EXAMPLE 3 Covering All Characteristics

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EXAMPLE 3 Covering All Characteristics

SOUTHWESTERN AIRWAYS needs to assign its crews to cover all its upcoming flights. We will focus on the problem of assigning three crews based in San Francisco to the flights listed in the first column of Table 12.4. The other 12 columns show the 12 feasible sequences of flights for a crew. (The numbers in each column indicate the order of the flights.) Exactly three of the sequences need to be chosen (one per crew) in such a way that every flight is covered. (It is permissible to have more than one crew on a flight, where the extra crews would fly as passengers, but union contracts require that the extra crews would still need to be paid for their time as if they were working.) The cost of assigning a crew to a particular sequence of flights is given (in thousands of dollars) in the bottom row of the table. The ob- jective is to minimize the total cost of the three crew assignments that cover all the flights.

Formulation with Binary Variables. With 12 feasible sequences of flights, we have 12 yes-or-no decisions:

INTRODUCTION TO OPERATIONS RESEARCH-0126

INTRODUCTION TO OPERATIONS RESEARCH-0127

This example illustrates a broader class of problems called set covering problems.3 Any set covering problem can be described in general terms as involving a number of po- tential activities (such as flight sequences) and characteristics (such as flights). Each ac- tivity possesses some but not all of the characteristics. The objective is to determine the least costly combination of activities that collectively possess (cover) each characteristic at least once. Thus, let Si be the set of all activities that possess characteristic i. At least one member of the set Si must be included among the chosen activities, so a constraint,

INTRODUCTION TO OPERATIONS RESEARCH-0128

so now exactly one member of each set Si must be included among the chosen activities. For the crew scheduling example, this means that each flight must be included exactly once among the chosen flight sequences, which rules out having extra crews (as passengers) on any flight.

3Strictly speaking, a set covering problem does not include any other functional constraints such as the last functional constraint in the above crew scheduling example. It also is sometimes assumed that every coefficient in the objective function being minimized equals one, and then the name weighted set covering problem is used when this assumption does not hold.

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