Supplemental Document to the Paper A Game-Theoretic Approach for Regulating Hazmat Transportation

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1 manuscript doi: /trsc (supplemental document) Supplemental Document to the Paper A Game-Theoretic Approach for Regulating Hazmat Transportation Lucio Bianco, Massimiliano Caramia, Stefano Giordani Dipartimento di Ingegneria dell Impresa, Università di Roma Tor Vergata, Via del Politecnico, Roma, Italy, bianco@diiuniroma2it, caramia@diiuniroma2it, stefanogiordani@uniroma2it Veronica Piccialli Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma Tor Vergata, Via del Politecnico, Roma, Italy, veronicapiccialli@uniroma2it This supplemental document contains two appendices to the main paper (available at The first appendix reports the proofs of two propositions in Section 4 of the main paper The second contains the complete list of the performance results obtained on test instance I 1 Appendix A: In this appendix we report the proofs of Propositions 1 and 2 in Section 4 of the main paper In order to facilitate the reading we rewrite next the Nash Equilibrium Problem (NEP) among the carriers k K, that appears in the main paper as NEP (5): min θ k (x;t,d) x s ij,s S(k),(i,j) A st x s ij for all k K (i,j) A + (i) x s ij 0, (j,i) A (i) x s ji =es i, i N, s S(k) (i,j) A, s S(k) (A1) 1

2 Bianco et al: Supplemental Document to the Paper A Game-Theoretic Approach for Regulating Hazmat Transportation 2 Article submitted to Transportation Science; manuscript no doi: /trsc (supplemental document) Proposition 1 (Section 4, main paper) The Nash equilibrium problem (A1) is equivalent to the Variational Inequality VI(D,F) 1, with D= p D k, k=1 x 1θ 1 (x;t,d) x 2θ 2 (x;t,d) F(x)= x pθ p (x;t,d) Furthermore, the Nash equilibrium problem (A1) admits at least one solution (A2) Proof: The objective function θ k (x;t,d) of player (carrier) k is a continuously differentiable quadratic function of x We now prove that it is convex with respect to the player s variables Indeed, the Hessian of θ k (x;t,d) with respect to x k is equal to the matrix where 2 x θ k k (x;t,d)= M sk 1 sk 1 M sk 2 sk 1 M sk 1 sk 2 M sk 1 sk ν k M sk 2 sk 2 M sk 2 sk ν k, M sk ν k s k 1 M sk ν k s k 2 M sk ν k s k ν k M ss =Diag ( 2d a (ρ a b s ) 2) m a=1, Msl =Diag ( 2d a ρ 2 ab s b l) m a=1, with s,l S(k)={s k 1,,sk ν k }, and where Diag(v) denotes the diagonal matrix having as elements on the diagonal the elements of vector v The Hessian matrix 2 x k θ k (x;t,d) can be rewritten as 2 x kθ k(x;t,d)=2 b k ( b k ) T H, (A3) where bk = b sk 1, H =Diag(d a ρ 2 a )m a=1, (A4) b sk ν k and the symbol denotes the Kronecker product 2 of two matrices By the the properties of the Kronecker product (see Horn and Johnson 1986) the Hessian matrix has eigenvalues equal to the product of the eigenvalues of H (that are equal to the non negative values d a ρ 2 a) and of the positive semidefinite rank one matrix b k ( b k ) T that are all non-negative Therefore, we have that the Hessian 2 θ x k k (x;t,d) is positive semidefinite, and hence the function θ k (x;t,d) is convex with respect x k It is well known (see, eg, Proposition 142 in Facchinei and Pang 2003) that given the convexity of θ k (x;t,d) and the convexity of the feasible sets D k, for each player (carrier) k = 1,,p, NEP (A1) is 1 A solution of the VI(D,F) is a point x D such that F(x) T (y x) 0 for all y D 2 Given a matrix A R m n and a matrix B R p q, A B is the block matrix of size mp nq, A B = A 11B A 1nB A m1b A mnb

3 Bianco et al: Supplemental Document to the Paper A Game-Theoretic Approach for Regulating Hazmat Transportation Article submitted to Transportation Science; manuscript no doi: /trsc (supplemental document) 3 equivalent to the variational inequality VI(D,F), where D is the Cartesian product of the feasible set of the players, and F is obtained by concatenating the gradients of the players objective functions with respect to their own variables In our case, the resulting D and F are given by (A2), where x kθ k (x;t,d) is a vector of mν k components whose generic element is given by θ k (x;t,d) x s a ( ) = c a b s +t a ρ a b s +d a ρ a b s ρ a b s x s + a ρ a b l x l a l S + l S(k),l =s d a ρ a b l ρ a b s x l a, (A5) with s S(k) and a A As for the second statement, since the sets D k are compact, and F(x) is a vector of continuous functions, a solution of the VI(D,F) (and, hence, an equilibrium of NEP (A1)) always exists Before giving the next proof we recall the expression of the Jacobian matrix JF(x), that is the constant (m S m S ) symmetric matrix given by H 11 H 12 H 1p H 21 H 22 H 2p JF(x)=, H p1 H p2 H pp where H kk, with k=1,,p, is the (mν k mν k ) matrix H kk = 2 x k θ k (x;t,d) (given by equation (A3)) and H kl, with k,l=1,,p and k l, is the (mν k mν l ) matrix b sk 1 b s l 1 H b s k 1 b s l ν l H H kl =H T lk = b sk ν k b sl 1 H b s k ν k b sl ν l H where H =Diag(d a ρ 2 a )m a=1 (A6) (A7) Proposition 2 (Section 4, main paper) For any value of the toll parameters t a and d a 0, with a = 1,,m: (i) the VI(D,F) is monotone; (ii) NEP (A1) is equivalent to the following convex optimization problem: min ϕ(x)=κ T x+ 1 2 xt JF(x)x st x s a x s a=e s i, i N, s S a A + (i) a A (i) x s a 0, a A, s S, (A8)

4 Bianco et al: Supplemental Document to the Paper A Game-Theoretic Approach for Regulating Hazmat Transportation 4 Article submitted to Transportation Science; manuscript no doi: /trsc (supplemental document) where JF(x) is defined by (A6), and κ= b 1 (c 1 +t 1 ρ 1 ) b 1 (c m +t m ρ m ) b S (c 1 +t 1 ρ 1 ) b S (c m +t m ρ m ) (iii) If ν k =1 for all k K and d a >0 for all a A, then there exists a unique equilibrium of NEP (A1), Proof: (i) The VI(D,F) is monotone on D if and only if the Jacobian matrix of F JF(x) is positive semidefinite for all x D Looking at expression (A6), it is obvious that it can be rewritten as: 2 b 1 ( b 1 ) T H b 1 ( b 2 ) T H b1 ( b p ) T H JF(x)= bp ( b 1 ) T H b p ( b 2 ) T H 2 b p ( b p ) T H where b k, with k=1,,p, and H are defined in (A4) Once defined the vector b1 b=, bp JF(x) can be rewritten as b1 ( b 1 ) T 0 0 JF(x)= ( B+bb T) 0 b2 ( b 2 ) T 0 H, B=, 0 0 b p ( b p ) T (A9) (A10) that is positive semidefinite again by the properties of the Kronecker product since H is positive semidefinite for all d a 0 and the matrix B+bb T is is a rank one update of the positive semidefinite matrix B Therefore, F is monotone (ii) By looking at expression (A6), it emerges that the Jacobian JF(x) is symmetric It is well known (see, eg, Theorem 131 in Facchinei and Pang 2003) that given a vector F(x) of continuously differentiable functions, if its Jacobian is symmetric, then there exists a real-valued function ϕ(x) such that ϕ(x) = F(x), and ϕ(x)= 1 0 F T (x 0 +t(x x 0 ))(x x 0 )dt In our case, by expression (A2) (and (A5)) of F(x), it follows that ϕ(x)=κ T x+ 1 2 xt JF(x)x, that is a quadratic function, with Hessian equal to JF(x) which has been proved to be positive semidefinite at point (i) Therefore, the well known minimum principle applied to problem (A8) is equivalent to solving the VI(D,F), and the thesis follows

5 Bianco et al: Supplemental Document to the Paper A Game-Theoretic Approach for Regulating Hazmat Transportation Article submitted to Transportation Science; manuscript no doi: /trsc (supplemental document) 5 (iii) The existence of an equilibrium is implied by Proposition 1 of the main paper As for the uniqueness, under the assumption ν k =1, for all k=1,,p, equation (A10) implies that 2(b 1 ) 2 b 1 b 2 b 1 b p b 2 b 1 2(b 2 ) 2 b 2 b p JF(x)= H b p b 1 b p b 2 2(b p ) 2 Therefore, this matrix can be equivalently rewritten as where the symbol denotes the Hadamard product 3 JF(x)= ( bb T +Diag(b b) ) H, Matrix H (defined in (A4)) is a diagonal matrix having eigenvalues equal to the elements of the diagonal, so that it is positive definite if we assume that all the diagonal elements of H are positive (ie, all the toll parameters d a and risk coefficient ρ a are positive) As for matrix bb T +Diag(b b), it is positive definite since it is a rank one update of the positive definite matrix Diag(b b) having as eigenvalues (b s ) 2 >0, for s S Therefore, since the Kronecker product of two positive definite matrices is positive definite, then matrix JF(x) is positive definite, implying strict monotonicity of the function F(x) that ensures uniqueness of the solution of the VI(D,F), and hence of the equilibria of the NEP (A1) Appendix B: For instance I 1, and for every set of weight parameters, Table B1 lists the results related to the optimal solution of model (1, main paper) of Marcotte et al (2009) used for the comparison, followed by the results of all the non-dominated solutions (ordered by non-decreasing carriers travel cost values) produced by our heuristic method (presented in Section 6 of the main paper) with carriers total travel cost less than four times the carriers total travel cost related to the optimal solution of problem (1, main paper) In particular, the table reports the performance ratios R tot /R min, Φ/Φ min (R min ), and C tot /C min, for each determined solution R tot Φ C tot Table B1: The results of test instance I 1 R tot R min Φ min (R min ) C min R min Φ min (R min ) C min R min Φ min (R min ) C min α = 075 2,16 1,12 1,05 α = 1 2,3 1,12 1,03 α = 05 1,94 1,12 1,07 S1 2,53 0,69 1,37 S2 2,53 0,69 1,37 S3 2,53 0,69 1,37 2,37 0,72 1,44 2,37 0,72 1,44 2,37 0,72 1,44 2,35 0,73 1,48 2,35 0,73 1,48 2,35 0,73 1,48 2,26 0,81 1,51 2,26 0,81 1,51 2,26 0,81 1,51 2,04 0,85 1,52 2,04 0,85 1,52 2,04 0,85 1,52 2,03 0,85 1,58 2,03 0,85 1,58 2,03 0,85 1,58 2,17 0,83 1,59 2,17 0,83 1,59 2,17 0,83 1,59 1,91 0,70 1,62 1,91 0,70 1,62 1,91 0,70 1,62 1,73 0,70 1,63 1,73 0,70 1,63 1,73 0,70 1,63 1,90 0,70 1,67 1,90 0,70 1,67 1,90 0,70 1,67 1,65 0,79 1,69 1,65 0,79 1,69 1,65 0,79 1,69 1,73 0,70 1,72 1,73 0,70 1,72 1,73 0,70 1,72 1,56 0,85 1,74 1,56 0,85 1,74 1,56 0,85 1,74 1,45 0,86 1,75 1,45 0,86 1,75 1,45 0,86 1,75 Φ C tot R tot Φ C tot 3 Given two matrices A and B of the same size the Hadamard product A B is the matrix having as elements the product of the elements of the two matices, ie (A B) ij =A ijb ij

6 Bianco et al: Supplemental Document to the Paper A Game-Theoretic Approach for Regulating Hazmat Transportation 6 Article submitted to Transportation Science; manuscript no doi: /trsc (supplemental document) 1,64 0,80 1,75 1,64 0,80 1,75 1,64 0,80 1,75 1,43 0,90 1,77 1,43 0,90 1,77 1,43 0,90 1,77 1,44 0,86 1,78 1,44 0,86 1,78 1,44 0,86 1,78 1,43 0,90 1,81 1,43 0,90 1,81 1,43 0,90 1,81 1,26 0,82 1,86 1,26 0,82 1,86 1,26 0,82 1,86 1,32 0,69 1,86 1,32 0,69 1,86 1,32 0,69 1,86 1,31 0,69 1,86 1,31 0,69 1,86 1,31 0,69 1,86 1,24 0,83 1,90 1,24 0,83 1,90 1,24 0,83 1,90 1,30 0,69 1,90 1,30 0,69 1,90 1,30 0,69 1,90 1,27 0,69 1,92 1,27 0,69 1,92 1,27 0,69 1,92 1,23 0,73 1,93 1,23 0,73 1,93 1,23 0,73 1,93 1,22 0,94 1,99 1,22 0,94 1,99 1,22 0,94 1,99 1,21 0,94 2,03 1,21 0,94 2,03 1,21 0,94 2,03 1,19 0,94 2,04 1,19 0,94 2,04 1,19 0,94 2,04 1,19 0,94 2,04 1,19 0,94 2,04 1,19 0,94 2,04 1,17 1,02 2,05 1,17 1,02 2,05 1,17 1,02 2,05 1,17 1,02 2,10 1,20 0,88 2,07 1,17 1,02 2,10 1,16 1,12 2,23 1,18 0,88 2,08 1,18 1,02 2,23 1,18 1,02 2,23 1,16 0,88 2,10 1,17 0,92 2,32 1,18 1,00 2,32 1,15 1,00 2,33 1,15 1,00 2,33 1,15 1,00 2,33 1,15 0,93 2,33 1,16 0,92 2,33 1,16 1,00 2,40 1,15 0,96 2,36 1,15 0,92 2,34 1,15 1,00 2,51 1,22 0,88 2,47 1,15 0,92 2,39 1,16 0,90 2,55 1,14 0,93 2,53 1,18 0,91 2,40 1,12 1,12 2,55 1,14 0,93 2,77 1,16 0,92 2,43 1,14 1,09 2,56 1,16 0,88 2,85 1,16 0,92 2,48 1,11 1,12 2,57 1,17 0,88 2,92 1,14 0,92 2,48 1,09 1,12 2,59 1,16 0,88 2,98 1,14 0,93 2,48 1,13 1,12 2,81 1,13 1,09 3,08 1,13 0,77 2,57 1,13 1,12 2,81 1,16 0,87 3,12 1,13 0,93 2,61 1,14 1,12 2,84 1,13 1,09 3,23 1,11 1,10 2,86 1,12 1,12 2,87 1,11 1,10 3,24 1,10 1,09 2,86 1,10 1,12 2,89 1,11 1,06 3,63 1,54 0,69 2,89 1,09 1,12 2,90 1,10 1,11 3,86 1,13 0,73 2,95 1,13 1,10 2,92 1,10 1,11 4,04 1,10 1,10 2,97 1,21 0,86 2,92 1,09 1,11 3,08 1,09 1,12 2,93 1,06 1,12 3,08 1,18 0,86 2,99 1,09 1,10 3,17 1,17 0,89 3,05 1,12 0,73 3,32 1,15 0,91 3,05 1,07 1,11 3,61 1,16 0,86 3,08 1,09 1,10 3,65 1,08 1,12 3,08 1,12 0,68 3,95 1,13 0,93 3,12 1,20 0,85 3,20 1,18 0,77 3,21 1,14 0,88 3,26 1,08 1,00 3,31 1,07 1,00 3,32 1,06 1,00 3,33 1,07 1,00 3,33 1,13 0,86 3,34 1,16 0,86 3,43 1,04 1,11 3,43 1,06 1,00 3,45 References Facchinei, F, J-S Pang 2003 Finite-Dimensional Variational Inequalities and Complementarity Problems Springer Verlag, New York, NY, USA Horn, Roger A and Johnson, Charles R 1986 Topics in Matrix Analysis Cambridge University Press Marcotte, P, A Mercier, G Savard, V Verter 2009 Toll Policies for Mitigating Hazardous Materials Transport Risk Transportation Science 43(2)