Linear Programming Assignment Problem Operations Research Jan Fbry
Linear Programming Assignment Problem ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Assignment Problem employees projects jobs service teams doctors Assignment „ 1 to 1“ jobs managers machines cars night shifts Objective: maximize the effect of assignment ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Assignment Problem Example – Prague Build, Inc. Ø Excavating shafts for basements (Michle, Prosek, Radlice, Trója) Ø Each excavation takes 5 days Ø 4 excavators stored in 4 separated garages (everyday‘s movement) Ø One excavator to one destination Ø Distances between garages and destinations Objective: minimize total distance necessary for all movements ______________________________________ Operations Research Jan Fábry
______________________________________ Operations Research Jan Fábry
Linear Programming Applications Assignment Problem Example – Prague Build, Inc. Distances Michle Prosek Radlice Trója Garage 1 5 22 12 18 Garage 2 15 17 6 10 Garage 3 8 25 5 20 Garage 4 10 12 19 12 ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Assignment Problem Example – Prague Build, Inc. Decision variables Michle Prosek Radlice Trója Garage 1 x 12 x 13 x 14 Garage 2 x 21 x 22 x 23 x 24 Garage 3 x 31 x 32 x 33 x 34 Garage 4 x 41 x 42 x 43 x 44 ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Assignment Problem Example – Prague Build, Inc. Decision variables xij = 1 if the excavator from the garage i goes to the destination j 0 otherwise Binary variable ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Assignment Problem Example – Prague Build, Inc. Optimal solution Michle Prosek Radlice Trója Garage 1 1 0 0 0 Garage 2 0 0 0 1 Garage 3 0 0 1 0 Garage 4 0 1 0 0 ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Assignment Problem Example – Prague Build, Inc. Optimal solution 1 movement Michle Prosek Radlice Trója Garage 1 5 km - - - Garage 2 - - - 10 km Garage 3 - - 5 km - Garage 4 - 12 km - - Minimal total distance 320 km ______________________________________ Operations Research Jan Fábry
______________________________________ Operations Research Jan Fábry
Network Models ______________________________________ Operations Research Jan Fábry
Network Models Ø Network Ø Nodes Ø Arcs UNDIRECTED j i UNDIRECTED NETWORK j i DIRECTED NETWORK ______________________________________ Operations Research Jan Fábry
Network Models Ø Path Sequence of arcs in which the initial node of each arc is identical with the terminal node of the preceding arc. 3 7 5 ______________________________________ Operations Research Jan Fábry
Network Models Ø Path 2 11 5 4 3 5 4 6 3 6 Open Path ______________________________________ Operations Research Jan Fábry
Network Models Ø Circuit (Cycle) Path starting and ending in the same node (closed path). 2 2 1 1 1 5 4 3 3 5 4 6 6 ______________________________________ Operations Research Jan Fábry
Network Models Ø Connected Network There is a path connecting every pair of nodes in the network. 2 1 5 4 3 3 5 4 6 6 ______________________________________ Operations Research Jan Fábry
Network Models Ø Unconnected Network 2 1 5 4 3 3 5 4 6 6 ______________________________________ Operations Research Jan Fábry
Network Models Ø Tree Connected network without any circuit. 2 3 3 4 4 6 4 5 4 1 Exactly 6 arcs (n-1) Removing 1 arc Adding 1 arc 7 Unconnected network Circuit in the network ______________________________________ Operations Research Jan Fábry
Network Models Ø Tree STAR „CHRISTMAS“ TREE SNAKE ______________________________________ Operations Research Jan Fábry
Network Models Ø Spanning Tree including all the nodes from the original network. 2 1 5 4 3 3 5 4 6 6 ______________________________________ Operations Research Jan Fábry
Network Models Ø Evaluated Network Values - distance - time - cost - capacity Arcs i Nodes i yi yij j j yj ______________________________________ Operations Research Jan Fábry
Network Models Basic Network Applications Ø Shortest Path Problem Ø Traveling Salesperson Problem (TSP) Ø Minimal Spanning Tree Ø Maximum Flow Problem Project Management Ø Critical Path Method (CPM) Ø Program Evaluation Review Technique (PERT) ______________________________________ Operations Research Jan Fábry
Network Models Shortest Path Problem ______________________________________ Operations Research Jan Fábry
Network Models Shortest Path Problem Shortest path between 2 nodes 2 14 1 1 18 2 25 3 30 4 15 3 5 23 12 10 5 4 15 6 16 6 ______________________________________ Operations Research Jan Fábry
Network Models Shortest Path Problem Shortest Paths Between All Pairs of Nodes 1 2 3 1 14 24 2 14 10 3 24 10 - 4 26 12 15 5 32 18 28 6 40 26 16 4 5 6 26 32 40 12 18 26 15 28 16 23 15 23 30 15 30 - ______________________________________ Operations Research Jan Fábry
Network Models Traveling Salesperson Problem ______________________________________ Operations Research Jan Fábry
Network Models Traveling Salesperson Problem (TSP) 2 Home city 1 11 14 18 2 5 23 12 10 25 3 30 4 15 3 Shortest tour 5 4 15 6 16 6 110 km ______________________________________ Operations Research Jan Fábry
Network Models Minimal Spanning Tree ______________________________________ Operations Research Jan Fábry
Network Models Minimal Spanning Tree Example - Exhibition Ø Exhibition area with 9 locations that need electricity power Ø Use cable for extensions Ø Price of cable = 10 CZK / 1 m Objective: minimize the cost of all the extensions ______________________________________ Operations Research Jan Fábry
Network Models Minimal Spanning Tree Example - Exhibition 88 Power 85 2 2 7 7 63 76 1 90 75 60 70 5 55 80 3 3 43 8 9 54 8 74 68 52 4 9 5 40 61 35 10 10 4 120 6 ______________________________________ Operations Research Jan Fábry 71 6
Network Models Minimal Spanning Tree Optimum 490 m Example - Exhibition 4 900 CZK 2 7 2 Power 7 61 76 1 9 5 5 40 52 4 4 43 8 3 60 55 9 68 35 10 10 6 ______________________________________ Operations Research Jan Fábry 6
Network Models Maximum Flow Problem Input Capacited network Source 7 9 Gas Fluid Traffic Information People 3 Sink Output ______________________________________ Operations Research Jan Fábry
Network Models Maximum Flow Problem UNDIRECTED ARC i DIRECTED ARC j i j Flow Capacity ______________________________________ Operations Research Jan Fábry
Network Models Maximum Flow Problem Mathematical Model Ø Flow through each arc Capacity of the arc Ø Quantity flowing out = Quantity flowing into (except the source and the sink) Ø Total flow into the source = 0 Ø Total flow out of the sink = 0 Ø Total flow out of the source = Total flow into the sink ______________________________________ Operations Research Jan Fábry
Network Models Maximum Flow Problem Example – White Lake City Ø The city is situated on the edge of a small lake Ø To minimize disruptive effects of possible flood Ø Reconstruction of drain system Ø 2 alternatives - Northern Channel & Southern Channel Objective: maximizing the quantity of water being pumped in one hour ______________________________________ Operations Research Jan Fábry
740 2 900 1100 280 720 7 400 300 Lake 8 410 700 3 550 4 300 5 270 370 800 Northern Channel 1510 130 6 220 Reservoir 780 2 6 800 470 3 520 230 370 800 1050 1400 840 4 250 660 8 290 700 5 420 7 Southern Channel ______________________________________ Operations Research Jan Fábry
Northern Channel 610 1 930 m 3 610 22 55 100 1 1100 Optimum 300 410 700 33 76 220 390 300 44 220 67 88 300 1500 130 9 ______________________________________ Operations Research Jan Fábry
Southern Channel 780 2 450 m 3 780 22 470 66 1050 470 33 9 200 1 800 400 Optimum 1400 840 44 88 250 210 55 360 150 77 ______________________________________ Operations Research Jan Fábry
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