73-220-Lecture11 - Applications:NetworkModels 73-220...

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1 Special Linear Programming  Applications: Network Models 73-220 Lecture 11
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2 Agenda Review Last Class Sensitivity analysis. Network models: Transportation,  Transshipment, and Assignment Next Class
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3 Objectives Understand formulation of the major  network applications transportation problem, assignment problem,  transshipment problem Thru personal homework, be able to use  Excel to formulate & solve such  applications Understand & be able to use notation Double summation, double subscripts
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4 Network Models Network model is one which can be  represented by a set of nodes, a set of arcs,  and functions (e.g. costs, supplies,  demands, etc.) associated with the arcs  and/or nodes. Transportation, assignment, and  transshipment problems are all examples of  network problems.
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5 Transportation, Transshipment &  Assignment Problems Can be formulated as LP & solved by  computer If RHS of LP formulations are all integers,  optimal solution will be also be integer  (special property!) Most s/w contains separate code for these  models (to take advantage of their network  structure)
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6 Transportation Problem Seeks to minimize total shipping costs of  transporting goods from  m  origins (each  with a supply  s i ) to  n  destinations (each with  a demand  d j ), when the unit shipping cost  from an origin,  i , to a destination,  j , is  c ij Network model for a transportation problem  with two sources and three destinations  follows  . ..
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7 Transportation Problem Network Representation 1 1 2 2 3 3 1 1 2 2 c 11 11 c 12 12 c 13 13 c 21 21 c 22 22 c 23 23 d 1 d 2 d 3 s 1 s 2 SOURCES DESTINATIONS DESTINATIONS s 2
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8 Transportation Problem LP Formulation: Formulated in terms of amounts shipped  from origins to destinations,  x ij                         Min    ΣΣ c ij x ij                       i j             s.t.     Σ x ij  <   s i   for each origin  i                       j                         Σ x ij  =  d j    for each destination  j                       i                         x ij  >  0 for all  i  and  j
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9 Notation Review Σ i Σ j c ij x ij   Σ  is summation symbol c ij  &   x ij   are “double subscripted” variables Work from inner  Σ  to outer  Σ; so vary “j”  terms fully for each value “i” takes on » i = 1:  c 11 x 11  + c 12 x 12  + c 13 x 13  + … » i = 2:  + c 21 x 21  + c 22 x 22  + c 23 x 23  + … »
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10 Transportation Problem (cont.) Special Cases - modifications to LP formulation: Minimum shipping guarantees from  i  to  j :  
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This note was uploaded on 11/28/2011 for the course FINANCE 101 taught by Professor Chan during the Spring '11 term at Aarhus Universitet.

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73-220-Lecture11 - Applications:NetworkModels 73-220...

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