Simulating realistic set covering problems with known optimal solutions
Introduction
The Set Covering Problem (SCP) is known to be an NP-hard optimization problem (Karp, 1972). There are two principal categories of SCPs. The first category is the unicost SCP where all the objective function coefficients have the same positive value. The second category is the non-unicost SCP where the objective function values are all positive but not identical. Non-unicost SCP instances are often computationally very demanding (Beasley, 1990). That is why they are the focus of this research.
Let A = (aij) be a binary m × n matrix and c = (cj) be a positive integer-valued vector. Let the indices of the rows and columns of A be represented by I = {1, 2, 3, … , m} and J = {1, 2, 3, … , n}, respectively. Each column of A represents a subset of I. Any column j ∈ J covers row i ∈ I if aij = 1. The cost of including subset j in the solution (or cover) is cj. The objective of SCP is to choose a minimum-cost collection of subsets whose union covers I. The decision variables in SCP are defined for all j ∈ J as follows:Then SCP may be written aswhere aij ∈ {0, 1}, ∀i ∈ I, j ∈ J.
Some of the application areas of SCP, as outlined in Balas and Padberg (1976), are scheduling, airline fleet scheduling, truck delivery, cutting stock, line and capacity balancing, facility location, capital investment, switching current design and symbolic logic, information retrieval, marketing, and political districting. Other applications include crew scheduling (Medard and Sawhney, 2007, Yan and Tu, 2002), bus crew scheduling (Smith & Wren, 1988), crew re-scheduling (Huisman, 2007), naval vessel scheduling (Brown et al., 1987, Fisher and Rosenwein, 1989), steel mill operations (Vasko, Wolf, & Stott, 1987), vehicle routing with time windows (Russell & Chiang, 2006), improving wireless sensor network lifetime (Cardei & Du, 2005), preference scheduling for nurses (Bard & Purnomo, 2005), and the lottery problem (Jans & Degraeve, 2008). The many SCP applications in diverse settings over many years indicate the importance of SCP.
Section snippets
Background
New algorithms and heuristics to solve optimization problems, including SCP, are developed on an ongoing basis. Any new solution method should be tested for its efficacy. If possible, comparative evaluations with other solution methods for the same class of optimization problems should be conducted to determine whether the newly coined solution procedure is trustworthy and how its performance compares to that of existing solution methods.
In general, testing of solution procedures is necessary
SCP generation procedure
Let be the optimal solution vector, J* be the set of positions (indexes) of 1s in x* and J⧹J* = J − J* be the set of indices of variables that have value 0 in the optimal solution. For example, if the optimal solution x* is (1 0 1 0 1 0), then J* = {1, 3, 5} and J⧹J* = {2, 4, 6}. Also let be the number of constraint matrix columns such that j ∈ J* and Aj = k and let nk be the number of constraint matrix columns such that Aj = k. For example, means there are three columns j ∈ J* with Aj = 2, and n5 =
Computational demonstration
The primary objective of this demonstration is to show that this SCP generation procedure yields a broad range of practical test problems with known optima enabling a researcher to test the efficacy of different SCP solution procedures. An implementation of the SCP generation procedure was developed using the MATLAB® language using a personal computer with Intel® Pentium® M processor with speed 1.73 GHz and 3 GB of RAM. A total of 525 SCP instances were generated and solved using three simple
Conclusions
SCP instances with known optimal solutions and with induced correlation between objective function coefficients and the column sums of constraint coefficients can now be simulated. In order to simulate SCP instances with known optima and specified coefficient correlation, the usual problem generation process must be modified significantly. For example, an unexpected finding of this research is that the range for the number of constraints for simulated SCP instances with known optimal solutions
References (32)
A column generation approach for the rail crew re-scheduling problem
European Journal of Operational Research
(2007)- et al.
A note on a symmetrical set covering problem: The lottery problem
European Journal of Operational Research
(2008) - et al.
New trends in exact algorithms for the 0–1 knapsack problem
European Journal of Operational Research
(2000) - et al.
Airline crew scheduling from planning to operations
European Journal of Operational Research
(2007) - et al.
Experiments with parallel branch-and-bound algorithms for the set covering problem
Operations Research Letters
(1993) - et al.
Scatter search for the vehicle routing problem with time windows
European Journal of Operational Research
(2006) - et al.
A bus crew scheduling system using a set covering formulation
Transportation Research Part A
(1988) - et al.
A network model for airline cabin crew scheduling
European Journal of Operational Research
(2002) - et al.
Generating traveling-salesman problems with known optimal tours
The Journal of the Operations Research Society
(1988) - et al.
Set partitionaing: A survey
SIAM Review
(1976)
Preference scheduling for nurses using column generation
European Journal of Operational Research
A Lagrangian heuristic for set covering problems
Naval Research Logistics
Scheduling ocean transportation of crude oil
Management Science
Improving wireless sensor network lifetime through power aware organization
Wireless Networks
An investigation of the relationship between problem characteristics and algorithm performance: A case study of the GAP
IIE Transactions
An interactive optimization system for bulk-cargo ship scheduling
Naval Research Logistics
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2015, European Journal of Operational ResearchCitation Excerpt :For example, it would make sense that activities associated with more resource units would be the activities that bring more profit. Computational studies on simulated instances of classical optimization problems, including KP01 (see, for example, Martello, Pisinger, & Toth, 1999, 2000; Martello & Toth, 1979, 1987,1988,1997; Pisinger, 1997), the Generalized Assignment Problem (GAP) (Amini & Racer, 1994; Cario et al., 2002; Martello & Toth, 1981), the Capital Budgeting (or Multidimensional Knapsack) Problem (Fréville & Plateau, 1994, 1996; Hill & Reilly, 2000b), and the Set Covering Problem (Rushmeier & Nemhauser, 1993; Sapkota & Reilly, 2011), show that correlation between key types of coefficients affects the performances of algorithms and heuristics. Based on results in Cario et al. (2002), there appears to be a relationship between the relative entropy of the joint distribution of GAP objective-function and capacity-constraint coefficient values and the performances of algorithms and heuristics.
Constrained covering solid travelling salesman problems in uncertain environment
2019, Journal of Ambient Intelligence and Humanized ComputingStaff assignment to multiple projects based on DEA efficiency
2019, Engineering EconomicsProof of covering minimality by generalizing the notion of independence
2017, Journal of Applied and Industrial MathematicsLocating distribution/service centers based on multi objective decision making using set covering and proximity to stock market
2016, International Journal of Industrial Engineering Computations
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