An Iterative Solution Approach to Process Plant Layout using Mixed Integer Optimisation

Transcription

1 17 th European Symposum on Computer Aded Process Engneerng ESCAPE17 V. Plesu and P.S. Agach (Edtors) 2007 Elsever B.V. All rghts reserved. 1 An Iteratve Soluton Approach to Process Plant Layout usng Mxed Integer Optmsaton Gang Xu, Lazaros G. Papageorgou Centre for Process Systems Engneerng, Department of Chemcal Engneerng, Unversty College London, Torrngton Place, London WC1E 7JE, Unted Kngdom Emal: l.papageorgou@ucl.ac.uk Abstract Ths paper presents an effcent soluton approach to tackle large-scale snglefloor process plant layout problems. Based on the mxed nteger lnear programmng (MILP) model proposed by Papageorgou and Rotsten [1], the fnal layout (.e. coordnates and dmensons) s determned from an ntal feasble soluton by an teratve mprovement procedure usng mxed nteger optmsaton. The applcablty of the soluton algorthm s demonstrated through two llustratve examples. Keywords: Process Plant Layout, Iteratve Soluton Approach, Mxed Integer Optmsaton 1. Introducton Plant layout s consdered as one of the mportant parts n the desgn stage of a chemcal plant. It deals wth the spatal arrangement of equpment tems and the requred connectons among them. The generaton of a good layout needs great ngenuty and experence because of ts sgnfcant mpact on process desgn and operaton. Engneerng, economc, safety and management ssues need to be consdered smultaneously and a reasonable balance must be acheved between these crtera.

2 2 G. Xu et al. A number of methodologes have been proposed to tackle the process plant layout problem. Intal approaches were based on heurstc rules and graph parttonng technques. Stochastc optmsaton technques [2] have been appled to obtan good qualty solutons. Fnally, mathematcal programmng models were presented to solve sngle and multple floor process plant layout problems. A mxed nteger nonlnear programmng (MINLP) approach [3] ntegrated safety and economc consderatons wth layout ssues. A dscretedoman MILP model was developed n [4]. A number of contnuous-doman MILP formulatons have been proposed to determne the land area, floor locaton and detaled layout of each process unt [1, -8]. It s wdely accepted that the optmal solutons for large-scale process plant layout problems are very dffcult to acheve usng current computatonal resources. The development of effcent soluton methods are of sgnfcant mportance snce t offers great opportuntes to obtan near optmal solutons wthn modest computatonal tmes. Effcent soluton approaches for sngle and multple floor cases were proposed n [9,10].The approach presented n ths paper s an teratve one where the soluton obtaned from prevous teraton s mproved by releasng and reallocatng a number of unts n the flowsheet. Ths s tested on two llustratve examples and some comparatve results are reported. 2. Problem Statement The sngle-floor process plant layout problem can be stated as follows: Gven () a set of equpment tems and ther dmensons, () the connecton costs among equpment tems; determne the allocaton of each equpment tem (.e., coordnates and orentatons); so as to mnmse the total connecton cost. In ths work, we adopt the contnuous-doman MILP model (named as LAYOUT; Papageorgou and Rotsten [1]) for the sngle-floor process plant layout problem where the optmal locaton of unt s determned by contnuous varables X and Y. Bnary varables E1 j and E2 j are used to avod overlappng between unts and j. Equpment tems are smplfed as rectangular shapes and the connectons among them are calculated as rectlnear dstances. 3. Iteratve Soluton Approach In ths secton, we present an teratve approach to tackle the sngle-floor process plant layout problem effcently. Accordng to ths approach, we start from the frst nteger soluton obtaned by solvng the LAYOUT model. Several unts are then selected and reallocated by solvng the reduced MILP model. The tems that are not released mantan ther relatve postons. Fnally, the approach termnates when no mprovement of the objectve functon value s observed after a prespecfed number of successve teratons. Next, the followng sets are defned for the descrpton of the teratve algorthm:

3 An Iteratve Soluton Approach to Process Plant Layout usng Mxed Integer Optmsaton 3 Sets I Δ Set of plant equpment unts consdered Set of unts released n the subproblems The steps of the proposed approach are shown below: Step 1: Intalse Δ = φ. Solve LAYOUT for every I to obtan the frst nteger soluton. Step 2: Fx E1 j and E2 j for every (, j) I. Step 3: Decde whch unts are released ether randomly or by probablstc rules (see Table 1). Update Δ. Step 4: Release E1 j and E2 j f and/or j Δ Step : Solve LAYOUT. If the objectve functon value over a prespecfed number of successve teratons remans the same, STOP. Otherwse, Δ = φ,go to Step 2. It s beleved that the selecton of released unts s of sgnfcant mportance to the fnal soluton qualty. Here, we propose random and probablstc selecton schemes as shown n Table 1. Table 1. Unt selecton probablty Approach Random_M Connect_M Cost_M Selecton probablty Unform dstrbuton = P P = NC NC j j ( C + C ) j ( C + C ) Cj Dj PL = D j Lnk_N C All algorthms used are named as Random_M, Connect_M, Cost_M and Lnk_N, where M and N represent the number of released unts and lnks, respectvely. Algorthm Random_M ndcates that M unts are chosen randomly. Alternatvely, equpment tems can be selected based on dfferent probablty dstrbutons. In algorthms Connect_M and Cost_M, selecton probabltes of each tem, defned as P, are assocated wth the number of connectons, NC, and the unt connecton costs of tem, respectvely. Algorthm Lnk_N attempts to release all pars of nodes that are connected by the N chosen lnks. The selecton probablty of each par, PL j, s related to the connecton costs between and j. j j j j j j

4 4 G. Xu et al. 4. Computatonal Results Two llustratve examples are nvestgated to demonstrate the applcablty of the proposed teratve approach. Tables 2 and 3 lst all the nput data for both examples (rmu stands for relatve monetary unts). Table 2. Dmensons of equpment unts for Examples 1 and 2 Unt Example 1 Example 2 α [m] β [m] Unt α [m] β [m] Unt α [m] β [m] Table 3. Connecton costs for Examples 1 and 2 Example 1 Example 2 Connecton Cost [rmu/m] Connecton Cost [rmu/m] Connecton Cost [rmu/m] (1,2) (1,18) 200 (3,7) 230 (1,) (18,7) 240 (4,9) 160 (2,3) (2,7) 230 (,6) 20 (3,4) (3,6) 400 (,9) 160 (4,) 8.3 (6,7) 230 (6,8) 170 (,6) 86.3 (7,10) 270 (7,14) 270 (,7) 82.8 (6,11) 280 (11,13) 300 (6,7) 6. (13,10) 170 (13,1) 170 (10,6) 300 (8,16) 20 (8,17) 20 (16,8) 140 (9,12) 170 (9,8) 17 (12,8) 17 Examples 1 and 2 are solved usng 8 teratve algorthms and model LAYOUT as shown n Table 4. All problems are mplemented n GAMS [11] usng CPLEX mxed nteger optmsaton solver wth 0% margn of optmalty. All runs are performed on an hp pavlon laptop wth seconds CPU lmt. The proposed approach termnates when the objectve functon can not be mproved after 20 successve teratons. Each algorthm s repeated 10 tmes and the best and medan objectve functon values are reported together wth the medan computatonal tmes. Example 1 consders a 7-unt ethylene oxde plant ntroduced by Penteado and Crc [3]. The optmal soluton s rmu acheved by model LAYOUT

5 An Iteratve Soluton Approach to Process Plant Layout usng Mxed Integer Optmsaton wthn 2.86 seconds. When applyng teratve algorthms wth dfferent values of M and N, all algorthms end up wth the optmal soluton thus llustratng the robustness of the proposed approach. Example 2 consders the layout desgn of an 18-unt ndustral mult-purpose batch plant presented by Georgads et al [4]. Wthn the prespecfed CPU lmt (10000s), model LAYOUT can not solve ths example to optmalty resultng n an nteger feasble soluton wth an objectve functon value of 320 rmu. The best result acheved through the teratve approach s rmu from Lnk_1 and Lnk_2, whch s 2.80% better than model LAYOUT. Also, note that the best medan results has been obtaned by Lnk_2 (31810 rmu), whch consttutes a 2.27% mprovement over the LAYOUT model. Table 4. Computatonal results for Examples 1 and 2 Example 1 Example 2 Approach Best Medan CPU Best Medan CPU Random_ Random_ Connect_ Connect_ Cost_ Cost_ Lnk_ Lnk_ LAYOUT * *Maxmum CPU lmt (10000s) The layouts for both examples assocated wth the best objectve functon values obtaned from the teratve approach are shown n Fgure 1. Example (OBJ= ) 40 Example 2 (OBJ=31640) Fgure 1. Best layout obtaned for Examples 1 and 2

6 6 G. Xu et al.. Conclusons In ths work, an teratve soluton approach has been proposed to solve largescale process plant layout problems. Accordng to the MILP formulaton [1], the soluton qualty has been mproved from an ntal feasble layout through an teratve process usng releasng and reallocaton schemes. Durng each teraton, process unts are selected ether randomly or based on specfc probablstc rules. Fnally, the applcablty of the proposed approach has been demonstrated by two llustratve examples. The results show that the teratve soluton approach has great potental to obtan good qualty solutons for process plant layout problems wth large szes usng modest computatonal requrements. Acknowledgements GX acknowledges support from the Centre for Process Systems Engneerng. References 1. L.G. Papageorgou, and G.E. Rotsten, Ind. Eng. Chem. Res., 37 (1998) C.M.L. Castell, R. Lakshmanan, J.M. Skllng and R. Banares, Comput. Chem. Eng., 22 (1998) S F.D. Penteado and A.R. Crc, Ind. Eng. Chem. Res., 3 (1996) M.C. Georgads, G. Schllng, G.E. Rotsten, S. and Macchetto, Comput. Chem. Eng., 23 (1999) D.I. Patsatzs and L.G. Papageorgou, Comput. Chem. Eng., 26 (2002) A. P. Barbosa-Povoa, R. Mateus and A.Q. Novas, Int. J. Prod. Res., 39 (2001) D.B. Ozyruth and M.J. Realff, AIChE J., 4 (1999) R. Gurardello and R.E. Swaney, Comput. Chem. Eng., 30 (200) D.I. Patsatzs and L.G. Papageorgou, Ind. Eng. Chem. Res., 42 (2003) G. Xu and L.G. Papageorgou, Ind. Eng. Chem. Res., 46 (2007) A. Brooke, D. Kendrck, A. Meeraus and R. Raman, GAMS: A user s gude GAMS development Corp. Washngton, DC (1998).