By Sven Erlander (auth.)
This ebook has grown out of a wish to discover the probabilities of utilizing optimizing types in transportation making plans. This procedure has been all through. types which mix descriptive and optimizing parts should not handled. The gravity version is the following studied because the technique to an optimizing version. having said that method, a lot of the fabric shoula be of basic curiosity. Algorithms are usually not mentioned. the writer has benefited from discussions with many colleagues. M. Florian recommended the time period "interacti vi ty". N. F. Stewart and P. Smeds gave many valu capable reviews on a primary draft. M. Beckmann made me imagine once again concerning the ultimate chapters. R. Grubbstrem and ok. Jornsten helped clarifYing a few issues within the related chapters. final insufficiencies are end result of the writer. Gun Mannervik typed with nice persistence. Linkoping in October 1979 Sven Erlander summary The e-book proposes prolonged use of optimizing versions in transportation plann ing. An entropy limited linear application for the journey distribution challenge is formulated and proven to have the ordinarJ doubly restricted gravity version as its answer. Entropy is right here used as a degree of interactivity, that is restricted to be at a prescribed point. during this means the adaptation found in the reference journey matrix is preserved. (The homes of entropy as a dispersion degree are almost immediately mentioned. ) The exact arithmetic of the optimum suggestions in addition to of sensitivity and duality are given.
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Extra resources for Optimal Spatial Interaction and the Gravity Model
O (l-a. ) ~O I: i*i O a. ]/(T+l) ~ /3. ~O + /3. J0 - I: i a. ~ J + = a. - I: /3. ]/(T+l). C(x*) [a. ~O + fL JO + nH(x*) - C(X*)]/(T+l). C(x*) ] (T+l) - C(x*) T =C(x*) = a. C(x*) (T+l) + /3. JO + n H(x*). e. 2) without entropy constraint, for land use and transport planning. The theorem above clearly shows that the value of the objective function at optimum is a continuous strictly increasing function of the entropy level HO as long as n > O. Intuitively it is also clear that n close to zero corresponds to the smallest value of C(x*) , where the entropy constraint is not constraining the feasible region very much.
LV .. J jJJ i~~ which is independent of X. Hence xli! J is the optimal solution (since 'l- it is a feasible solution), and for this optimal solution the entropy attains its maxiu:um (Theorem 2). Similarly, xli!. 5 ), which means that 'l- is an optimal solu- c Theorem 5. 3) such that the entropy constraint is active. Then X* is regular - There are inner points. Proof. First assume that x· is regular. • ,m; j=l, ••• ,n, by Xab' From Theorem follows that X* Xab' Take a convex combination of x* and xab with 0 < p < 1 and form * H(pX* + (l-p)Xab ) > pH(x*) + > (l-p) H (x ab ), since H is strictly concace.
Ln}. 7) However, Tl = 0 would imply that the entropy constraint is inactive and can be removed. 5). Hence xLP is an optimal solution and it follows that = H(X LP') = H(X*} HO . mln _> HO. But this contradicts the assumption HO. 7 ) holds with n > O. HO. Hence n = 0 must be excluded, < From Theorem 3 follows that H(x*} = HO. e. = HO. 3), g(x* + h} =- HO. Let the components of h be small enough and let h .. 1-J > 0 x"!. = O. 1-J if We shall later show that we can find such h. We obtain g(x* + h} = = -H(X* + hJ - L L (x~ .