aircraft.gms : Aircraft Allocation Under Uncertain Demand

**Description**

The objective of this model is to allocate aircrafts to routes to maximize the expected profit when traffic demand is uncertain. Two different formulations are used, the delta and the lambda formulation.

**Reference**

- Dantzig, G B, Chapter 28. In Linear Programming and Extensions. Princeton University Press, Princeton, New Jersey, 1963.

**Small Model of Type :** LP

**Category :** GAMS Model library

**Main file :** aircraft.gms

$Title Aircraft allocation under uncertain demand (AIRCRAF,SEQ=8) $Ontext The objective of this model is to allocate aircrafts to routes to maximize the expected profit when traffic demand is uncertain. Two different formulations are used, the delta and the lambda formulation. Dantzig, G B, Chapter 28. In Linear Programming and Extensions. Princeton University Press, Princeton, New Jersey, 1963. $Offtext Set i aircraft types and unassigned passengers / a, b, c, d / j assigned and unassigned routes / route-1, route-2, route-3, route-4, route-5 / h demand states / 1*5 / Alias(h,hp); Table dd(j,h) demand distribution on route j 1 2 3 4 5 route-1 200 220 250 270 300 route-2 50 150 route-3 140 160 180 200 220 route-4 10 50 80 100 340 route-5 580 600 620 Table lambda(j,h) probability of demand state h on route j 1 2 3 4 5 route-1 .2 .05 .35 .2 .2 route-2 .3 .7 route-3 .1 .2 .4 .2 .1 route-4 .2 .2 .3 .2 .1 route-5 .1 .8 .1 Table c(i,j) costs per aircraft (1000s) route-1 route-2 route-3 route-4 route-5 a 18 21 18 16 10 b 15 16 14 9 c 10 9 6 d 17 16 17 15 10 Table p(i,j) passenger capacity of aircraft i on route j route-1 route-2 route-3 route-4 route-5 a 16 15 28 23 81 b 10 14 15 57 c 5 7 29 d 9 11 22 17 55 Parameter aa(i) aircraft availability / a 10, b 19, c 25, d 15 / k(j) revenue lost (1000 per 100 bumped) / (route-1,route-2) 13, (route-3,route-4) 7, route-5 1 / ed(j) expected demand gamma(j,h) probability of exceeding demand increment h on route j deltb(j,h) incremental passenger load in demand states; ed(j) = sum(h, lambda(j,h)*dd(j,h)); gamma(j,h) = sum(hp$(ord(hp) ge ord(h)), lambda(j,hp)); deltb(j,h) = (dd(j,h)-dd(j,h-1))$dd(j,h); Display ed, gamma, deltb; Positive Variables x(i,j) number of aircraft type i assigned to route j y(j,h) passengers actually carried b(j,h) passengers bumped oc operating cost bc bumping cost Variable phi total expected costs Equations ab(i) aircraft balance db(j) demand balance yd(j,h) definition of boarded passengers bd(j,h) definition of bumped passengers ocd operating cost definition bcd1 bumping cost definition: version 1 bcd2 bumping cost definition: version 2 obj objective function; ab(i).. sum(j, x(i,j)) =l= aa(i); db(j).. sum(i, p(i,j)*x(i,j)) =g= sum(h$deltb(j,h), y(j,h)); yd(j,h).. y(j,h) =l= sum(i, p(i,j)*x(i,j)); bd(j,h).. b(j,h) =e= dd(j,h) - y(j,h); ocd.. oc =e= sum((i,j), c(i,j)*x(i,j)); bcd1.. bc =e= sum(j, k(j)*(ed(j)-sum(h, gamma(j,h)*y(j,h)))); bcd2.. bc =e= sum((j,h), k(j)*lambda(j,h)*b(j,h)); obj.. phi =e= oc + bc ; Model alloc1 aircraft allocation version 1 / ab, db, ocd, bcd1, obj / alloc2 aircraft allocation version 2 / ab, yd, bd, ocd, bcd2, obj / ; * version 1 requires upper bounds on y y.up(j,h) = deltb(j,h) Solve alloc1 minimizing phi using lp; Display y.l; y.up(j,h) = +inf; Solve alloc2 minimizing phi using lp; Display y.l;