Continuous transportation as a material distribution topology optimization problem
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Abstract
The problem of moving a commodity with a given initial mass distribution to a prespecified target mass distribution so that the total work is minimized can be traced back at least to Monge’s work from 1781. Here, we consider a version of this problem aiming to minimize a combination of road construction and transportation cost by determining, at each point, the local direction of transportation. This paper covers the modeling of the problem, highlights how it can be formulated as a material distribution topology optimization problem, and shows some results.
Keywords
Topology optimization Continuous transportation Road design Largescale problems1 Introduction and background
Transportation problems have a long history in science. Already in 1781, Monge (1781) studied the problem of how to minimize the work required to move a commodity with a given initial mass distribution to a prespecified target mass distribution. Monge’s problem formulation is general in that it considers the computation of transport paths, which distinguishes it from route planning problems that are restricted to an existing network. In contrast, flow network problems are by far the most investigated domain of transportation theory, and have resulted in a row of now mature tools from linear, integer, and constraint programming. This should come as no surprise, since most transportation is undertaken on an existing infrastructure. From an economics perspective, it is on the other hand of interest to target not only transportation cost, but also the cost for road construction. Such considerations come into play also on a smaller scale, for instance in agriculture or forestry, when temporary or otherwise designated roads must be paved. For general consideration of distribution of matter, it is natural to employ a fluid dynamics formalism. An early attempt to do so was made by Beckmann (1952), who minimizes transportation cost under a conservation of matter constraint. Beckmann seems to have been unfamiliar with the Monge–Kantorovich problem at the time, but there is a close link between the formulations, noted by Igbida (2013), among others. For a survey of continuous transportation modeling (see for instance Puu 2009).
The literature on the Monge problem and its derivation is extensive (albeit small compared to network counterparts), but considers mainly the theoretical properties of the problem formulations and its solutions. Much less seems to have been written regarding solution methods applicable for practical transportation problems. The intention of this paper is to bridge this gap so that practically useful algorithms can be formulated ultimately.
Here, we aim to solve numerically a continuous transportation problem by using material distribution based topology optimization. The rationale for this is that road design is effectively a material distribution problem, and transportation is nothing but flow of matter. The material distribution method was originally introduced by Bendsøe and Kikuchi (1988), who considered optimization problems in solid mechanics. The monograph by Bendsøe and Sigmund (2003) gives a comprehensive presentation of material distribution based topology optimization techniques and highlights many applications. The review by Sigmund and Maute (2013) gives an overview of different approaches to topology optimization and compares their strengths and weaknesses. The “flagship problem” for topology optimization is to minimize the compliance of an elastic body subject to a constraint on the amount of available material. There are many wellestablished techniques to solve this problem and some of these are currently used in the design process of advanced components in the automotive and aeronautical industries. Today, largescale material distribution problems have many millions or even billions of design variables (Aage et al. 2017; Schmidt and Schulz 2011; Wadbro and Berggren 2009). For other problems, such as those with state constraints or those concerning wave propagation, the methodologies are still maturing. During the last decades, much work has been focused on developing the methodologies (Le et al. 2010) and extending the ideas to other fields, such as fluid flow (Borrvall and Petersson 2003), fluid–structure interaction (Andreasen and Sigmund 2013; Yoon 2010), and acoustic (Dühring et al. 2008; Wadbro 2014; Wadbro and Berggren 2006) as well as electromagnetic (Aage and Johansen 2017; Andkjær and Sigmund 2011; Erentok and Sigmund 2011; Nomura et al. 2007; Hassan et al. 2014) wave propagation. Although various kinds of systems including transportation processes (thermal, electric, convective, etc.) have been targeted for topology optimization in the past, the application of topology optimization to continuous logistic transportation models is, to the best of our knowledge, novel. The contribution closest in style to our work is that of Ryu et al. (2012), who used topology optimization techniques to attack a path planning problem for a robot that moves in an environment with obstacles.
2 Problem description
In this section, we present a model that accounts for the road construction cost as well as the transport cost of a single commodity in a steady state setting. Since, the model is new and the modeling includes many different symbols, we start by providing the following nomenclature list:
 x

Position (x = (x_{1},x_{2}))
 ρ

Density distribution of commodity
 q

Rate of production/consumption of the commodity
 u

Transport velocity
 v

Transport speed (u = v)
 Φ

Potential
 κ

Conductivity
 α

Road design
 s

A function specifying the relation between α and v
 \({\kern 2.3pt}\mathcal {F}\)

Set of feasible conductivities
 \({\kern 2.3pt}\mathcal {A}\)

Set of feasible road designs
 J _{ T }

Transport cost
 \(J_{T}^{\epsilon }\)

𝜖relaxed, differentiable version of transport cost
 J _{ R }

Road cost
 β

Tradeoff parameter.
2.1 Transportation of a product
On a microscopic level, transportation of goods is a discrete process where point charges (representing trucks, backhaulers etc.) move along certain paths (on roads or offroad) between sources and destinations. On a macroscopic level, and seen over long periods of time, it is relevant to model transportation as a continuous process in the same way as continuous fields are used to describe electric current or gas flow that is in reality movement of microscopic particles. This view on transportation enables the use of a mathematically tractable formalism from continuum mechanics. The model we propose is similar in spirit to that of Beckmann (1952), who also considers flow under a conservation of matter condition. While (Beckmann 1952) set the theoretical foundation for continuous transportation modeling with the aim of minimizing transportation, our aim is to consider also road construction cost, and to employ topology optimization in order to find solutions for general scenarios.
2.2 Potential approach
Remark 1
2.3 Adding road design
3 Numerical treatment
In our conceptual optimization (23), the road design α is constrained to only attain the values 0 and 1 almost everywhere. Design optimization problems with this type of binary constraints are often associated with various issues. From a mathematical viewpoint, these problems are typically illposed in that the problem lacks solutions within the set of feasible designs; and from a computational point of view the problem is a largescale nonlinear mixed integer program and hence computationally intractable. We remark that for reallife problems there are often requirements, such as a minimal width of structural members of the design, which assures that the problem has solutions within the space of admissible designs.
That is, the problem is to determine both α and κ to minimize a weighted sum of road construction cost \({J_{R}^{h}}\) and transportation cost \({J_{T}^{h}}\). In the numerical experiments, the method of moving asymptotes (MMA) (Svanberg 1987) solves problem (34). The required gradients are computed by using the expressions provided in the Appendix and the chain rule to take the filtering into account.
4 Numerical experiments

In TC1, there are three separate supply positions, located at (3/4,1/4), (3/4,2/4), and (3/4,3/4) and a single demand position located at (1/4,1/2). The supply and demand are distributed over a radius of 1/32 centered at the positions above.

In TC2, the demand is the same as in TC1, but the supply is uniformly distributed over a rectangular region whose lower left and upper right corners are located at (1/2,1/8) and (7/8,7/8), respectively.

In TC3, the supply is uniformly distributed over a rectangular region whose lower left and upper right corners are located at (3/4,1/2) and (7/8,7/8), respectively, and the demand uniformly distributed over a rectangular region whose lower left and upper right corners are located at (1/8,1/8) and (1/2,1/4).
Put simply, the initial guess for both α_{h} and κ_{h} is 1/2 over the whole unit square, except from a border of width 1/16, where it is 0. The reason that we set both these variables to 0 around the edge of the domain is that we expect that there will be no benefit from placing neither roads nor adding any conductivity to these parts. However, since we do not want to influence the design between the supply and demand positions, we let the initial road and conductivity distributions all be equal to 1/2 in a region including these locations and their nearest surrounding for all test cases. For all experiments, we set the parameters controlling the transportation speed, as given by expression (31) as: v_{min} = 1, v_{max} = 5, and p = 3. That is, the transportation speed is five times as large on roads as outside them and it is possible to transport goods even if no roads are placed in Ω. In all experiments, we set the filter radii for the road design and conductivity to τ_{R} = τ_{C} = 1/128.
Figure 6 shows results from TC1 obtained with parameter β = 0.18 (bottom row) and β = 0.68 (top row). For large values of β (β ≥ 0.87 for this experiment) in problem (34), the road construction cost dominates completely and no roads are placed inside Ω. Similarly for β = 0 (road construction cost does not affect the objective), essentially the full domain Ω is filled with roads. For small to intermediate values of β (0 < β ≤ 0.58 for this experiment), we obtain road designs with three main roads from the supply positions to the demand position, the width of these roads is larger for relatively small values of β, as illustrated in the top row of Fig. 6; the road’s width decreases as β increases until it is comparable with the filter size τ_{R} (cf. the top row in Fig. 3; the results on both resolutions share the same main characteristics). Finally, as β becomes larger, the road construction cost gets increasingly costly compared to the transportation cost and the optimized road design no longer connects the supply and demand, so an increasing share of the transportation needs to be carried out offroad. This is illustrated by the optimized road design obtained for β = 0.68, the bottom left image in Fig. 6, where the highspeed road material only covers a small area close to the demand location.
Figure 7 shows results from TC2 obtained with parameter β = 0.16 (bottom row) and β = 0.54 (top row). Just as for TC1, when β is large (β ≥ 0.87 for this experiment) in problem (34), the road construction cost dominates completely and no roads are placed inside Ω, and when β = 0 essentially the full domain Ω is filled with roads. For very small values of β the road design covers the convex envelope of the supply and demand regions, but as β increases, roads start to form. The bottom row shows the road design and mass flux for β = 0.16. In this case, the road design reminds of a root structure with wide roots near the demand positions that branches out to more and finer roots close and over the supply region. As β increases further, the overall root structure remains. However, the individual roots become thinner and the distance between subsequent branch points increases. This procedure continues until the road tree only has few main roots or roads from the demand region to the edge of the supply region. For this test case, this occurs at about β = 0.54, corresponding to the results in the top row of Fig. 7. Increasing β further results in that the roads become shorter from the end closest to the supply region.
Figure 8 shows results from TC3 obtained with parameter β = 0.24 (bottom row) and β = 0.73 (top row). Just as for TC1 and TC2, when β is large (β ≥ 0.74 for this experiment) in problem (34), the road construction cost dominates completely and no roads are placed inside Ω, and when β = 0 essentially the full domain Ω is filled with roads. Moreover, just as for TC2, for very small values of β the road design covers the convex envelope of the supply and demand regions but as β increases, roads start to form. For this experiment, the road design for small to intermediate values of parameter β (0.16 ≤ β ≤ 0.23 for this experiment), the road design does not fully cover the convex envelope of the supply and demand regions, and smaller roads start to form near the remote edges of these regions. For intermediate values of β (0.24 ≤ β ≤ 0.73 for this experiment), the road design consists of one connected component that branches out to come close to all parts of the supply and demand regions. As β increases, the width of the individual roads, as well as the number of road segments in each branch, decreases. The bottom and top row in Fig. 8 show the road design and mass flux for β = 0.24 and β = 0.73. In contrast to the two other test cases, the road design connects the supply and demand regions up until the point when β becomes large enough not to motivate the construction of any roads. This is likely due to the fact that in this case both supply and demand are distributed, so that there is no location where one can place a small piece of road that would decrease the transportation cost for most of the goods; for the optimized road designs at high road cost TC1 and TC2 all goods was moved through a small region just in front of the demand region. Perhaps one could find a particular range of values of parameter β so that the optimized road design only consists of a small part of the region between supply and demand, but the numerical experiments presented in this section suggest that if such a parameter region exist it would be very narrow.
5 Discussion
This paper models and solves continuous transportation problems as material distribution topology optimization problems. The end goal is to find the optimal placement of roads (material) in a region as well as the local transportation direction for a commodity. Here, the road layout determines the local transport speed (we have a high transport speed on the roads and a lower transport speed off the roads). In contrast to typical material distribution topology optimization approaches, the original method, proposed in this paper, uses two design fields that represent the road design and the local transportation direction, respectively.
Given a road design, the suggested potential approach yields problem (19)—a problem that is related to the minimum heat compliance problem, which is a standard test problem for material distribution topology optimization. Both problems are governed by Poisson’s equation, but with different source terms and boundary conditions. However, there are two major differences. The first is that the objective in our setting is to minimize a weighted L^{1}norm of the gradient of the solution field of the governing equation (13), while the aim in the minimum heat compliance problem is to minimize a weighted L^{2}norm of the temperature field. The second difference is that in the minimum heat compliance problem, one sets out to place two materials with conductivities \(\underline {\kappa }\) and 1 so the original problem is to determine for each point which material type it should contain; here, on the contrary, the conductivities may attain any value in the continuous range from \(\underline {\kappa }\) to 1. (Recall that for the transportation problem, the road design is determined by α—which at each point holds the value 1 or 0 to signify whether the point is occupied by a road or not, respectively—controls the transportation speed, while the conductivities determine the local transportation direction.) The focus on this paper is to simultaneously find the best road design and local transport direction to minimize a total cost including the transport cost and the road construction cost.
By using the strategy proposed in this paper, we have successfully optimized road designs for a few test cases, in which the supply and demand positions are concentrated around given points as well as distributed over given regions. By solving problem (34) for a sequence of values for parameter β, we obtain a sequence of road network designs corresponding to different tradeoffs between the road construction cost and the transportation cost. These solutions and the corresponding relative road construction and transportation cost may provide guidance when making decisions regarding if and where to construct roads to increase transportation efficiency.
An interesting tradeoff problem occurring in many practical situations is when, given a supply and demand function q as well as an existing road network covering the region Ω_{R} ⊂Ω, one faces the decision if new roads should be constructed and if so where. To solve this problem within the current framework, the only change that needs to be done is to modify the road construction cost, so that the integral in definition (22) is taken over Ω ∖Ω_{R} instead of over Ω and modify the set of admissible road designs to ensure that α ≡ 1 in Ω_{R}.
Notes
Acknowledgments
This work is financially supported by the Swedish Research Council on Environment, Agricultural Sciences and Spatial Planning (No. 942201562), and the Swedish Foundation for Strategic Research (No. AM130029). The numerical experiments were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the High Performance Computing Center North (HPC2N).
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