Scheduling under Uncertainty using MILP Sensitivity Analysis

Transcription

1 Schedulig uder Ucertaity usig MILP Sesitivity Aalysis M. Ierapetritou ad Zheya Jia Departmet of Chemical & Biochemical Egieerig Rutgers, the State Uiversity of New Jersey Piscataway, NJ Abstract The aim of this paper is to develop a itegrated framework i order to address the issue of ucertaity i short-term schedulig. The idea of iferece-based sesitivity aalysis for MILP problem is employed withi a brach ad boud solutio framework to determie the importace of differet parameters ad costraits ad to provide a set of alterative schedules for the rage of ucertai parameters uder cosideratio. A illustrative example is cosidered usig the proposed approach ad the results are compared with parametric programmig ad robust optimizatio Keywords: ucertaity, sesitivity aalysis, cotiuous time formulatio, schedulig 1. Itroductio I the studies of short-term schedulig, the problem data are mostly assumed to be determiistic. However, i real plats, parameters like processig times, market requiremets vary with respect to time ad are ofte subject to uexpected deviatios. Therefore, the cosideratio of ucertaity i schedulig problem becomes of great importace i order to preserve plat feasibility ad viability durig operatios. A umber of approaches exist i the literature to address the problem of ucertaity. Due to space limitatios the iterested reader is referred to a recet paper by Jia ad Ierapetritou (2003) for a detailed literature review i the subject. Most of the existig approaches however, ca hadle oly a certai type of ucertai parameters, ad more importatly, the additioal complexity costitutes them ifeasible for realistic applicatios. I this paper, a ovel framework is proposed for ucertaity aalysis of schedulig problems based o the ideas of sesitivity aalysis of the correspodig MILP problem. The paper is orgaized as follows. The determiistic model ad the robustess metric that are adopted i this work are briefly described i the ext sectio. I sectio 3, the basic backgroud of sesitivity aalysis is provided, followed by the proposed approach for schedulig uder ucertaity. Sectio 4 presets the results of oe example to illustrate the applicability of the proposed approach. Author to whom correspodece should be adressed : mariath@sol.rutgers.edu

2 2. Determiistic Schedulig Formulatio ad Robustess Metric I this paper, the mathematical model used for batch plat schedulig follows the mai idea of cotiuous time formulatio proposed by Ierapetritou ad Floudas (1998). The model ivolves the followig costraits: miimize H or maximize price(s)d(s, ) (1) subject to wv(i,j,) 1 i I j s p + c i Ij j Jj i Ij j Jj st(s,) = st(s,-1)-d(s,) + p (s,i) b(i, j, 1) p (s,i) b(i, j, ) (3) st(s,) stmax(s) Vmi(i,j)wv(i,j,) b(i,j,) Vmax(i,j)wv(i,j,) (5) d(s, ) r(s)tf(i,j,)=ts(i,j,)- α(i,j)wv(i,j,)+ β(i,j)b(i,j,) (6) Ts(i,j,+1) Tf(i,j,)-U(1-wv(i,j,)) (7) Ts(i,j,) Tf(i,j,)-U(1-wv(i,j,)) (8) Ts(i,j,) Tf(i,j,)-U(1-wv(i,j,)) (9) Ts(i,j,+1) Tf(i,j,),Tf(i,j,+1) Tf(i,j,) (10) Tf(i,j,) H,Ts(i,j,) H (11) (2) (4) where U deotes a upper boud of the makespa, for the cases where the objective is the miimizatio of makespa. For profit maximizatio, U=H i costraits (8) to (10). I geeral, the objective fuctio is to miimize the makespa as show i (1) or to maximize the total profit. The rest of the model ivolves the allocatio costraits (2), material balaces (3), capacity costraits (4), (5), demad costraits (6) ad the timig costraits (7)-(11). The detail explaatio of the model is provided i Ierapetritou ad Floudas I order to improve the schedule flexibility prior to its executio, it is importat to measure the performace of a determiistic schedule uder chagig coditios due to ucertaity. Vi ad Ierapetritou (2001) proposed a robustess metric SD corr takig ito cosideratio the ifeasible scearios. Their proposed robustess metric is defied as: corr p H 2 p act avg p avg (p p tot 1) tot (H H ) SD = ; H = where H avg is the average makespa over all the scearios p tot, while H act = H p, if sceario p is feasible ad H act = H corr, if sceario p is ifeasible. 3. Schedulig uder Ucertaity 3.1 MILP sesitivity aalysis The formulatio preseted i sectio 2 correspods to Mixed Iteger Liear Programmig (MILP) problem where the biary variables wv(i,j,) deote the

3 assigmet of tasks (i) to uits (j) at evet poit (), respectively, throughout the time horizo. Therefore, the effects of operatio parameters o the plat performace ca be ivestigated through the sesitivity aalysis of the MILP model of the determiistic schedulig problem. Although sesitivity aalysis theory is well developed i liear programmig, efforts are still beig made i order to hadle the iteger programmig case maily due to lack of optimality criteria for the iteger optimizatio problems. Most of the existig approaches address the sesitivity aalysis for pure 0-1 iteger programmig problems. A method of sesitivity aalysis for mixed iteger liear programmig is preseted by Dawade ad Hooker (2002), based o the idea of iferece duality. It reveals that ay perturbatio that satisfies a certai system of liear iequalities will reduce the optimal value o more tha a prespecified amout. The iferece-based sesitivity aalysis cosists of two parts: dual aalysis that determies how much the problem ca be perturbed while keepig the objective fuctio value i a certai rage, while primal aalysis gives a upper boud o how much the objective fuctio value will icrease if the problem is perturbed by a certai amout. More specifically the mai results of the iferece-based sesitivity aalysis are summarized below. For the geeral mixed iteger problem: miimize z=cx subject to Ax a 0 x h,x it eger, j = 1, Kk j (13) Assumig a perturbatio of all problem parameters such that: miimize z = (c+ c)x subject to (A+ A)x a+ a 0 x h,x iteger,j=1, K,k j (14) If there exist s 1 p,...,s p that satisfy the followig set of iequalities, the costrait z z * - z remais valid: -for the perturbatios A ad a i the parameters ivolved i the left ad right had side of the costraits: p p p p p i j ij j + j j j λ p i j= j= 1 s A u s (u u ) a p p p p = j j + λ p + p j= 1 r q u a z z p p p p j j ij j j s s A,s q,j= 1, K, r p (15) -for a perturbatio c of the coefficiets of the objective fuctio: j= p p p p j j j j j p cu s (u u ) r p p p j j j j s c,s q, j= 1, K, (16)

4 where q j p = λ i p A ij - λ 0 p c j ; p correspods to the leaf ode where the dual variable of the objective fuctio (λ 0 p ) equals to 1, whereas u j p ad ū jp deote the lower ad upper boud of x j at ode p, respectively; z p is the objective value at ode p; ad z p = z * - z p. Thus by utilizig costraits (18), (19) i the schedulig problem, the rage of parameters where the objective remais withi certai limits ca be idetified ad used to evaluate alterative schedules at the brach ad boud tree. Moreover the importace of differet costraits ad parameters are determied ad ca be utilized to improve future plat operability. 3.2 Proposed ucertaity aalysis approach The basic idea of the proposed approach is to utilize the iformatio obtaied from the sesitivity aalysis of the determiistic solutio to determie (a) the importace of differet parameters ad costraits ad (b) the rage of parameters where the optimal solutio remais uchaged. The mai steps of the proposed approach are show i Figure 1. More specifically, there are two parts i the proposed aalysis. I the first part, importat iformatio about the effect of differet parameters is extracted followig the sesitivity aalysis step, whereas i the secod part alterative schedules are determied ad evaluated for differet ucertaity rages. First, the determiistic schedulig is solved at the omial values usig a brach ad boud solutio approach, ad the dual multiplier λ p is collected at each leaf ode p. The the iferece-based sesitivity aalysis as described i previous sectio is performed for all the importat schedulig parameters, icludig demads, prices, processig times ad capacities. Note that, oly the dual iformatio of the odes that correspod to ozero dual variables is required. Usig the results of this aalysis, a umber of questios, such as how the capacity of the uits affects the objective, ca be aswered regardig the plat robustess to parameter chages. I the secod part of the aalysis, the sesitivity iformatio is used to defie the rage of ucertai parameters where the schedule is optimal ad idetify alterative schedules at differet ucertaity rages. The set of costraits (18), (19) are used to determie the rage of ucertai parameters for certai chages i the objective fuctio. The brach ad boud procedure is the cotiued usig the odes with the objective value withi the predicted limits to idetify ew optimal solutios. The alterative schedules are evaluated usig the robustess metric (SD corr ) as defied i sectio 2, the average ad the omial performace of the objective fuctio. Sice the etire aalysis is based o oe out of a large umber of possible brach ad boud trees that ca be used to solve the MILP, it provides coservative sesitivity rages. Theoretically, the exact sesitivity rages ca be obtaied by ivestigatig a expoetial umber of brach ad boud trees. However, usig the above aalysis, useful iformatio ca be extracted regardig the approximate rage of parameter chage for certai objective chage, plat robustess to parameter chages, as well as to determie the importace of differet parameters.

5 Figure 1: Flow chart of the Proposed Approach 4. Illustrative Example The illustrative example (Ierapetritou ad Floudas, 1998) cosiders two differet products produced through five processig stages: heatig, reactios 1, 2 ad 3, ad separatio of product 2 from impure E. The problem is solved with the objective of maximizig the profit withi the time horizo of 12 hours. After the sesitivity aalysis is performed, the followig iformatio is obtaied. It is foud that the most critical task of the productio lie is reactio 2. By decreasig the processig capacity of reactio 2 i reactor 1 or reactor 2 by 11 uits, the profit will be reduced by 5%, whereas very small chage or o chage at all is observed at the objective fuctio with reactio 1 or reactio 3 processig capacity chage i both reactors. The objective value is also ot sesitive to the chage of other parameters; for example, the processig capacity of separatio i separator ca drop by up to 120 uits without decreasig the profit. Aother importat modelig issue that ca be addressed is the questio of costrait redudacy. Here the importace of storage costraits are ivestigated ad it is foud that these costraits are redudat sice they are ot active i ay of the solutio brach ad boud odes. More iterestigly, the duratio costraits are also foud to be redudat, which meas that the maximum processig capacities are already reached with the curret processig times, so that the profit caot be improved eve with zero processig times assumig fixed umber of evet poits. The demad of product 2 is cosidered to be the ucertai parameter varyig withi the rage of [20, 80] ad the objective fuctio is modified to miimize the makespa. A brach ad boud tree is costructed at omial poit r('p2')=50 ad the dual iformatio is stored at each ode. Applyig the iferece duality sesitivity aalysis, the followig expressio is obtaied regardig the rage of demad chage followig specific objective chage ( H): d H, which meas that if the demad is icreased by d, the ew makespa becomes at most $H om d. Whe r('p2') is icreased from 50 to 80, schedule 1 becomes ifeasible. Usig the previous iequality, we solve the LP problem at each leaf ode with the demad of 80 ad check the leaf odes with the objective

6 value below 7.89 i the brach ad boud tree. The ew optimal solutio is foud to be schedule 2 ad schedule 3 is oe feasible solutio. The schedules are the evaluated with respect to the mea ad omial makespa ad stadard deviatio withi the demad rage of [20, 80] ad the values are show i Table 1. Comparig with schedule 2, schedule 3 has larger mea makespa but lower stadard deviatio, which meas higher robustess, therefore depedig o the decisio maker's attitude towards risk ad the expected demad growth oe ca choose schedule 3 compared to schedule 2, whereas schedule 1 remais a valid alterative if the demad is expected to remai costat. Nomeclature Idices i = tasks; j = uits; = evet poits; s = states; k = scearios Parameters p k = probability of sceario k; price(s) = price of state s ρ p (s,i), ρ c (s,i) = proportio of state s produced, cosumed from task i, respectively r(s) = market requiremet for state s at the ed of the time horizo Vmi(i,j), Vmax(i,j) = miimum ad maximum capacity of uit j whe processig task i stmax(s) = maximum storage capacity of state s Variables wv(i,j,) = biary variables that assig the begiig of task i i uit j at evet poit b(i,j,) = amout of material udertakig task i i uit j at evet poit d(s,) = amout of state s beig delivered to the market at evet poit H = time horizo st(s,) = amout of state s at evet poit Ts(i,j,), Tf(i,j,) = startig ad fiishig time of task i i uit j at evet poit Refereces Acevedo A. ad E.N.Pistikopolous, 1997, Id.Eg.Chem.Res. 36, 717. Dawade M.W. ad J.N.Hooker, 2000, Oper.Res., 48, 623. Ierapetritou M.G. ad C.A.Floudas, 1998, Id.Eg.Chem.Res. 37, Jia Z. ad M.G.Ierapetritou, 2003, accepted for publicatio, Id.Eg.Chem.Res. Vi J.P. ad M.G.Ierapetritou, 2001, Id.Eg.Chem.Res., 40, 4543.