . .R. . . . . . .1. . . . . . . . .

FLOOR

2

Z68 m 11.42 m

5.22 m I

Z 68 m _.................................

2

1

8:___n_l_ 4 8 .........................................................

Figure 2" Optimal Layout 4. ILLUSTRATIVE E X A M P L E

Consider the ethylene oxide plant (see Figure 1), derived from the case study presented in [3]. Three potential floors are assumed to be available. Connection and pumping cost data are given in Table 1. The annualised floor construction cost parameters, FC1 and FC2, are 3330rmu and 6.7rmu/m 2, respectively and the annualised land cost parameter, LC, is 26.6rmu/m 2, where rmu stands for relative money units. Table 1" Connection and Pumping Costs Connection C~j [rmu/m] C~ [rmu/m] C~.j[rmu/m] (1,2) 200 400 4000 (2,3) 200 400 4000 (3,4) 200 300 3000 (4,5) 200 300 3000 (5,1 ) 200 100 1000 (5,6) 200 200 2000 (6,7) 200 150 1500 (7,5) 200 150 1500 The above example was modelled in the GAMS system [ 11], using the CPLEX 6.5 optimiser for the solution of the MILP model with a 5% margin of optimality. The resulting mathematical model includes 434 constraints, 113 integer and 119 continuous variables. The equipment dimensions and the optimal layout are shown in Figure 2. The optimal solution (equipment orientation and location, equipment-floor allocation) is given in Table 2. It should be noted that

480 two out of the three initially available floors have been chosen. The total plant layout cost is 50817 r m u with the following breakdown: 23% for connection cost; 32.5% for horizontal and vertical pumping costs and 44.5% for land and construction costs. The optimal land area is 400 m 2 ( x max 20m, ymax 2Ore). =

:

Equipment 1 2 3 4 5 6 7

Table 2: Optimal Solution Orientation Location li [m] di [m] xi [m] Yi [m] 5.22 5.22 14.29 5.97 11.42 11.42 14.29 14.29 7.68 7.68 14.29 14.29 8.48 8.48 14.29 6.21 7.68 7.68 6.21 6.21 2.60 2.60 6.21 11.35 2.40 2.40 3.71 11.25

Allocation Floor 2 2 1 1 1 1 1

5. CONCLUDING REMARKS In this paper, the optimal multi-floor process plant layout problem has been considered. A general mathematical framework has been described, which determines simultaneously the number of floors, land area, optimal equipment-floor allocation and location (i.e. coordinates and orientation) so as to minimise the total plant layout cost. The current model can easily accommodate various considerations related to space restrictions and/or safety. The resulting optimisation problem corresponds to an MILP model. Current work focuses on testing the framework to larger examples and investigating alternative solution procedures (e.g. decomposition schemes). REFERENCES

1. J. C. Mecklenburgh, Process Plant Layout, Institution of Chemical Engineers: London, 1st edition, 1985. 2. S. Jayakumar and G. V. Reklaitis, Comp. Chem. Eng., 18 (1994) 441. 3. E D. Penteado and A. R. Ciric, Ind. Eng. Chem. Res., 35 (1996) 1354 4. C. M. L. Castel, R. Lakshmanan, J. M. Skilling and R. Banares-Alcantara, Comp. Chem. Eng., 22 (1998) $993. 5. L. G. Papageorgiou and G. E. Rotstein, Ind. Eng. Chem. Res., 37 (1998) 3631. 6. D. B. Ozyruth and M. J. Realff, AIChE J., 45 (1999) 2161. 7. A. Suzuki, T. Fuchino, M. Muraki and T. Hayakawa, Kagaku Kogaku Ronbunshu, 17 (1991) 1110. 8. S. Jayakumar and G. V. Reklaitis, Comp. Chem. Eng., 20 (1996) 563. 9. M. C. Georgiadis, G. E. Rotstein and S. Macchietto, Ind. Eng. Chem. Res., 36 (1997), 4852. 10. M. C. Georgiadis, G. Schilling, G. E. Rotstein and S. Macchietto, Comp. Chem. Eng., 23 (1999) 823. 11. A. Brooke and D. Kendrick, A. Meeraus and R. Raman, GAMS: A Users's Guide, The Scientific Press, 1998.