Chapter 4 Conflict Resolution and Sector Workload Formulations

Transcription

1 Equation Chapter 4 Section Chapter 4 Conflict Reolution and Sector Workload Formulation 4.. Occupancy Workload Contraint Formulation We tart by examining the approach ued by Sherali, Smith, and Trani [47] to generate workload contraint for the AM model. Eentially, thi approach determine the maximum number of aircraft that will imultaneouly occupy each ector, given the election of any et of urrogate flight plan, and impoe a correponding penalty tructure in the objective function. Conider the total collection of flight plan f =,..., F. Thee plan involve traveral between certain pair of fixe that might interect in ome ector =,..., S under preent conideration. Define the occupancy workload for any uch ector at any moment in time to be the number of aircraft reident within that ector at that given intant in time. To characterize thi type of workload for each ector =,..., S, we can examine the occupancy duration of the variou flight within over the time horizon H. The model AOM of Sherali et al. [48] provide thi information by contructing a Gantt chart of flight plan occupancy interval for each ector. In practice, ATC operator routinely monitor everal aircraft that are imultaneouly travering their repective ector. Naturally, when the occupancy workload (maximum imultaneou occupancy of a number of aircraft) become too high, a potentially dangerou or untenable ituation can arie. f 65

2 66 For each ector S, let i =,, I index the collection of maximal overlapping et C i of flight plan (f,p), where an overlapping et of flight plan i called maximal if it i not a trict ubet of another overlapping et. For example, examining Figure 4-, we have I = 4 maximal et. Hence, we have, C i th ( f, p) : flight plan ( f, p) belong to the i =, i =,..., I, S. (4.) maximal overlapping et for ector An efficient algorithm for determining thee et i decribed in Sherali and Brown [45]. Note that i poible that (f,p ) and (f 2,p 2 ) C i for ome and i, with f =f 2 (i.e. the pair correpond to the ame flight), although in thi cae, the plan would pertain to two ditinct urrogate. Figure 4-: Gantt Chart for Formulating Workload Contraint Now define the variable n to repreent the maximum number of overlapping flight plan within each ector =,..., S. Note that n i given by the larget number of flight plan elected from any of the maximal overlapping et C i, i =,, I, i.e.,

3 67 n = max fp i=,..., I ( f, p) Ci x, (4.2) becaue any other overlapping et i a ubet of ome maximal overlapping et. In the model formulation, the variable n i bounded on a uitable interval [, n ], and furthermore, it value i penalized in the objective function uing a penalty factor that increae nonlinearly in an appropriate fahion with an increae in thi type of workload. The motivation here i that if the maximum number of aircraft being imultaneouly monitored in a ector increae from one to three, for example, the aociated penalty hould likely more than triple. Moreover, there hould be ome abolute maximum number n of overlapping flight at any point in time, a determined by the capacity of ector. Thi workload meaure and the aociated penalty tructure can be modeled without the need to dicretize time. Toward thi end, define the binary variable y n if the occupancy workload in ector i n =, S, n =,..., n (4.) 0 otherwie and let µ n be the aociated penalty for having y n =. We aume that µ µ, and µ 2µ µ j =,..., n. (4.4) 2 j ( j ) ( j 2) Note that the condition (4.4) implie that 0 ( µ µ ) ( µ µ )... ( µ µ ). (4.5) 2 2 n ( n )

4 68 Figure 4-2: Illutration of a Convex Sector Workload enalty Structure Figure 4-2 illutrate the implied convex nature of thi penalty tructure. Oberve that by enforcing n, we alway incur a workload cot of at leat µ, even when no aircraft are being monitored over the horizon. Thi i appropriate ince there alway exit a fixed monitoring cot. More importantly, by avoiding a cot of zero correponding to n =0, there i greater flexibility in conidering practical workload cot that would atify (4.4). For example, we might have 0 < µ = µ 2 =... = µ for ome τ threhold number τ of aircraft being monitored, after which the cot might increae at an increaing rate a in (4.5). Thi i the cot tructure that arie in practice, and the model aume that thi hold true. The penalty tructure i then incorporated into the contraint a follow, noting (4.2). x fp n 0 i=,..., I, =,..., S (4.6a) ( f, p) Ci n n = nyn, =,..., S (4.6b) n= n yn =, =,..., S (4.6c) n= y 0, n=,..., n, =,..., S. (4.6d) n The following term i included in the objective function:

5 69 (min)... n + µ y. (4.6e) S n= n n Note that both n and y n,, n, have been declared a continuou variable in (4.6). Sherali et al. [47] prove that the binary retriction on y hold automatically at optimality in the model, and hence, o do the integrality and bounding retriction on the variable n, =,..., S Alternative Occupancy Workload Contraint Formulation The foregoing approach focue on the maximum ector monitoring workload for ATC peronnel. However, thi doe not capture the total, or average workload requirement, or the peritence of peak period. Hence, we offer the following additional modeling contruct. The model AOM [48] determine the ector occupancie and the repective occupancy duration for each flight plan. Define Ω a the et of flight plan (f,p) that occupy the ector =,..., S interval for flight plan p of flight f in ector i given by during the horizon H, where the total occupancy time t fp. The average workload for ector can then be decribed a w = t x, =,..., S, (4.7) fp fp H ( f, p) Ω where H i the length of the horizon being conidered. Oberve that if t fp i the total airborne time correponding to flight plan (f,p), we mut have tfp = tfp, ( f, p). (4.8) S The average occupancy workload i penalized in the objective function in a linear fahion. The rationale here i that the average occupancy workload repreent the nominal tate of monitoring activity. A uch, peronnel and equipment can be

6 70 cheduled a a direct function of the expected amount of work to be performed (auming, of coure, that the expected workload i within the ector capacity). Hence, we include in the objective function (min)... + γ w, (4.9) S where γ i a uitable contant penalty factor. Recalling (4.2), and letting n be bounded above by ome maximum number, n, of imultaneouly occupying aircraft, a before, we can characterize the peak occupancy workload in each ector via n x, i, =,..., S, and n n, =,..., S. (4.0) fp ( f, p) Ci With repect to workload, an operation tempo that i contant and predictable i preferred to one that i either erratic or not predictable. When the ATC workload varie ignificantly, additional peronnel and equipment reource are required that might remain idle during non-peak period. To accommodate thi feature, we hall aign a penalty in the objective function correponding to the maximal variability defined a the difference between the peak occupancy workload and the nominal (average) occupancy workload over the horizon. Thi maximal variability, n w, i computed via n w = ny, =,..., S, (4.) ( ) n n n= 0 where the y variable repreent convex combination weight, atifying n yn =, yn 0, n= 0,..., n, for each =,..., S. (4.2) n= 0

7 7 Oberve that the lower bound for n in (4.2) i zero. Thi correpond to the cae where the occupancy workload i contant over the horizon H (i.e. the peak workload equal the average workload). Furthermore, note that the quantity ( n w ) not necearily integral. Remark: If n(t) i the number of aircraft overlapping at time t, note that i H H n w = n() t dt dt H H n 0 0 =, (4.) and o, we have (n - w ) 0 a expected. If we define the penalty tructure for µ n a a function of ( n w ) in a imilar manner a in (4.4) and (4.5), we can rely on the reulting implied convex tructure to enure that, at optimality, at mot two y n -variable will be non-zero, and, if y n and are two uch non-zero variable, then n and n 2 are adjacent, o that the aociated penalty i a convex combination of µ n and µ n 2. Accordingly, we modify the contraint (4.6) a follow: y n 2 w = t x, =,..., S (4.4a) fp fp H ( f, p) Ω x fp n 0, i =,..., I, =,..., S (4.4b) ( f, p) Ci n n w = ny, =,..., S (4.4c) n n= 0 n yn =, =,..., S (4.4d) n= 0 n n, =,..., S, and y 0, n= 0,..., n, =,..., S. (4.4e) n The related occupancy workload term in the objective function are given by:

8 72 (min)... + γ w + n nyn S S n= 0 µ, (4.4f) where µ µ, and µ 2 µ µ, j = 2,..., n. (4.5) 0 j ( j ) ( j 2) Oberve that the penalty term of (4.4f) correpond to the tre placed on the ATC ytem a a reult of the election and execution of a particular et of flight plan. In a CDM environment, thee ytem cot will be traded off againt airline cot (e.g. fuel and delay cot) to elect an optimal et of flight plan. Thee airline cot are dicued in detail in Chapter Conflict Reolution Workload Formulation Reolving conflict between aircraft travering a ector impoe a conflict reolution workload that i in addition to the occupancy workload dicued in the foregoing ection. For example, an ATC controller mut contact conflicting aircraft, direct new vector for them, and ubequently monitor compliance, to enure that the required eparation between aircraft pair i maintained. Accordingly, we acribe a uitable penalty ϕ Q in the objective function for each conflict that mut be reolved, correponding to the flight plan and Q, i.e., (min)... + ϕ QzQ, (4.6) ( Q, ) A where A i the et of pair of flight plan that potentially conflict in the overall airpace under conideration, during any point in time within the horizon. Remark: Uing the tructure of (4.6), we can acribe a unique workload penalty for each conflict, baed upon the geometry of the conflict itelf, a a meaure of the difficulty, or intenity, of the required conflict reolution action. For example, two aircraft that are approaching each other head-on might require a quicker ATC repone

9 7 than two aircraft traveling near parallel trajectorie. Recall that the AEM compute conflict geometry information for each conflicting pair of flight plan, (,Q), including their relative trajectorie and the minimum projected eparation ditance Dicretized Time Slot Workload Formulation We ummarize the trategy employed by Sherali, Smith and Trani [47] to formulate conflict contraint for the AM model. Firt, the time horizon i dicretized into uniform-length egment, t =,..., T, for each ector =,..., S, where the duration of the egment i dependent on the ector conflict reolution capacity. Baed on it conflict analyi, AM then generate a conflict graph G ( N, A ) for each ector and t t t each time egment t, where N t i the et of node repreenting all flight plan travering ector plan are in conflict, then The graph during egment t, and A t i the et of edge uch that if two flight A t include an edge connecting the correponding node. G t i typically a collection of dijoint component. Figure 4- how an intance of uch a conflict graph where the aircraft pair (,Q), (,R), (Q,R), and (Q,W) are in conflict. To facilitate the model generation proce, an overall conflict graph GNA (, ) and A i alo contructed, where i the et of node repreenting all flight plan, i the et of edge correponding to pair of flight plan that are in conflict in any ector during any point in time within the time horizon. N Figure 4-: Example Conflict Graph G t (N t,a t ) The AM model impoe the workload retriction that no more than one conflict may occur in each ector edge of during any time egment t. The algorithm examine the A t, in a pairwie fahion, and contruct contraint of the form,

10 74 x Sk, (4.7) Sk where Sk i the et of node at which the k th pair of edge i incident. Oberve that S k equal three or four, depending on whether the pair of edge i adjacent. The equential proce adopted by AM to generate contraint of the type (4.7) automatically preclude the creation of redundant contraint. In an alternative formulation AM2, Sherali, Smith, and Trani [47] defined z Q a a binary variable that take a value of one whenever conflicting flight plan and Q are elected (by including the contraint z x + x, z 0 ), and precribed the Q Q Q following retriction to repreent the above conflict workload contraint, in lieu of (4.7): zq, t =,..., T, =,..., S (4.8) ( Q, ) At ome ection. One advantage of uing (4.8) i that it readily generalize to the cae where r imultaneouly occurring conflict are permitted, a dicued in the next 4.5. Continuou Time Workload Formulation with Conflict Buffer Suppoe that while running AEM, whenever we encounter a conflict between a pair of flight plan and Q, < Q, over ome duration [ t, t, uch that thi conflict i 2 ] to be reolved in ector, we record thi interval along with the conflicting pair ( Q, ) in a Conflict Gantt Chart for ector, denoted CGC. Note that if and Q lie in different ector over the duration [ t, t 2 ], the reolution of thi conflict i aigned to the ector that contain the focal aircraft at time t, with (unlikely) tie being broken arbitrarily by aigning the conflict to the ector that contain the maller-indexed () flight plan.

11 75 In addition, depending on the ector and the everity and characteritic of the identified conflict, we define a prep-buffer duration b Q > 0 that erve to repreent a preparatory duration required by the air traffic controller in ector to addre the imminent conflict between flight plan and Q. We augment the interval [ t, t ] 2 repreented in the Having contructed the CGC by adding the prep-buffer, i.e., by replacing reolution workload contraint that the t with t. CGC for all ector, we now impoe a conflict b Q # of imultaneou augmented conflict to be reolved in at any time r, (4.9) where r i ome deignated workload parameter for ector, =,..., S. Uing the algorithm precribed by Sherali and Brown [45], uppoe that we identify the entire collection of maximal overlapping et M k, k,..., K compried of pair of conflicting flight plan (,Q), < Q, in =, for CGC, where each M k i CGC that overlap at ome time, and i maximal in the ene that it i not a trict ubet of any other et of imultaneouly occurring pair of conflicting flight plan. Figure 4-4 illutrate a hypothetical CGC and it aociated maximal overlapping et Oberve that, a in the cae of the pair ( AB), M k, k,..., K =. in Figure 4-4, diconnected conflict interval for any pair of flight plan are treated a eparate conflict in thi definition.

12 76 Figure 4-4: Conflict Gantt Chart CGC and it Aociated Maximal Overlapping Set Accordingly, we can formulate (4.9) by impoing ( Q, ) Mk z r, k =,..., K, =,..., S, (4.20) Q where recall that z Q equal one if flight plan and Q are elected, and equal zero otherwie. Oberve that (4.20) i a valid repreentation of (4.9), ince any et of overlapping conflict mut be a ubet of ome M, { } k k,..., K, where the et M k themelve contain overlapping conflict. Let u refer to the collection of contraint (4.20) for each,..., S a M = inequalitie. Recall from Section 4. that in the previou formulation of conflict reolution workload contraint, given any ector, we dicretized the horizon into lot t =,..., T, each of duration t, and we contructed the conflict graph Gt( Nt, A t) for each lot t =,..., T, eentially baed on conflict that (partially) overlapped lot t in CGC (without any prep-buffer). Similar to (4.20), we then impoed the retriction (uing the ame generalized parameter r ): ( Q, ) At z r, t =,..., T, =,..., S. (4.2) Q

13 77 D Let u refer to the collection of contraint in (4.2) for each inequalitie, for =,..., S. a ropoition 4- (a) Suppoe that b 0 (, Q) A. Then for any value of t 0, the D -inequalitie imply the Q M -inequaltie. (b) Converely, uppoe that given a value of t > 0 for each correponding lot t =,..., T, we compute > { } τ () t = max 0, t2() t t() t (4.22) where { } t ( ) min : [, ] i a conflict interval for ome (, ) t = t t t Q At (4.2) { } t ( ) max : [, ] i a conflict interval for ome (, ) 2 t = t t t Q At. (4.24) Then, τ ( t) t t =,..., T, and moreover, for b τ ( t), (, Q) A, t =,..., T, (4.25) Q t we have that the correponding M -inequalitie imply the D -inequalitie. roof (a) Conider an arbitrary M k and it correponding inequality in (4.20). Since all the conflict between pair ( Q, ) Mk occur imultaneouly at ome point in time (with b Q = 0, (, Q) A ), there exit ome time lot t, for which the aociated graph G t contain all thee conflict, i.e., At M k. Hence, we have that

14 78 z r z r, (4.26) Q Q ( Q, ) At ( Q, ) Mk i.e., the correponding D -inequality in (4.2) implie thi M -inequality in (4.20). (b) Converely, given the hypothei of part (b), conider any lot t {,..., T } and the correponding inequality (4.2) for the accompanying graph G. If t 2 () t t ( t) t a depicted in Figure 4-5(a), then τ () t 0 t. Moreover, all the conflict in A t overlap at ome point in time, becaue if not, then ome conflict in A t end trictly before another begin, whence we would have τ () t > 0. Conequently, there exit an M k, k {,..., K }, uch that M k At, and o, z r z r. (4.27) Q Q ( Q, ) Mk ( Q, ) At On the other hand, if t2() t > t() t a depicted in Figure 4-5(b), let t () t and t () t 2 repectively correpond to ome conflicting pair of flight plan ( GH, ) and ( UV, ). Since the conflict ( UV, ) begin after ( GH, ) end, and ince they both overlap time lot t, we have that both thee event occur within time lot t, and o, τ t2() t t() t t. Moreover, examining the time t () t, for any conflict ( Q, ) At that occur over ome duration [ t, t ], ay, we have by (4.22) that (i) if t < t () t, a occur with conflict ( AB), in Figure 4-5(b), then ince t t () t, we get t τ () t t () t t (4.28) and (ii) if t t () t, a occur with conflict ( CD, ) in Figure 4-5(b), then ince t t () t 2, we again get that (4.28) hold true becaue t τ () t = t t2() t + t() t t() t t < t. But (4.28) aert that under (4.25), all the conflict pair in A t would overlap at time t () t in

15 79 CGC, and o, there exit an M k, k {,..., K}, for which M k At. Therefore, we again have that (4.27) hold true, and thi complete the proof. Figure 4-5: Illutration for ropoition 4- M Oberve from ropoition 4- that if we ue a prep-buffer duration of zero, i.e., we do not augment the conflict interval, then the M D -inequalitie impoe a more tringent et of conflict reolution workload contraint (for any lot duration) than the -inequalitie. In thi cae, the lot-baed retriction contrain not only the imultaneouly occurring conflict, but alo recognize that non-overlapping conflict that might occur in relatively quick ucceion impoe a treful workload on the ATC controller. Indeed, a ropoition 4- illutrate, a the prep-buffer begin to increae, the condition (4.9) related to the -inequalitie begin to accommodate the conideration of conflict that occur in relatively quick ucceion within the workload formulation, and the M -inequalitie all imply the D -inequalitie once the prep-buffer become ufficiently large. ropoition 4- demontrate that thi occur at or before a prep-buffer value of t, the lot duration for the D -inequalitie. The foregoing dicuion erve a two-fold purpoe. Firt, it lend further inight into the lot-baed conflict reolution contraint, and econd, it offer an alternative modeling of uch contraint within the framework of AM. Oberve that each inequality of the type (4.20) i induced by an underlying conflict graph G ( N, A ), k k k where the edge et Ak M, and where N i the et of node at which the edge k k

16 80 M are incident. Thi i analogou to the conflict graph G ( N, A ) for the lot-baed k t t t formulation. Moreover, the overall conflict graph i given by G( N, A) in either cae. Conequently, replacing (, t ) with (, k) appropriately, or vice vera, the reult obtained uing one of thee modeling contruct can be tranported to the other in an analogou fahion. In the equel, we hall focu on developing further reult in the context of the M -inequalitie framework Valid Inequalitie Alternative Conflict Modeling Strategie For each ector, let G ( N, A ) be the conflict graph that i contructed for the k k k overlapping et M k, k =,..., K, where N k i the et of node repreenting the k overlapping et of flight plan travering ector during the time horizon, and Ak M k th i the et of edge connecting the correponding node in N k repreenting imultaneouly occurring conflict between pair of flight plan within p p < p M k. Recall from Section.4 and 4.2 that thee conflict interval are baed upon ome threhold probabilitie, and 2 and include the repective prep-buffer duration,, for conflict of everity level one and two, repectively, b Q > 0. A in Sherali, Smith, and Trani [47], we define FC a the et of flight plan pair and Q that poe a fatal conflict rik and we impoe the contraint x + x, (, Q) FC. (4.29) Q Accordingly, the graph G k and the entire dicuion in the equel below i concerned with the et of non-fatal or reolvable conflict. A derived in Sherali, Smith, and Trani [47], and working with r contraint in the x-pace a given by =, for now, let denote the et of conflict C

17 8 C = x: x Sk k KNR, x b inary, Sk (4.0) where each S k i a et of node in the conflict graph (repreenting flight plan) at which deignated pair of edge (in M k ) are incident (whence S k equal three or four), and where K NR record the collection of all uch non-redundant contraint over the entire et of conflict graph G k (, k). Recall that Sherali, Smith, and Trani [47] alo propoe an alternative repreentation in a higher-dimenional pace ( x, z ), where each z -variable repreent the quadratic product xx Q. Thi reult in the conflict contraint et z Q C 2 ( xz, ): zq ( k, ) ( Q, ) Mk zq x + xq (, Q) A =, (4.) zq 0 (, Q) A x binary where A i the arc et of the overall conflict graph. The experiment conducted by Sherali, Smith, and Trani reveal that C i preferable for pare conflict ituation, while C 2 i preferable for relatively more dene conflict graph. We hall now augment C 2 in order to derive a provably tronger repreentation than that given by C, and motivate our conideration of thi revied repreentation, denoted, defined below. Subequently, we will further augment C with ome additional clae of valid inequalitie. C To motivate the derivation of, conider the following imple example. C Example 4- Conider the following conflict graph.

18 82 The contraint (4.) defining C for thi intance are a given below. 2 z Q + z, R z x + x, Q Q z x + x, R R z 0, x binary. (4.2) If we maximize { x xq xr} + + ubject to the continuou relaxation of (4.2), we obtain the fractional extreme point olution z, Q = zr = x, 2 = x 2 Q = xr =. However, oberve that the contraint x + xq + xr 2 (4.) that would be inherent in the definition of C delete thi fractional olution. Motivated by thi example (a well a by ropoition 4-2 below) let u define T NC to be a et of triplet ( QR,, ) that do not admit a clique in any conflict ubgraph More preciely, let G k. ( QR,, ), < Q< R: for ome ( k, ), a ubgraph of Gk that i TNC = induced by the node, Q, and R contain preciely two edge,. (4.4) but no uch ubgraph for any ( k, ) contain three edge

19 8 Accordingly, let u define C ( xz, ): zq, (, k) ( Q, ) Mk = x + xq + xr 2, (, Q, R) T NC. zq x + xq, (, Q) A z 0, xbinary (4.5a) (4.5b) (4.5c) (4.5d) Conider the following reult, where for any et Ci, the et C i denote it continuou relaxation, obtained by replacing x binary with 0 x e, where e will alway denote a (conformable) vector of one. ropoition 4-2 C yield a tighter repreentation of the conflict contraint than doe C in the ene that the contraint defining C imply thoe defining C. roof Conider any contraint in C where k S =, with S {, Q, R} k =, ay, and where < Q< R. If ( QR,, ) TNC, then the correponding contraint in (4.0) i directly preent in (4.5b). Otherwie, there exit ome (, k) for which on the node et { QR,, }. Accordingly, (4.5c) contain the contraint G k contain a clique z x + x, z x + x, and z x + x (4.6) Q Q QR Q R R R while (4.5a) for thi (, k) implie zq + zqr + zr. (4.7) Summing the three contraint in (4.6) and uing (4.7), we get

20 84 zq + zqr + zr x + xq + xr ) (4.8) 2( which implie x + xq + xr 2. Hence again, the correponding contraint in C i implied in the continuou ene. Now conider a contraint in C for which S k = 4, and i given by x + xq + xr + xs (4.9) baed on edge implie the contraint ( Q, ) and ( R, S ) in M k for ome graph k G. For thi ( k),, (4.5a) z Q + z (4.40) RS while (4.5c) contain the relationhip zq x + xq and zrs xr + xs. (4.4) Summing (4.40) and uing (4.4), we have a before that (4.9) i implied. Thi complete the proof. than C. The next example illutrate that C can yield a trictly tighter repreentation

21 85 Example 4-2 Conider the tar conflict graph hown below. The correponding contraint of C are given by x + x + x 2, x + x + x 2, x + x + x 2, and 0 x e. (4.42) Q R Q W R W The olution ( ) (, Q, R, W,,, ) x x x x = i a vertex of (4.42) a evidenced by the four linearly independent contraint, given by x =, and the three tructural inequalitie in (4.42) being active. However, note that the contraint of C imply ( ) ( ) ( ) ( ) z + z + z x + x + x + x + x + x, Q R W p Q R W ie.. x + x + x + x 4. Q R W (4.4) The above fractional olution i deleted by (4.4), and o in general, C can provide a trictly tighter repreentation than C. However, to generate the tar-graph facet 2x ( x Q x R x W ) for the underlying et C (ee Sherali and Smith [46]), we would need to add zq + zr + Section 4.4 below. zw x to C. Thi extenion i conidered in

22 An Efficient Scheme for Generating C In order to devie an efficient cheme for generating the conflict contraint C defined in (4.5), uppoe that we commence by contructing the overall conflict graph GNA (, ). Here, N i the et of node repreenting all the flight plan, and A i the et of arc uch that if flight plan and Q conflict at any point in time over the horizon (with probability exceeding p for everity level one conflict and p2 p < for everity level two conflict), then we have an edge connecting and Q. Let u index the edge ( Q, ) A,( < Q), a c =,...,c. Let u alo define an indexing cheme for pair cc ( ) of conflict a,...,ς, where ς =, and where the two-tuple for any given pair of 2 ll ) l < l } {,...,c} conflict (,, for, with { ll, ς ( ll, ): {,...,c} {,..., c} {,..., ς } given by, i repreented by the function ς l ( l )( 2c l) ( l, l ) = ( c i) + ( l l) + ( l l) i= 2. (4.44) Let u now define a one-dimenional vector initialized at zero, and where for any pair of edge {, } { } nonadjacent, then we leave E ( l, l ) edge involving the node, Q, and ς 0 permanently, while if R, ay, then E ς ( l, l ) E having ς component that are ll,..., c, l< l, if l and l are l and l are adjacent might be revied to a value of or 2 according to the following cae. Whenever we detect ( QR,, ) a a potential candidate for the et ( ) TNC given by (4.4) baed on ome conflict graph G k, we et E ς l, l =, and if QR,, i known not to belong to T NC (perhap at ome later ( ) tage after being identified a a candidate for ) then we et E ς l,l = 2. Note that T ( ) for any triplet ( Q,,R), if all the three poible edge l, l, and l induced by thee node exit in the edge et NC M k for any conflict graph G k, then we enforce

23 (, ) = (, ) = (, ) E ς l l E ς l l E ς l l = 2, where the notation ( l, l 2) denote that thi twotuple i (, ) if l and it i (, ) if of l l l 2 Conider any < l l l2 < l. G k being examined in turn, and uppoe that we order the edge M k according to their aigned indice in {,...,c }. For each l M examined in thi k order, conider each M k l > l in and do the following If l and l are not adjacent, then reiterate. Ele, let the incident node for thi adjacent pair ( ll, ), l < l be given by ( QR,, ). Conider the following three poible cae: Cae (i) ( ) If E ς l,l = 2, then reiterate. Cae (ii) ( ) (,, ) If E ς l,l =, then we have previouly identified thi for incluion in T NC ) M k QR a a candidate. Hence, we check if thi property till hold true a follow. If the third edge induced by ( QR,, exit in and i given by l > l (thi edge triplet ha not a yet been conidered), then put ( ) ( ) (,l ) E ς l, l = E ς l, l = E ς l =2, (4.45) ) (i.e. ( QR,, form a clique and i hence no longer part of ), and reiterate. T NC

24 88 Cae (iii) ( ) 0 (,, ) If E ς l,l = and if the third edge induced by QR exit in Ak ( l ) and i given by l > l, then execute (4.45) and reiterate. Otherwie, put E ς,l = and reiterate. Once we have examined G k for all (4.5b), i.e., x + xq + xr 2, for each (, ) (, k), we may then generate the contraint ll where E ( l, l ) ς =. A we proceed equentially for ς =,..., ς, we generate only non-redundant contraint by checking againt previouly generated contraint Additional Valid Inequalitie We now preent an example to motivate a further tightening of C via the generation of ome additional valid inequalitie. Example 4- Conider the following conflict graph, where < Q < R< W, ay, Noting that the triplet ( ) ( ) RW,, and QRW,, TNC a defined by (4.4), the contraint defining C (ee (4.5)) are given a follow:

25 89 z + z + z + z + z, Q R W QR QW z x + x, z x + x, z x + x, Q Q R R W W z x + x, z x + x, QR Q R QW Q W x + x + x 2, x + x + x 2, R W Q R W z 0, 0 x e. (4.46) The problem maximize { x + x + x + x : contraint (4.46)} (4.47) Q R W yield the olution x, =, 0, and = xq = xr = x 2 W zq = zr = zqr = zw = zqw = (4.48) 2 with objective value equal to 2.5. However, ince the x -variable are binary, we can impoe the contraint x + xq + xr + xw 2.5 = 2, (4.49) which delete the fractional olution. In fact, incorporating (4.49) within (4.47) now yield a binary extreme point olution. In thi cae, (4.49) i actually an even-hole facet-defining contraint for the correponding et C (ee Sherali and Smith [46]). Although we could olve a problem of type (4.47) to generate a cut of type (4.49) for each conflict graph G k, thi might be burdenome. Hence, we propoe the generation of the following two type of valid inequalitie. Firt, imilar to (4.47), but uing the total et of conflict contraint C, we olve the linear programming problem

26 90 ν C = max { x : contraint C in (4.5), xi = i }. (4.50) If ν C i fractional, we impoe the valid inequality x ν C (4.5) within C. Second, we conider the linear program ν = min { cx : contraint C in (4.5), C2 xi =, i, contraint (4.5) (if generated) } (4.52) where c = c, or c i an integerized (rounded up, ay) coefficient of x containing ome of the other model objective term a well. In any cae, if ν C 2 i fractional, we impoe cx ν C2 (4.5) within C. We can further tighten (4.5) in one of two way. Firt, if 2 i a (highet) common factor of c ν C, where 2 i fractional, we can replace (4.5) by the tronger valid inequality c ν C 2 x. (4.54) Alternatively, we can derive a valid inequality (tronger than even (4.54)) by olving the following 0- knapack problem

27 9 νc = min cx : πx π0, xbinary (4.55) where π x π 0 i the tronget urrogate contraint for (4.52) derived by urrogating the tructural contraint in (4.52) uing the optimal dual multiplier obtained while olving thi linear program. Baed on (4.55) we can impoe cx ν C (4.56) within. We will experiment with the ue of C augmented with the foregoing valid C inequalitie in order to precribe a viable trategy Alternative Conflict Contraint Formulation Motivated by Example 4-2, let u further tighten the repreentation of C by replacing the T NC contraint a explained below. -cut by higher-dimenional underlying tar graph convex hull For any ( k, ), conider the conflict graph G ( N, M ). Examine the edge in i M k that are incident at any given node Nk k k k, and define { } J ( ) = j N : ( Q) M k k k, (4.57) where ( Q) denote ( Q, ) if < Q, and ( Q, ) if Q<, noting our convention that for any edge ( Q, ) Mk, we have < Q. Likewie, we denote by ( Q) either Q if < Q, or Q otherwie. Conider the following reult:

28 92 ropoition 4- Let G ( N, M ) be the k th conflict graph for ector, and for any k k k N k, let Jk ( ) be given by (4.57). Then, the following i a valid inequality for the conflict contraint: Q Jk ( ) z ( Q) x. (4.58) roof If x = 0, then (4.58) implie that ( Q) 0 z = Q J ( ) (ince z 0 ), which i k valid becaue z ( Q) repreent the product xxq. If x =, then thi i again valid becaue the conflict contraint aert that we mut have xq when Q Jk ( ) x =. Thi complete the proof. Baed on ropoition 4-, conider the following collection of contraint of the type (4.57), where we have impoed the condition Jk ( ) 2 becaue, a we hall how in the equel, the contraint for the cae Jk ( ) = are inconequential with repect to the projection onto the original X-pace. z( Q) x, Nk uch that Jk ( ) 2, (, k ). (4.59) Q Jk ( ) Note that if we conider the overall conflict graph G( N, A), and accordingly define, imilar to (4.57), { } J( ) = Q: ( Q) A N, (4.60)

29 9 then everal ubet of J( ) will be ued in the different graph G k to generate contraint of the type (4.59) for any N. Naturally, if J ( ) J 2 ( ), then the k k2 contraint (4.59) that i baed on J ( ) k. Hence, let u define 2 2 J ( ) k may be dropped, ince it i implied by that for = { : at leat one contraint of the type (4.59) i g enerated } (4.6) * I N * and for each I, let Jr ( ), for r =,..., r, be the collection of et of type Jk ( ) that yield a nonredundant ytem of contraint in (4.59). Accordingly, let u define the conflict contraint et C ( xz, ): z Q, ( k, ) ( Q, ) Mk * z( Q) x, r =,..., r, I = Q J. r zq x + xq, (, Q) A z 0, x binary 4 ( ) (4.62a) (4.62b) (4.62c) (4.62d) Conider the following reult, which jutifie the omiion of contraint (4.59) correponding to Jk ( ) = in (4.62). ropoition Let C be defined a in C with the additional contraint 4 4 z( ) x, N uch that J ( ) { Q}, (, k ). (4.6) Q k k Accordingly, define X { x: ( x, z) C 4 } = and X { x: ( x, z) C 4 } + + =, where C and C 4 repectively denote the continuou relaxation of C 4 and C 4. Then X = X +.

30 94 roof It i clear that X + X becaue of the additional contraint (4.6) defining C + 4. Hence, uppoe that x X a evidenced by ( x, z) C4, and let u how that x X + by demontrating that there exit a ẑ for which ( x, zˆ ) C + 4. Toward thi end, conider { } zˆ max 0, x + x, (, Q) A. (4.64) Q Q Oberve from (4.62c) and (4.62d) that we mut have { } ˆ z max 0, x + x = z (, Q) A. (4.65) Q Q Q Hence, ince ( x, z) C4, we alo have that ( x, z) C4 ˆ by virtue of (4.65). Furthermore, conidering any contraint of type (4.6), we have that (, ˆ) contraint a well becaue x z atifie thi { } ˆ( ) x 0 and x x + x x max 0, x + x = z. (4.66) Q Q Q Therefore, we have that x X +, or that X X +. Thi complete the proof. We now exhibit that C provide a tighter repreentation than C. 4 ropoition 4-5 C4 provide a tighter repreentation for the conflict contraint that doe C in the ene that C C. 4

31 95 roof x, z Let ( ) 4 C. Noting the tructure of ufficient to demontrate that x atifie any T any uch inequality x, z C C, in order to how that ( ) NC, it i -inequality. Toward thi end, conider x + xq + xr 2 (4.67) where, without lo of generality, uppoe that { QR} N with { } ome { } (, k), r,, k QR, J ( ) for. Hence, there exit a contraint of the type (4.62b) defining C for which QR J(. ) From thi contraint, uing (4.62c), and z 0, we get 4 k ( ) ( ) x z z + z x + x + x + x R, (4.68) ( Q) ( Q) ( R) Q Q Jr ( ) which implie (4.67). Thi complete the proof. To illutrate that C4 can provide a trictly tighter repreentation than C, conider the conflict graph of Example 4-2. For thi graph, the et C i given by C ( xz, ): zq + zr + zw x + xq + xr 2, x + xq + xw 2, x + xr + xw 2, =. (4.69) zq x + xq, zr x + xr, zw x + xw, z 0, 0 x e The even tructural inequalitie defining C are linearly independent, and their interection yield the feaible fractional vertex z Q = zr = zw =, x = xq = xr = xw = 2. (4.70)

32 96 However, note that C 4 ha the contraint zq + zr + zw x (4.7) of type (4.62b) that delete thi fractional olution. Oberve that (4.7) along with (4.62 c) implie the valid inequality ( ) ( ) ( ) x z + z + z x + x + x + x + x + x, (4.72) Q R W Q R W that i, x + ( x + x + x ), (4.7) 2 Q R W which alo delete the olution (4.70). In fact, a hown by Sherali and Smith [46], (4.7) i a facet for the underlying tar conflict graph, and that in general for tar graph, the contraint of C 4 characterize the complete convex hull repreentation for the conflict contraint, implying an exponential collection of facet in the original projected x -pace repreentation. Hence, thi repreentation C for the conflict contraint might be particularly ueful. We hall conduct experiment uing the alternative repreentation C or C (or both) of the conflict contraint, and examine their relative performance Generalized M -inequalitie Let u reviit the cae where the number of conflict to be allowed imultaneouly in any ector i retricted to be no more than ome r in general. The extended form of C i then given a follow: 4

33 97 ( xz, ): z (4.74a) Q r, ( k, ) ( Q, ) Mk z (4.74b) ( Q) r x, Nk uch that Jk ( ) r +, (, k) C4 = Q J ( ). k (4.74c) zq x + xq, (, Q) A z 0, x binary (4.74d) Note that a in ropoition 4-4, we can how that the projection of x -pace remain unaltered if we include the contraint of the type (4.62b) that correpond to Jk( ) r, even if we tighten (4.62b) in that cae to C 4 onto the Q Jk ( ) z J ( ) ( Q) k x. (4.75) A in the proof of ropoition 4-4, we obtain x ˆ z( Q) Q Jk( ), which upon (, ˆ) umming over all Q J ( ), implie that (4.75) i atified by the k x z ued in that proof. Furthermore, a in (4.62b), we can identify and eliminate redundant contraint (4.74b). In particular, if for ome (, k ) and ( ) k (where poibly = ), we have 2, 2 2 for ome I * defined imilar to (4.6) that J ( ) J ( ) and r r, (4.76) k 2k2 2 then the contraint in (4.74b) correponding to (, k ) may be dropped for thi. * Hence, canning (4.74b) in thi fahion for each I for which at leat two contraint of type (4.74b) are generated, we can eliminate the implied inequalitie from thi et. Note again that a hown by Sherali and Smith [46], C 4 yield a convex hull repreentation for the cae of tar graph.

34 Additional Conflict Modeling Contruct An advantage of uing the (or alternatively, C ) conflict contraint defined C4 over the ( x, z) -pace, aide from the tighter repreentation that it provide a elucidated in ropoition 4-2 and ropoition 4-5, i that it permit the conideration of more detailed modeling trategie. We decribe two uch additional modeling contruct below. Firt, note that ince we have a pecific variable z Q that denote the exitence of a conflict between elected flight plan and Q whenever z =, we can acribe a Q pecific cot ϕ Q to thi variable that reflect a penalty baed on the (expected) duration and everity of thi conflict. Thi provide a greater flexibility in filtering the et of conflict that would need reolution, in addition to imply limiting the number of imultaneou (generic) conflict. Second, we can modify the contraint (4.62a) by intead impoing zq + η, where 0 η ( r ), (4.77) ( Q, ) Mk where η equal the additional conflict permitted beyond. Accordingly, in the objective function, we can accommodate the term µ η, (4.78) where the coefficient µ penalize the exceive workload impoed via admitting more than one imultaneou conflict in ector. (Recall that the coefficient ϕ Q defined above penalize the pecific conflict themelve baed on duration and everity.) Including (4.77) often (4.62a) and avoid blatant infeaibilitie in tight ituation. Oberve alo that in lieu of the linear penalty function (4.78), we can impoe an increaing-rate penalty function by repreenting η in term of it pecific poible

35 value over the interval [ 0, ] r uing binary variable, and then by aociating pecific penaltie to each of thee value-repreenting binary variable. We will invetigate the foregoing two trategie outlined in thi ection at a ubequent tage, once the primary model ha been analyzed. 99