Robust Optimization Model for a Dynamic Network Design Problem Under Demand Uncertainty Authors: Byung Do Chung, Tao Yao, Chi Xie, and Andreas Thorsen The final publication is available at Springer via htp:/dx.doi.org/10.1007/s11067-010-9147-2. Chung, Byung Do, Tao Yao, Chi Xie, and Andreas Thorsen. “Robust Optimization Model for a Dynamic Network Design Problem Under Demand Uncertainty.” Networks and Spatial Economics 11, no. 2 (September 4, 2010): 371–389. doi:10.1007/s11067-010-9147-2. Made available through Montana State University’s ScholarWorks scholarworks.montana.edu Robust Optimization Model for a Dynamic Network Design Problem Under Demand Uncertainty Byung Do Chung&Tao Yao&Chi Xie& Andreas Thorsen AbstractThis paper describesa robust optimization approach for a network design problem explicitly incorporating trafic dynamics and demand uncertainty. In particular, we consider a cel transmission model based network design problem of the linear programming type and use box uncertainty sets to characterize the demand uncertainty. The major contribution of this paper is to formulate such a robust network design problem as a tractable linear programming model and demonstrate the model robustness by comparing its solution performance with the nominal solution from the coresponding deterministic model. The results of the numerical experiments justify the modeling advantage of the robust optimization approach and provide useful managerial insights for enacting capacity expansion policies under demand uncertainty. 1 Introduction Network design consists of a broad spectrum of problems, each coresponding to diferent sets of objectives, decision variables and resource constraints, implying diferent behavioral and system assumptions, and possessing varying data require- ments and capabilities in terms of representing network supplies and demands. Network design models have been extensively used as various types of strategic, tactical and operational decision-making tools and spanned over a variety of applications in, for example, transportation, production, distribution, and communi- cation fields. In a transportation network, trafic congestion has long been a major concern of the network operator, which occurs when trafic volumes exceed the road capacity. Network design problems (NDP) for transportation networks in general aim at minimizing network trafic congestions (or minimizing some general network-wide traveler costs) through implementing an optimal capacity expansion policy in the network. An optimal capacity expansion policy, however, may not be reached without properly considering the behavioral nature of travel demands, which are inherently time-variant and uncertain. Travel demands are an aggregate result of individual travel activities, which are determined by various observed and unobserved socioeconomic factors and subject to geographical, technological and temporal constraints. The vast body of the literature has focused on static deterministic NDPs (see, for example, Magnanti and Wong1984; Minoux1989; Yang and Bel1998). A major limitation of static network design models is the inability to capture trafic dynamics, such as trafic shockwave propagation and the build-up and avoidance of queues. Dynamic models, on the other hand, alow us to model the time-dependent variation of trafic flows and travel behaviors and hence beter describe trafic evolution and interaction phenomena over the network (Peeta and Ziliaskopoulos 2001). Travel demand uncertainty is not only the underlying characteristic of travel activities but also a likely result of our inaccurate or inconsistent travel demand estimation procedures. Without explicit and rigorous recognition of uncertainty in travel demands, any transportation network development plans and policies may take on unnecessary risk and even result in misleading outcomes (Zhao and Kockelman2002). In terms of their mathematical functional forms, dynamic trafic assignment (DTA) based NDPs can be classified into two major groups: single-level models and bi-level models (see the discussion in Lin et al.2008). The focus of this paper is on an application of robust optimization (RO) for dynamic NDPs under demand uncertainty, or more succinctly, a robust dynamic NDP (RDNDP), which has a single-level strcture. The single-level structure provides an easier way to manipulate robust counterpart and make RDNDP to be computationaly tractable. The research community has observed a number of recent network design studies that explicitly incorporate demand uncertainty into NDPs with time-varying flows (see Waler and Ziliaskopoulos2001; Karoonsoontawong and Waler2007; Ukkusuri and Waler2008; Karoonsoontawong and Waler2008). The common feature of these problems is that time-varying flows are described by the cel transmission model (CTM) (Daganzo1994,1995) and the network flow patern is then characterized by CTM-based DTA methods, under either the system-optimal (Ziliaskopoulos2000) or user-optimal assignment mechanism (Ukkusuri and Waler 2008). The demand uncertainty of these problems is accommodated by a chance constraint seting, a two-stage recourse model, or a scenario-based simulation method. These techniques, however, sufer from deficiencies related to lack of data availability and problem tractability, which limit their applicability to a broad range of applications. Resulting models from these stochastic modeling methods are often computationaly intractable and require known probability distributions. We folow a similar fashion to form our RDNDP using the CTM-based system- optimal DTA model, but employ the RO approach to account for demand uncertainty. Given the fact that the CTM-based DTA model has a linear programming (LP) formulation, we use the set-based RO method (Ben-Tal and Nemirovski1998,1999,2000,2002) to form a tractable LP model for the RDNDP, which overcomes the limitations of previous stochastic optimization methods. Specificaly, in our RDNDP, no probability distribution is presumed; instead, we only need to simply specify an uncertain set, which is readily available in most applications. The solution feasibility is guaranteed by the RO method through the use of the prescribed uncertainty set and can be readily made computationaly tractable through an appropriate reformulation. We highlight the main contributions of our work at a glance below: & We develop an RO framework for the SO-DTA based RDNDP. For simplicity, we present our RO model only for single-destination, system-optimal networks. However, the basic RO counterpart formulation method can be readily transfered to the multi-destination problem case. This work adds to the body of knowledge in the dynamic network design by presenting an emerging method related to the solution robustness. & An appealing feature of our robust counterpart problem is that it stil has an LP formulation, so it is in general computationaly tractable and can be solved in polynomial time by a few wel-known solution algorithms. & Our numerical experiments demonstrate the value of RO in the context of dynamic trafic assignment and network design problems. The computational viability is ilustrated for the proposed modeling framework. The numerical analysis for the impact of the investment budget bound and the demand uncertainty level on network design solutions justifies the solution robustness. The remaining part of this paper is structured as folows. Section2provides a discussion of the relevant literature. In Section3, we generalize the formulation given by Ukkusuri and Waler (2008) as a CTM-based deterministic dynamic NDP (DDNDP). We then in Section4propose a robust counterpart formulation of the DDNDP to account for demand uncertainty, which we name the RDNDP. Computational experiments and result analyses from applying the RDNDP model to a few numerical examples are elaborated in Section5. Finaly, Section6 concludes the paper and indicates some future research directions. 2 Literature review Numerous NDPs for transportation applications have been presented in the past three decades (see Magnanti and Wong1984; Minoux1989; Yang and Bel1998). These NDPs are distinguished by a variety of problem setings and supply and demand assumptions. The literature review presented below by no means provides a comprehensive survey to general network design problems or to network design applications in the transportation field; instead, our discussion is focused on those network design models and solution methods with data uncertainty, particularly network design problems with time-varying flows. A great amount of atention has been paid to NDPs with data uncertainty in past years and various modeling techniques are used for dealing with uncertain input data and parameters. The main approaches can be classified into two groups: stochastic programming (SP) and robust optimization (RO). The SP approach requires known probability distributions of the uncertain data and includes techniques such as the Monte Carlo sampling approach and chance-constrained programming. For example, Waler and Ziliaskopoulos (2001) solved a NDP under uncertain demands where the probability distributions of demand rates are known a priori. They used a CTM- based system-optimal NDP formulation with chance constraints. Ukkusuri and Waler (2008) extended the CTM to model both the system-optimal and user-optimal NDPs and presented the formulations of a chance-constrained NDP model and a two-stage resource NDP model to account for demand uncertainty. Mulvey et al. (1995) proposed a scenario-based RO approach for general LP problems. Karoonsoontawong and Waler (2007) applied this approach to a CTM- based dynamic NDP with stochastic demands under both the system-optimal and user-optimal conditions. A similar RO model formulation approach was employed by Ukkusuri et al. (2007), in which a scenario-based robust NDP with discrete decision variables was tackled by a genetic algorithm. The limitations of scenario- based RO approach are similar to stochastic programming in that we must know the probability of each scenario in advance and it is computationaly expensive when there are a large number of scenarios. Recently, a variety of papers have used the set-based RO technique to characterize optimization models with data uncertainty. Interested readers are refered to Ben-Tal and Nemirovski (2002) and Bertsimas et al. (2007) for reviews of the set-based RO methods. For NDPs with uncertain demands, Yin and Lawpongpanich (2007) considered a static continuous equilibrium NDP under demand uncertainty. Ordonez and Zhao (2007) formulated and solved a static multicommodity NDP with demand and travel time uncertainties bounded by polyhedral sets. Mudchanatongsuk et al. (2008) extended the work by Ordonez and Zhao by considering some generalized assumptions on demand uncertainty, in which they discussed a path-constrained NDP and introduced a column generation method to solve the robust NDP with polyhedral uncertainty sets. Ban et al. (2009) considered a robust road pricing problem (which is an NDP in the broader definition) that contains multiple trafic assignment solutions. Atamturk and Zhang (2007) formulated and solved a NDP by using the two-stage RO method and taking advantage of the network structure for its solutions. To characterize their uncertainty sets, they used a budget of uncertainty which limits the number of observed demand values that can difer from nominal values. They also discussed the numerical results for a simple location-transportation problem and compared the two-stage robust approach with the single-stage robust approach as wel as two-stage scenario-based stochastic programming. There have also been approaches where the set-based RO approach is used to construct discrete network design models. For example, Lou et al. (2009) described a discrete NDP with user-equilibrium flows based on the concept of uncertainty budget and proposed a cuting-plane method for problem solutions; Lu (2007) addressed a discrete user-equilibrium NDP with polyhedral uncertainty sets using the RO approach and used an iterative solution algorithm to solve the problem. To the best of our knowledge, no work has been done in applying the set-based RO technique to investigate a NDP with dynamic flows and uncertain demands. In this paper, our efort is given to analyticaly developing and numericaly analyzing the robust counterpart model of such an NDP in the context of transportation network design. 3 Deterministic model This section presents the deterministic version of the dynamic NDP model we have discussed, or the DDNDP model in abbreviation, which provides the basic modeling platform and functional form for the RDNDP model we wil introduce in the next section. For discussion convenience, let us first present the notation used throughout these models (see Table1). Table 1 The notation Sets Description τ Set of discrete time intervals, {1,..,T} C Set of cels, {1,..,I}, including the set of sink cels (CS) and the set of source cels (CR) CE Set of cels that can be expanded,CE⊂C\CS A Adjacency matrix,A={aij}, where each (i,j) component, aij, equals 1 if celiis connected to celj, and equals 0 otherwise Parameters Description dti Demand generated in cel iat timet cti Travel cost in celiat timet Nti Capacity in celiat timet Qti Inflow/outflow capacity of celiat timet dti Ratio of the free-flow speed over the backward propagationspeed of celiat timet bxi Initial number of vehicles of celi B Total investment budget available for capacity expansion fi Conversion coeficient of investment cost of celifor a unit increase ofbi χi Increase in capacity of celifor a unit increase ofbi 8i Increase in inflow/outflow capacity of celifor a unit increase ofbi Variables Description bi Investment cost spent on celi xti Number of vehicles staying in cel iat timet yti Number of vehicles moving from cel ito celjat timet The network design problem aims at minimizing the sum of the total system travel cost and the capacity expansion cost. To penalize the unmet demand by the end of planning horizon, the travel cost in celiat timet,cti, is set as folows: cti¼ 1 i2CnCS;t6¼T M i2CnCS;t¼T ( whereMis a suficiently large, positive number. The big-Mvalue can also simply serve as a penalty cost, for example, in emergency evacuation networks, representing the potential loss of life and property caused by vehicles that do not arive at the destination by the end of the time horizon. Use of the penalty cost has the efect of minimizing the number of vehicles staying in an evacuation network. By assuming the system-optimal principle and the linear relationship between investment and capacity increase, the DDNDP model can be writen as a LP program with the notation listed in Table1: minx;y;b X t2t X i2CnCS ctixtiþ X i2CE fibi subject to xti xt1i X k2C akiyt1ki þ X j2C aijyt1ij ¼dt1i 8i2C;8t2t ð1Þ X k2C akiytki Qti8i2CnCE;8t2t ð2 1Þ X k2C akiytki Qtiþbiϕi8i2CE;8t2t ð2 2Þ X k2C akiytkiþdtixti dtiNti8i2CnCE;8t2t ð3 1Þ X k2C akiytkiþdtixti dtiðNtiþbi#iÞ8i2CE;8t2t ð3 2Þ X j2C aijytij Qti8i2CnCE;8t2t ð4 1Þ X j2C aijytij Qtiþbiϕi8i2CE;8t2t ð4 2Þ X j2C aijytij xti 08i2C;8t2t ð5Þ X i2CE bi B ð6Þ x0i¼bxi8i2C ð7Þ y0ij¼0 8ði;jÞ2C C ð8Þ xti 0 8i2C;8t2t ð9Þ ytij 0 8ði;jÞ2C C;8t2t ð10Þ bi 0 8i2CE ð11Þ The objective function includes both the travel cost and expansion cost.1The coeficientficonverts the investment cost (money measure) to the travel cost (time measure). In fact, such coeficient is the reciprocal of value of time, which can be measured by empirical methods (Wardman (1998). Note here that the expansion cost appears in the objective function and is subject to the investment budget constraint, which makes this formulation diferent from the traditional charge design problem (where the expansion cost term is only included in the objective function) and budget design problem (where the expansion cost terms only appears in the investment budget constraint). The constraint set of the DDNDP model specifies the capacity expansion limit, flow conservation and propagation relationships, initial network conditions and flow non-negativity conditions. The flow conservation constraint (i.e., Eq. (1) for celiat timetcan be generalized by setingdtito be zero in ordinary and sink cels.Constraints (2-1) and (2-2) are the bounds for the total inflow rate of non-expandable and expandable celiat timet, respectively. Similarly, the total outflow rate of celi at timetis restricted by constraints (4-1) and (4-2). Constraints (3-1) and (3-2) bounds the total inflow rate into a cel by its remaining space. Constraint (5) bounds the total outflow rate of a cel by its curent occupancy, and constraint (6) sets the upper bound on the sum of capacity investments over al cels. The remaining constraints from Eq. (7) to Eq. (11) set initial network conditions and flow non- negativity conditions. 4 Robust formulation Now we develop the robust counterpart of the DDNDP model, which incorporates the demand uncertainty into a LP program via the RO approach. In the deterministic 1Costs in general do not vary linearly with respect to the transportation facility capacity or size. Typicaly, scale economies or diseconomies exist. Abdulaal and LeBlanc (1979) discussed the cases of linear relationship, scale economies and scale diseconomies in the context of transportation network design problems. If the average investment cost per unit of capacity is declining, then scale economies exist. Empirical data are needed to establish the economies of scale for road construction. This paper assumes a linear relationship between the investment cost and the capacity, for the reasons of simplicity and the requirement of the linear model. The linear case can be regarded as an approximation to the case of scale economies in an expected capacity-increasing range. version, Eq. (1) is the only set of constraints related to the demand generation. This equality constraint can be rewriten as an inequality constraint (Waler and Ziliaskopoulos2006), xti xt1i X k2C akiyt1ki þ X j2C aijyt1ij dt1i : ð12Þ It is assumed that al possible demand instances ofdtibelong to a box uncertaintyset, Udti¼ d t i1 qti;dti1þqti h i ð13Þ wheredtiis the nominal demand level andqtiis the demand uncertainty level. Then,the robust counterpart of the Eq. (12) with demand uncertainty becomes xti xt1i X k2C akiyt1ki þ X j2C aijyt1ij dt1i ; dt1i 2Udt1i : ð14Þ This is equivalent to the folowing inequality, xti xt1i X k2C akiyt1ki þ X j2C aijyt1ij maxdt1i 2Udt1i dt1i ; ð15Þ which becomes the flow conservation constraint for the RDNDP model. The above conversion of the flow conservation constraint leads the RDNDP to be in a deterministic functional form with the maximum possible demand in the box uncertainty set. Given that other constraints can be directly transfered from the DDNDP model to the RDNDP model, the RDNDP formulation can be writen into the folowing LP form: minx;y;b X t2t X i2CnCS ctixtiþ X i2CnCS fibi subject to xti xt1i X k2C akiyt1ki þ X j2C aijyt1ij ¼dt1i ð1þqt1i Þ8i2C;8t2t ð16Þ X k2C akiytki Qti8i2CnCE;8t2t X k2C akiytki Qtiþbi8i8i2CE;8t2t X k2C akiytkiþdtixti dtiNti8i2CnCE;8t2t X k2C akiytkiþdtixti dtiðNtiþbi#iÞ8i2CE;8t2t X j2C aijytij Qti8i2CnCE;8t2t X j2C aijytij Qtiþbiϕi8i2CE;8t2t X j2C aijytij xti 08i2C;8t2t X i2CE bi B y0ij¼08ði;jÞ2C C xti 08i2C;8t2t ytij 08ði;jÞ2C C;8t2t bi 08i2CE In Eq. (16), the value ofdt1i 1þqt1i is the maximum possible demand in celiat timet–1,accordingtotheuncertaintysetUdt1i , which represents the worst-case scenario. Therefore, the optimal solution wil remain feasible for al instances of demand. In other words, we wil obtain an optimal solution with the cel capacity values that are adequate for any realized demand scenarios within the uncertainty set Udt1i .We make the folowing observation between the optimal objective value and the total budget levelBfrom the RDNDP model. The implication of this property is that the network designers should consider the budget level as large as possible even if the objective function minimize the money used for network expansion together with the travel cost. Property 1The optimal objective function value of the RDNDP monotonicaly decreases with respect to the investment budget level. ProofLet the objective function of RDNDP bez»rðBÞ, given the total budget levelB.Without loss of generality, we assume that two budget levelsB1andB2are given as B1