Set Covering Problem

Consider an m x n) matrix A, called the set covering matrix, whose elements Uij s are either 0 or 1. If a j = 1, we say that column covers row /. If not, = 0. The set covering problem is to select the minimum number of columns such that every row is covered by at least one column. 

To formulate the set covering problem as an integer programming (IP) model, we define a binary variable for each column such that 

Constraints denoted by Equation 5.22 guarantee that every row is covered by at least one column. In other words, for row i, when at least one a,y = 1, the corresponding Xj must be one. The objective function, given by Equation 5.21 guarantees that the minimum number of columns is selected to cover all the rows. 

Equation 5.23 guarantees that row 1 will be covered by at least one column. Similarly, Equations 5.24,5.25, and 5.26 guarantee that rows 2,3, and 4 will be covered respectively by at least one column. 

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Set-covering problems The set covering problem finds a set of facilities of minimum cost from a finite set of candidate facilities (each with a given cost) so that every demand node is covered. A node is considered covered if at least one facility is located within a given distance of the node. The set-covering problem does not account for possible congestion in the facilities since it does not consider the number of demand nodes that are served by each fadlity or the size of the demand of each node. 

Reverse auction. Combinatorial auctions are also proposed for procurement problems in markets with one buyer and multiple sellers. The reverse combinatorial auction is formulated as a set covering problem rather than a set packing problem. An interesting (and complicating) issue that arises in this setting is that there are various business rules that are used to constrain the choice of winners. These business rules appear as side constraints in the winner determination problem. The winner determination problem with no side constraints can be written as ... 

The winner determination for supply curve auctions can also be written as a set covering problem as shown Eq. (5.11) using a Dantzig-Wolfe type decomposition [39]. To use a set covering model we introduce the concept of supply... 

In the set covering problem every row has to be covered by at least one column. In the set partitioning problem, every row has to be covered by exactly one column. Otherwise the two problems are the same. Thus, the only change in the IP model is that constraints given by Equation 5.22 will now become equalities ... 

The warehouse location problem is basically a set covering problem. The first step is to define the set covering matrix (A) based on the network configuration shown in Figure 5.3. The rows of the matrix will be the nine retail outlets and the columns will be the four potential warehouse locations. The elements of matrix A, a, will be set to 1 if retailer i (R ) can be supplied by warehouse location j (Wj), that is, there is a direct link between R and W. Otherwise, we set a, = 0. 

Note that if we relax the t binary variables by the inequalities 0 < y < 1, then (3-110) becomes a linear program with a (global) solution that is a lower bound to the MILP (3-110). There are specific MILP classes where the LP relaxation of (3-110) has the same solution as the MILP. Among these problems is the well-known assignment problem. Other MILPs that can be solved with efficient special-purpose methods are the knapsack problem, the set covering and set partitioning problems, and the traveling salesperson problem. See Nemhauser and Wolsey (1988) for a detailed treatment of these problems. 

One problem highlighted by several reviewers [14,20] is that datasets like the Huuskonen set cover unnecessarily large ranges of solubility. The Huuskonen set covers the range log S (log of solubility in mol/1) from —11.62 to +1.58, which converts approximately to 9.6 x 10 7-1.5 x 107pg/ml for a MW of 400 Da. 

The large room acts like a blackbody at 20°C, so for analysis purposes we can assume the hole is covered by an imaginary black surface S at 20°C. We shall set the problem up for a numerical solution for the radiosities and then calculate the heat-transfer rates. After that, we shall examine an insulated-surface case for this same geometry. 

Disappointingly, we found that a 2-opt optimization of the greedy TSP-tour leads to only 0.3% improvement (Table 1). We tried a few approaches to further improve the TSP solution (or to verify its proximity to the optimal solution), in particular, solving minimum length cycle cover problem with further merging of cycles in the cover into a hamiltonian path. A cycle cover is a set of (simple) cycles such that each vertex in the graph is contained in exactly one cycle. The... 

Attack by insects presents a completely diflFerent set of problems. Insects can often be controlled by lowering the moisture content, but some woodboring beetles and dry-wood termites can survive at moisture contents as low as 6-8%. Their control requires special techniques. These techniques are discussed in Chapter 12 and by Erickson (6) and therefore will not be covered in detail here. 

In applying the set covering model to the warehouse location problem in supply chain, we treat the potential warehouse locations as "colunms" and the customer regions as "rows" of the set covering matrix A. We construct the matrix A, by setting its elements as follows ... 

HiU, Philip, and Garl Peterson. Mechanics and Thermodynamics of Propulsion. 2d ed. Upper Saddle River, N.J. Prentice Hall, 1991. A classic textbook on propulsion that covers the basic science and engineering of jet and rocket engines and their components. Also gives excellent sets of problems with answers. 

The central notion of CBR is a case. The main role of a case is to describe and to remember a single event from past experience where a problem or problem situation was solved. A case is made up of two components problem and solution. Typically, the problem description consists of a set of attributes and their values. Many cases are collected in a set to build a case library (case base). The library of cases must roughly cover the set of problems that may arise in the considered domain of application. 

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