×

A generalized two-agent location problem: Asymmetric dynamics and coordination. (English) Zbl 1213.91077

Summary: We generalize a static two-agent location problem into dynamic, asymmetric settings. The dynamics is due to the ability of the agents to move at limited speeds. Since each agent has its own objective (demand) function and these functions are interdependent, decisions made by each agent may affect the performance of the other agent and thus affect the overall performance of the system. We show that, under a broad range of system’s parameters, centralized (system-wide optimal) and non-cooperative (Nash) behavior of the agents are characterized by a similar structure. The timing of these trajectories and the intermediate speeds are however different. Moreover, non-cooperative agents travel more and may never rest and thus the system performance deteriorates under decentralized decision-making. We show that a static linear reward approach, recently developed in [B. Golany and U. G. Rothblum, Nav. Res. Logist. 53, No. 1, 1–15 (2006; Zbl 1112.90040)], can be generalized to provide coordination of the moving agents and suggest its dynamic modification. When the reward scheme is applied, the agents are induced to choose the system-wide optimal solution, even though they operate in a decentralized decision-making mode.

MSC:

91B26 Auctions, bargaining, bidding and selling, and other market models
91A23 Differential games (aspects of game theory)
91A10 Noncooperative games

Citations:

Zbl 1112.90040
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Cachon, G.P.: Competitive supply chain inventory management. In: Tayur, S., Ganesham, R., Magazine, M. (eds.) Quantitative Models for Supply Chain Management. Kluwer International, Dordrecht (1999) · Zbl 0978.90001
[2] Cachon, G.P.: Supply chain coordination with contracts. In: Graves, S., de Kok, T. (eds.) The Handbook of Operations Research and Management Science: Supply Chain Management. Kluwer Academic, Dordrecht (2003)
[3] Taylor, T.A.: Supply chain coordination under channel rebates with sales effort effects. Manag. Sci. 48(8), 992–1007 (2002) · Zbl 1232.90198 · doi:10.1287/mnsc.48.8.992.168
[4] Naor, P.: The regulation of queue size by levying tolls. Econometrica 37, 15–24 (1969) · Zbl 0172.21801 · doi:10.2307/1909200
[5] Dolan, R.J.: Incentives and mechanisms for priority queuing problems. Bell J. Econ. 9, 421–436 (1978) · doi:10.2307/3003591
[6] Mendelson, H., Whang, S.: Optimal incentive-compatible priority pricing for the M/M/1 queue. Oper. Res. 38(5), 870–883 (1990) · Zbl 0723.90023 · doi:10.1287/opre.38.5.870
[7] Avi-Itzhak, B., Golany, B., Rothblum, U.G.: Strategic equilibrium vs. global optimum for a pair of competing servers. J. Appl. Probab. 43, 1165–1172 (2006) · Zbl 1169.90333 · doi:10.1239/jap/1165505215
[8] Beckmann, M., McGuire, C.B., Winsten, C.B.: Studies in the Economics of Transportation. Yale University Press, New Haven (1956)
[9] Cole, R., Dodis, Y., Roughgarden, T.: Pricing network edges for heterogeneous selfish users. In: Proceedings of the 35th Annual ACM Symposium on Theory Computing (STOC), pp. 521–530 (2003) · Zbl 1192.68032
[10] Golany, B., Rothblum, U.G.: Inducing coordination in supply chains through linear reward schemes. Nav. Res. Logist. 53(1), 1–15 (2006) · Zbl 1112.90040 · doi:10.1002/nav.20117
[11] Ferentinos, K.P., Arvanitis, K.G., Sigrimis, N.: Heuristic optimisation methods for motion planning of autonomous agricultural vehicles. J. Glob. Optim. 23(2), 155–170 (2002) · Zbl 1175.90095 · doi:10.1023/A:1015527207828
[12] Shima, T., Rasmussen, S.J., Sparks, A.G., Passino, K.M.: Multiple task assignments for cooperating uninhabited aerial vehicles using genetic algorithms. Comput. Oper. Res. 33, 3252–3269 (2006) · Zbl 1113.90088 · doi:10.1016/j.cor.2005.02.039
[13] Hotelling, H.: Stability in competition. Econ. J. 39(153), 41–57 (1929) · doi:10.2307/2224214
[14] Munson, C.L., Hu, J., Rosenblatt, M.J.: Teaching the costs of uncoordinated supply chains. Interfaces 33(3), 24–39 (2003) · doi:10.1287/inte.33.3.24.16009
[15] Başar, T., Olsder, G.L.: Dynamic Noncooperative Game Theory. Academic Press, London (1982) · Zbl 0479.90085
[16] Feichtinger, G., Jørgensen, S.: Differential game models in management science. Eur. J. Oper. Res. 14, 137–155 (1983) · Zbl 0519.90103 · doi:10.1016/0377-2217(83)90308-9
[17] Kogan, K., Tapiero, C.S.: Supply Chain Games: Operations Management and Risk Valuation. Springer, Boston (2007) · Zbl 1156.91015
[18] He, X., Prasad, A., Sethi, S.P., Gutierrez, G.J.: A survey of Stackelberg differential game models in supply and marketing channels. J. Syst. Sci. Syst. Eng. 16(4), 385–413 (2007) · doi:10.1007/s11518-007-5058-2
[19] Bess, R.: New Zealand seafood firm competitiveness in export markets: the role of the quota management system and aquaculture legislation. Mar. Policy 30(4), 367–378 (2006) · doi:10.1016/j.marpol.2005.06.011
[20] Carafano, J.J., Walsh, B.W., Muhlhausen, D.B., Keith, L.P., Gentilli, D.D.: Better, faster and cheaper border security, a policy paper. The Heritage Foundation (2006)
[21] Wheatley, E., Doty, R.: Outsourcing the Dirty Work: The Use of Private Security Firms at the Mexico/US Border. Paper presented at the annual meeting of the ISA’s 49th Annual Convention, Bridging Multiple Divides, San Francisco (2008)
[22] Cooper, L.: Solutions of generalized locational equilibrium models. J. Reg. Sci. 7(1), 1–18 (1967) · doi:10.1111/j.1467-9787.1967.tb01419.x
[23] Tapiero, C.S.: Transportation-location-allocation problems over time. J. Reg. Sci. 14(3), 377–384 (1971) · doi:10.1111/j.1467-9787.1971.tb00268.x
[24] Tapiero, C.S., Soliman, M.A.: Multi-commodity transportation problems over time. Networks 2, 311–327 (1972) · Zbl 0257.90009 · doi:10.1002/net.3230020405
[25] Cavalier, T.M., Sherali, H.D.: Sequential location-allocation problems on chains and trees with probabilistic link demands. Math. Program. 32(3), 249–277 (1985) · Zbl 0569.90019 · doi:10.1007/BF01582049
[26] Rothblum, U.G.: Optimality vs. equilibrium: inducing coordination by linear rewards and penalties. Unpublished manuscript (2005)
[27] Sethi, S.P., Thompson, G.L.: Optimal Control Theory: Applications to Management Science and Economics, 2nd edn. Kluwer Academic, Dordrecht (2000) · Zbl 0998.49002
[28] Sydsaeter, K.: Topics in Mathematical Analysis for Economists. Academic Press, Dordrecht (1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.