Linear Programming and Network Flows

The foUovdng examples illustrate thispoint.

Single Duty Crew Scheduling. In this formulation the binary variable x: Observe that the ones in each column ofA occur in consecutive rows because each driver 's duty contains a single work shift nosplit shifts or work breaks. We show that this problem is a shortest path problem. Tomake this identification, we perform the following operations: This transformation does not change the solutionto the system. Now add a redundant equation equal to minus the sums of all theequations in the revised system. Therefore, the problem is to ship one unit of flow from node 1 to node 9 at minimumcost in the network given in Figure 1.

If instead of requiring a single driver to be on duty in each period, we specify anumber to be on duty in each period, the same transformation would produce anetwork flow problem, but in this case the right hand side coefficients supply anddemands could be arbitrary. Therefore, the transformed problem would be a generalminimum cost network flow problem, rather than a shortest p ath problem.

In construction and many other project planning applications, workers need tocomplete a variety of tasks that are related by precedence conditions; for example, inconstructing a house, a builder must pour the foundation before framing the house andcomplete the framing before beginning to install either electrical or plumbing fixtures. On the surface, this problem, which is a linear program in the variables s: Note, however, that if we move thevariable Sj to the left hand side of the constraint, then each constraint contains exactlytwo variables, one with a plus one coefficient and one with a minus one coefficient.

Thelinear programming dual of this problem has a familiar structure. If we associate a dualvariable xj: This longest path has thefollowing interpretation. Since delaying any job in this sequence must necessarily delaythe completion of the overall project, this path has become known as the critical path andthe problem has become known as the critical path problem.

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This model heis become aprincipal tool in project management, particularly for managing large-scale corwtructionprojects. The critical path itself is important because it identifies those jobs that requiremanagerial attention in order to complete the project as quickly as possible. Researchers and practitioners have enhanced this basic model in several ways. For example, if resources are available for expediting individual jobs, we could considerthe most efficient use of these resources to complete the overall project as quickly aspossible.

Certain versions of this problem can be formulated as minimum cost flowproblems. The open pit mining problem is another network flow problem that arises fromprecedence conditions. Consider the open pit mine shown in Figure 1. As shown inthis figure, we have divided the region to be mined into blocks. The provisions of anygiven mining technology, and perhaps the geography of the mine, impose restrictionson how we can remove the blocks: Suppose now that eachblock j has an associated revenue n e.

This network will also have a dummy "collection node" withdemand equal to minus the sum of the rj's, and an arc connecting it to node j that is. The dual problem is one of finding a network flow that minimizes ths sum offlows on the arcs incident to node 0. The critical path scheduling problem and open pit mining problem illustrate oneway that network flow problems arise indirectly.

Whenever, two variables in a linearprogram are related by a precedence conditions, the variable corresponding to thisprecedence constraint in the dual linear program vll have a network flow structure. Ifthe only constraints in the problem are precedence constraints, the dual linear programwill be a network flow problem. Matrix Rounding of Census Information. Census Bureau uses census infonnation to construct millions of tablesfor a wide variety of purposes.

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By law, the Bureau has an obligation to protect the sourceof its information and not disclose statistics that can be attributed to any particularindividual. It can attempt to do so by rounding the census information contained in anytable.



Consider, for example, the data shown in Figure 1. Since the upper leftmostentry in this table is a 1, the tabulated information might disclose information about aparticular individual. We might disguise the information in this table as follows;round each entry in the table, including the row and column sums, either up or dovn tothe nearest multiple of three, say, so that the entries in the table continue to add to the rounded row and column sums, and the overall sum of the entries in the new tableadds to a rounded version of the overall sum in the original table.

The problem can be cast as findinga feasible flow in a network and can be solved by an application of the maximum flowalgorithm.

Linear Programming and Network Flows, 4th Edition

The network contains a node for each row in the table and one node for eachcolumn. It contains an arc connecting node i corresponding to row i and node j corresponding to column j: In addition, we add a supersource s to thenetwork connected to each row node i: Similarly, we add a supersink t with the arc connecting eachcolumn node j to this node; the flow on this arc must be the j-th column sum, roundedup or down.

We also add an arc connecting node t and node s; the flow on this arcmust be the sum of all entries in the original table rounded up or down. If we rescale all the flows, meeisuring them in integral units of the rounding base. Parte 1 de 3 Dewey Ravindra K.

Operations Research 08F: Maximum Flow Problem Formulation

We have divided the discussioninto the following broad major topics: For the purposes of this discussion, we will consider four different types ofnetworks arising in practice: Network flow models are also used for several purposes: In matrix notation, we represent the minimum cost flow problem in terms of a node-arc incidence matrix N. Route Networks Route networks, which are one level of abstraction removed from physicalnetworks, are familiar to most students of operations research and management science. Derived Networks This category is a "grab bag" of specialized applications and illustrates thatsometimes network flow problems arise in surprising ways from problems that on thesurface might not appear to involve networks.

Shortest path formulation of the single duty scheduling problem. Critical Path Scheduling and Networks Derived from Precedence Conditions In construction and many other project planning applications, workers need tocomplete a variety of tasks that are related by precedence conditions; for example, inconstructing a house, a builder must pour the foundation before framing the house andcomplete the framing before beginning to install either electrical or plumbing fixtures.

ISYE 671 - Linear Programming and Network Flows

This type of application can be formulated mathematically as follows. Levent Mollamustafaoglu rated it really liked it Jan 31, Sarah sa rated it liked it Feb 23, Juan Pablo rated it really liked it Dec 10, Mahmood rated it it was amazing May 14, Fahimeh rated it it was amazing Jun 17, Eskarlet Montiel rated it it was amazing Jan 02, Sunny rated it liked it Nov 30, Nuray rated it liked it Oct 02, Fatemeh Safinejad rated it really liked it Nov 25, Pooria Paridar rated it it was amazing Feb 26, Camilo rated it liked it Oct 13, Ehsan rated it it was amazing Oct 26, Bino Bandar rated it it was ok Nov 17, Lambada Roeun rated it really liked it Oct 09, Mahdi Bajestan rated it liked it Oct 18, Ema Jones rated it really liked it Mar 29, Iman rated it really liked it Oct 02, An emphasis is placed on providing geometric viewpoints and economic interpretations as well as strengthening the understanding of the fundamental ideas.

Each chapter is accompanied by Notes and References sections that provide historical developments in addition to current and future trends. Updated exercises allow readers to test their comprehension of the presented material, and extensive references provide resources for further study.



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Linear Programming and Network Flows, Fourth Edition is an excellent book for linear programming and network flow courses at the upper-undergraduate and graduate levels. It is also a valuable resource for applied scientists who would like to refresh their understanding of linear programming and network flow techniques. Request an Evaluation Copy for this title. Contact your Rep for all inquiries.

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