Supply Network Protection under Capacity Constraint

Transcription

1 Supply Network Protecton under Capacty Constrant Naj Brcha Mustapha Nourelfath January 2014 CIRRELT

2 Naj Brcha *, Mustapha Nourelfath Interunversty Research Centre on Enterprse Networks, Logstcs and Transportaton (CIRRELT) and Department of Mechancal Engneerng, 1065, avenue de la Médecne, Unversté Laval, Québec (Québec), G1V 0A6 Abstract. Ths artcle develops a game-theoretcal model to deal wth the protecton of facltes, n the context of the capactated fxed-charge locaton and capacty acquston problem. A set of nvestment alternatves are avalable for drect protecton of facltes. Furthermore, extra-capacty of neghbourng functonal facltes can be used after attacks to avod the backlog of demands and backorders. The proposed model consders a noncooperatve two-perod game between the players, and an algorthm s presented to determne the equlbrum soluton and the optmal defender strategy under capacty constrants. A method s developed to evaluate the utltes of the defender and the attacker. The beneft of the proposed approach s llustrated usng a numercal example. The defence strategy of our model s compared to other strateges, and the obtaned results ndcate clearly the superorty of our model n fndng the best trade-off between drect protecton nvestment and extra-capacty deployment. Keywords. Capacty, protecton, faclty, attack, damage, game theory, vulnerablty, contest. Acknowledgements. The authors would lke to acknowledge the fnancal support of the Natural Scences and Engneerng Research Councl of Canada (NSERC) and the Fonds de recherche du Québec - Nature et technologes (FRQNT) for a Doctoral research scholarshp to Naj Brcha. Results and vews expressed n ths publcaton are the sole responsblty of the authors and do not necessarly reflect those of CIRRELT. Les résultats et opnons contenus dans cette publcaton ne reflètent pas nécessarement la poston du CIRRELT et n'engagent pas sa responsablté. * Correspondng author: Naj.Brcha@crrelt.ca Dépôt légal Bblothèque et Archves natonales du Québec Bblothèque et Archves Canada, 2014 Brcha, Nourelfath and CIRRELT, 2014

3 1. Introducton Crtcal nfrastructures such as supply networks represent enormous nvestments consecrated to the dstrbuton of goods and servces. These nvestments requre large captal outlays. Even a mnor dsrupton can degrade the system performance. Reductons of capacty can ntroduce sgnfcant delays n gettng back to the planned producton schedule and nflct substantal losses. Such supply networks can be vctm of dfferent threats, such as accdental falures, natural catastrophes, terrorst attacks and sabotages, fre, ndustral accdents, tsunams, earthquakes, floods and cyclones. Recent events have shown that f one or more enttes of a crtcal logstcal network (key facltes, bottlenecks, crtcal lnks, etc.) are damaged due to an accdent or an ntentonal attack, the network s paralyzed and the damage would be enormous, resultng n a negatve mpact at the socal, poltcal and economc levels. Attacks can also have a serous mpact on the health and safety or the system effectve functonng. There s a relevant lterature on the defence of networks nfrastructures aganst ntentonal attacks. In [1], the authors analyze the strategc defence and attack of complex networks and systems wth components n seres, parallel, nterlnked, nterdependent, ndependent, or combnatons of these. The authors of [2] have recently developed a method based on a Monte Carlo smulaton approach for evaluatng the expected damage related to nodes deprvaton of supply of commodtes n multcommodty networks as a consequence of ntentonal attack on arbtrarly chosen network lnks. In [3], the authors showed that scale free networks are robust aganst random falures but fragle to ntentonal attacks. In [4], crtcal locatons susceptble to terrorst attacks are determned by decson makers on base of geographc regons classfcaton. In [24] the authors study the effects of ntentonal attacks on transportaton networks of two arcs that are subject to traffc congeston. The authors of [27] provde optmal protecton confguratons for a network wth components vulnerable to an nterdctor wth potentally dfferent attackng strateges. Ref. [28] develops an ordnal optmzaton based method to dentfy top contrbutors to power networks falure when consderng cascade falure events. There s also a mature lterature on faclty desgn wth probablstc falure of components [5,29,30]. However, the possblty of ntentonal strkes or attacks s not normally taken nto account n ths lterature, except n [6] where the authors developed a game-theoretcal model to protect facltes aganst ntentonal attacks n the context of the uncapactated fxed-charge locaton problem. Ths artcle deals wth the protecton of network logstc facltes n the context of the capactated fxedcharge locaton and capacty acquston problem (CFL & CAP). Faclty locaton and capacty CIRRELT

4 acquston are of vtal mportance to supply chan management [7-9]. Here, we consder the CFL & CAP to deal wth resource allocaton of protecton and allocaton of the extra-capacty among facltes. The am of the CFL & CAP s to decde smultaneously on the optmal locaton and capacty sze of each new faclty to be establshed [8-13]. In ths problem, we are gven a set of customer locatons wth known demands and a set of potental faclty locatons. If we decde to locate a faclty wth a chosen capacty at a canddate ste, we ncur a known fxed locaton cost. There s a known unt cost of shppng between each canddate faclty ste and each customer locaton. The problem s to fnd the locatons of the facltes and the shpment pattern between the facltes and the customers, to mnmze the sum of the faclty locaton and shpment costs, subject to constrants that all demands must be served, facltes capactes must not be exceeded, and customers can only be served from open facltes. For the protecton of network logstc facltes n the context of the CFL & CAP, we consder that not only a set of nvestment alternatves are avalable for drect protecton of facltes, but also extracapactes of neghbourng functonal facltes can be used after attacks for ndrect protecton. Extracapacty s among the dfferent strateges to deal wth the rsk of uncertan producton capacty. It can be used, after a capacty shock, to quckly brng back the producton on schedule and to avod the backlog of demands [14]. In the case of demand growth, facltes of supply network mght hold extracapacty aganst demand varablty [15]. In our case extra-capacty of neghbourng functonal facltes s used after attacks, n order to satsfy all customers demand and to avod the backorders. The CFL & CAP problem assumes that, once constructed, the facltes chosen wll always operate as planned. However, f a faclty s attacked, t may become unavalable and customers must be served from others functonal facltes of the supply network that are farther than ther regular facltes, but subject to constrants that functonal facltes capactes must not be exceeded. To satsfy customers demand and to avod backorders, the amount produced by dsabled faclty must be allocated optmally among the functonal facltes. Ths reduces the cost of delayed producton after attacks, but may lead to excessve addtonal transportaton costs. The strategc decson dealt wth here s how to allocate optmally the protectve resources and the extra-capacty among the facltes, knowng that these facltes are exposed to external ntentonal attacks. In other words, gven a set of nvestments alternatves for protectng the facltes and set of extra capactes, we want to determne how much to nvest optmally n drect protecton of facltes and n ndrect protecton by extra capacty, whle takng nto account that both the defender and the attacker are fully optmzng agents. The dea of 2 CIRRELT

5 usng extra-capacty to ndrectly protect supply networks aganst ntentonal attack s used n ths paper to develop a game-theoretc model wth the objectve of fndng the best trade-off between drect nvestments n protecton and ndrect protecton by extra-capactes deployment. The remander of the paper s organzed as follows. Secton 2 presents the mathematcal model of capactated fxed-charge locaton and capacty acquston problem. Secton 3 formulates the studed problem as a two-perod non-cooperatve game. Secton 4 evaluates the players utltes. Secton 5 develops an algorthm to solve the game. Secton 6 presents a numercal example. Secton 7 concludes the paper. 2. The faclty locaton and capacty acquston problem The CFL & CAP consders the problem of locatng facltes to mnmze the sum of the faclty locaton costs, the costs of capacty acquston assocated wth the sze of open faclty and the shppng costs from open facltes to customers subject to constrants that all demands must be served, faclty capactes must not be exceeded, and customers can only be served from open facltes [9]. The CFL &CAP has been wdely studed n the lterature and appled n a varety of domans, and s known to be NP-hard [16]. Let n denote the number of customers (ndexed by j) and m denote the number of alternatve faclty locatons (ndexed by ). The followng notatons are also used n the mathematcal model: I the set of canddate faclty locatons, ndexed by, J the set of customer locatons, ndexed by j, g the fxed cost of locatng a faclty at canddate ste, h (.) the total capacty acquston cost at faclty, c j the unt cost of shppng between canddate faclty ste and customer locaton j, D j the demand at customer locaton j, CAP the maxmum capacty that can be bult-n at canddate ste. The decson varables are: Z j the quantty shpped from canddate faclty ste to customer locaton j, CIRRELT

6 Y 1 f a faclty s to be located at canddate ste, and 0 otherwse. The problem can be modeled as follows: Mnmze g Y h Z c Z m n n j j j 1 j1 j1, (1) Subject to m Z j=d j j 1, 2,..., n, (2) 1 0 Z Y D 1, 2,..., m, j 1, 2,..., n, (3) j j n Zj CAP 1, 2,..., m. (4) j1 Y 0, 1 1, 2,..., m. (5) The objectve functon (1) mnmzes the sum of the fxed faclty locaton costs, the costs of capacty acquston assocated to open faclty, and the shpments or transportaton costs. Constrant (2) ensures that each customer s demand wll be fully satsfed. Constrant (3) s a smple nonnegatvty constrant and t guarantees that customers receve shpments only from open facltes. Capacty constrant (4) ensures that an open faclty does not supply more than ts capacty CAP, and constrant (5) requres the locaton varables to be bnary. We assume that h (.) s a lnear functon, then the form of the objectve functon wll become g Y Ac Z c Z m n n j j j 1 j1 j1, where Ac s the capacty acquston cost at faclty locaton per unt. To hedge aganst facltes unavalablty, t s possble to acqure at the begnnng capactes that are hgher than the optmal values obtaned from the model (1)-(5). Our objectve s to fnd the best defence and capacty acquston strateges, knowng that the facltes are subject to ntentonal attacks. As already explaned n the ntroducton, gven a set of nvestments alternatves for protectng the 4 CIRRELT

7 facltes and set of extra capactes, we want to determne how much to nvest optmally n drect protecton of facltes and n ndrect protecton by extra capacty, whle takng nto account that both the defender and the attacker are fully optmzng agents. The next secton presents our game-theoretc model developed to fnd the best trade-off between drect nvestments n protecton and ndrect protecton by extra-capactes deployment. 3. Problem formulaton usng game theory We consder a system contanng m facltes (targets) desgned usng the optmzaton model (1)- (5). Our model consders two players: the defender and the attacker. We are gven a set of nvestment alternatves for protectng the facltes (we call ths drect protecton), and a set of extra-capacty optons for each faclty (we call ths ndrect protecton). The objectve s to determne how to allocate optmally (drect) protectve resources and (ndrect) extra-capactes among the facltes takng nto account the attacker strategy. For the attacker, there are also many ways to attan the facltes performance wth a set of nvestment alternatves. We consder a two-perod game between the defender who selects a strategy n the frst perod that mnmzes the maxmum loss that the attacker may cause n the second perod. Ths means that the defender determnes n the frst perod, ts 2m free choce varables smultaneously and ndependently, and the attacker determnes n the second perod, ts m free choce varables smultaneously and ndependently. For each player strategy, a utlty functon s assocated to each game concluson. The defender maxmzes hs utlty by mnmzng the expected damage of the system and the nvestment expendture ncurred to extra-capacty and to protect the system. The attacker maxmzes hs utlty also, calculated as the expected damage mnus the attacks expendtures. To better understand, the game, the defence and attack choces are detaled n what follows The defender We consder that for each faclty there exsts a set of avalable types of protectons aganst the dentfed threat. Each protecton type s ndcated by ndex p (p = 0, 1, 2,, ). p = 0 means that no defence s used. Let Fp be the defender nvestment expendture n dollar terms. We express Fp as the CIRRELT

8 product of two parts, F p and f p. That s, F p = f p F p where F p s an nvestment effort ncurred by the defender at unt cost f p (f p >0) to protect a faclty located at ste usng protecton type p. We consder a vector P = whch represents a protecton strategy of the m facltes such as takes values from p = 1, 2,,. For example, P = means that we have faclty 1 s protected usng type 2 protecton, faclty 2 s protected usng type 1 protecton and faclty 3 s protected usng type 3 protecton. To each protecton strategy P, corresponds a vector of nvestments F = F. For example, when P = 2 1 3, we have F = F F F Let p be a bnary varable whch s equal to 1 f a protecton of type p s selected for faclty, and 0 otherwse. We assume that only one type of protecton of faclty s used: p 1,. (6) p1 Let be the number of avalable extra-capacty optons for each faclty. Let e (e = 1, 2,, be an ndex that ndcates each extra-capacty opton. The defender ncurs an nvestment of proporton of the capacty acqured e assocated of the faclty located at ste usng extra-capacty opton e. Let * C denote the capacty acqured. We assume that h (.) the capacty acquston cost at faclty locaton s a lnear functon. We then consder that the nvestment of extra-capactyce e assocated of the faclty located at ste usng extra-capacty opton e, measured n dollar terms, s gven by * e e CE Ac C, where Ac s the capacty acquston cost at faclty locaton per unt. We represent an extra-capacty strategy of the n facltes by a vector E = e = 0, 1,, facltes. ), takes values from. For example, E = means that opton 2 extra-capacty s selected for the 3 To each extra-capacty strategy E, corresponds a vector of nvestments T = when E = 2 2 2, we have T=,, For example, 6 CIRRELT

9 Let us ntroduce a bnary varable e whch s equal to 1 f an extra-capacty of type e s selected for faclty. Assumng that one type of extra-capacty s used, we have: e 1,. (7) e The attacker The attacker seeks to attack the system to ensure that t does not functon relably. He (she) has a set of avalable attack actons aganst any faclty. Each attack type s ndcated by ndex g (g = 0, 1, 2,, ). g = 0 ndcates the absence of an attack. Analogously, the attacker ncurs an effort Q g at unt cost q g to attack faclty located at ste j usng attack acton g. The neffcency of nvestment s q g, and 1 q g s the effcency. Its nvestment expendture, n dollar terms, s Q g = qq g g,, where q g >0. We consder a vector G = whch represents an attack strategy aganst the m facltes, takes values from g = 1, 2,,. For example, G = means that we have 2 facltes that can be attacked usng attacks of type 3 and faclty 2 that can be attacked usng attack of type 1. Let g be a bnary varable whch s equal to 1 f an attack of type g s used for faclty, and 0 otherwse. We assume that only one type of attack of faclty s used: g 1,. (8) g 1 We suppose that successful or faled attacks aganst dfferent facltes are ndependent. We also assume that each faclty can be attacked by the attacker only once, and that many facltes can be attacked at the same tme Vulnerablty of facltes In game theory, nteracton between 2 conflcted players (here the defender and the attacker) can be modeled by ntroducng the concept of the contest success functon commonly used n the rent seekng CIRRELT

10 lterature [17, 18, 19, 20]. The vulnerablty or the probablty of a successful attack on faclty s defned by ts destructon probablty pg. The vulnerablty of the attacked faclty s usually determned by the rato form of the attacker defender contest success functon [21, 22, 23]. The vulnerablty of any faclty by a contest success functon s: pg Qg Fp Qg, (9) where / Q > 0, / pg g F < 0, and 0 s a parameter that expresses the ntensty of pg p the contest. We assume that ɛ does not depend on p and g. On the one hand, f the attacker exerts hgh offensve effort, t s lkely to wn the contest whch s expressed by hgh vulnerablty; on the other hand, f the defender exerts hgh defensve effort, t s lkely to wn the contest whch s expressed by low vulnerablty [18, 19, 21, 23] The game Havng the vulnerabltes of facltes as functons of the attacker s and the defender s efforts, both agents can estmate the expected damage caused by the attack for any possble dstrbuton of these efforts. The defender s objectve s to maxmze ts utlty functon by mnmzng the expected damage and weghng aganst protecton and extra-capacty expendtures. The attacker s objectve s to maxmze the expected damage whle weghng aganst the attacks expendtures [25]. Facltes are usually bult over tme by the defender. The attacker takes t as gven when he (she) chooses hs (her) attack strategy. Therefore, we consder a two-perod mn-max game where the defender nvests n the frst perod, and the attacker moves n the second perod. Ths means that the defender selects a strategy n the frst perod that mnmzes the maxmum loss that the attacker may cause n the second perod. The utltes of each player are evaluated n the next secton, whle the game wll be solved wth backward recurson, n whch the second perod s solved frst n Secton 5. 8 CIRRELT

11 4. Evaluaton of the players utltes The damage caused by an attack s assocated wth two knds of damage. Damage 1: The expected cost requred for restorng the attacked facltes. If R s the cost requred to restore the attacked faclty, ths cost depends on the defence and attack strateges P and G, and t s gven by: C R (P,G) = m p g pg R. (10) 1 g0 p1 Damage 2: The second expected cost s assocated wth two terms: - The cost ncurred because of the ncreasng n transportaton cost after attacks. When one or several facltes are unavalable, to avod the backlog of demands and backorders, avalable extra-capactes of neghbourng functonal facltes are used. Addng the avalable extra-capacty to ntal capacty, customers could be served and may receve shpments from these facltes, whch are anyway farther away (subject to constrants that ther total capacty must not be exceeded). As a matter of fact, the transportaton cost wll ncrease as customers wll be reassgned. - The backorder cost ncurred when the demands cannot be satsfed. Ths wll happen ether when the entre system s dsabled, or when even the avalable extra-capactes are not enough to fulfl the demands. Whle damage 1 has been expressed by equaton (10), the evaluaton of damage 2 terms s provded n what follows. Each faclty can be ether Dsabled or Functonal. Let S = {S k } be a set of possble combnatons when consderng all facltes. For m facltes, there are 2 m possble combnatons. For example for three facltes denoted by Fac1, Fac2 and Fac3, there are 8 possble combnatons for the facltes (k = 0, 1,, 7) as llustrated n Table 1. CIRRELT

12 Let us denote by Ct k the cost ncurred because of the ncrease n transportaton cost,.e. the cost under combnaton S k mnus the cost n a normal stuaton and B k the backorder cost when the combnaton s S k. Let us denote by B mg the brand mage of the company, YD k the annual unmet demand and Ac the average of the capacty acquston costs per unt. We suppose that the backorder denoted by B k s computed per week, based on 20% of the average of the capacty acquston costs and t s gven by B k = B mg + (0.20 Ac YD k )/52. Here, 52 s the number of weeks durng 1 year. The evaluaton of the cost Ct k and the backorder cost B k for all combnatons except combnaton 0 where all facltes are functonal, may requre the soluton of (2 m -1) optmsaton model (1)-(5). Each soluton of the optmsaton model (1)-(5) corresponds to a combnaton S k. In the Table 1, f we consder attack combnaton 2 where only facltes 1 and 3 are operatonal, the model (1)-(5) s solved to determne the cost Ct 2 and B 2. The cost Ct k and the backorder cost B k are related to all the possble outcomes. Let denote by the bnary varable, whch s equal to 1 f the cost Ct k ncurred because of the ncrease n transportaton cost or 0 f the backorder B k cost s ncurred. Let us denote by T k (E) whch can be the cost Ct k or the backorder cost B k such as T ( E) Ct ( E) (1 ) B. Let pge k k k k C be an attack outcomes, wth p, g and E gven and k varyng from 0 to 2 m -1. Table 1 llustrates the calculaton of the attack outcomes C pge k. For example f we consder attack combnaton 2 where faclty 2 s dsabled and facltes 1 and 3 are operatonal, we have T ( E) Ct ( E) (1 ) B and C T E pge 2 ( ) 2. 2 pg 10 CIRRELT

13 Table 1 Possble combnatons: case of three facltes Combnaton of dsabled facltes Index k Attack outcome C k All facltes are Functonal 0 C pge 0 0 Only Fac1 s Dsabled 1 C 1 T( E) 1 pge Only Fac2 s Dsabled 2 C 2 T ( E) 2 pge Only Fac3 s Dsabled 3 C 3 T ( E) 3 pge pge pg pg pg Fac1 and Fac2 are Dsabled, Fac3 s Functonal Fac1 and Fac3 are Dsabled, Fac2 s Functonal Fac2 and Fac3 are Dsabled, Fac1 s Functonal 4 C pge 4 T1 E pg pg T2 E pg pg T ( E) 1 2 ( ) 1 (1 2 ) ( ) 2 (1 1 ) 4 pg pg 5 C pge 5 T1 E pg pg T3 E pg pg T ( E) ( ) 1 (1 3 ) ( ) 3 (1 1 ) pg pg 6 C pge 6 T2 E pg pg T3 E pg pg T ( E) ( ) 2 (1 3 ) ( ) 3 (1 2 ) pg pg All facltes are Dsabled 7 C pge 7 T1 E pg pg pg T2 E pg pg pg T3 E pg pg pg T4 ( E) pg 1 pg 2 (1 pg 3 ) T5 ( E) pg 1 pg 3 (1 pg 2 ) T6 ( E) pg 2 pg 3 (1 pg 1 ) T ( E) ( ) 1 (1 2 )(1 3 ) 7 ( ) 2 (1 1 )(1 3 ) ( ) 3 (1 1 )(1 2 ) pg pg pg CIRRELT

14 by: The expected cost assocated wth the transportaton cost ncrease and the backorder cost s gven TB(P,G,E) = m 2 1 CpgE k. (11) k1 The expected damage s then: m j j m 2 1 D(P,G,E) = p g pg R C pge k (12) 1 g0 p1 k1 The defender expected utlty s: U d (P,G,E) = D (P,G,E) m B CE p p e e 1 p1 1 e0 m R C k B CE m m 2 1 m m = E p g pg pg p p e e 1 g0 p1 k1 1 p1 1 e0 (13) The attacker expected utlty s: U a (P,G,E) = D(P,G,E) m 1 g0 Q g g m m g0 p1 k1 = p g pg R CpgE k m Q (14) 1 g0 g g 12 CIRRELT

15 5. Soluton of the game We analyze a two perod game where the defender moves n the frst perod, and the attacker moves n the second perod [18, 26]. Ths means that the defender selects a strategy n the frst perod that mnmzes the maxmum loss that the attacker may cause n the second perod. In order to fnd the equlbrum, the game s solved wth backward nducton n whch the second perod s solved frst usng the followng algorthm, whch s an adaptaton of the algorthm n [6] to take nto account the extra-capacty dmenson as a mean for ndrect protecton: 1) Inputs A system of m facltes ( = 1, 2,,m) located by solvng the optmsaton model (1)-(5). A set of protecton types for each faclty (p = 1, 2,, ). A set of attack types per faclty (g = 0, 1,, ). A set of Parameters: extra-capacty optons per faclty (e = 0, 1, 2,, - Protecton nvestment efforts F p ; - Unt costs of protecton efforts f p ; - Attack nvestment efforts Q g ; - Unt costs of attack efforts q g ; - Capacty acqured * C ; - Capacty acquston cost per unt Ac ; - Proporton of the capacty acqured e ; - Contest ntenstes ɛ ; and - Restoraton costs R. ). 2) Intalzaton Assgn U mn = (U mn s the defender mnmal utlty); Assgn U max = 0 (U max s the attacker maxmal utlty). CIRRELT

16 3) Determnaton of the optmal attack strategy (.e., the strategy that maxmses the attacker utlty) For each protecton strategy P = 3.1. For each attack strategy G = Construct a matrx = p such as 1 p 0 f =p, and otherwse p 1, ; p Construct a matrx = g such as g 1 0 f =g, and otherwse g 1, ; g Determne the matrx (P,G) = pg () evaluated by usng equaton (9); Calculate the costs C R (P,G) by usng equatons (10); For each extra-capacty strategy E = For each combnaton k = 1, 2,, 2 m -1 such as each element () s Solve the model (1)-(5) to determne the cost T k (E); Calculate the expected cost TB(P,G, E) by usng equaton (11); Calculate the attacker utlty U a (P,G, E) by usng equaton (14); pg If U a (P,G, E) > U max assgn U max = U a (P,G, E), G opt = G = opt, Q opt = Q opt ; 4) Determnaton of the optmal defence strategy (.e., maxmsng the defender utlty) For each protecton strategy P = 4.1. Assgn opt = g such as g 1 0 f opt g ; otherwse 14 CIRRELT

17 4.2. Determne the matrx (P,G opt ) = opt p such as each element () s evaluated by usng equaton (9) wth = p and under attack strategy G opt,.e., opt p Q opt Fp Q opt ; 4.3. Calculate the costs C R (P,G opt ) by usng equatons (10) (under attack strategy G opt ); 4.4. For each extra-capacty strategy E = For each combnaton k = 1, 2,, 2 m Solve the model (1)-(5) to determne the cost T k (E); Calculate the expected cost TB(P,G opt,e) by usng equaton (11); pg Calculate the defender utlty U d (P,G opt,e) by usng equaton (13); If -U d (P,G opt,e) < U mn assgn U mn = -U d (P,G opt,e), opt opt P opt = P =, F opt = opt F, E opt = E = opt, T opt = T =. 6. Illustratve example In ths secton, a smple example s presented to llustrate the model. The defender optmal strategy obtaned s compared to some defence strateges. The ssues of lmted budgets and the nfluence of contest ntenstes are also dscussed. The model s used to fnd the best trade-off between drect nvestments n protecton and extra-capactes deployment Input data We consder 3 facltes and 5 demand nodes. Table 2 presents the maxmum capacty and the fxed costs of locatng facltes. Table 3 provdes the yearly demands, and Table 4 gves the unt costs of producng and shppng between faclty stes and customer locatons. We assume that the total capacty acquston cost at faclty locaton, h (.), s a lnear functon. Table 5 gves the capacty acquston cost at faclty locaton per unt. CIRRELT

18 When there s no attack, the optmal CFL & CAP soluton for ths problem s shown n Table 6 and pctured n Fgure 1. Ths soluton entals a fxed cost of $6,300,000 and a transportaton cost of $3,812,000 over year. Table 2 Maxmum capacty and fxed cost of locatng a faclty at canddate ste Ste Fxed cost (n $) Maxmum Capacty ,100,000 2,400,000 1,800,000 45,000 61,200 38,700 Table 3 Demand per year at customer locaton Customer locaton j Demand 25,000 21,000 13,000 11,000 10,500 Table 4 Unt cost (n $) of shppng from canddate faclty ste to customer locaton j Customer locaton j Ste CIRRELT

19 Table 5 Capacty acquston cost at faclty locaton per unt Ste Cost (n $) per unt (Ac ) Table 6 Optmal CFL & CAP soluton Customer locaton j Capacty acqured Ste , , , , ,500 * C 25,000 34,000 21,500 Cost (n $) of capacty acqured 275, , ,000 Fg. 1. The optmal CFL & CAP soluton CIRRELT

20 Table 7 provdes the protecton nvestment efforts F p, and the unt costs of protecton efforts f p. Analogously, the attack nvestment efforts Qg and the unt costs of attack efforts are gven n Table 8. Table 9 presents the extra-capacty parameters. Table 10 gves the restoraton costs R by week. We consder that all contest ntenstes are equal to 1, except n Secton 6.5. Table 7 Defence parameters Protecton types p Unt costs f p Protecton efforts F p Table 8 Attacker parameters Attack types m Unt costs q g Attack efforts Q g Table 9 Extra-capacty parameters Extra-capacty optons e Proporton of the capacty acqured e 1 32% 2 68% 3 80% Table 10 Restoraton costs of dsabled facltes (n $) Dsabled faclty Restoraton costs R 1 16, , , CIRRELT

21 6.2. Evaluaton of the costs Ct k and B k for all combnatons of dsabled facltes We consder the case of E = (0 0 0) and E = (2 1 3) to evaluate the costs Ct k and B k for all combnatons of dsabled facltes as llustrated n Table 11 and Fgure 2. If for example extra-capacty strategy s E = (0 0 0), the annual unmet demand s 25,000 for combnaton of dsabled facltes S 1 (faclty 1 s dsabled by an attack), and 34,000 for combnaton S 2, etc. In ths case, the backorders are ncurred. If we assume that the cost of the brand mage of the company (B mg ) s $200,000 and the average of the capacty acquston costs at faclty locaton per unt Ac s $11, then the backorder B 1 s equal to200,000 25, /52=200,055. If for example extra-capacty strategy s E = (2 1 3), when faclty 1 s dsabled by an attack, the quantty shpped from dsabled faclty to customers can be assgned to functonal facltes 2 and 3. Recall that Ct k s the cost ncurred because of the ncrease n transportaton cost when the combnaton s S k,.e. the cost under combnaton S k by solvng the new resultng CFL & CAP whch leads to an optmal soluton mnus the cost n a normal stuaton. Snce the restoraton tme s one week, each cost Ct k s evaluated as the excess n transportaton cost durng ths week. For example, when extra-capacty strategy s E = (2 1 3) and faclty 1 s dsabled, the cost Ct 1 s gven by the cost under combnaton S1 ($ 4,277,200) mnus the cost ($ 3,812,000) n a normal stuaton durng one week. That s, Ct 1 s equal to (4,277,200-3,812,000)/52= $8,946. Here, 52 s the number of weeks durng 1 year. Table 11 The costs Ct k and B k for all possble combnatons S k per week for E = (0 0 0) and E = (2 1 3) k Dsabled E = (0 0 0) E = (2 1 3) facltes B k (n $) Ct k (n $) B k (n $) Ct k (n $) , , , , , , , 2 200, , , 3 200, , , 3 200, , , 2, 3 200, ,163 0 CIRRELT

22 Fg. 2. The Optmal CFL & CAP soluton f faclty 1 s dsabled and the extra-capacty opton s E = (2 1 3) 6.3. Determnaton of the optmal attack strategy The attacker strategy that maxmses hs utlty by applyng step 3 of the algorthm (.e, the second perod of the game s solved frst) s G opt = (3 3 1). Ths means that facltes 1and 2 are dsabled usng type 3 attacks; and faclty 3 s dsabled usng type 1 attack. The maxmum loss s $1,393,785 and the correspondng attacker utlty s U max = $1,349, Determnaton of the optmal defender strategy By applyng step 4 of the algorthm and to fnd the equlbrum, takng nto account the attacker strategy above (.e, the attacker maxmzes hs utlty), the frst perod of the game s solved n order to maxmze the defender utlty and consequently to allocate optmally the protectve resources and extracapacty among the facltes. The obtaned soluton corresponds to the followng strategy: P opt = (3 1 3): Ths means that facltes 1 and 3 are protected usng type 3 protectons; and faclty 2 s protected usng type 1 protecton. E opt = (2 1 3): Ths means that faclty 1 extra-capacty s 68% of the faclty 1 capacty acqured, faclty 2 extra-capacty s 32% of the faclty 2 capacty acqured and 80% of the faclty 3 capacty acqured s the faclty 3 extra capacty. The total cost of extra-capacty s $489, CIRRELT

23 The loss s $459,915 and the correspondng defender utlty s U mn = $1,008, The optmal defender and attacker strategy as a functon of the contest ntensty Fgure 3 shows graphcally the defender and attacker utlty as a functon of the contest ntensty. We remark agan that, the greatest contest ntensty s more benefcal for the defender than for the attacker. Defender and attacker utlty (n $1,000) U max U mn ɛ Fg. 3. Defender and attacker utlty as a functon of the contest ntensty 6.6. Comparson We compare the defender optmal strategy obtaned by our model wth some defence strateges to show that our method s better than others, when the attacker tres to maxmze ts utlty. In our example, the facltes that have hgher fxed costs are ranked as follows (see Table 2): Fac2, Fac1 and CIRRELT

24 Fac3. On the other hand, the protecton types (p) are ranked accordng to ther nvestment expendtures as follows (see Table 7): 2, 3, 1 and the extra-capacty optons (e) are ranked accordng to ther nvestment expendtures as follows (see Table 9): 1, 2, 3. The strateges consdered n ths comparson are as follows: - Strategy 1: Protecton of facltes by more expensve protecton types and hgher extra-capacty In ths strategy, we consder that each faclty s protected usng type 2 protecton and the hghest extra-capacty avalable. That s, the protecton strategy corresponds to P 1 = (2 2 2) and the extra-capacty strategy corresponds to E 1 = (3 3 3). The correspondng defender utlty s U mn =$1,383, Strategy 2: Protecton of facltes by more expensve protecton types and usng lower extracapactes We consder that each faclty s protected usng type 2 protecton and the lowest extra-capacty avalable. That s, the protecton strategy corresponds to P 2 = (2 2 2) and the extra-capacty strategy corresponds to E 2 = (1 1 1). The correspondng defender utlty s U mn =$1,549, Strategy 3: Protecton of facltes by cheaper protecton types and usng hgher extra-capactes We consder here that all facltes are protected usng type 1 protectons and usng the lowest extracapactes. That s, the protecton strategy corresponds to P 3 = (1 1 1) and the extra-capacty strategy corresponds to E 3 = (3 3 3). The correspondng defender utlty s U mn =$1,432, Strategy 4: Protecton of facltes by cheaper protecton types and usng lower extra-capactes We consder that all facltes are protected usng type 1 protectons and usng an the lowest extracapactes avalable. That s, the protecton strategy corresponds to P 4 = (1 1 1) and the extracapacty strategy corresponds to E 4 = (1 1 1). The correspondng defender utlty s U mn =$1,680, Strategy 5: Facltes wth hgher fxed costs are protected by more expensve protecton types and usng lower extra-capacty 22 CIRRELT

25 Ths means that we consder that: faclty 1 s protected usng type 3 protecton and usng extracapacty opton 2; faclty 2 s protected usng type 2 protecton and extra-capacty opton 1; faclty 3 s protected usng type 1 protecton and extra-capacty opton 3. That s, the protecton strategy corresponds to P 4 = (3 2 1) and the extra-capacty strategy corresponds to E 4 = (2 1 3). The correspondng defender utlty s U mn =$1,109,842. The obtaned results ndcate that the defender strategy obtaned by our model s: % better than Strategy 1; % better than Strategy 2; % better than Strategy 3; % better than Strategy 4; and % better than Strategy 5. Our model gves better results as t s desgned to fnd the best trade-off between drect protecton nvestment and extra-capacty deployment, unlke the above fve strateges Lmted budget We assume that the defender strategy optmzaton problem s solved for a lmted defender s budget. Table 12 presents the best trade-off between extra-capacty optons and protecton strateges for dfferent budgets. We remark from ths table that for a lower protecton budget, the expected damage s ndeed hgher. The nvestment sze n faclty protecton (ncludng the use of extra-capacty) affects the facltes defence. In fact, for each ncrease n the defender budget, t s mportant to quantfy the resultng reducton n the expected damage, n order to decde f t s worth nvestng more n protectng facltes and usng extra-capacty. CIRRELT

26 Table 12 Obtaned defence and extra-capacty strateges for dfferent budgets (ɛ j = 1) Defender Protecton Extra Expected Extra-capacty Defence strategy budget cost capacty damage strategy P (n $) (n $) Cost (n $) (n $) opt E opt 25,000 21, ,393,785 ( ) ( ) 50,000 40, ,329,245 ( ) ( ) 100,000 59, ,177,915 ( ) ( ) 265,000 59, ,177,915 ( ) ( ) 360,000 59, ,177,915 ( ) ( ) 400,000 59, , ,708 ( ) ( ) 450,000 78, , ,787 ( ) ( ) 500,000 78, , ,787 ( ) ( ) 550, , , ,770 ( ) ( ) 600,000 59, , ,915 ( ) ( ) 650,000 59, , ,915 ( ) ( ) 700,000 59, , ,915 ( ) ( ) The expected damage cost as a functon of the defender budget (.e, budget for the protecton and for usng the extra-capacty) s presented n Fgure 4. Ths curve s drawn by evaluatng the optmal protecton and extra-capacty strategy for each budget, consderng that the attacker chooses the harmful strategy. Ths curve shows graphcally the relatonshp between the nvestment szes of protecton and extra-capacty and the effect on facltes defence. From ths curve, we remark that the budget greater than $600,000 cannot reduce the expected damage. 24 CIRRELT

27 Expected damage (n $1,000) Defender budget (n $1,000) Fg. 4. Expected damage costs as a functon of defender budget (ɛ j = 1) 7. Concluson Ths artcle deals wth the protecton of supply network n the context of the capactated fxedcharge locaton and capacty acquston problem. Faclty locaton and capacty acquston are of vtal mportance to supply chan management. We consder that not only a set of nvestment alternatves are avalable for drect protecton of facltes, but also extra-capactes of neghbourng functonal facltes can be used after attacks for ndrect protecton. The strategc decson dealt wth s how to allocate optmally the protectve resources and the extra-capacty among the facltes, knowng that these facltes are exposed to external ntentonal attacks. The dea of usng extra-capacty to ndrectly protect supply networks aganst ntentonal attack s then used to develop a game-theoretc model, wth the objectve of fndng the best trade-off between drect nvestments n protecton and ndrect protecton by extra-capactes deployment. In ths model, a non-cooperatve two-perod game s analyzed. The attacker tres to maxmze, n the frst perod, hs utlty functon by maxmzng the expected damage wth several alternatves of attack. On the other hand, the defender attempts to mnmze, n the second perod, the expected damage caused by attacks wth several alternatves of CIRRELT

28 facltes protecton and extra-capacty optons and consequently to maxmze hs utlty functon, where resources for drect protecton and extra-capactes are lmted. A method s developed to evaluate the utltes of the players. The expected costs evaluated by our method nclude the cost ncurred because of the ncreasng n transportaton cost after attacks, the cost necessary to restore dsabled facltes and the backorder cost. An algorthm s presented for determnng the equlbrum soluton and the optmal defender strategy. The defence strategy obtaned by our model s compared to fve strateges, and the obtaned results ndcate clearly the superorty of our model n fndng the best trade-off between drect protecton nvestment and extra-capacty deployment. The developed approach gves mportant manageral nsghts for the protecton of located facltes under capacty constrants, whle usng extra-capacty optons for protecton purpose. We are currently workng on the modelng and analyss of nterdependences between facltes whle consderng mult-echelon supply chan networks. References [1] Hausken K. Strategc Defence and Attack of Complex Networks. Internatonal Journal of Performablty Engneerng 2009;5(1): [2] Levtn G, Gertsbakh I, Shpungn Y. Evaluatng the damage assocated wth ntentonal supply deprvaton n mult-commodty network. Relablty Engneerng and System Safety 2013; 119: [3] Albert R, Barabas AL. Statstcal Mechancs of Complex Networks, Revews of Modern Physcs 2002;74: [4] Patterson SA, Apostolaks GE. Identfcaton of Crtcal Locatons Across Multple Infrastructures for Terrorst Actons. Relablty Engneerng and System Safety 2007;92 (9): [5] Colbourn CJ. The Combnatorcs of Network Relablty. New York: Oxford Unversty Press; [6] Brcha N, Nourelfath M. Crtcal supply networks protecton aganst ntentonal attacks: a game-theoretcal model. Relablty Engneerng and System Safety 2013;119:1 10. [7] Rajagopalan S, Andreas CS. Capacty Acquston and Dsposal wth Dscrete Faclty Szes. Management Scence 1994;40(7): CIRRELT

29 [8] Verter V, Dncer C. An Integrated Evaluaton of Faclty Locaton, Capacty Acquston and Technology Selecton for Desgnng Global Manufacturng Strateges. European Journal of Operatonal Research 1992;60(1):1 18. [9] Verter V, Dncer C. Faclty locaton and capacty acquston: an ntegrated approach. Nav Res Logst 1995;42: [10] Dasc A, Laporte G. An Analytcal Approach to the Faclty Locaton and Capacty Acquston Problem under Demand Uncertanty. Journal of the Operatonal Research Socety 2005;56: [11] L S, Trupat D. Dynamc capacty expanson problem wth multple products: Techology selecton and tmng of capacty expanson. Operatons Research 1994;42(5): [12] Shulman A. An Algorthm for Solvng Dynamc Capactated Plant Locaton Problems wth Dscrete Expanson Szes. Operatons Research 1991;39: [13] Venables H, Moscardn A. Ant Based Heurstcs for the Capactated Fxed Charge Locaton Problem. Ant Colony Optmzaton and Swarm Intellgence, lecture Notes n Computer Scence 2008;5217: [14] Chan F, Wang Z, Zhang J. A two-level hedgng pont polcy for controllng a manufacturng system wth tme-delay, demand uncertanty and extra capacty. European Journal of Operatonal Research 2007;176(3): [15] Davd SW, Erkoc M, Karabuk S. Managng capacty n the hgh-tech ndustry: a revew of lterature. The engneerng economst, Insttute of Industral Engneers 2005;50: [16] Garey M, Johnson D. Computers and Intractablty. A Gude to the Theory of NP- Completeness. Freemann, San Francsco; [17] Hausken K. Producton and Conflct Models versus Rent-Seekng Models, Publc Choce, Sprnger 2005;123(1): [18] Hausken K, Levtn G. Mnmax defence strategy for complex mult-state systems. Relablty Engneerng and System Safety 2009;94(2): [19] Ntzan S. Modellng rent-seekng contests. European Journal of Poltcal Economy 1994;10(1): [20] Tullock G. Effcent rent seekng. In: Buchanan JM, Tollson RD, Tullock G, edtors. Toward a theory of the rent-seekng socety. College Staton, TX: Texas A & M Unversty Press; p CIRRELT

30 [21] Levtn G, Hausken K. Redundancy vs. Protecton vs. False Targets for Systems under Attack. IEEE Transactons on Relablty 2009;58(1): [22] Skaperdas S. Contest success functons. Econ Theory 1996;7(2): [23] Hausken K. Protectng complex nfrastructures aganst multple strategc attackers. Internatonal Journal of Systems Scence 2011;42(1): [24] Ber VM, Hausken K. Defendng and attackng a network of two arcs subject to traffc congeston. Relablty Engneerng and System Safety 2013;112: [25] Azaez N, Ber VM. Optmal resource allocaton for securty n relablty systems. European Journal of Operatonal Research 2007;181: [26] Levtn G, Hausken K. Redundancy vs. protecton n defendng parallel systems aganst unntentonal and ntentonal mpacts. IEEE Transactons on Relablty 2009;58(4): [27] Ramrez-Marquez JE, Rocco CM, Levtn G. Optmal network protecton aganst dverse nterdctor strateges. Relablty Engneerng and System Safety 2011;96: [28] Rocco CM, Ramrez-Marquez JE. Identfcaton of top contrbutors to system vulnerablty va an ordnal optmzaton based method. Relablty Engneerng and System Safety 2013;114: [29] Scaparra M, Church R. A blevel mxed-nteger program for crtcal nfrastructure protecton plannng. Computers and Operatons Research 2008;35(6): [30] Snyder LV. Faclty locaton under uncertanty: a revew. IIE Transactons 2006;38(7): CIRRELT

31 Annex Nomenclature n number of customers n the system m number of facltes n the system jth potental faclty locaton, = 1, 2,, m j th demand locaton, j = 1, 2,, n g h (.) c j D j CAP Z j fxed cost of locatng a faclty at canddate ste total capacty acquston cost at faclty unt cost of shppng between canddate faclty ste and customer locaton j demand at customer locaton j maxmum capacty that can be bult-n at canddate ste quantty shpped from canddate faclty ste to customer locaton j Y bnary varable, whch s equal to 1 f a faclty s to be located at canddate ste, and 0 otherwse p F p f p number of protecton types for faclty ndex of protecton type, p = 1, 2,, nvestment effort to protect a faclty located at ste usng protecton type p unt cost of effort to protect a faclty located at ste usng protecton type p F p nvestment expendture to protect a faclty located at ste usng protecton type p value from p = 1, 2,, opt optmal defence strategy value from p = 1, 2,, P vector of protecton strategy, P= opt P opt vector of the optmal protecton strategy, P opt = F vector of nvestments to protecton strategy P, F= F F opt vector of nvestments to protecton strategy P opt, F opt = F opt F element of nvestments vector F CIRRELT

32 F opt p element of nvestments vector F opt bnary varable whch s equal to 1 f a protecton of type p s used for faclty matrx, = p number of extra-capacty optons for each faclty e e * C Ac CEe ndex of extra-capacty optons, e = 1, 2,, proporton of the capacty acqured assocated of the faclty located at ste usng extra- capacty opton e capacty acqured assocated of the faclty located at ste capacty acquston cost at faclty locaton per unt nvestment of extra-capacty assocated of the faclty located at ste usng extra- capacty opton e, * e e CE Ac C E vector of extra-capacty strategy, E = values from e = 0, 1,, T vector of nvestments to each extra-capacty strategy E, T = e bnary varable whch s equal to 1 f an extra-capacty opton e s selected for faclty number of attack types aganst any faclty g ndex of attack type (g = 0, 1,, ) Q g attack effort to attack faclty located at ste usng attack acton g q g unt cost to attack faclty located at ste usng attack acton g Q g nvestment expendture to attack faclty located at ste usng attack acton g value from g = 0, 1,, opt value from g of the optmal attack strategy G vector of attack strategy, G= 30 CIRRELT

33 G opt vector of the optmal attack strategy, G opt = opt Q opt vector of attack effort of the optmal attack strategy, Q opt = Q opt Q opt g element of attack effort vector Q opt bnary varable whch s equal to 1 f a type g attack s used for faclty matrx, g opt matrx, opt = g pg destructon probablty of a faclty opt destructon probablty of a faclty for the optmal defence strategy p (P,G) matrx, (P,G) = (P,G opt ) matrx,(p,g opt ) = opt ɛ C R (P,G) C R (P,G opt ) R pg p parameter that expresses the ntensty of the contest concernng faclty expected cost requred to restore the attacked facltes whch depends on P and G expected cost requred to restore the attacked facltes whch depends on P and G opt cost requred to restore the attacked faclty k combnatons ndex, (k = 0,, 2 m 1) S k combnatons of dsabled and functonal facltes for the facltes S set of combnatons of dsabled and functonal facltes, S = {S k } Ct k Ac B mg YD k cost ncurred because of the ncrease n transportaton cost when the combnaton s S k average of the capacty acquston costs per unt brand mage of the company annual unmet demand B k pge k backorder cost when the combnaton s S k bnary varable, whch s equal to 1 f the cost Ct k ncurred and 0 f the backorder B k cost s ncurred C attack outcomes of combnaton k whch depends on p, g and E CIRRELT

34 TB(P,G,E) expected cost assocated wth the transportaton cost ncrease and the backorder cost whch depends on P, G and E D(P,G,E) expected damage whch depends on P, G and E U d (P,G,E) defender expected utlty whch depends on P, G and E U d (P,G opt,e) defender expected utlty whch depends on P, G opt and E U a (P,G,E) attacker expected utlty whch depends on P, G and E U mn U max defender mnmal utlty attacker maxmal utlty 32 CIRRELT