Mathematical Solutions for Solving Periodic Railway Transportation

Algorithm 1

The mathematical model of the railway scheduling problem.

(1) Min∑^𝑖=𝑛_𝑖=1(𝑇_𝑖𝐴_𝑟−𝑇_𝑖𝐷₁)+∑^𝑖=𝑚_𝑗=1(𝑇_𝑗𝐴₁−𝑇_𝑗𝐷_𝑟);

Subject To

/frequency constraint   ∀𝑖=1⋯𝑛,∀𝑘=1⋯𝑟

(2) 𝑇_𝑖+1𝐷_𝑘−𝑇_𝑖𝐷_𝑘=Freq;

/Time constrains ∀𝑖=1⋯𝑛,∀𝑘=1⋯𝑟

(3.1) 𝑇_𝑖𝐴_𝑘+1−𝑇_𝑖𝐷_𝑘=Time𝑖_{𝑘−(𝑘+1)};

(3.2) 𝑇_𝑗𝐴_𝑘−𝑇_𝑗𝐷_𝑘+1=Time𝑗_{𝑘−(𝑘+1)};

/Stations time constrains   ∀𝑖=1⋯𝑛,∀𝑘=1⋯𝑟

(4) 𝑇_𝑖𝐷_𝑘−𝑇_𝑖𝐴_𝑘−𝑇𝑆𝑖_𝑘=𝐶𝑆𝑖_𝑘;

/Constrains to limit journey time   ∀𝑖=1⋯𝑛,∀𝑗=1⋯𝑚

(5.1) 𝑇_𝑖𝐴_𝑟−𝑇_𝑖𝐷₁≤(1+𝛿/100)∗Time𝑖_1−𝑟;

(5.2) 𝑇_𝑗𝐴₁−𝑇_𝑗𝐷_𝑟≤(1+𝛿/100)∗Time𝑗_𝑟−1;

/Crossing constrains   ∀𝑖=1⋯𝑛,∀𝑗=1⋯𝑚,∀𝑘=1⋯𝑟

(6.1) 𝑇_𝑗𝐴_𝑘−𝑇_𝑖𝐷_𝑘<=𝑈𝐵∗𝑌_{𝑖−𝑗;𝑘−(𝑘+1)};

(6.2) 𝑇_𝑖𝐴_𝑘+1−𝑇_𝑗𝐷_𝑘+1<=𝑈𝐵∗(1−𝑌_{𝑖−𝑗;𝑘−(𝑘+1)});

/Expedition time constrains   ∀𝑖=1⋯𝑛,∀𝑗=1⋯𝑚,∀𝑘=1⋯𝑟

(7.1) 𝑇_𝑗𝐴_𝑘−𝑇_𝑖𝐷_𝑘−𝑈𝐵∗(𝑋_𝑖−𝑗−𝑌_{𝑖−𝑗;𝑘−(𝑘+1)}+𝑌_{𝑖−𝑗;(𝑘+1)−(𝑘+2)}−1)+𝐸𝑇_𝑖<=0;

(7.2) 𝑇_𝑖𝐴_𝑘−𝑇_𝑗𝐷_𝑘−𝑈𝐵∗(𝑋_𝑖−𝑗−𝑌_{𝑖−𝑗;𝑘−(𝑘+1)}+𝑌_{𝑖−𝑗;(𝑘+1)−(𝑘+2)}−2)+𝐸𝑇_𝑗<=0;

/Reception time constrains   ∀𝑖=1⋯𝑛,∀𝑗=1⋯𝑚,∀𝑘=1⋯𝑟

(8.1) 𝑇_𝑖𝐴_𝑘−𝑇_𝑗𝐴_𝑘−𝑈𝐵∗(𝑋_𝑖−𝑗−𝑌_{𝑖−𝑗;𝑘−(𝑘+1)}+𝑌_{𝑖−𝑗;(𝑘+1)−(𝑘+2)}−1)+𝑅𝑇_𝑖<=0;

(8.2) 𝑇_𝑗𝐴_𝑘−𝑇_𝑖𝐴_𝑘−𝑈𝐵∗(𝑋_𝑖−𝑗−𝑌_{𝑖−𝑗;𝑘−(𝑘+1)}+𝑌_{𝑖−𝑗;(𝑘+1)−(𝑘+2)}−2)+𝑅𝑇_𝑗<=0;

/Binary constraints

𝑋_𝑖−𝑗;∀𝑖=1⋯𝑛,∀𝑗=1⋯𝑚

𝑌_{𝑖−𝑗;𝑘−(𝑘+1)};∀𝑖=1⋯𝑛,∀𝑗=1⋯𝑚,∀𝑘=1⋯𝑟