資源配置問題是經典的應用問題，通常被描述為在競爭活動之間分配有限的資源，並在非常廣泛的領域中找到應用。在本文中，我們討論了兩個分配問題，並將這兩個問題構建成為整數規劃模型。問題之一是中風患者的醫療資源再分配問題。如今，急救醫療技術人員決定根據辛辛那提院前卒中量表將疑似卒中患者送往初級卒中中心或可進行血管內血栓切除術的醫院。然而，如果大血管閉塞的急性缺血性卒中患者首先被送往僅可施打血栓溶解劑的醫院，然後需要轉移到同時有能力進行血管內治療合併機械取栓術與施打血栓溶解劑的醫院，則獲得最終治療的時間會大大增加。為了解決這個問題，需要實施一個血管內治療合併機械取栓術的資源分配計劃。我們為每個僅可施打血栓溶解劑的醫院制定了一個均衡規劃模型來決定是否擴增血管內治療合併機械取栓術。另一個問題是航空公司之間的機場時間帶競爭。在航空公司市場上，各家公司通過航班頻率和機場時間帶的競爭來追求更高的利潤。我們構建了納什均衡規劃模型來計算確切的航班頻率，包括每個航空公司都是玩家的基本模型和每個聯盟都是玩家的聯盟模型；但是，由於此問題可能不存在純納許均衡解。因此，我們採用混合策略模型。我們期望獲得每個玩家的預期總利潤和聯盟中每個航空公司的代碼共享航班的建議百分比。

論文的最後一部分著重於解決多面體策略集上的整數值廣義納什均衡問題的算法。使用一些線性規劃技術，我們可以將每個玩家可能悖離的策略集構建為一個多面體集。採用有理生成函數的概念，我們可以推導出包含純納什均衡的集合的短有理生成函數。當策略維度固定時，我們可以在多項式時間內進一步檢查集合中整數點的數量。此外，如果該問題存在至少一個純納什均衡，我們就有一個多項式延遲演算法來枚舉所有納什均衡解。本文實現了三種經典博弈，即範式博弈、整數規劃博弈和多面體上的整數值廣義納什均衡，用以測試所提出算法的有效性。測試結果證實，所提出的算法確實找到了這些博弈的所有納什均衡解。也驗證了該算法最耗時的過程是計算整個策略集減去每個玩家的悖離集的並集的短有理生成函數，以及整個演算法的運行時間取決於玩家策略的數量和策略的維度。

Resource allocation is a classical issue in practical applications, commonly described as the allocation of limited resources among competing activities. It has a wide range of applications, and in this thesis, we discuss two allocation problems and formulate them as integer equilibrium programming models. One of the problems we address is the medical resource redistribution problem for stroke patients. Currently, emergency medical technicians decide whether to send a suspected stroke patient to a primary stroke center (PSC) or an endovascular thrombectomy (EVT)-capable hospital based on the Cincinnati Prehospital Stroke Scale. However, if an acute ischemic stroke patient with large vessel occlusion is first sent to a PSC and then needs to be transferred to an EVT-capable hospital, the time to get definitive treatment is significantly increased. To solve this problem, an EVT-resource allocation plan needs to be implemented. We propose an equilibrium programming model for each PSC to decide whether to expand EVT-resource. The other problem we address is the frequency competition among airlines. In the airline market, each company competes on flight frequencies and airport slots to pursue higher profits. We constructed Nash-equilibrium programming models to compute the exact flight frequencies, including a basic model where each airline is a player and an alliance model where each alliance is a player. However, a pure strategy may not exist, so we adopt a mixed strategy instead. Our goal is to obtain the expected total profits for each player and a suggested percentage of code-share flights for each airline in an alliance.

The last part of the thesis focuses on an algorithm for solving the integer-valued generalized Nash equilibrium problem over a polyhedral strategy set. Using some linear programming techniques, we can formulate the set of profiles that each player could deviate as a polyhedron. Adopting the notion of short rational generating functions, we can derive a short generating function of the set including pure Nash equilibria when exist. We can further check the number of integer points in the set in polynomial-time when the profile dimension is fixed. Moreover, if there exists at least one pure Nash equilibrium, we have a polynomial-delay algorithm to enumerate through all Nash equilibria. Three classic games, namely, normal-form game, integer programming game and integer-valued generalized Nash equilibrium on a polyhedron, are implemented to test the effectiveness of the proposed algorithm. Testing results confirm that the proposed algorithm indeed find all Nash equilibria of these games. It is also verified that the most time-consuming process of the algorithm is on calculating the generating function of the set which is the entire profile set subtracting the union of each player's deviate set, and the total running time depends on the number of player profile and the profile dimension.

Abstract (Chinese) I
Abstract III
Contents V
List of Figures VIII
List of Tables IX
1 Introduction 1
1.1 Resource allocation problem 2
1.1.1 The medical resource redistribution problem for stroke patients 3
1.1.2 The frequency competition among airlines 7
1.2 Literature review 11
1.2.1 The medical resource redistribution problem for stroke patients 12
1.2.2 The frequency competition among airlines 13
1.2.3 Method for solving generalized Nash equilibrium problems 16
1.3 Thesis outline 18
2 Optimization of the hospital selection strategy and the competition of endovascular thrombectomy resources 19
2.1 Hospital selection strategy for suspected stroke patients with minimization of the expected time-to-receivedefinitive-treatment 22
2.2 Endovascular thrombectomy (EVT) resource competition among
primary stroke centers 23
2.3 Case study 25
2.3.1 Study setting 25
2.3.2 Comparison of sending strategies 26
2.3.3 Redistribution of EVT-resource 27
3 Frequency competition among airlines on coordinated airports
network 29
3.1 Mixed-strategy game 39
3.1.1 Mixed-strategy game formulation in the airline scenario 39
3.2 Mixed-strategy game formulation in the alliance scenario 41
3.3 Empirical analysis 43
3.3.1 Data sources 44
3.3.2 Feasible strategy generation 45
3.3.3 Parameter estimation 49
3.3.4 Mixed-strategy Nash equilibrium 50
3.3.5 Profit comparison under status quo, centralized strategy and
mixed Strategy 52
3.3.6 Passengers comparison under status quo, centralized strategy,
and mixed strategy 54
4 Finding integer-valued generalized Nash equilibrium solutions over
a polyhedron using short rational generating functions 56
4.1 Rational generating functions and integer points in polytope 57
4.2 Nash equilibrium of integer-valued generalized Nash equilibrium
problem on a polyhedron 60
4.2.1 Integer-valued generalized Nash equilibrium problem on a polyhedron with a payoff function being a min-affine function in integer coefficients 60
4.2.2 Normal-form game 64
4.3 Proposed algorithm for solving integer-valued generalized Nash equilibrium problems on a polyhedron with a payoff function being a min-affine function in integer coefficients 66
4.3.1 Algorithm 66
4.3.2 Implementation for three commonly seen games 67
5 Conclusions 73
Bibliography 76

Adler, N. (2001). Competition in a deregulated air transportation market. European Journal of Operational Research, 129 , 337–345.

Aguirregabiria, V., & Ho, C.-Y. (2012). A dynamic oligopoly game of the us airline industry: Estimation and policy experiments. Journal of Econometrics, 168, 156–173.

Ali, A., Zachrison, K. S., Eschenfeldt, P. C., Schwamm, L. H., & Hur, C. (2018). Optimization of prehospital triage of patients with suspected ischemic stroke. Stroke, 49 , 2532–2535.

Arrow, K. J., & Debreu, G. (1954). Existence of an equilibrium for a competitive economy. Econometrica, 22 , 265–290.

Aubin, J. P. (1982). Mathematical Methods of Game and Economic Theory. Elsevier.

Ball, M., Barnhart, C., Nemhauser, G., & Odoni, A. (2007). Air transportation: Irregular operations and control. Handbooks in Operations Research and Management Science, 14 , 1–67.

Barvinok, A. I. (1994). A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. Mathematics of Operations Research, 19, 769–779.

Barvinok, A. I., & Pommersheim, J. E. (1999). An algorithmic theory of lattice points in polyhedra. Mathematics, 38 , 91–147.

Barvinok, A. I., & Woods, K. (2003). Short rational generating functions for lattice point problems. Journal of the American Mathematical Society, 16 , 957–979.

Basar, T., & Geert, J. O. (1999). Dynamic Noncooperative Game Theory. Society for Industrial and Applied Mathematics.

Belobaba, P., Odoni, A., & Barnhart, C. (2015). The Global Airline Industry. John Wiley & Sons.

Bensoussan, A. (1974). Points de Nash dans le cas de fonctionnelles quadratiques et jeux differentiels lineaires a n personnes. SIAM Journal on Control, 12, 460–499.

Berkhemer, O. A., Fransen, P. S. S., Beumer, D., Van den Berg, L. A., Lingsma, H. F., Yoo, A. J., Schonewille, W. J., Vos, J. A., Nederkoorn, P. J., & Wermer, M. J. (2014). A randomized trial of intraarterial treatment for acute ischemic stroke. New England Journal of Medicine, 372 , 11–20.

Bissessur, A., & Alamdari, F. (1998). Factors affecting the operational success of strategic airline alliances. Transportation, 25 , 331–355.

Bureau of Transportation Statistics (2016a). Transtats. retrieved from air carrier financial reports: Schedule p-5.2. https://www.transtats.bts.gov/Fields.asp?Table_ID=297.

Bureau of Transportation Statistics (2016b). Transtats. retrieved from air carrier statistics: T-100 domestic segment (u.s. carriers). http://www.transtats.bts.gov/Fields.asp?Table_ID=259.

Bureau of Transportation Statistics (2016c). Transtats. retrieved from origin and destination survey: Db1bmarket. https://www.transtats.bts.gov/Fields.asp?Table_ID=247.

Campbell, B. C. V., Mitchell, P. J., Kleinig, T. J., Dewey, H. M., Churilov, L., & Yassi, N. (2015). Endovascular therapy for ischemic stroke with perfusionimaging selection. New England Journal of Medicine, 372, 1009–1018.

Carvalho, M., Lodi, A., & Pedroso, J. P. (2022). Computing equilibria for integer programming games. European Journal of Operational Research, 303, 1057–1070.

Cenedese, C., Cucuzzella, M., Scherpen, J. M. A., & Grammatico, S. (2021). Highway traffic control via smart e-mobility - part i: Theory. ArXiv preprint arXiv:2102.09354.

Czerny, A. (2018). Optimal use-it-or-lose-it rules for airport slot management. Available at SSRN 3243395.

Dobson, G., & Lederer, P. J. (1993). Airline scheduling and routing in a hub-andspoke system. Transportation Science, 27, 281–297.

English, J. D., Yavagal, D. R., Gupta, R., Janardhan, V., Zaidat, O. O., & Xavier, A. R. (2016). Mechanical thrombectomy-ready comprehensive stroke center requirements and endovascular stroke systems of care: Recommendations from the endovascular stroke standards committee of the society of vascular and interventional neurology (SVIN). Interventional Neurology, 4, 138–50.

Fabiani, F., & Grammatico, S. (2020). Multi-vehicle automated driving as a generalized mixed-integer potential game. IEEE Transactions on Intelligent Transportation Systems, 21 , 1064–1073.

Facchinei, F., & Christian, K. (2007). Generalized Nash equilibrium problems. 4OR, 5, 173–210.

Facchinei, F., Fischer, A., & Piccialli, V. (2009). Generalized Nash equilibrium problems and newton methods. Mathematical Programming, 117, 163–194.

Forsyth, P., Gillen, D., Muller, J., & Niemeier, H.-M. (2016). Airport Competition: The European Experience. Routledge.

Fourer, R., Gay, D. M., & Kernighan, B. W. (2002). AMPL: A Modeling Language for Mathematical Programming: Second edition.

Fransen, P. S. S., Berkhemer, O. A., Lingsma, H. F., Beumer, D., Van den Berg, L. A., & Yoo, A. J. (2016). Time to reperfusion and treatment effect for acute ischemic stroke: A randomized clinical trial. JAMA Neurology, 73, 190–196.

Fukui, H. (2010). An empirical analysis of airport slot trading in the united states. Transportation Research Part B: Methodological, 44, 330–357.

Gabriel, S. A., Conejo, A. J., Ruiz, C., & Siddiqui, S. A. (2013a). Solving discretely constrained, mixed linear complementarity problems with applications in energy. Computers & Operations Research, 40, 1339–1350.

Gabriel, S. A., Siddiqui, S. A., Conejo, A. J., & Ruiz, C. (2013b). Solving discretely-constrained Nash–cournot games with an application to power markets. Networks and Spatial Economics, 13, 307–326.

Garcia, B. L., Bekker, R., van der Mei, R. D., Chavannes, N. H., & Kruyt, N. D. (2021). Optimal patient protocols in regional acute stroke care. Health Care Management Science, 24.

Goh, K., & Uncles, M. (2003). The benefits of airline global alliances: an empirical assessment of the perceptions of business travelers. Transportation Research Part A: Policy and Practice, 37, 479–497.

Goyal, M., Demchuk, A. M., Menon, B. K., Eesa, M., Rempel, J. L., & Thornton, J. (2015). Randomized assessment of rapid endovascular treatment of ischemic stroke. New England Journal of Medicine, 372, 1019–1030.

Goyal, M., Jadhav, A. P., Bonafe, A., Diener, H., Pereira, V. M., & Levy, E. (2016). Analysis of workflow and time to treatment and the effects on outcome in endovascular treatment of acute ischemic stroke: results from the swift prime randomized controlled trial. Radiology, 279, 888–97.

Grauberger, W., & Kimms, A. (2014). Computing approximate nash equilibria in general network revenue management games. European Journal of Operational Research, 237, 1008–1020.

Gunes, E. D., & Nickel, S. (2015). Location Problems in Healthcare. Springer. Gunes, E. D., & Yaman, H. (2010). Health network mergers and hospital replanning. Journal of the Operational Research Society, 61 , 275–283.

Hane, C. A., Barnhart, C., Johnson, E. L., Marsten, R. E., Nemhauser, G. L., & Sigismondi, G. (1995). The fleet assignment problem: Solving a large-scale integer program. Mathematical Programming, 70, 211–232.
Hansen, M. (1990). Airline competition in a hub-dominated environment: An application of noncooperative game theory. Transportation Research Part B: Methodological, 24, 27–43.

Hsu, C.-I., & Wen, Y.-H. (2003). Determining flight frequencies on an airline network with demand–supply interactions. Transportation Research Part E: Logistics and Transportation Review, 39, 417–441.

IATA (2016). Strategizing for success. https://airlines.iata.org/analysis/joint-ventures-help-airlines-deliver-choice-to-consumers.

IBM (2017). Ilog cplex 12.6 user’s manual. http://www.ilog.com/products/cplex/.

International Air Transport Association (2020). Worldwide airport slots. https://www.iata.org/en/policy/slots/. Accessed: 2020-07-31.
Jacquillat, A., & Odoni, A. R. (2015). An integrated scheduling and operations approach to airport congestion mitigation. Operations Research, 63, 1390–1410.

John, G. B. (2018). Frequency has been part of airlines’strategy. https://www.ft.com/content/1c4f74f4-38cc-11e8-8b98-2f31af407cc8.

Jumaa, M. A., Castonguay, A. C., Salahuddin, H., Shawver, J., Saju, L., & Burgess, R. (2020). Long-term implementation of a prehospital severity scale for EMS triage of acute stroke: a real-world experience. Journal of Neurointerventional Surgery, 12, 19–24.

Krawczyk, J. B., & Uryasev, S. (2000). Relaxation algorithms to find Nash equilibria with economic applications. Environmental Modeling and Assessment, 5, 63–73.

Koppe, M., Ryan, C. T., & Queyranne, M. (2011). Rational generating functions and integer programming games. Operations Research, 59, 1445–1460.

Liu, P.-C. B., Hansen, M., & Mukherjee, A. (2006). Scenario-based management of air traffic flow: Developing and using capacity scenario trees. Transportation Research Record, 1951, 113–121.

Loera, J. A. D., Hemmecke, R., & K¨oppe, M. (2009). Pareto optima of multicriteria integer linear programs. Operations Research, 21, 39–48.

Luo, Z.-Q., Pang, J.-S., & Ralph, D. (1996). Mathematical Programs with Equilibrium Constraints. Cambridge University Press.

McKenzie, L. W. (1959). On the existence of general equilibrium for a competitive market. Econometrica, 27, 54–71.

Menon, B. K., Sajobi, T. T., Zhang, Y., Rempel, J. L., Shuaib, A., & Thornton, J. (2016). Analysis of workflow and time to treatment on thrombectomy outcome in the endovascular treatment for small core and proximal occlusion ischemic stroke (escape) randomized, controlled trial. Circulation, 133, 2279–2286.

Nabetani, K., Tseng, P., & Fukushima, M. (2011). Parametrized variational inequality approaches to generalized Nash equilibrium problems with shared constraints. Computational Optimization and Applications, 48, 423–452.

Narasimhan, C. (1988). Competitive promotional strategies. Journal of business, (pp. 427–449).

Nash, J. (1951). Non-cooperative games. Annals of mathematics, (pp. 286–295).

Nedjati, A., & Valipour, M. (2013). Solving health care facility location problems with new heuristic algorithm method. International Journal of Modeling and Optimization, 3, 12–14.

Nikaido, H., & Isoda, K. (1955). Note on non-cooperative convex games. Pacific Journal of Mathematics, 5, 807–815.

Nocedal, J. (2006). Knitro: An integrated package for nonlinear optimization. In Large-Scale Nonlinear Optimization (pp. 35–60). Springer.

Odoni, A. R. (1987). The flow management problem in air traffic control. In Flow control of congested networks (pp. 269–288). Springer.

Panicucci, B., Pappalardo, M., & Passacantando, M. (2009). On solving generalized Nash equilibrium problems via optimization. Optimization Letters, 3, 419–435.

Ribeiro, N. A., Jacquillat, A., Antunes, A. P., Odoni, A. R., & Pita, J. P. (2018). An optimization approach for airport slot allocation under iata guidelines. Transportation Research Part B: Methodological, 112, 132–156.

Richards, C. T., Huebinger, R., Tataris, K. L., Weber, J. M., Eggers, L., & Markul, E. (2018). Cincinnati prehospital stroke scale can identify large vessel occlusion stroke. Prehospital Emergency Care, 22, 312–318.

Rosen, J. B. (1965). Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 33, 1445–1460.

Ryan, C. T., Jiang, A. X., & Leyton-Brown, K. (2010). Computing pure strategy Nash equilibria in compact symmetric games. In Proceedings of the 11th ACM Conference on Electronic Commerce (pp. 63–72).

Sagratella, S. (2016). Computing all solutions of Nash equilibrium problems with discrete strategy sets. SIAM Journal on Optimization, 26, 2190–2218.
Sagratella, S. (2017). Algorithms for generalized potential games with mixedinteger variables. Computational Optimization and Applications, 68, 689–717.Sagratella, S., Schmidt, M., & Sudermann-Merx, N. (2019). Multi-vehicle automated driving as a generalized mixed-integer potential game. European Journal of Operational Research, 284, 373–382.

Saver, J. L., Fonarow, G. C., Smith, E. E., Reeves, M. J., Grau-Sepulveda, M. V., Pan, W., Olson, D. M., Hernandez, A. F., Peterson, E. D., & Schwamm, L. H. (2013). Time to treatment with intravenous tissue plasminogen activator and outcome from acute ischemic stroke. International Journal of Modeling and Optimization, 309, 2480–2488.

Saver, J. L., Goyal, M., Bonafe, A., Diener, H.-C., Levy, E. I., & Pereira, V. M. (2015). Stent-retriever thrombectomy after intravenous t-PA vs. t-PA alone in stroke. New England Journal of Medicine, 372, 2285–95.

Scheitz, J. F., Abdul-Rahim, A. H., MacIsaac, R. L., Cooray, C., Sucharew, H., & Kleindorfer, D. (2017). Clinical selection strategies to identify ischemic stroke patients with large anterior vessel occlusion. Stroke, 48, 290–297.

Schipper, Y., Nijkamp, P., & Rietveld, P. (2007). Deregulation and welfare in airline markets: An analysis of frequency equilibria. European Journal of Operational Research, 178, 194–206.

Schlemm, E., Ebinger, M., Nolte, C. H., Endres, M., & Schlemm, L. (2017). Optimal transport destination for ischemic stroke patients with unknown vessel status. Stroke, 48, 2184–2191.

Schoenfelder, J., Zarrin, M., Griesbaum, R., & Berlis, A. (2022). Stroke care networks and the impact on quality of care. Health Care Management Science, 25, 24–41.

Sheth, S. A., Jahan, R., Gralla, J., Pereira, V. M., Nogueira, R. G., Levy, E. I., & SWIFT-STAR trialists (2015). Time to endovascular reperfusion and degree of disability in acute stroke. Annals of Neurology, 78, 584–93.

Smith, B. C., & Johnson, E. L. (2006). Robust airline fleet assignment: Imposing station purity using station decomposition. Transportation Science, 40, 497–516.

Suen, W. (2005). Non-Cooperation—The Dark Side of Strategic Alliances.
Springer.

Tom, B. (2018). The 3 major airline alliances: Star alliance, oneworld and skyteam-why are they good? https://reurl.cc/KM7aWm.

United States Senate (1999). Competitive implications of domestic and international alliances among airlines: Hearing before the Subcommittee on Aviation of the Committee on Commerce, Science, and Transportation, United States Senate, One Hundred Fifth Congress. Washington: U.S. G.P.O.

Valouxis, C., & Housos, E. (2000). Hybrid optimization techniques for the workshift and rest assignment of nursing personnel. Artificial Intelligence in Medicine, 20, 155–175.

Vaze, V., & Barnhart, C. (2012a). Modeling airline frequency competition for airport congestion mitigation. Transportation Science, 46, 512–535.

Vaze, V., & Barnhart, C. (2012b). Modeling airline frequency competition for airport congestion mitigation. Transportation Science, 46, 512–535.

Vaze, V., & Barnhart, C. (2015a). The price of airline frequency competition. In Game Theoretic Analysis of Congestion, Safety and Security (pp. 173–217). Springer.

Vaze, V., & Barnhart, C. (2015b). The price of airline frequency competition. In Game Theoretic Analysis of Congestion, Safety and Security (pp. 173–217). Springer.

Wang, C.-H., Chang, Y.-C., Yang, Y., Chiang, W., Tang, S., Tsai, L., Lee, C., Jeng, J., Ma, H., Matthew, Hsieh, M., & Lee, Y. (2022a). Prehospital-strokescale parameterized hospital selection protocol for suspected stroke patients considering door-to-treatment durations. Health Care Management Science, 11, e023760.

Wang, C.-H., Liu, T.-Y., Chiang, W., Tang, S., Tsai, L., Lee, C., Lin, Y.-H., Jeng, J., Ma, H., Matthew, Hsieh, M., & Lee, Y. (2022b). Expanding resources of endovascular thrombectomy: An optimization model. Journal of the Formosan Medical Association, 121, 978–985.

Wei, W., & Hansen, M. (2007). Airlines’ competition in aircraft size and service frequency in duopoly markets. Transportation Research Part E: Logistics and Transportation Review, 43, 409–424.

Wen, Y.-H., & Hsu, C.-I. (2006). Interactive multiobjective programming in airline network design for international airline code-share alliance. European Journal of Operational Research, 174, 404–426.

Xu, Y., Parikh, N. S., Jiao, B., Willey, J. Z., Boehme, A. K., & Elkind, M. S. V. (2019). Decision analysis model for prehospital triage of patients with acute stroke. Stroke, 50, 970–977.

Yang, W., Su, Q., Huang, S. H., Wang, Q., Zhu, Y., & Zhou, M. (2019). Simulation modeling and optimization for ambulance allocation considering spatiotemporal stochastic demand. Journal of Management Science and Engineering, 4, 252–265.

Zaffar, M. A., Rajagopalan, H. K., Saydam, C., Mayorga, M. E., & Sharer, E. (2016). Coverage, survivability or response time: A comparative study of performance statistics used in ambulance location models via simulation–optimization. Operations Research for Health Care, 11, 1–12.

Zografos, K. G., Madas, M. A., & Androutsopoulos, K. N. (2017). Increasing airport capacity utilisation through optimum slot scheduling: review of current developments and identification of future needs. Journal of Scheduling, 20, 3–24.