ECE 307 – Techniques for Engineering Decisions Transshipment and Shortest Path Problems George Gross Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved TRANSSHIPMENT PROBLEMS ‰ We consider the shipment of a homogeneous commodity from a specified point or source to a particular destination or sink ‰ In general, the source and the sink need not be directly connected; rather, the flow goes through the transshipment points or the intermediate nodes ‰ The objective is to determine the maximal flow from the source to the sink © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved FLOW NETWORK EXAMPLE s t © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved TRANSSHIPMENT PROBLEMS  nodes 1, 2, 3, 4, and are the transshipment points  arcs of the network are ( s, ), ( s, ), ( 1, ), ( 1, ), ( 2, ), ( 3, ), ( 3, ), ( 4, ), ( 5, ), ( 4, t ), ( 5, t ) ; the existence of an arc from to and from to allows bidirectional flows between the two nodes  each arc may be constrained in terms of a limit on the flow through the arc © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved MAX FLOW PROBLEM ‰ We denote by f ij the flow from i to j and this equals the amount of the commodity shipped from i to j on an arc ( i , j ) that directly connects the nodes i and j ‰ The problem is to determine the maximal flow f from s to t taking into account the flow limits k ij of each arc ( i , j ) ‰ The mathematical statement of the problem is © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved MAX FLOW PROBLEM max Z = f s.t ≤ f ij ≤ k ij ∀ arc ( i , j ) that connects nodes i and j ∑ f si = f at source s i ∑i f it = ∑i f ij f = at sink t ∑k f jk conservation of flow relations ⎫⎪ ⎬ at each transshipment node j ⎪⎭ © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved MAX FLOW PROBLEM ‰ While the simplex approach can solve the max flow problem, it is possible to construct a highly efficient network method to find f directly ‰ We develop such a scheme by making use of network or graph theoretic notions ‰ We start by introducing some definitions © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved DEFINITIONS OF NETWORK TERMS ‰ Each arc is directed and so for an arc ( i , j ), f ij ≥ ‰ A forward arc at a node i is one that leaves the node i to some node j and is denoted by ( i , j ) ‰ A backward arc at node i is one that enters node i from some node j and is denoted by ( j , i ) © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved DEFINITIONS OF NETWORK TERMS ‰ A path connecting node i to node j is a sequence of arcs that starts at node i and terminates at node j  we denote a path by P = { ( i, k ), ( k, l ), , ( m, j ) }  in the example network •( 1, ), ( 2, ), ( 5, ) is a path from to •( 1, ), ( 3, ) is another path from to © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved DEFINITIONS OF NETWORK TERMS ‰ A cycle is a path with i = j , i.e., P = { ( i, k ), ( k, l ), , ( m, i ) } ‰ We denote the set of nodes of the network by N  the definition is N = { i : i is a node of the network }  In the example network N = { s , 1, 2, 3, 4, 5, t } © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 10  determines the shortest distance from to every other node EXAMPLE : FIVE – NODE NETWORK © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 67 EXAMPLE : FIVE – NODE NETWORK 0 3 2 10 4 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 68 APPPLICATION : EQUIPMENT REPLACEMENT PROBLEM ‰ We consider the problem of replacing old equipment or continuing its maintenance ‰ As equipment ages, the level of maintenance required increases and typically, this results in increased operating costs ‰ O&M costs may be reduced by replacing aging equipment; however, replacement requires additional capital investment and so higher fixed costs © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 69 APPPLICATION : EQUIPMENT REPLACEMENT PROBLEM ‰ The problem is how often to replace equipment so as to minimize the total costs given by total costs = capital costs fixed + O&M costs variable © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 70 EXAMPLE: EQUIPMENT REPLACEMENT ‰ Equipment replacement is planned during the next years ‰ The cost elements are p j = purchase costs in year j s j = salvage value of original equipment after j years of use c j = O&M costs in year j of operation of equipment with the property that … cj < cj + < cj + < … ‰ We formulate this problem as a shortest route problem on a directed network © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 71 EQUIPMENT REPLACEMENT PROBLEM end of d13 d12 d15 d14 period d36 d35 d23 d34 d24 start of planning d16 d45 d25 d56 d46 d26 planning period © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 72 APPPLICATION : EQUIPMENT REPLACEMENT PROBLEM where, the “distances” d ij are defined to be finite if i < j , i.e., year i precedes the year j , with j−i d ij = pi − s j−i + ∑ cτ j > i τ =1   purchase salvage value O&M costs price in after j – i for j – i years year i years of use of operation © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 73 APPPLICATION : EQUIPMENT REPLACEMENT PROBLEM ‰ For example, if the purchase is made in year d 16 = p − s + c ∑ τ =1 τ ‰ The solution is the shortest distance path from year to year ; if for example the path is { ( 1, 2) , (2, 3) , (3, 4) , (4, 5) , (5, 6) } then the solution is interpreted as the replacement of the equipment each year with total costs = ∑ i =1 p i − s + 5c © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 74 COMPACT BOOK STORAGE IN A LIBRARY ‰ This problem concerns the storage of books in a limited size library ‰ Books are stored according to their size, in terms of height and thickness, with books placed in groups of same or higher height; the set of book heights { Hi } is arranged in ascending order with H1 < H2 < … < Hn © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 75 COMPACT BOOK STORAGE IN A LIBRARY ‰ Any book of height Hi may be shelved on a shelf of height at least Hi , i.e., Hi , Hi+1 , Hi+2 , ‰ The length Li of shelving required for height Hi is computed given the thickness of each book; the total shelf area required is ∑ Hi Li i  if only height class [ corresponding to the tallest book ] exists, total shelf area required is the total length of the thickness of all books times the height of the tallest book © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 76 COMPACT BOOK STORAGE IN A LIBRARY  if or more height classes are considered, the total area required is less than the total area required for a single class ‰ The costs of construction of shelf areas for each height class Hi have the components si fixed costs [ independent of shelf area ] ci variable costs / unit area © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 77 COMPACT BOOK STORAGE IN A LIBRARY ‰ For example, if we consider the problem with height classes Hm and Hn with Hm < Hn  all books of height ≤ Hm are shelved in shelf with the height Hm  all the other books are shelved on the shelf with height Hn ‰ The corresponding total costs are ⎡ ⎢ sm + c m H m ⎢⎣ ⎤ ⎡ L j ⎥ + ⎢ sn + c n H n ⎥⎦ ⎢⎣ j =1 m ∑ ⎤ Lj ⎥ ⎥⎦ j = m +1 n ∑ © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 78 ... Gross, University of Illinois at Urbana-Champaign, All Rights Reserved FLOW NETWORK EXAMPLE s t © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved TRANSSHIPMENT... s to t is equal to the minimal cut, i.e., the cut S , T with the smallest capacity ‰ The max-flow min-cut theorem allows us, in principle, to find the maximal flow in a network by finding the... Urbana-Champaign, All Rights Reserved 19 EXAMPLE ‰ Consider the simple network with the flow capacities on each arc indicated s t © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign,