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IEOR 4601 Assignment 9 1. In class we considered a network problem with two resources and three itineraries with two fares per itinerary and illustrated a variety of heuristics for that problem. Here we will consider an expanded model with the same three itineraries but with ve fares per itinerary. Itinerary one consumes one unit of resource one and has ve fares: 185; 160; 150; 130; 100. Itinerary two consumes one unit of resource two and has ve fares: 130; 110; 95; 80; 75. Itinerary three consumes one unit of each resource and has ve fares: 260; 240; 220; 190; 170. Using the single index model the incidence matrix A is given by A = 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 and the fare vector by p = ( 185 160 150 130 100 130 110 95 80 75 260 240 220 190 170 ) : Assume a discrete time model with T = 1; 000 periods and time varying arrival rates given by t = ( :00 :00 :015 :036 :054 :00 :00 :015 :04 :06 :00 :00 :01 :03 :03 ) for 501 t 1000 and t = ( :02 :04 :015 :00 :00 :03 0:04 :015 :00 :00 :04 :02 :01 :00 :00 ) for 1 t 500. The initial vector of capacities is c = ( 100; 120 ) : All vectors should be interpreted as column vectors (although for convenience we write them as row vectors). a) Find the aggregate arrival rate Tj = R T 0 tjdt; j = 1; : : : ; 15 for the 15 ODFs. b) Write the corresponding Deterministic Linear Program (dual of the ADP formula- tion) to determine how much capacity y should be allocated to each ODF. 1
c) Use Excel solver or any other linear programming solver to determine an optimal capacity allocation yj ; j = 1; : : : ; 15 and use the sensitivity report to obtain the dual or shadow prices z = z(T; u) for the two resources. Describe how you would implement the unmodied bid-price heuristic? d) Partition the 15 ODFs into sets F = fj : yj = Tjg P = fj : 0

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