Network Optimization Models for Resource Allocation in Developing Military Countermeasures

Transcription

1 OPERATION REEARCH Vol. 60, No. 1, January February 2012, pp IN X (print IN (online INFORM Network Optiization Models for Resource Allocation in Developing Military Countereasures Boaz Golany Technion Israel Institute of Technology, Haifa 32000, Israel, Moshe Kress Operations Research Departent, Naval Postgraduate chool, Monterey, California 93940, Michal Penn, Uriel G. Rothblu Technion Israel Institute of Technology, Haifa 32000, Israel A ilitary ars race is characterized by an iterative developent of easures and countereasures. An attacker attepts to introduce new weapons in order to gain soe advantage, whereas a defender attepts to develop countereasures that can itigate or even eliinate the effects of the weapons. This paper addresses the defender s decision proble: given liited resources, which countereasures should be developed and how uch should be invested in their developent to iniize the daage caused by the attacker s weapons over a certain tie horizon. We forulate several optiization odels, corresponding to different operational settings, as constrained shortest-path probles and variants thereof. We then deonstrate the potential applicability and robustness of this approach with respect to various scenarios. ubject classifications: ars race; network optiization; constrained shortest path. Area of review: Optiization. History: Received January 2010; revisions received August 2010, October 2010, February 2011, April 2011; accepted June Introduction The ter ars race is typically used to describe ilitary buildup efforts by countries that are in conflict with one another (e.g., U versus the UR during the Cold War era, India versus Pakistan, Greece versus Turkey, etc.. tudies of such phenoena typically address strategic issues and are coonly found in the political, econoics, and strategic planning literature see, for exaple, the recent study of ilitary technology races in Koubi (1999. In contrast, the present paper addresses operational aspects that arise in ars races. In particular, it focuses on how uch resources should be invested by a defender in such a race and how to tie these investents. We consider an ars race between two asyetric parties: Red (R and Blue (B. R is the attacker, who is trying to develop an assortent of new weapons to attack the defender B. Being aware of R s capabilities, intentions, and activities, B is trying to develop countereasures (CMs that will itigate, or even neutralize, the effects of R s weapons. The CMs ay be technological, tactical, or both. If R copletes the developent of a certain weapon and akes it operational before B is ready with appropriate CMs, then R inflicts a certain daage on B (typically easured in casualties and econoic daages per each tie-unit until an appropriate CM becoes operational. If B wins the race and a CM is operational before R deploys a weapon, then the daage to B is saller when that weapon becoes available. If B s CMs are perfectly effective against that weapon (see Etcheson 1989, the daage to B can be as low as zero. Given a set of existing and potential weapons to be deployed by R, the proble that B faces is how to utilize its liited resources to develop the ost effective ix of CMs a ix that iniizes total daage. An exaple of the settings addressed in this paper is the counterinsurgency warfare faced by coalition forces in Iraq and Afghanistan ( where the insurgents develop and deploy new types of iprovised explosive devices (IED, with ever-increasing lethal capability, while the coalition forces continue to develop technologies, tactics, techniques, and procedures to respond to that threat (see, e.g., Neuneck Ars race probles are related to a broader class of probles addressing investent rates in R&D projects that are carried out in copetitive arket environents (see, e.g., Golany and Rothblu 2008, Kaien and chwartz 1971, and pector and Zuckeran Most of the articles that have appeared in this literature have assued the winner-takes-all hypothesis whereby the first party that achieves an advantage aintains it indefinitely and all other parties lose. 48

2 Operations Research 60(1, pp , 2012 INFORM 49 In a recent paper, Golany et al. (2012 analyze a stochastic version of the ars race proble of the kind described above. In contrast with the coon winner-takes-all assuption, the odels presented in Golany et al. (2012 address situations in which any advantage gained by one of the parties participating in the race is teporary in nature and is lost once another party overtakes the lead. Two types of odels are presented in Golany et al. (2012: optiization odels that derive optial resource allocation schees, and gae-theoretic odels that derive Nashequilibriu solutions. pecifically, resources invested by B in developing CMs deterine, probabilistically, the tie when these CMs are ready and operational, and thus also deterine the expected daage caused to B by R. Assuing a predeterined developent policy of CMs in parallel or sequentially corresponding convex prograing probles are forulated and solution ethods are discussed. The stochastic odels in Golany et al. (2012 enable B to deterine optial investent schees while capturing uncertain durations of R&D activities and liited intelligence about R s capabilities. The approach taken in this paper is quite different because we focus on developing deterinistic odels to address B s resource allocation probles. The deterinistic approach is justified in settings where (1 the CM developent efforts do not involve a significant research eleent and are ainly coposed of a sequence of engineering stages whose durations can be forecasted with reasonable accuracy and (2 when there are reliable intelligence reports regarding R s capabilities, intentions, and possible hostile actions. The ain contribution of this paper is in extending the operational situation described in Golany et al. (2012 in three ways: (a assuing arbitrary CM developent policies (not necessarily parallel or sequential; (b introducing teporal budget constraints, which are quite realistic in defense contracting; (c allowing for a wide variety of inconsistent CMs in the sense that a certain CM ay be ore effective against weapon I than weapon II, whereas the reverse is true for another CM. Also, unlike the continuous investent levels considered in Golany et al. (2012, the forulation presented herein restricts the investent levels to a finite nuber of discrete values. iilar discretization was ipleented by Burnett et al. (1993 to analyze investent levels aong alternative projects related to the natural gas industry in the United tates. We odel the decision proble of B as a variant of a resource-constrained shortest-path (RCP proble, where the constraints capture global or teporal budgetary constraints. The RCP proble is known to be NPcoplete, in the ordinary sense see Garey and Johnson (1979, and we show that our variant is also NP-coplete. RCP probles have been addressed by any authors including Beasley and Christofides (1989, Duitrescu and Boland (2001, Hassin (1992, and Mehlhorn and Ziegelann (2000. In particular, RCP probles of liited size can be solved through special-purpose algoriths such as those developed in Carlyle and Wood (2003 and Gellerann et al. (2005, or through efficient generalpurpose algoriths available in coercial optiization software packages. To deonstrate the potential usefulness of our RCP odels and analyze their robustness to sall data perturbations, we conducted an extensive coputational study in which we eployed the solver in the MOEK optiization package. This solver was proven to be quite efficient for realistically sized proble instances of our RCP odels. The rest of the paper is organized as follows. In 2 we introduce notation and state the decision proble forally. In 3 and 4 we forulate several variants of the proble, addressing single and ultiple weapons and CMs, as constrained network optiization odels. ection 5 deonstrates the usefulness of the odels by presenting the results of extensive nuerical experients. Finally, 6 suggests soe directions for future research. 2. Proble Forulation The weapons that R develops are indexed by w = 1. For each w, let s w denote the tie when weapon w becoes operational; the s w s are obtained or estiated by B s intelligence agencies. In particular, s w = 0 eans that weapon w is already operational at tie 0. If s w > 0, then weapon w does not contribute to the daage inflicted on B until tie s w. We consider a finite tie horizon of length T. Without loss of generality, we assue that T is large enough such that all weapons would becoe operational before tie T and that the weapons are indexed in increasing order of the s w s; that is, 0 s 1 s T. Absent any CM, the daage rate per unittie inflicted by weapon w on B is d0 w 0. Absent any weapon, the daage rate per unit-tie is 0, independent of the available CMs. To itigate the effect of R s weapons, B develops CMs. These CMs are indexed by M = 1 M, and we use the notation CM to refer to the th CM. For each w and, let d w 0 be the daage rate caused by weapon w when only CM is operational. When a set of CMs is available, their effect is not cuulative the daage rate of weapon w is deterined by the ost effective CM that is available at that tie. Although in soe cases there ay be cuulative effects of CMs, e.g., when one CM is a detection device and another CM is a neutralization device, we focus in this paper on a single faily of CMs (e.g., interception systes or bob neutralization systes that evolves and iproves over tie and whose ebers differ in their capabilities. Therefore, when a set M M of CMs is available, the daage rate by weapon w, if operational, is dm w in M d w ; when no CM is available, the daage rate is d w dw 0. We will find it useful to apply the notation d M dm 1 d M for M M, d d for M and d 0 d.

3 50 Operations Research 60(1, pp , 2012 INFORM The daage rate caused by a group of weapons is represented by a onotonically increasing function D, which converts daage rates of individual weapons into a total daage rate. By onotonically increasing function D x 1 x W, we ean that if x w y w for all w = 1 W, then D x 1 x W D y 1 y W ; and if x w > y w for all w = 1 W, then D x 1 x W > D y 1 y W. For exaple, D can be the su of the individual daage rates or their axiu. When a weapon w W is not operational, its individual contribution to the total daage is set to 0 when evaluating the function D. Given a set W, let I W 0 1 be the indicator vector of W, i.e., I W w = 1 if w W and I W w = 0 otherwise. The daage rate caused by the set W of operational weapons when the set of available CMs is M is then expressed by d W M = D I W d M where stands for the Hadaard product. That is, for vectors x y, the Hadaard product x y has x y w = x w y w for each w W. In particular, because each d w represents the daage rate of a certain weapon w in the presence of CM, we have that D I w d = d w. Also, D 0 = 0. It is assued that each CM can be developed at any one of several levels of intensity that are indexed by k = 1. 1 A higher intensity level of developent has two effects: first, the developent tie of the CM is shorter, and therefore it becoes operational sooner, and second, the associated cost is higher. We assue that the intensity of developing a CM does not affect its effectiveness. Intensity 0 indicates no developent. For M and k, let t k 0 denote the tie it takes to coplete CM when developed at intensity level k and let c k 0 be the corresponding cost. As higher intensity levels are associated with shorter developent ties and higher costs, we index the intensities so that t c t 1 t 1 and c 1 c 1 for each M (1 To avoid degenerate situations, we assue throughout that there are no ties aong the d w s, tk s and ck s. The proble that B faces is to decide which CMs to develop, at what ties to start developent, and at what intensity levels. The goal is to iniize the cuulative daage over the tie horizon subject to budgetary constraints. The siplest budgetary constraint is a global one where an upper bound, say C, is prescribed on the total funds that can be spent in developing the CMs. In this case, the entire budget is available at t = 0. Because there is no reason to defer the developent of any CM, the starting tie for all the CMs occurs at tie t = 0. However, budgetary constraints ay also be teporal where there is an upper bound on expenditure in a certain tie period. In such cases, starting ties of the developent of the CMs becoe decision variables. 3. Consistent CMs In this section we analyze variants of the resource allocation proble described in 2 under a consistency assuption, forally introduced in 3.2. The probles are forulated as constrained network optiization odels, which facilitates their efficient solution. In 3.1 we consider the case of a single weapon (which is trivially consistent under a global budgetary constraint. In 3.2 we extend the analysis to the case of ultiple consistent weapons and in 3.3 we address teporal budgetary constraints ingle Weapon Here we assue that there is a single weapon and no teporal budget constraints; in this case, the convention that d 1 = D I 1 d for M assures that D is the identity, thus, dm 1 = D d M = d M for every M M. Henceforth, in this subsection we suppress the index w = 1. Also, we rank the CMs by their effectiveness (against the single weapon, that is, d 0 > d 1 > > d M 0. Because there are no teporal budget constraints, it can be assued that the developent of all CMs starts at 0. We first assue that the single weapon is already operational at tie 0. The following definitions and notation are used throughout the paper. A CM-developent policy, henceforth called siply a policy, is a set of CMs along with corresponding developent intensities. Thus, a policy is represented by a set of pairs k where the values of are distinct. We order the pairs in by their first coordinate such that the sequence 1 k 1 2 k 2 p k p satisfies 1 < 2 < < p. Clearly, there is no point in developing a CM that becoes available after a ore effective CM is already operational because such a CM cannot reduce the daage caused by the weapon. Consequently, one can restrict attention only to policies that have the following property: if k and k with < are in, then t k < tk. We refer to policies that satisfy this condition as effective policies. A path in a graph is an ordered set of vertices where each consecutive pair is an edge. A path that starts at vertex a and ends at vertex b is referred to as an a b -path. For a path, V is the set of vertices in, excluding the end vertices, and E is the set of the corresponding edges. Effective policies can be represented by paths in the acyclic (directed graph G = V E, where V k M and k (2 and E k k < and t k < tk (3 Next, we augent the graph G with a single origin and a single destination, denoted O and D, respectively, and with edges O D, O k, and k D for every k V. The resulting augented graph is referred to as the Feasible CM Developent chedule (FCD graph, and its vertex and edge sets are denoted V and E, respectively.

4 Operations Research 60(1, pp , 2012 INFORM 51 Figure 1. An FCD graph for a single weapon. O (1,2 (2,2 (1,1 (2,1 D It is convenient to represent the vertices of an FCD graph on the interval 0 T with vertices of V represented by their corresponding copletion ties t k, O corresponding to 0 and D corresponding to T. In this representation, all edges have orientation fro left to right, assuring that the FCD graph is acyclic and therefore its paths are siple. Figure 1 presents a situation where developing CM 2 at intensity 2 takes less tie than developing CM 1 at intensity 1, and therefore is not an edge in the corresponding FCD graph. However, if the sae intensity level is applied to CM 1 and CM 2, then CM 1 is copleted before CM 2 hence, and are edges in the FCD graph. The discussion above iplies that we have a one-to-one correspondence between the set of effective policies and the set of (O-D-paths in the FCD graph. For exaple, in Figure 1, the effective policy is represented by the path O D, but there is no path for the noneffective policy We define two transforations denoted policy and path, which convert paths to effective policies and vice versa. These transforations are inverses of each other, i.e., if is a path in an FCD and is a policy, then policy ( = if and only if path( = ; in particular, policy path = and path policy =. Although the definitions of policy and path ay see trivial, they will be particularly useful in 4. The FCD graph has M + 2 vertices and at ost 2( M M + 1 edges. The bound on the nuber of edges is attained when the (tie intervals t t1 M are pairwise disjoint, that is, when the developent ties of the various CMs are highly variable. The actual nuber of edges of the FCD graph depends on the t k values and, in general, the graph ay be sparse because of overlaps of the aforeentioned tie intervals. That is, the copletion tie of soe highly effective CMs developed at high intensity levels ay be shorter than that of lesseffective CMs that are developed at a low intensity level. A nonepty effective policy, represented by a path O 1 k 1 2 k 2 p 1 k p 1 p k p D of the FCD graph, defines a partition of 0 T into the tie intervals 0 t k 1 1 t k 1 1 t k 2 2 t k p 1 p 1 t k p p t k p p T. During the tie interval t k j j t k j+1 j+1 the available protection against the weapon is that of CM j, that is, the daage rate inflicted by the weapon is d j. The total daage during that period is d j t k j+1 j+1 t k j j. iilarly, the total daage inflicted during the interval 0 t k 1 1 is d 0 t k 1 1, and the total daage inflicted during the interval t k p p T is d p T t k p p. Thus, by assigning to each edge e of the FCD graph a daage value d e given by d t k tk if e = k k d 0 t k 0 if e = O k d e d T t k d 0 T 0 if e = k D if e = O D the total daage inflicted in the tie interval 0 T when an effective policy is ipleented is expressed by d d e (5 e E path The next lea records an opposite triangular inequality that the d e s (the daage values satisfy, which we will use later in 4. Lea 1. uppose 1 2 and 2 3 E. Then, 1 3 E and d 1 3 > d d 2 3 (6 Proof. We prove only for V 1, e.g., i = i k i for i=1,2,3. Trivially, the assuptions iply 1 < 2 < 3 and t k 1 1 < t k 2 2 < t k 3 3. To verify (6, note that d 1 k 1 3 k 3 = d 1 t k 3 3 t k 1 1 = d 1 t k 3 3 t k d 1 t k 2 2 t k 1 1 > d 2 t k 3 3 t k d 1 t k 2 2 t k 1 1 = d 1 k 1 2 k 2 +d 2 k 2 3 k 3 (the last inequality follows fro d 1 > d 2. (4

5 52 Operations Research 60(1, pp , 2012 INFORM The cost c e of an edge in the FCD graph represents the resources needed for developing the CM corresponding to the end-vertex of that edge, that is, c k if the end-vertex of e is k c e (7 0 if the end-vertex of e is D The cost of ipleenting an effective policy is then expressed by c c e (8 e E path Due to (7, (8, and the one-to-one correspondence of effective policies and paths in the FCD graph, the proble of selecting the effective policy that iniizes the daage subject to budget constraint C reduces to the proble of finding an (O D-path in the FCD graph that iniizes the d-length (5 subject to the c-length (8 being bounded by C, that is, in s t e E e E Forally, d e (9 c e C is an O D -path in the FCD graph Proposition 1. In the case of a single weapon, the vertices k on a path that is optial for (9 deterine an (effective optial policy that iniizes the daage that R causes B subject to the budget constraint. Unfortunately, the next result shows that (9 is theoretically hard. Lea 2. The constrained shortest-path proble on an FCD graph is NP coplete. Proof. ee the appendix. We next relax the assuption that the (single weapon is available at tie 0 and assue that it becoes operational at tie 0 < s < T. In this case, the total daage inflicted during a tie interval a b in which only CM is available is given by d ax b s ax a s, and the sae forula applies with = 0 if no CM is available. It follows that the total daage inflicted by the weapon when an effective policy is used is the length of the corresponding path where edge lengths are given by the odification of (4 obtained by replacing: t k ax tk s 0 s for each k V (10 No change is needed in the representation of the cost associated with developing effective policies. It follows that the proble still reduces to the constrained shortest-path proble of (9, and Proposition 1 extends to the case where s Multiple Weapons and Consistent CMs Consider the case where there are ultiple weapons threatening B. We say that the CMs are consistent if the rankings of their effectiveness against all the weapons coincide. Forally, this eans that the CMs can be indexed so that d w 0 > dw 1 > > dw M for each w (11 In particular, for any subset M M, arg in M d w is invariant of w, iplying that for every W, d W M = D I W d M = in M D I W d (12 The consistency property is applicable when the weapons are siilar (e.g., various types of roadside IEDs and the differences aong the CMs are only anifested in the extent of their daage reduction. Henceforth, in this section, we assue that the CMs are consistent. Also, no teporal budget constraints are iposed and the developent of all CMs starts at 0. Recall that a policy is a collection of pairs k with distinct values of. The definition of effective policies given in 3.1 relies on the ranking of the effectiveness of the CMs. Because the rankings with respect to all weapons are the sae, it follows that if CM is ore effective than CM with respect to one weapon, then this effectiveness doinance applies to all weapons. Consequently, if a policy is effective with respect to one weapon, then it is effective with respect to all weapons. Again, we start off with the assuption that all the weapons are operational at tie 0. The daage inflicted by the weapons when an effective policy is ipleented is a onotone-increasing function of the daages inflicted by the individual weapons. pecifically, and siilarly to the analysis in 3.1 (see (4, the total daage inflicted during the tie interval 0 T when an effective policy is ipleented is expressed by (5, where D I d t k tk D I d 0 t k 0 d e D I d T t k D I d 0 T if e = k k if e = O k if e = k D if e = O D (13 The expression for the cost c of ipleenting an effective policy reains unchanged and is expressed by (7 (8. As in 3.1, it follows that the proble of selecting the best effective policy reduces to the constrained shortest-path proble of (9. Forally, Proposition 2. In the case of ultiple weapons and consistent CMs, the vertices k on a path that is optial for (9 (with the d e values given by (13 deterine an optial effective policy that iniizes the daage caused by R to B subject to the total budget constraint.

6 Operations Research 60(1, pp , 2012 INFORM 53 Next, assue that soe weapons are not operational at tie 0 and weapon w becoes available at tie s w 0. Recall (fro 2 that the weapons are indexed in increasing order of the s w s, i.e., 0 s 1 s T ; thus, potential sets of operational weapons that B ay encounter are 1 w, where w. Using (12, we next observe that B can continue to restrict attention to effective policies. We adopt the convention that a b = if b < a and r + = ax r 0 for a real nuber r. Now, if an effective policy is ipleented and k k E path, then for M and w, daage rate d 1 w is inflicted during the tie interval t k tk s w s w+1 = ax t k s w in t k s w+1, whose length is in t k s w+1 ax t k s w + ; when the edge eanates fro O, t k is replaced by 0, and when it terinates at D, is replaced by T. Therefore, for e E and w, let t k e w and d e in t k s w+1 ax t k s w + if e = k k in t k s w+1 s w + if e = O k in T s w+1 ax t k s w + if e = k D in T s w+1 s w + if e = O D w w D I 1 w d e w if e eanates fro k D I 1 w d 0 e w if e eanates fro O (14 (15 The total daage inflicted when an effective policy is used is then the length of the corresponding path where edge lengths are given by (15. No changes are needed in the representation of the cost associated with the effective policy. Therefore, the proble is still reduced to the constrained shortest-path proble of (9 and Proposition 2 extends to the case where the s w s are not necessarily Multiple Weapons and Consistent CMs with Teporal Budget Constraints In this subsection we consider the situation exained in 3.2 with additional constraints that restrict periodical expenditures. We specify H tie intervals (subsets of 0 T as: I 1 = T 1 T 1 I 2 = T 2 T 2 I H = T H T H, where T h < T h and assue that there is a bound C h on the expenditure during tie interval I h, for each h = 1 2 H, in addition to the global budget constraint. The tie intervals are not necessarily disjoint. There are two special cases of particular interest. In the first case, the intervals I h are disjoint and thus partition 0 T. In this case the periodic budget constraints represent strict cash flow constraints (e.g., typical to the U.. governent budgeting rules in which excess funds in one period cannot be utilized in the next period. Clearly, for the global constraint to be eaningful, the data ust satisfy H h=1 Ch > C. The second case involves relaxed cash flow constraints where budget overflows are allowed to be used in future periods. In this case, the tie epochs 0 = T 1 < T 2 < < T H < T H+1 = T are given and I h = 0 T h+1 for h = 1 H. Here, for the data to be eaningful it ust satisfy C 1 < C 2 < < C H < C (the constraint corresponding to 0 T H+1 is left out as it is the global budget constraint. We assue that costs are incurred continuously and at a unifor rate. Also, the developent of CMs is carried out without planned interruptions (an assuption that is quite reasonable because in reality, disrupting a project ay incur high set-up cost when developent is resued. In the presence of bounds on periodic expenditures, the tie at which the developent of each CM starts becoes a decision variable. To address these additional decision variables, we extend the definition of intensity and refer to plans of CM developent projects. Each plan consists of a pair, where k is the intensity of developing the CM and is the start tie. For siplicity of exposition, we focus on the case where neither the effectiveness of a CM nor the cost and duration of developing it are affected by ; relaxation of these assuptions is briefly discussed at the end of this subsection. As discussed above, when there are no teporal budgetary constraints, there is no reason to defer the developent of any CM, and the t k values represent both the developent duration and the copletion tie. However, this identity does not hold when the starting ties of developing the CMs are not 0. pecifically, if CM is developed using plan, then its copletion tie is t +t k. To avoid irrelevant situations, we assue that t < T for all, k, and. A policy is now defined as a set of triplets k with distinct values of, and effective policies are defined in ters of the t (the copletion ties rather than the t k s (the duration ties. Consider the odification of the FCD graph, where vertices and edges are defined in ters of the triplets k and the copletion ties t instead of the pairs k and the duration ties t k. We refer to the resulting graph as the Tiing-Feasible CMs Developent chedule (T-FCD graph. Note that if the developent of all CMs can start at any one of q potential tie periods and is the set of potential intensity levels, then the T-FCD graph has q M + 2 vertices and at ost q 2( M 2 + 2q M + 1 edges. However, unlike the FCD graph, the T-FCD graph ay be quite dense in real-world applications. Because an effective policy is such that for any two nodes k and k <, in the T-FCD graph we have that t < t k, it follows that, as before, there is a one-to-one correspondence between the set of effective policies and the set of (O-D- paths in the T-FCD graph.

7 54 Operations Research 60(1, pp , 2012 INFORM uppose that the developent plan of CM is. During the developent period t a cost ĉ k c k /tk per unit tie is incurred. The total expenditure on CM during the tie interval I h = T h T h is then the length of the (possibly epty interval I h t ties the per unit-tie cost ĉ k. We note that I h t = ax T h in T h t. Consequently, the expenditure during the tie interval I h on CM is ĉ k in T h t ax T h +. For each edge e of the T-FCD graph and h = 0 1 H, let ĉ k in T h t ax T h + c h e if e terinates at k 0 if e terinates at D (16 The total cost associated with effective policy during the tie-interval I h is then expressed by e E path c h e and this su is subject to the corresponding teporal budget constraint C h. We next consider the total cost associated with ipleenting effective policy. iilarly to the case of a single weapon presented in 3.1, here the total cost is expressed by (8, with the c e s given by (7, except that k is replaced by k. Now, consider the total daage associated with an effective policy. First, assue that all weapons are operational at tie 0. For each edge e of the T-FCD graph, let d e be given by the variant of (13 in which k and k are replaced by and k, respectively. The total daage associated with policy is then expressed by (5. When weapon w becoes operational at tie s w > 0, let d e be given by (14 and (15, with replacing k. The above discussion deonstrates that the proble of selecting a policy that iniizes the total daage subject to total and teporal budget constraints reduces to the proble of finding an (O D-path in the T-FCD graph that iniizes the d-length subject to corresponding constraints on the c-length and the c h -lengths, that is, in s t e E e E c h e e E Forally, d e (17 c e C Ch for h = 1 H is an O D -path in the T-FCD graph Proposition 3. In the case of ultiple weapons and consistent CMs with teporal budget constraints, the vertices k on a path that is optial for (17 deterine an optial effective policy that iniizes the daage caused by R to B subject to the total and teporal budget constraints. We note that our odel can be easily odified to capture situations where costs and developent ties of CMs depend on the starting ties of their developent; all that is needed is to replace the paraeters c k and tk by start-tiedependent counterparts c k and tk, respectively (in k which case t = + tk. Also, allowing effectiveness of the CMs to depend on the starting ties can be captured by replacing with in the odification of ( Inconsistent CMs In this section we relax the consistency assuption, allowing for one CM to be ore effective than another with respect to weapon w, whereas the reverse holds for weapon w w. Inconsistency is present, for exaple, when the weapons of R are not technologically or operationally siilar. We analyze the inconsistent case, only when the function D is the suation function, in which case D I W d M = w W dm w. We start our analysis under the assuptions that all weapons are operational at tie 0 and there are no teporal budgetary constraints. Without the consistency assuption, the CMs can no longer be ranked uniforly according to their effectiveness against the weapons. till, each weapon has its own total order regarding the effectiveness of the CMs, which we call w-doination and denote by w ; thus, we write w if d w < dw Policies and Their Effective Parts We return to the definition in which a policy is a set of pairs k with distinct values of. A policy is w-effective if it is effective in the sense of the definition in 3.1 for weapon w. Given a policy and a weapon w, the w-effective part of, denoted w, is the subset of obtained by reoving all pairs k for which there exists k with w and t k < tk Because a policy applies at ost one intensity level for each CM, the w s ust satisfy the following coupling requireent : k w and k w k = k (18 The total cost of ipleenting a policy is expressed by c c k (19 k 4.2. The Multi-FCD Graph Each weapon w defines an FCD graph, denoted FCD w, where < in (3, with respect to the CM indices, is replaced by w. The sets of vertices and edges of the FCD w graph are denoted V w and E w, respectively. Each V w is a replica of U O D k M k ; in particular, the eleents of V w are denoted O w, D w, and k w. Because the CMs are inconsistent and an edge k w k w exists in E w if and only if w and t k < tk, it is possible that k w k w E w but k w k w E w for w w (this will happen

8 Operations Research 60(1, pp , 2012 INFORM 55 Figure 2. A ulti-fcd graph for two weapons. O 1 (1,2 1 (2,2 1 (1,1 1 (2,1 1 D 1 O 2 (1,2 2 (2,2 2 (1,1 2 (2,1 2 D 2 if and only if t k < tk, w, and w. As k w k w E w iplies that w, the FCD w graphs are acyclic, and therefore their paths are siple. In order to refer to the FCD w graphs jointly, we define the ulti-fcd graph V E whose vertex and edge sets are, respectively, V w V w and E w E w. The FCD w subgraphs are disjoint coponents of the ulti- FCD graph, in particular, (O w D w -paths of the FCD w graph are identified with the (O w D w -paths of the ulti- FCD graph. Figure 2 presents a ulti-fcd graph for two weapons. As in Figure 1, here too, t1 2 < t2 2 < t1 1 < t1 2. However, whereas CM 1 is ore effective than CM 2 with respect to weapon 1, the reverse is true for weapon 2. Results in 3.1 show that there is a one-to-one correspondence of w-effective policies and (O w D w -paths of the FCD w graph, which are the (O w D w -paths of the ulti- FCD graph. Following the notation introduced in 3.1, the (O w D w -path corresponding to a w-effective policy w is denoted path w. Figure 3 deonstrates the w-effective parts of a policy. Here, = ; its 1-effective part is =, whereas its 2-effective part is only 2 2. For each e E, let d w tk tk if e = k w k w d0 w d e tk if = Ow k w d w T tk if e = (20 k w D w d0 wt if e = Ow D w iilar to (5 in 3.1, we then have that the daage that weapon w causes when policy is ipleented is expressed by d w e E path w d e. The total daage associated with the ipleentation of policy (caused by all weapons is then expressed by d d w = w w 4.3. Configurations e E path w d e (21 Unlike the one-to-one correspondence of effective policies and paths in , the relation between policies and paths in the inconsistent case is ore coplex because the w s of a policy can differ fro each other. To overcoe this difficulty, we define a configuration to be a collection = w w, where each w is an (O w D w -path in the FCD w subgraph of the ulti-fcd graph. uch a configuration is called plausible if it satisfies the coupling requireent k w V w and k w V w k = k (22 Note that (18 iplies that the paths corresponding to the w-effective parts of a policy for a plausible configuration. On the other hand, given a plausible configuration = w w, we define policy w V w. Finally, we point out that policy consists of vertices fro

9 56 Operations Research 60(1, pp , 2012 INFORM Figure 3. A representation of the w-effective parts of a policy. O 1 (1,2 1 (2,2 1 (1,1 1 (2,1 1 D 1 O 2 (1,2 2 (2,2 2 (1,1 2 (2,1 2 D 2 Policy ( 1 2 the ulti-fcd graph that ay be replications of k pairs, e.g., correspondence between configurations and vectors k w and k w, w = w. The plausibility of assures that policy is indeed a policy. The transforations path w w and policy are not inverses of each other. 2 For a policy we define policy path w w (23 note that and c = k c k k c k = c (24 strict inclusion and strict inequality in (24 are possible when contains superfluous k s. Also, for each path w of a configuration, V w policy w and therefore, by the opposite triangular inequality (Lea 1, e E w d e e E path policy w d e (25 Cobining (25 with (23 shows that for every configuration = w w : d policy = w w d e e E path policy w e E w d e (26 Figure 4 deonstrates a situation with strict inequality in (26. The figure illustrates a plausible configuration with 1 = O D 1 and 2 = O D 2 (the edges of these paths are arked in bold. The policy corresponding to is = and the path corresponding to the 1-effective part of is O D 1, with edges O and (arked by dotted arrows replacing edge O of 1. Due to the opposite triangular inequality, provides better protection against weapon 1 than 1 1 the 1-effective policy corresponding to Network Optiization with ide Constraints For an edge e and vertex v of the ultiple-fcd, we write e v if e eanates fro v and e v if e terinates at v. tandard results of network odeling iply a oneto-one correspondence between configurations and vectors x = x e e E that satisfy: x e = x e for each M k e k w e k w x e = 1 = e O w x e 0 1 e D w x e for each w for each e E and w (27 In particular, if configuration = w w corresponds to x satisfying (27, then d e = d e x e (28 w e E w e E

10 Operations Research 60(1, pp , 2012 INFORM 57 Figure 4. Deonstrating opposite triangular inequality. O 1 (1,2 1 (2,2 1 (1,1 1 (2,1 1 D 1 O 2 (1,2 2 (2,2 2 (1,1 2 (2,1 2 D 2 Policy ( Plausible configuration Iproved plausible configuration Given such a solution x, let y x = y x k k U be defined by y x k = ax for each k U w e k w x e equivalently, y x is the unique solution of y x k for each k U and w y x k w e k w x e y x k 0 1 e k w x e for each k U for each k U We note that the variable y x k gets the value 1 if for any weapon w, CM is developed at level k under x when considering the w-subgraph of the ulti-fcd graph: in all other cases y x k gets the value 0. Consider a configuration corresponding to the vector x = x e e E that satisfies (27. Evidently, is plausible if and only if k y x k 1 for each M; an equivalent condition is that for soe vector y = y k k U, x and y satisfy y k for each k U and w e k w x e y k w y k 1 k y k 0 1 e k w x e for each k U for each M for each k U (29 Thus, we have a one-to-one correspondence between the set of plausible configurations and the set of solutions of (27 and (29. Further, if x y satisfying (27 and (29 corresponds to a plausible configuration, then policy = k y k = 1 and c policy = k U c k yk (30 the budget constraint on policy is then expressed by k U c k yk C (31 We next show that the proble of selecting an optial policy reduces to solving the optiization proble: in x y d e x e (32 e E s t x y satisfies and 31 Proposition 4. In a setting that possibly involves inconsistent weapons, let x y be an optial solution of (32 and k U y k = 1. Then is a policy that iniizes the daage caused by R to B subject to the total budget constraint. Proof. Fro the definition of, c = k U c k y k C (33

11 58 Operations Research 60(1, pp , 2012 INFORM assuring that satisfies the budget constraint. Next, let OPT be the optial value of (32. With = w w as the plausible configuration corresponding to x y, we have that = policy ; it then follows fro (26 and (28 that d = d policy d e = d e x e = OPT (34 w e E w w e E To see that is optial, consider an arbitrary policy that satisfies the total budget constraint. Let x y be the solution of (27 and (29 that corresponds to the plausible configuration path w w. Then policy = and (30 and (24 iply that k U c k yk = c c C assuring that x y satisfies (31. Therefore, x y is feasible for (32, and therefore OPT e E d e x e. It now follows fro (34, (28, and (21 that d OPT d e x e = e E w e E path w d e = d We next consider the case where weapons are not necessarily operational at tie 0, and for each w, weapon w becoes operational at tie 0 s w T. The situation is handled by adjusting the definition of the daage coefficients given in (20 (resebling the use of (10 to odify (4 in 3.1. pecifically, for M, k, and w, let t k w ax tk s w and for each e E, let d w tk w tk w if e = k w k w d w w 0 tk s w if e = O w k w d e d w w T tk if e = k w D w d0 w T s w if e = O w D w With this adjustent, the total daage inflicted when an effective policy is used is the su of the length of the corresponding paths, the decision proble of B reduces to (32 and Proposition 4 applies. It is also possible to generalize (32 to the case where periodic budget constraints are iposed (see 3.3. Although the odel reains essentially the sae, its size increases significantly. The nuerical experients reported in the next section apply to this ore general case with cash flow budget constraints. That is, the intervals I h partition 0 T with tie epochs 0 = T 1 < T 2 < < T H < T H+1 = T, and each interval I h corresponds to the period T h T h Nuerical Experients In order to exaine the applicability and robustness of our constrained network odel, we conducted an extensive coputational study. In this study, we ipleented a odel with inconsistent weapons (see 4, ultiperiod teporal budget constraints (see 3.3 where the teporal budget constraints correspond to H disjoint intervals that partition 0 T and D as the suation function (see the last paragraph of 4.4. A detailed forulation of the odel ipleented in our nuerical study is given in the appendix. The case study consisted of 10 base cases and the generation and solution of 100 instances for each of the base cases. Each instance represented a sall perturbation of the developent copletion ties of the CMs. The specific odel we address is provided in the appendix. All coputational tests were carried out on a laptop coputer with a Genuine Intel 1 GHz T2500 processor and 1 GB RAM, running the Red Hat Linux 5 operating syste. The code was built with C++ version and linked with glibc and Mosek 5.0. Each one of the 10 base cases represented 10 types of (inconsistent weapons = 10, 10 possible CMs M = 10, 3 levels of intensity = 3, and 4 tie periods H = 4. Consequently, each integer prograing forulation has about 38,000 variables and 73,000 constraints and the corresponding ulti-fcd graph has about 120 vertices (one vertex for each possible copletion tie t and 4,000 edges. Although the probles are quite large, the odel proved to be coputationally efficient; the average running tie of each instance was about 15 inutes (the shortest tie was just a few seconds and the longest tie was a little over 4 hours Paraeters etting The CMs were divided into three groups according to their copletion ties: the first group (consisting of CMs indexed by = had short developent ties, the second group (consisting of CMs indexed by = had ediu developent ties, and the third group (consisting of CMs indexed by = had long developent ties. The tie units were easured in onths, and each tie period was set to be 12 onths. Therefore, the first period is 0 12, the second 12 24, etc.; thus, T 2 = 12, T 3 = 24, and T 4 = 36. The tie horizon, T, was defined to be T 4 + ax k t k. For siplicity, in our coputational testings we liit, the optional starting ties of developing the CMs to take the values of 0 = T 1, T 2, T 3, or T 4. The copletion ties, the t k s, were sapled uniforly fro the data listed in Table 1. For each CM, the developent copletion ties were ordered according to the three intensities such that t 1 > t 2 > t3. The daage rates, the dw s, were sapled fro a tried Noral distribution with paraeters, = 2, = 1 5 and truncated by 10 and 0 as upper and lower bounds, respectively. Recall that in the case of inconsistent

12 Operations Research 60(1, pp , 2012 INFORM 59 Table 1. Values for the t k s CMs, the d w s do not have the sae order of the s for all w W ; still, for each w d0 w is the axiu over all dw. In addition, s w, the tie weapon w is expected to be operational, was sapled fro the Unifor distribution on the interval 0 15, and these ties were sorted in an ascending order. The developent costs c k were sapled fro the Unifor distribution on the interval 3 10 and for each, the c k s were ordered to be nondecreasing in k. The budget teporal constraints for all tie periods were set to be identical for all h, so that C h = Ĉ for all h. Ĉ was calculated as the average costs ultiplied by twice the average period s duration, that is: Ĉ = k M c k /tk 2 M k where = H h=1 T h+1 T h /H The budget constraint for the entire tie period T was taken as C = Ĉ H Testing Robustness As indicated before, an experient was carried out in order to test the robustness of the solutions to ild perturbations in the data. In particular, we investigated the sensitivity to sall deviations in the developent copletion ties. First, the 10 base cases were solved. We next generated 100 instances for each base case, with each instance representing a ild perturbation of the copletion ties of the base case. The instances were generated by sapling fro a tried Noral distribution. pecifically, for the copletion tie of the th CM under the kth intensity we used a Noral distribution with = t k and deterined in the following way: for = 1 2 3: = ; for = 4 5 6: = ; and for = : = The range of the tried Noral distribution was The copletion ties were not reordered to be onotone in the developent intensity level. The tie intervals and the daage costs, which are functions of the copletion ties, were recalculated in the sae way as in the base case, whereas the C i s were not recalculated. Each of the 100 instances of each base case was solved separately. Figure 5. Histogra of the data in Table 2. Frequency Histogra of (PertBasisOpt PertOpt/ PertBasisOpt Measureents Bin We tested the robustness of the solution relative to each base case j 1 10 and each perturbation i For each j and i, let PertOpt ji be the optial value (total daage of the ith perturbation of the jth base proble, let X opt j Y opt j be an optial solution of the jth base case, and let PertBasisOpt ji be the total daage incurred if the actual developent copletion ties are as in the jith instance and X opt j Y opt j is used. We consider the following easure of robustness: ji = PertBasisOpt ji PertOpt ji PertBasisOpt ji Note that X opt j Y opt j ight be an infeasible solution to the ith perturbation of the jth base case. Infeasibility ight be of two types violation of the topology or violation of the teporal budget constraints. Topological violation can occur if soe of the edges in the solution corresponding to X opt j Y opt j do not exist in the graph corresponding to the jith instance. uch violations were handled by direct calculation of the actual value of the solution of the jith instance. We did not calculate the ji s for instances that were deterined to be infeasible due to violations of the teporal budget constraints Nuerical Results Table 2 and its histogra in Figure 5 illustrate the frequencies of the values obtained fro 792 feasible solutions out of the 1,000 runs (in 208 runs there were ild violations in the budget constraints, as indicated in Tables 3 and 4. We further exained the effect of the teporal budget violations in the following way. For each infeasible Table 2. Frequency of the values in the original runs. Value ore Frequency

13 60 Operations Research 60(1, pp , 2012 INFORM Table 3. values with the original teporal budget constraints. Count Average TD Max Min perturbation, we reran the odel with an increase of 5% in each violated teporal budget constraint. The purpose of this exercise was twofold first, to see whether a relatively sall increase in the budget (5% is sufficient to restore feasibility, and second, to see what effect that increase will have on the values. Table 4 and its histogra in Figure 6 provide the resultant values after the 5% increase was ipleented where necessary, and Table 5 provides the new suary statistics. Coparing Table 3 to Table 5, we observe that the sall increase in budget was sufficient to eliinate alost all the cases of infeasibility (979 out of the 1,000 instances are now feasible. Thus, we conclude that the violations of the teporal budget constraints were of sall agnitude. Also, we note that the average value reains rather sall ( This exercise deonstrates the robustness of our odel. 6. Concluding Rearks and Extensions This paper addresses a resource allocation proble faced by a defense agency charged with developing CMs in a dynaic ars race against an adversary that seeks to cause as uch daage as possible. uch scenarios are becoing ore and ore relevant in asyetric wars between governent forces and insurgents. The ain contribution in this paper is the forulation of tractable network optiization odels that encopass the essential eleents of the proble. The odels we develop are deterinistic, and as such they can be criticized for failing to address the (obvious uncertainty that exists in the developent ties of the CMs. However, the extensive nuerical analysis that is presented in 5 deonstrates the robustness of the odels whose outcoes reain stable when there is soe noise in the data. The ethods we developed apply to a odification of our odel where the data consists of daage rates d W for M and W and the daage rate in the presence of a set of weapons W and a set of CMs M is DM W in M d W (in the presence of consistency, (12 shows that the odel we study is an instance of the above. When all weapons are operational at tie 0, this odel reduces Figure 6. Histogra of the data in Table 4. Frequency Histogra of (PertBasisOpt PertOpt/ PertBasisOpt More Bin to the one studied in 3.1 by looking at as a single weapon. When the weapons are not necessarily operational at tie 0 and weapon w becoes available at tie s w 0, with 0 s 1 s, we can ipose a odified consistency assuption that asserts that the ranking of the CMs against the sets of the for 1 w, w, is the sae. Under this assuption, the analysis of 3.2 applies, and the decision proble can be reduced to (9. In future research we intend to explore dynaic versions of the odels developed here. In particular, we intend to look at ultistage odels with recourse. That is, in each period, B will be able to observe new data that were realized since his previous decisions were ade and adjust the decisions accordingly. We plan to investigate such dynaic odels in both rolling and folding horizon fraeworks (in the rolling horizon fraework, each decision epoch covers a fixed nuber of periods in the future, whereas in the folding horizon fraework, we advance towards a given target date and so the decision epochs correspond to an ever-decreasing set of periods. Appendix Proof of Lea 2 The decision version of the constrained shortest-path proble for the FCD graphs defined in this paper can be stated as follows: Given positive nubers R and C, deterine whether there is an (O D-path in the FCD graph with d-length R and c-length C. This decision proble is clearly in NP because the length of a path in a graph can be deterined in polynoial tie. We next prove that this decision proble is NP-coplete by showing that any instance of the partition proble known to be NP-coplete Table 4. values with 5% increase of the violated teporal budgets constraints. Value Frequency Value ore Frequency