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January  2022, 18(1): 297-314. doi: 10.3934/jimo.2020154

## Modelling and computation of optimal multiple investment timing in multi-stage capacity expansion infrastructure projects

 1 Department of Mathematics, Tianjin University of Commerce, Tianjin 300134, China 2 Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China

* Corresponding author: Jinghuan Li

Received  January 2020 Revised  August 2020 Published  January 2022 Early access  October 2020

Fund Project: This project was supported by the Scientific Research Plan of Tianjin Municipal Education Commission (2017SK076)

So far, the optimal investment timing to maximize the total profit of multi-stage capacity expansion infrastructure projects is not clear. In the case of uncertain demands, the optimal multiple stopping time theory is adopted to model the optimal decision-making of investment timing for multi-stage expansion infrastructure projects in a finite time horizon. In this context, the first-stage of the project involves a dedicated asset investment for later expansion, and the capacity of the project at each stage is constrained, which makes the cash flow of the project exhibit the characteristic of bull call spread. The upwind finite difference method and multi-least squares Monte Carlo simulation are combined to solve the project value and determine the optimal exercise boundaries at all stages described by a sequence of demand thresholds. A multi-stage power plant project is taken as an example to validate the model. Through the example, the optimal investment strategies and the value of the multi-stage project are provided; the effects of the dedicated asset and capacity constraint are illustrated. This study novelly reveals the effect of the capacity constraints on the project value using the bull call spread theory.

Citation: Jinghuan Li, Shuhua Zhang, Yu Li. Modelling and computation of optimal multiple investment timing in multi-stage capacity expansion infrastructure projects. Journal of Industrial and Management Optimization, 2022, 18 (1) : 297-314. doi: 10.3934/jimo.2020154
##### References:
 [1] L. E. Brandao and E. Saraiva, The option value of government guarantees in infrastructure projects, Constr. Manage. Econ., 26 (2008), 1171-1180.  doi: 10.1080/01446190802428051. [2] G. Cortazar, M. Gravet and J. Urzua, The valuation of multidimensional American real options using the LSM simulation method, Comput. Oper. Res., 35 (2008), 113-129.  doi: 10.1016/j.cor.2006.02.016. [3] E. Dahlgren and T. Leung, An optimal multiple stopping approach to infrastructure investment decisions, J. Econ. Dyn. Control, 53 (2015), 251-267.  doi: 10.1016/j.jedc.2015.02.001. [4] T. Dangl, Investment and capacity choice under uncertain demand, Eur. J. Oper. Res., 117 (1999), 415-428.  doi: 10.1016/S0377-2217(98)00274-4. [5] R. De Neufville, S. Scholtes and T. Wang, Real options by spreadsheet: Parking garage case example, J. Infrastruct. Syst., 12 (2006), 107-111.  doi: 10.1061/(ASCE)1076-0342(2006)12:2(107). [6] A. K. Dixit and R. S. Pindyck, Investment under Uncertainty, Princeton University Press, Princeton, NJ, 1994.  doi: 10.1515/9781400830176. [7] P. Doan and K. Menyah, Impact of irreversibility and uncertainty on the timing of infrastructure projects, J. Constr. Eng. Manage., 139 (2013), 331-338.  doi: 10.1061/(ASCE)CO.1943-7862.0000615. [8] U. Dörr, Valuation of Swing Options and Examination of Exercise Strategies by Monte Carlo Techniques, Master Dissertation, Christ Church College, University of Oxford, 2003. [9] C. F. Fisher, J. S. Paik and W. R. Schriver, Power Plant Economy of Scale and Cost Trends-Further Analyses and Review of Empirical Studies, University of Tennessee, 1986. doi: 10.2172/5508075. [10] B. Flyvbjerg, M. Holm and S. Buhl, How common and how large are cost overruns in transport infrastructure projects?, Transp. Rev., 23 (2003), 71-88. [11] C. C. Gkochari, Optimal investment timing in the dry bulk shipping sector, Transp. Res. Part E, 79 (2015), 102-109.  doi: 10.1016/j.tre.2015.02.018. [12] K. C. Han and A. Heinemann, A bull call spread as a strategy for small investors, J. of Pers. Fin., 6 (2008), 108-127. [13] H. B. Herath and C. Park, Multi-stage capital unvestment opportunities as compound real options, Eng. Econ., 47 (2002), 1-27.  doi: 10.1080/00137910208965021. [14] Y. L. Huang and C. C. Pi, Valuation of multi-stage BOT projects involving dedicated asset investments: A sequential compound option approach, Constr. Manage. Econ., 27 (2009), 653-666.  doi: 10.1080/01446190903002789. [15] B. Klein and K. B. Leffler, The role of market forces in assuring contractual performance, J. Polit. Econ., 89 (1981), 615-641.  doi: 10.1086/260996. [16] N. A. Kr$\mathrm{\ddot{u}}$ger, To kill a real option - Incomplete contracts, real options and PPP, Transp. Res. Part A, 46 (2012), 1359-1371.  doi: 10.1016/j.tra.2012.04.009. [17] Y. Lai, Z. Li and Y. Zeng, Control variate methods and applications to Asian and basket options pricing under jump-diffusion models, IMA J. Manage. Math., 26 (2015), 11-37.  doi: 10.1093/imaman/dpt016. [18] J. Li, Y. Li and S. Zhang, Optimal expansion timing decisions in multi-stage PPP projects involving dedicated asset and government subsidies, J. Ind. Manage. Optim., 16 (2020), 2065-2086.  doi: 10.3934/jimo.2019043. [19] Y. P. Lin, Upwind finite difference schemes for linear conservation law with memory, Numer. Meth. Part. Diff. Equ., 10 (1994), 475-489.  doi: 10.1002/num.1690100406. [20] F. A. Longstaff and E. S. Schwartz, Valuing American options by simulation: A simple least squares approach, Rev. Financ. Stud., 14 (2001), 113-147.  doi: 10.1093/rfs/14.1.113. [21] E. Lukas, S. Mölls and A. Welling, Venture capital, staged financing and optimal funding policies under uncertainty, Eur. J. Oper. Res., 250 (2016), 305-313.  doi: 10.1016/j.ejor.2015.10.051. [22] J. Martins, R. C. Marques and C. O. Cruz, Maximizing the value for money of PPP arrangements through flexibility: An application to airports, J. Air Transp. Manage., 39 (2014), 72-80.  doi: 10.1016/j.jairtraman.2014.04.003. [23] M. Marzouk and M. Ali, Mitigating risks in wastewater treatment plant PPPs using minimum revenue guarantee and real options, Util. Policy, 53 (2018), 121-133.  doi: 10.1016/j.jup.2018.06.012. [24] J. Paslawski, Flexible approach for construction process management under risk and uncertaity, Proc. Eng., 208 (2017), 114-124.  doi: 10.1016/j.proeng.2017.11.028. [25] P. C. Pendharkar, Valuing interdependent multi-stage IT investments: A real options approach, Eur. J. Oper. Res., 201 (2010), 847-859.  doi: 10.1016/j.ejor.2009.03.037. [26] A. M. P. Santos, J. P. Mendes and C. Guedes Soares, A dynamic model for marginal cost pricing of port infrastructures, Marit. Policy Manage., 43 (2016), 812-829.  doi: 10.1080/03088839.2016.1152404. [27] N. Song, Y. Xie, W. Ching and et al., A real option approach for investment opportunity valuation, J. Ind. Manage. Optim., 13 (2017), 1213-1235.  doi: 10.3934/jimo.2016069. [28] Z. Tan and H. Yang, Flexible build-operate-transfer contracts for road franchising under demand uncertainty, Transp. Res. Part B, 46 (2012), 1419-1439.  doi: 10.1016/j.trb.2012.07.001. [29] S. Wang, L. S. Jennings and K. L. Teo, An upwind finite-difference method for the approximation of viscosity solutions to Hamilton-Jacobi-Bellman equations, IMA J. Math. Control Inf., 17 (2000), 167-178.  doi: 10.1093/imamci/17.2.167. [30] O. E. Williamson, The Economic Institutions of Capitalism, The Free Press, New York, 1985. [31] T. Zhao and C. L. Tseng, Valuing flexibility in infrastructure expansion, J. Infrastruct. Syst., 9 (2003), 89-97.  doi: 10.1061/(ASCE)1076-0342(2003)9:3(89).

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##### References:
 [1] L. E. Brandao and E. Saraiva, The option value of government guarantees in infrastructure projects, Constr. Manage. Econ., 26 (2008), 1171-1180.  doi: 10.1080/01446190802428051. [2] G. Cortazar, M. Gravet and J. Urzua, The valuation of multidimensional American real options using the LSM simulation method, Comput. Oper. Res., 35 (2008), 113-129.  doi: 10.1016/j.cor.2006.02.016. [3] E. Dahlgren and T. Leung, An optimal multiple stopping approach to infrastructure investment decisions, J. Econ. Dyn. Control, 53 (2015), 251-267.  doi: 10.1016/j.jedc.2015.02.001. [4] T. Dangl, Investment and capacity choice under uncertain demand, Eur. J. Oper. Res., 117 (1999), 415-428.  doi: 10.1016/S0377-2217(98)00274-4. [5] R. De Neufville, S. Scholtes and T. Wang, Real options by spreadsheet: Parking garage case example, J. Infrastruct. Syst., 12 (2006), 107-111.  doi: 10.1061/(ASCE)1076-0342(2006)12:2(107). [6] A. K. Dixit and R. S. Pindyck, Investment under Uncertainty, Princeton University Press, Princeton, NJ, 1994.  doi: 10.1515/9781400830176. [7] P. Doan and K. Menyah, Impact of irreversibility and uncertainty on the timing of infrastructure projects, J. Constr. Eng. Manage., 139 (2013), 331-338.  doi: 10.1061/(ASCE)CO.1943-7862.0000615. [8] U. Dörr, Valuation of Swing Options and Examination of Exercise Strategies by Monte Carlo Techniques, Master Dissertation, Christ Church College, University of Oxford, 2003. [9] C. F. Fisher, J. S. Paik and W. R. Schriver, Power Plant Economy of Scale and Cost Trends-Further Analyses and Review of Empirical Studies, University of Tennessee, 1986. doi: 10.2172/5508075. [10] B. Flyvbjerg, M. Holm and S. Buhl, How common and how large are cost overruns in transport infrastructure projects?, Transp. Rev., 23 (2003), 71-88. [11] C. C. Gkochari, Optimal investment timing in the dry bulk shipping sector, Transp. Res. Part E, 79 (2015), 102-109.  doi: 10.1016/j.tre.2015.02.018. [12] K. C. Han and A. Heinemann, A bull call spread as a strategy for small investors, J. of Pers. Fin., 6 (2008), 108-127. [13] H. B. Herath and C. Park, Multi-stage capital unvestment opportunities as compound real options, Eng. Econ., 47 (2002), 1-27.  doi: 10.1080/00137910208965021. [14] Y. L. Huang and C. C. Pi, Valuation of multi-stage BOT projects involving dedicated asset investments: A sequential compound option approach, Constr. Manage. Econ., 27 (2009), 653-666.  doi: 10.1080/01446190903002789. [15] B. Klein and K. B. Leffler, The role of market forces in assuring contractual performance, J. Polit. Econ., 89 (1981), 615-641.  doi: 10.1086/260996. [16] N. A. Kr$\mathrm{\ddot{u}}$ger, To kill a real option - Incomplete contracts, real options and PPP, Transp. Res. Part A, 46 (2012), 1359-1371.  doi: 10.1016/j.tra.2012.04.009. [17] Y. Lai, Z. Li and Y. Zeng, Control variate methods and applications to Asian and basket options pricing under jump-diffusion models, IMA J. Manage. Math., 26 (2015), 11-37.  doi: 10.1093/imaman/dpt016. [18] J. Li, Y. Li and S. Zhang, Optimal expansion timing decisions in multi-stage PPP projects involving dedicated asset and government subsidies, J. Ind. Manage. Optim., 16 (2020), 2065-2086.  doi: 10.3934/jimo.2019043. [19] Y. P. Lin, Upwind finite difference schemes for linear conservation law with memory, Numer. Meth. Part. Diff. Equ., 10 (1994), 475-489.  doi: 10.1002/num.1690100406. [20] F. A. Longstaff and E. S. Schwartz, Valuing American options by simulation: A simple least squares approach, Rev. Financ. Stud., 14 (2001), 113-147.  doi: 10.1093/rfs/14.1.113. [21] E. Lukas, S. Mölls and A. Welling, Venture capital, staged financing and optimal funding policies under uncertainty, Eur. J. Oper. Res., 250 (2016), 305-313.  doi: 10.1016/j.ejor.2015.10.051. [22] J. Martins, R. C. Marques and C. O. Cruz, Maximizing the value for money of PPP arrangements through flexibility: An application to airports, J. Air Transp. Manage., 39 (2014), 72-80.  doi: 10.1016/j.jairtraman.2014.04.003. [23] M. Marzouk and M. Ali, Mitigating risks in wastewater treatment plant PPPs using minimum revenue guarantee and real options, Util. Policy, 53 (2018), 121-133.  doi: 10.1016/j.jup.2018.06.012. [24] J. Paslawski, Flexible approach for construction process management under risk and uncertaity, Proc. Eng., 208 (2017), 114-124.  doi: 10.1016/j.proeng.2017.11.028. [25] P. C. Pendharkar, Valuing interdependent multi-stage IT investments: A real options approach, Eur. J. Oper. Res., 201 (2010), 847-859.  doi: 10.1016/j.ejor.2009.03.037. [26] A. M. P. Santos, J. P. Mendes and C. Guedes Soares, A dynamic model for marginal cost pricing of port infrastructures, Marit. Policy Manage., 43 (2016), 812-829.  doi: 10.1080/03088839.2016.1152404. [27] N. Song, Y. Xie, W. Ching and et al., A real option approach for investment opportunity valuation, J. Ind. Manage. Optim., 13 (2017), 1213-1235.  doi: 10.3934/jimo.2016069. [28] Z. Tan and H. Yang, Flexible build-operate-transfer contracts for road franchising under demand uncertainty, Transp. Res. Part B, 46 (2012), 1419-1439.  doi: 10.1016/j.trb.2012.07.001. [29] S. Wang, L. S. Jennings and K. L. Teo, An upwind finite-difference method for the approximation of viscosity solutions to Hamilton-Jacobi-Bellman equations, IMA J. Math. Control Inf., 17 (2000), 167-178.  doi: 10.1093/imamci/17.2.167. [30] O. E. Williamson, The Economic Institutions of Capitalism, The Free Press, New York, 1985. [31] T. Zhao and C. L. Tseng, Valuing flexibility in infrastructure expansion, J. Infrastruct. Syst., 9 (2003), 89-97.  doi: 10.1061/(ASCE)1076-0342(2003)9:3(89).
The operation load of the $i$-th stage project
The comparisons between multi-stage and single period investments
The distribution of investment times from one million sample path
The optimal exercise boundaries for the i-th investment
The influences of the dedicated asset ratio
The net revenue of the i-th stage project for different demand levels
The impacts of the demand volatility on the project value
Default parameters used in the calculations
 Parameter Symbol Value Unit Investment period $T$ 10 Year Planned investment times $N$ 3 time Construction period $\nu$ 1.5 Year Refraction time $\delta$ 2 Year Capacity of i-th stage $m_{i}$ $6\times10^{4}$ MW$\cdot$h/year Unit price $p$ $5\times10^{-4}$ million CNY/MW$\cdot$h Unit operational cost $c$ $4\times10^{-5}$ million CNY/MW$\cdot$h Construction cost parameter $\lambda$ $1\times10^{-2}$ million CNY/MW$\cdot$h Construction cost parameter $\beta$ 0.9 Drift $\alpha$ 6% Volatility rate $\sigma$ 15% Discount rate $r$ 8% Dedicated asset ratio $\eta$ 0.1
 Parameter Symbol Value Unit Investment period $T$ 10 Year Planned investment times $N$ 3 time Construction period $\nu$ 1.5 Year Refraction time $\delta$ 2 Year Capacity of i-th stage $m_{i}$ $6\times10^{4}$ MW$\cdot$h/year Unit price $p$ $5\times10^{-4}$ million CNY/MW$\cdot$h Unit operational cost $c$ $4\times10^{-5}$ million CNY/MW$\cdot$h Construction cost parameter $\lambda$ $1\times10^{-2}$ million CNY/MW$\cdot$h Construction cost parameter $\beta$ 0.9 Drift $\alpha$ 6% Volatility rate $\sigma$ 15% Discount rate $r$ 8% Dedicated asset ratio $\eta$ 0.1
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