Fairness considerations in network flow problems

2015 IEEE 54th Annual Conference on Decision and Control (CDC) December 15-18, 2015. Osaka, Japan Fairness Considerations in Network Flow Problems Ermin Wei∗, Chaithanya Bandi† Abstract at each node correspond to how much each agent should pay/receive for one unit of flow. Due to the cost structure In most of the physical networks, such as power, wa- of the arcs, at the optimal solution, a very wide range of ter and transportation systems, there is a system-wide objective function, typically social welfare, and an underlying physics constraint governing the flow in the networks. The standard economics and optimization theories suggest that at optimal operating point, the price in the system should correspond to the optimal dual variables associated with those physical constraint. While this set of prices can achieve the best social welfare, they may feature significant differences even for neighboring agents in the system. This work addresses fairness considerations in network flow problems, where we not only care about the standard social welfare maximization , but also distribution of prices. We first interpret the network flow problem as an economic market problem. We then show that by tuning a design parameter, we can achieve a spectrum of price-fairness, where the gap between prices satisfy certain design objective. We derive the required physical means to implement the fairness adjustment and show that the adjusted optimal solution depends on the original network topology. optimal prices may arise, resulting in drastically differences from suppliers and consumers as well as among the suppliers and consumers. The important aspect of fairness in the system is often neglected in the existing studies. We first present some background on the studies of fairness and then highlight some of our contributions in this work. 1.1. Literature Review The question of fair allocation of resources has been extensively studied through the years in many areas, notably in social sciences, welfare economics, and engineering. A variety of ways to measure what is fair have been proposed, and no single principle has come out to be universally accepted. However, there are general theories of justice and equity on which most fairness schemes have been based. In this section, we briefly review the most important theories and define special notions of fairness called proportional and max-min fairness, the two criteria that emerge from these foundational theories, which are also widely used in practice. For more details, see [10] and [5]. 1. Introduction Among the most prominent theories of justice is Aristotle’s equity principle, according to which resources should Network flow problem naturally emerges from many important applications, such as communication networks, be allocated in proportion to some preexisting claims, or transportation networks and energy supply networks [7], rights to the resources that each player has. Another theory, [11], [4], [6], [1]. This problem of optimally routing and widely considered in economics in the 19th century, is clas- distribution the flow to minimize the transportation cost has sical utilitarianism, which dictates an allocation of resources been well studied. In this work, we look at the economical that maximizes the sum of utilities. A third approach is due interpretation of this problem, where prices associated with to [9], where the key idea is to give priority to the players the flows arise from the optimal solution as a dual variable that are the least well off, so as to guarantee the highest min- associated with the flow constraints. In a competitive mar- imum utility level that every player derives. Finally, Nash ket implementation of the network flow problem, the prices introduced the Nash standard of comparison, which is the ∗ Department of Electrical Engineering and Computer Science, Northwestern University † Managerial Economics and Decision Sciences, Northwestern University percentage change in a player’s utility when he receives a 978-1-4799-7886-1/15/$31.00 ©2015 IEEE small additional amount of the resources. A transfer of resources between two players is then justified if the gainer’s 6909 Basic Notation: utility increases by a larger percentage than the loser’s utility decreases. In addition to using theories of justice, there A vector is viewed as a column vector. For a matrix A, has been literature studying the axiomatic foundations of the ifies properties that a fairness scheme should ideally satisfy. we write [A]i to denote the ith column of matrix A, and [A] j to denote the jth row of matrix A. For a vector x, xi denotes the ith component of the vector. We denote by I(n) the iden- The main work in this area is within the literature of fair bargains in economics (see [14] and references therein). tity matrix of dimension n by n. We use x′ and A′ to denote the transpose of a vector x and a matrix A respectively. We In this work, we emphasize the fairness in terms of unit use standard Euclidean norm (i.e., 2-norm) unless otherwise
1 noted, i.e., for a vector x in Rn , ||x|| = ∑ni=1 xi2 2 . The no- concept of fairness. In particular, this set of literature spec- price equality in the framework of network flow problem with cost minimization objective. This work is closely related to [8] and [13]. In [8], the authors proposed axiomatic approach to qualify various fairness metrics in the network resource allocation problem, where the fairness is measured in terms of each user’s utility, whereas in our work, we focus on a per-unit cost fairness, which we believe is also a natural consideration. The work in [13] also aims at controlling price differences in a system, but the goal there is to control inter-temporal fluctuations. rected arcs. We use the notation of an ordered pair (i, j) to denote the arc going from node i to node j. Each arc is as- 1.2. Contributions sociated with a positive cost of c̃i j , which represents the cost of transporting one unit of flow over the arc. Each node i is tation x ≥ 0 is used to denote element-wise nonnegativity. 2. Model of Network Flow Problem We consider an incapacitated linear network flow problem over an underlying directed graph denote by G̃ = (Ñ, Ã). Set Ñ is the set of nodes, {1, 2 . . . , ñ} and à is the set of di- associated with a scalar b̃i indicating the external flow coming into node i, where a negative value of bi is used when Our main contributions are twofold in this paper. We first show that network flow problems can be interpreted as there is outgoing flow from node i. The set of nodes with an economic market interaction, with producers and consumers interacting. Based on this observation, we then de- bi > 0 are called sources, denoted by S+ , while the set with bi < 0 are called sinks, denoted by S− . The global objective rive the market clearing prices, and introduce a systematic here is to route the incoming flow at the sources to the set framework to address the general issue of price fairness by of sinks with minimal cost. We can write the problem in the giving the designer an option to balance welfare and fair- following mathematical form, where we introduce the deci- ness. In particular, we refer to fairness in the sense that the sion variable x̃ = [x̃i j ]i j to denote the flow across arc (i, j), prices across from consumers and suppliers should not feamin ture big differences. In this sense, no one is being treated drastically unequally from the counterparts, and the suppliers are not making a huge profit over the consumers. Motivated by the max-min fairness, we have also done a case study where the max-min price difference is considered by the planner and given closed form solution to a physical implementation. x̃≥0 s.t. ∑ (1) c̃i j x̃i j (i, j)∈à b̃i + ∑ k,(k,i)∈à ∑ x̃ki − x̃i j = 0, for all i ∈ Ñ. j,(i, j)∈à We adopt the following two standard assumptions: Assumption 1. Total incoming and outgoing external flows sum to zero, i.e., The rest of the paper is organized as follows, Section 2 contains the description of the original problem formulation. In Section 3 we derive a simpler equivalent representations of the problem formulation and show that it has an market interaction interpretation, based on which we derive the fairness adjusted formulation and analyze it in Section 4. Section 4 also contains our case study of max-min price fairness. Section 5 contains our concluding remarks. ñ ∑ b̃i = 0. i=1 Assumption 2. The graph is connected. The first assumption is necessary for the feasibility of the system. The second assumption is natural also, since otherwise the problem can be broken down to multiple smaller problems to solve. 6910 Lemma 3.1. Problems (1) and (3) are equivalent, i.e., the optimal objective functions values are equal. 3. Equivalent Transformations In this section, we introduce some equivalent transfor- The proof is based on contradiction and is omitted here due to space constraint. mations of the original problem. The first one in Section 3.1, reduces the complicated network G̃ to a much simpler Remarks: Observe that due to the linearity nature of the problem, we can also deduce that any optimal solution to form, where only sources and sinks are present. This simplification will enable us to focus our development of fairness on a simpler form of the problem. Section 3.2 contains the economic interpretation of the network flow problem, which will be the basis for the fairness considerations in the next section. problem (3) is a tree solution, i.e., at most only n − 1 arcs will have positive flows. See [3] for details. 3.2. Welfare Equivalence In this section, inspired by the concept of shadow-price, 3.1. Transportation Reformulation we use the primal-dual frame work to show that the network flow problem can be equivalently viewed as a social welfare maximization problem with consumer-supplier interactions. This framework will be used as the foundation to develop fairness considerations in the next section. In this section, we show that the problem in form (1) can be equivalently formulated in a transportation problem format, where all nodes with bi = 0 are removed from the network. We construct a new graph G = (N, A). The nodes We first consider the following consumer-supplier interaction, where each source node can be viewed as a supplier and each sink node can be viewed as a consumer. On the supply side, each supplier (or producer) i in S+ receives a price of pi for one unit of flow transmitted over and has a supply of bi units needs to be transmitted. The supply side in set N = S+ S− are indexed {1, . . . n}, which includes all the sources and sinks from the original graph G̃, each assoS ciated with an external flow bi = b̃i . For each pair of source i and sink j, an arc is present and the cost associated with arc ci j is the minimal cost associated with any path from i to j in the original graph. Formally, we denote the set of paths from i to j by Πi j , where each element is represented by a problem can be written as follows, for each i ∈ S+ , the decision variable is yi j for all j, indicating the amount transmitted from supply i to consumer j. set of ordered arcs {ail , . . . ak j } ∈ Πi j starting from node i to j. Then the cost ci j is defined by max ci j = min ∑ {ail ,...ak j }∈Πi j a∈{a ,...a } il kj c̃a . yi j ≥0 (2) pi ∑ yi j (4) j∈S− ∑ s.t. yi j = b i , j∈S− If the set Πi j is empty in the original graph, then ci j = ∞. The new problem can be represented as follows, where For each consumer (or demand) j in S− , the decision the decision variable x = [xi j ]i j for i in S+ and j in S− repre- variables are di j , which represents the amount of flow con- sents the flow from source i to sink j. sumer j is buying from supplier i. The consumer pays a cost of p j for consuming each unit of flow. Hence the consumer ∑ min x s.t. (3) ci j xi j side problem can be written as i∈S+ , j∈S− ∑ xi j = b i , for all i ∈ S+ max j∈S− − ∑ xi j = b j , di j ≥0 for all j ∈ S− s.t. i∈S+ − pj ∑ di j (5) i∈S+ − ∑ di j = b j , i∈S+ x ≥ 0. The market reaches equilibrium when both producers and consumers locally solve their problems (4-5) and the market clears, i.e., yi j = di j . We note that due to the lin- The next lemma shows that this transformation is equivalent to the original problem and we will be working with problem (3) for the rest of the paper, due to its simple form. ear constraints, the objective value of problems (4-5) are uniquely determined. They are, nevertheless, of interest due 6911 to their interpretation associated with the supply and demand 4. Fairness Adjusted Formulation and Analysis interaction as demonstrated below. We now show that the market interaction is an equivalent way to implement the original network flow problem. This section is built upon the previous discussion where We refer to the sum of the objective functions in problems prices for accessing the market naturally arises as the dual (4) and (5), i.e.,∑i∈S+ (pi ∑ j∈S− yi j )− ∑ j∈S− (p j ∑i∈S+ di j ), as the social welfare. variable to problem (3). Since these prices are the optimal dual variables associated with an equality constraint, they are unconstrained and a price where one consumer pays can Theorem 3.1. The optimal primal-dual solution pair to differently from another. Similarly, the producers can face problem 3 and the market equilibrium of problems (under significantly gap in prices and the prices between producer market clearing condition) (4) and (5) are equivalent with and consumer can also be very different, based on the dis- pi − p j = ci j for xi j > 0. tribution of ci j . While the optimal solution for problem (3) does solve the social welfare problem optimally (based on Proof. Due to the equality constraints in problems (4) problems (4), (5)), it ignores an important issue in the soci- and (5), for each price vector p, the objective function value is fixed. The market equilibrium emerges ety, fairness. In the long run, this difference in prices may when we have market clearing, i.e., di j = yi j . We note sink paths or even relocation of producers and consumers. that the optimal dual variable q associated with equality constraints of problem (3) solves the following problem, However, in the short run, this can be perceived as unfair- minx≥0 ∑i∈S+ , j∈S− ci j xi j − ∑i∈S+ qi (∑ j∈S− xi j − bi ) − systematic way of reducing the price spread in the system maxq give economic incentives to lower the cost of certain source- ness in the system. Hence, our goal here is to introduce a ∑ j∈S− q j (− ∑i∈S+ xi j − b j ). and derive a physical implementation. Section 4.1 derives By rearranging the terms, we have that the optimal primal-dual pair solves maxq ∑i∈S+ qi bi + ∑i∈S− q j b j + the problem formulation when we add a general form of price fairness consideration into the previous problem. Sec- minx≥0 ∑i∈S+ , j∈S− xi j (ci j − qi + q j ). From LP duality, we have that the above expression h implies that ci j −qi +q j = 0 tion 4.2 is our case study, where we analyze the case when we penalize the maximum difference in price inequalities. for any xi j > 0. Hence, if we assign pi = qi and p j = q j to 4.1. Price Fairness Aware Formulation problems (4) and (5), we have that at market equilibrium, the social welfare is equivalent to the optimal objective function We will work with the original problem (3) and based value of problem (3), also they share the same solution. on the equivalence established in the previous section, the From complementary slackness for problem (3), we also have that ci j ≥ pi − p j , for all pairs of i, j. result here will shed light on how to implement the fairnessaware system in the market environment. We first introduce some short hand notations. We use vectors x = [xi j ]i j and We note that the equality constraint in the supplier and c = [ci j ]i j to represent all the flows and cost in the system. consumer problems can be applicable in many real time transportation systems, where the supply and demand has We also combine the two equality constraint into a compact matrix form of Bx = b, with B in R|N|×|A| composed of ele- to be met strictly. Examples include taxi dispatch system, ments 0, 1, −1. Hence, problem (3) can be compactly writ- water, oil and natural gas flow systems. ten as minx≥0,Bx=b c′ x. During the preceding analysis, we interpreted the dual The price vector p now is associated with the constraint variable associated with the equality constraints in prob- Bx = b. In order to study fairness in the system, we need to lem (3) as the price and derived the producer price to sell rewrite the price vector p. We focus on the case where an optimal primal-dual solution (x, p) to the original problem and consumer cost for using based on these dual variables. Based on these equivalent transformations, we have that the ing market dynamics with appropriate prices for accessing (3) is known and study the effect of fairness in price consideration on the problem. Without loss of generality, we order the nodes by decreasing order of optimal dual variables pi . the market. We let pn be the price associated with the last node n. We network flow problem can be equivalently implemented us- 6912 denote by e the vector of all 1 in R|N| and by E the matrix in R|N|×(|N|−1) mizer. Therefore problem (7) is equivalent to with  1 1 ...  0 1   E =   0 0 ... .. . ... 0 0 ... max ∆p  1  1   , 1   1 0 min c′ x + ||∆p||w ( E ′ (Bx − b) x≥0 q − S). Since ∆p is unconstrained, the scalar ||∆p|| can take any (6) nonnegative values, which yields another equivalent formulation of min c′ x + α( E ′ (Bx − b) max α ≥0 such that p = epn + E∆p. x≥0 q − S). If ||E ′ (Bx − b)||q − S ≥ 0, then the problem attains value of To introduce a fairness consideration in the system, we follow the same approach as in our previous works [13] and infinity and thus cannot be optimal, and therefore the pre- [12]. We introduce an additional term for fairness in addition ceding formulation is equivalent to (8). to the standard welfare consideration and show how this can be implemented. The preceding theorem shows that adding a term to control price spread is equivalent to relax the constraint of Theorem 4.1. The following two problems are equivalent Bx = b, where the design parameter S is used to control how min c′ x − (epn + E∆p)′ (Bx − b) − S ||∆p||w . max x≥0 pT ,∆p (7) much emphasis the fairness consideration term obtains compared with the original social welfare. A value of S = 0 recovers the original problem, and S = ∞ puts entire weight min c′ x, (8) x≥0 s.t. E ′ (Bx − b) q ≤ S, on the fairness and ignores the welfare. The equality constraint can be viewed as strictly satisfying supply and de- e′ (Bx − b) = 0, mand, while the relaxed version can be viewed as if there where the two norms ||·||w and ||·||q are dual operators and S is a nonnegative scalar. is storage (with nonempty initial level) to compensate for Proof. We first rearrange the terms in (7) as entire system, total incoming and outgoing flows are still max pT ,∆p the supply-demand (incoming-outgoing) mismatch at each node. The constraint e′ (Bx − b) = 0 implies that across the balanced. min c′ x − pn e′ (Bx − b) − ∆p′ E ′ (Bx − x) − S ||∆p||w . x≥0 4.2. Minimize Maximum Price Inequality By Saddle Point Theorem (Proposition 2.6.4 in [2]) and linearity of the problem, we can exchange the minimization To study this fairness adjustment and the physical im- and maximization, to express the problem equivalently as min x≥0 plementation thereof further, for the rest of the paper, we focus on the scenario where w-norm is the L1 and q-norm is max c′ x − pn e′ (Bx − b) − ∆p′ E ′ (Bx − x) − S ||∆p||w . its dual norm, L∞ norm. This is a natural choice, since the pT ,∆p nodes are ordered in the decreasing order of prices and L1 norm corresponds to For any solution pair with e′ (Bx − b) 6= 0, there exists pn such that the value of the previous problem attains infinity and thus cannot be the optimal solution. Therefore we con- ||∆p||1 = p1 − p2 + p2 − p3 + . . . + pn−1 − pn = p1 − pn . clude e′ (Bx − b) = 0. We next analyze the terms involving ∆p: −∆p′ E ′ (Bx − x) − S ||∆p||w . From Höler’s inequality, we have −∆p′ E ′ (Bx − b) − S ||∆p||w ≤ ||∆p||w ( E ′ (Bx − b) q A penalty on ||∆p||1 aims at reduce the difference between the maximum and the minimum prices in the system. The problem considered in this section is given by − S), min c′ x, (9) x≥0 −∆p′ E ′ (Bx = i.e., max∆p − b) − S ||∆p||w ′ ∗ ∗ ||∆p ||w (||E (Bx − b)||q − S), where ∆p is the maxi- s.t. 6913 E ′ (Bx − b) ∞ ≤ S, e′ (Bx − b) = 0, Recall definition of matrix E in Eq. (6), the ith eleE ′ (Bx − b) social welfare maximization), we also considered the issue [E ′ (Bx − b)] ment of the vector is given by i = i ∑t=1 [Bx − b]t , which represent the accumulative incoming- of fairness in the price inequality. We first presented an eco- outgoing flow mismatch starting from 1 up to i. This can be physically implemented by introduce a storage of size 2S introduced a penalty term of price inequality in the system nomic interpretation of the network flow problem and then at each node i, with level S without any knowledge of the original welfare maximization problem. We next show that wide objective and derived an equivalent physical implementation. We analyzed the case of minimizing maximum price inequality as a case study. Future work directions in- we can derive how to implement this price fairness aware clude to characterize storage distribution to general setup system with much smaller storage requirement, if we know (other variations of price inequality and network topology) the optimal primal-dual pair (x, p) to the original problem and the original problem has either only one source or one and nonlinear objectives, where the allocation of storage will depend on the underlying network topology. sink. 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