___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Assignment Problem Linear Programming

___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Assignment Problem Assignment „1 to 1“ employeesjobs machinesjobs projectsmanagers service teams cars doctors night shifts Objective: maximize the effect of assignment

Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Example – Prague Build, Inc. Objective: minimize total distance necessary for all movements Assignment Problem  Excavating shafts for basements (Michle, Prosek, Radlice, Trója)  Each excavation takes 5 days  4 excavators stored in 4 separated garages (everyday‘s movement)  One excavator to one destination  Distances between garages and destinations

___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry

Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Example – Prague Build, Inc. Assignment Problem MichleProsekRadliceTrója Garage Garage Garage Garage Distances

Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Example – Prague Build, Inc. Assignment Problem MichleProsekRadliceTrója Garage 1 x 11 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 Decision variables

Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Example – Prague Build, Inc. Assignment Problem Decision variables x ij = 1 if the excavator from the garage i goes to the destination j 0 otherwise Binary variable

Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Example – Prague Build, Inc. Assignment Problem Optimal solution MichleProsekRadliceTrója Garage Garage Garage Garage

Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Example – Prague Build, Inc. Assignment Problem Optimal solution 1 movement MichleProsekRadliceTrója Garage 1 5 km --- Garage km Garage km - Garage km -- Minimal total distance 320 km

___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry

___________________________________________________________________________ Operations Research  Jan Fábry Network Models

___________________________________________________________________________ Operations Research  Jan Fábry  Nodes  Arcs j jj j i j jj j iUNDIRECTEDDIRECTED UNDIRECTED NETWORK DIRECTED NETWORK  Network

Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Path Sequence of arcs in which the initial node of each arc is identical with the terminal node of the preceding arc

Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Path Open Path 1 6 

Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Circuit (Cycle) Path starting and ending in the same node (closed path) 

Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Connected Network There is a path connecting every pair of nodes in the network

Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Unconnected Network

Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Tree Connected network without any circuit. Exactly 6 arcs (n-1) Removing 1 arc Unconnected network Adding 1 arc Circuit in the network

Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Tree STAR „CHRISTMAS“ TREE SNAKE

Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Spanning Tree Tree including all the nodes from the original network

Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Evaluated Network - distance - time - cost - capacity ValuesArcs Nodes j i ii i j i ii i y ij yiyiyiyi yjyjyjyj

Network Models ___________________________________________________________________________ Operations Research  Jan Fábry Basic Network Applications Project Management  Shortest Path Problem  Traveling Salesperson Problem (TSP)  Minimal Spanning Tree  Critical Path Method (CPM)  Maximum Flow Problem  Program Evaluation Review Technique (PERT)

___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Shortest Path Problem Network Models

___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Shortest Path Problem Shortest path between 2 nodes

Network Models ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Shortest Path Problem Shortest Paths Between All Pairs of Nodes

___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Traveling Salesperson Problem Network Models

___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Traveling Salesperson Problem (TSP) Home city Shortest tour 110 km

___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Minimal Spanning Tree Network Models

___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Minimal Spanning Tree Example - Exhibition  Exhibition area with 9 locations that need electricity power  Use cable for extensions  Price of cable = 10 CZK / 1 m Objective: minimize the cost of all the extensions

Network Models ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Minimal Spanning Tree Example - Exhibition Power

Network Models ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Minimal Spanning Tree Example - Exhibition Power Optimum 490 m CZK

Network Models ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Maximum Flow Problem Input Output Capacited network Gas Fluid Traffic Information People Source Sink

Network Models ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Maximum Flow Problem j i i j UNDIRECTED ARC DIRECTED ARC Flow Flow Capacity

Network Models ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Maximum Flow Problem Mathematical Model  Flow through each arc  Capacity of the arc  Quantity flowing out = Quantity flowing into (except the source and the sink)  Total flow into the source = 0  Total flow out of the sink = 0  Total flow out of the source = Total flow into the sink

Maximum Flow Problem Example – White Lake City  The city is situated on the edge of a small lake  To minimize disruptive effects of possible flood  Reconstruction of drain system  2 alternatives - Northern Channel & Southern Channel Objective: maximizing the quantity of water being pumped in one hour Network Models ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry

___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Lake Reservoir Northern Channel Southern Channel

___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Northern Channel Optimum m 3

___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Southern Channel Optimum m