Optimal allocation of safety and security resources

Transcription

1 397 A publcaton of VOL. 31, 2013 CHEMICAL ENGINEERING TRANSACTIONS Guest Edtors: Eddy De Rademaeker, Bruno Fabano, Smberto Senn Buratt Copyrght 2013, AIDIC Servz S.r.l., ISBN ; ISSN The Italan Assocaton of Chemcal Engneerng Onlne at: Optmal allocaton of safety and securty resources Genserk L.L. Reners* a,b, Kenneth Sörensen a a Unverstet Antwerpen, Antwerp Research Group on Safety and Securty (ARGoSS), Prnsstraat 13, 2000 Antwerp, Belgum b Hogeschool-Unverstet Brussel, KULeuven, Research Group CEDON, Stormstraat 2, 1000 Brussels, Belgum genserk.reners@ua.ac.be A model was developed to eecute a cost-effcency analyss for takng preventon nvestment decsons. Usng the knapsack problem solvng technque well-known n the feld of operatonal research, the approach suggests how to prortze bundles of preventon measures based on ther costs and benefts wthn a predefned preventon budget. Ths way, an effcent allocaton of safety and securty measures s acheved usng the suggested methodology. 1. Introducton It s essental for companes to be able to reduce and control the wde varety of estng occupatonal rsks n a cost-effcent way. Such rsks are reduced through safety management, rsk reducton polces and preventon measures. To determne an optmal set of preventon measures, organzatons need to take nto account n a systematc way the measures costs as well as ther (hypothetcal) benefts. Preventon benefts can be calculated by determnng the dfference between hypothetcal accdent costs before and after mplementng precauton measures. In other words, the avoded accdent costs are calculated, or the fnancal benefts of accdents not happenng. The model bulds on these nsghts. A concept well-known n organzatons throughout the world for rankng rsks s the rsk matr, allowng to make a classfcaton of rsks n a systematc and transparent way (CCPS, 1992; Mddleton, 2001; Cook, 2007; Wlknson and Davd, 2008; Co, 2008 and 2011; Smth et al., 2009). Ths method can be used to measure and categorze rsks on an nformed judgment bass as to both lkelhood and consequence and as to a relatve level of mportance. Dfferent possble ways to use and refne the rsk matr are possble, as Garvey (2009) eplans. Despte the fact that the rsk matr merely allows to roughly assess rsks, many decson takers and consultants are convnced that the method s very useful to make a qualtatve dstncton between dfferent levels of rsks. The qualtatve dfference between rsks whch s based on the rsk matr s preferable over ad random decson takng (Co, 2008). A rsk matr whch s dvded nto four consequence grades and fve lkelhood grades, can for eample be employed. Consequence grades can be epressed n fnancal terms, whle lkelhood grades are epressed n the number of tmes per year that a rsk leads to an accdent n an organzaton. Table 1 llustrates the rsk matr used n the remanng of ths artcle. Every cell of the rsk matr corresponds to a rsk class. Per cell (thus per rsk class), the fnancal consequence value s multpled by the lkelhood value, and the total yearly costs per rsk class are determned and shown.

2 398 Table 1: Rsk matr used n ths artcle Lkelhood [year -1 ] Cell assgnments (n /year) > 1 7,500 75, ,000 2,500,000 > ,500 75, ,000 > ,500 25,000 > ,500 > Consequence classes / fnancal mpact [ ] < 7,500 75, ,000 < 2,500,000 Co (2008) ndcates that a certan rsk n each of the cells of any rsk matr s not equally large (or small) due to the classfcaton nto rsk classes. Hence, a rsk cell may contan dfferent varetes of rsks. Ths does not pose a problem n our research, snce to decde on bundles of preventon measures, we am at comparng bundles of rsks, not ndvdual rsks. A dscretzaton of the rsk matr nto n cells s llustrated n Fgure 1. Every rsk cell s numbered from 1 to n (n our eample, n = 20). Fgure 1: Dscretzaton of the rsk matr

3 399 The rsk matr can be refned by relatng the rsk classes to a cost-benefts analyss. Ths way, a decson support nstrument can be developed whch can be used to determne, takng a certan safety budget nto account, the rsk reducng measures or precauton measures leadng to the most optmal and cost-effcent result wthn an organzaton. 2. Basc model development In ndustral practce, companes are confronted wth budget lmtatons. In ths paper, we call the avalable yearly budget for safety preventon Bu tot. Let us assume that; when possble preventon nvestments eceed ths budget, they cannot be carred out. Therefore, only the preventatve measures havng a cost wthn Bu tot wll be consdered n the model. To employ a model for takng cost-effcent preventon decsons, certan actons should have been carred out by the user and certan nput nformaton s needed. All rsks should have been classfed nto one of the rsk cells of the rsk matr. Every cell corresponds to a potental cell cost C, determned by: C l c (1) wth: C = Costs resultng from an accdent related to a rsk from rsk cell l = lkelhood correspondng to rsk cell c = fnancal mpact (consequences) correspondng to rsk cell Table 1 llustrates the cost fgures (epressed n per year) for ths artcle s rsk matr. Other rsk matr confguratons are of course possble n real ndustral practce. When precauton nvestments are made to decrease rsks stuated wthn cell towards cell j (remark that j s characterzed wth lower consequences and/or lkelhood), the potental cell costs become C j. Hypothetcal benefts n that case can be calculated as C - C j. The requred nformaton for applcaton of the model s shown n Table 2. Table 2: Requred nformaton for applcaton of the model n Number of cells where rsks do est for the organzaton (= Nc; C Bu tot Costs of Preventon for gong from rsk cell to rsk cell j (CoP j ), Nc n ), j whereby Nc When all these data are known, t s possble to use the suggested approach to determne the most costeffcent preventon measures. The frst step of the model s the categorzaton of rsks nto the rsk classes of the rsk matr. We assume that Nc rsk cells (out of the n rsk cells n total) contan one or more rsks. Preventon costs to go from rsk cell to rsk cell j (remark that j < ) are wrtten as CoP j. If the preventon costs are hgher than the yearly preventon budget Bu tot, no nvestment wll be made n these preventon measures, hence these preventon costs are ecluded at the begnnng of model eecuton. The hypothetcal benefts correspondng to a decrease n rsk cell from to j are calculated by subtractng C j from C. Furthermore, a lst of preventon measures wll have been drawn up. Usng ths lst, the optmal rsk portfolo can be determned usng optmzaton. In ts smplest form, determnng the optmal rsk portfolo s equal to solvng a knapsack problem. The knapsack problem derves ts name from the fact that a person havng to fll hs fed sze knapsack wth the most valuable tems faces a smlar problem. The knapsack problem s one of the most fundamental problems n combnatoral optmzaton and has many applcatons, e.g., n stock portfolo management, as well as many etensons. In the basc verson of ths problem, a set of decson varables s defned where varable (correspondng to measure ) takes on value 1 f ths measure s chosen as part of the portfolo and 0 f t s not. A mathematcal formulaton of the knapsack problem s the followng:

4 400 mab s. t. C Bu {0, 1} tot (2) (3) (4) The frst equaton (2) epresses the total beneft from the selected portfolo, whch should be mamzed. The second equaton (3) epresses the fact that the total cost of the selected measures should not eceed the budget. The thrd constrant (4) mples that a measure s ether fully taken or not taken at all. A number of assumptons are mplctly taken n ths formulaton: A measure s ether taken or not (t cannot be partally taken); The total beneft of all measures taken s the sum of the ndvdual benefts of the chosen measures; The total cost of all measures taken s the sum of the costs of the ndvdual measures; Measures can be ndependently mplemented, wthout consequences for the other measures. Some of these assumptons are not entrely realstc. In the followng secton, we wll suggest some possble model refnements. Although the knapsack problem s NP hard 1, t can be solved effcently even for very large nstances (Martello et al., 2000). The advantage of usng the knapsack-based formulaton s that t can be solved by standard off-the-shelf commercal software for med-nteger programmng, such as CPLEX (ES1) or Gurob (ES2) or ther open source counterparts such as GLPK (ES3) or lpsolve (ES4). Moreover, even spreadsheet software such as Ecel or LbreOffce nclude a solver that can be used to model and optmze the safety measures portfolo usng the method descrbed n ths paper. In ndustral practce, an optmal allocaton of safety measures wth mamum one preventon measure from each of the rsk cells whch can be assgned, needs to be determned by usng the model. As eplaned before, to solve ths problem, four condtons have to be met: () the total beneft of measures taken, needs to be mamzed; () the avalable budget constrant needs to be respected; () mamum 1 decrease per rsk cell s allowed; and (v) a measure can be taken, or not. These condtons translate nto the followng mathematcal epressons: ma, j j B j j CoP j Bu tot, (6) j j 1 j 0,1 Solvng these equatons for a certan concrete problem yelds the optmal soluton for the allocaton of safety and securty measures. 3. Model refnements In general, the portfolo of safety measures chosen by a company s subject to a number of etra constrants, that epress relatonshps between these measures. Fortunately, these relatonshps are generally easly added to the knapsack-based model, usually by ntroducng addtonal constrants. For eample bnary relatonshps are possble: f rsk cell r s decreased, rsk cell t also has to be decreased and vce versa. Ths stuaton occurs when measures are mutually dependent on each other and takng one measure wthout the other makes no sense. An eample s when the use of a new devce that enhances safety requres tranng. It does not make sense to nstall the devce wthout the tranng, and t does not make sense to gve the tranng wthout nstallng the devce. (5) (7) (8) 1 An optmzaton problem s NP hard f the runnng tme of the fastest known algorthm to solve t ncreases eponentally n the problem sze.

5 401 Ths relatonshp between rsk cell decreases from r to s and from t to u can be epressed n the model by the etra constrant rs tu (9) Another stuaton whch mght occur s the followng: f rsk cell r s decreased, rsk cell t also has to be decreased, but the reverse s not true. As an eample, to prevent fre from spreadng between departments, a company s consderng nstallng a fre-resstng door. The tme the door ressts fre can be ncreased by addng an etra layer of freproof coatng to t. Clearly, nstallng the coatng and not the door makes no sense, but the reverse does. The relatonshp between rsk cell decrease rs (nstallng the door) and rsk cell decrease tu (nstallng the freproof coatng) can be epressed as: rs tu (10) Yet another possble stuaton s that ether rsk cell r or rsk cell t needs to be decreased, but not both rsk cells at the same tme. Ths stuaton can occur f two measures duplcate each others effects and the company judges t superfluous to nvest n both measures smultaneously. E.g., a machne can be protected by a concrete casng or a steel casng, but not by both. Ths can be mathematcally epressed as follows: rs 1 tu (11) Yet other bnary relatonshps are possble, and a mathematcal answer for solvng the knapsack problem can usually easly be found. Actually, n prncple, all relatonshps between measures can be epressed as constrants n the knapsack problem. However, for some stuatons, the benefts or costs of measures are not smply addtve. Suppose e.g., that two fre doors can be nstalled n seres to prevent fre from spreadng to the net room. Clearly, the effect of nstallng one door nstead of none wll be larger than the effect of nstallng two doors nstead of one. In other words, there wll be a dmnshng rate of return on the second door. Ths can be easly handled by dentfyng such stuatons and creatng vrtual measures n the cost-beneft table to represent the acton of takng both measures. To ensure that each measure s only taken once, some addtonal constrants are also necessary. 4. Conclusons and recommendatons Preventng occupatonal accdents n the ndustry, s an mportant ependture on a yearly bass, for any organzaton. Optmzng preventon nvestments and makng nvestment decsons n a cost-effcent way s therefore essental. To ths end, we suggest a user-frendly knapsack-based model for takng cost-effcent preventon decsons. The model uses some data that can easly be determned by any organzaton and that can be dsplayed usng a rsk matr. The most cost-effcent preventve measures are determned followng the knapsack algorthm, gven a certan preventon budget avalable. References CCPS (Center for Chemcal Process Safety) (1992). Gudelnes for Hazard Evaluaton Procedures, 2nd Ed., New York: Amercan Insttute of Chemcal Engneers. Cook, R. (2007). Smplfyng the creaton and use of the rsk matr. In F. Redmll & T. Anderson (Eds.). Improvements n System Safety (p ). Brstol: Sprnger. Co, L. A., Jr. (2008). What s wrong wth rsk matrces. Rsk Analyss 28(2). Co, L.A., Jr. (2011). Evaluatng and mprovng rsk formulas for allocatng lmted budgets to epensve rsk-reducton opportuntes. Accepted for publcaton (2011) n Rsk Analyss. DOI: /j ES1 (2012). Accessble va: ES2 (2012). Accessble va: ES3 (2012). Accessble va: ES4 (2012). Accessble va:

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