is an expertly guided tour of combinatorial optimization. The theme of the tour is the traveling salesman problem (TSP). In the course of exploring different facets of TSP research and results, our guides expose concepts, methods, and ideas from computational complexity, graph theory, probabilistic analysis of algorithms, statistical methods for comparing heuristics, polyhedral theory, and more generally, the entire field of combinatorial optimization. This well-written book more than fulfills the editors' hopes of providing comprehensive coverage of the techniques of combinatorial optimization as well as being a state-of-the-art survey of the traveling salesman problem. We highly recommend this book to advanced students and researchers in operations research, computer science, discrete mathematics, and quantitative business studies.The TSP is a classic 'hard' problem: easy to describe and difficult to solve. Precisely how difficult became clearer as computational complexity theory developed. Chapter 1 (A.J. Hoffman and P. Wolfe) explores the history of the problem. In Chapter 2, R.S. Garfinkel discusses motivations for the problem and some of its many applications and generalizations. Chapter 3 (D.S. Johnson and C.H. Papadimitriou) introduces computational complexity theory and shows how the difficulty of the TSP optimization problem can be specified as NP-hard. This chapter delivers the bad news of the TSP's intractability. P.C. Gilmore, E.L. Lawler, and D.B. Shmoys survey well-solved special cases of the TSP in Chapter 4. Several results are new. The authors categorize these special cases (those with restrictions on arc lengths and those with network structures) and show how the theory of subtour patching is used to give optimal solutions to several special cases. Knowing the well-solved

The delivery of goods from a warehouse to local customers is a critical aspect of a material logistics system. A strategic decision must be made periodically (e.g., once a year) whether to maintain a private delivery fleet, to employ outside commercial carrier services, or to use a combination of both options. We seek to develop a methodology to address this long range planning decision. Our model considers a geographic area, with random daily demands, served by a single warehouse. The costs considered include the fixed and variable (per mile) costs of a private vehicle and the outside carriers' delivery charges. A private vehicle is constrained by the length of the work-day, since it returns to the warehouse only after completing all its deliveries. Since actual customer locations change from day to day, for planning purposes we divide the geographic area into sectors and decide how best to serve each sector. The model determines the private fleet size and the specific assignment of each sector to a private vehicle or to an outside carrier. The centerpiece of our solution approach consists of a single-source capacitated facility location formulation, in which each “customer” (sector) is served by a single “facility” (private vehicle or outside carrier). Computational results are reported.

is an expertly guided tour of combinatorial optimization. The theme of the tour is the traveling salesman problem (TSP). In the course of exploring different facets of TSP research and results, our guides expose concepts, methods, and ideas from computational complexity, graph theory, probabilistic analysis of algorithms, statistical methods for comparing heuristics, polyhedral theory, and more generally, the entire field of combinatorial optimization. This well-written book more than fulfills the editors' hopes of providing comprehensive coverage of the techniques of combinatorial optimization as well as being a state-of-the-art survey of the traveling salesman problem. We highly recommend this book to advanced students and re-

Test Instance Generators (TIG's) are important to evaluate heuristic procedures for N P -hard problems. We analyze a TIG in use for the TSP. This TIG, due to Pilcher and Rardin, is based on a random cut method. We show that it generates a class of instances of intermediate complexity: not as hard as the entire TSP class unless N P = co(N P ); not as easy as P unless N P = P . Since the upper bound on complexity must hold for any efficient TIG, our analysis verifies that this random cut TIG is, in a sense, as good as possible a TIG for the TSP. This suggests that the random cut method may be a good basis for constructing TIG's for other problems.

We detail a random cut concept for generating instances of discrete optimization problems based on a partial description of the polytope of solutions. We show how implementations of this approach have the useful properties that an optimal solution and the form of valid equalities required to solve the problem by cutting methods are both known at the completion of generation. The former makes possible large‐scale testing of heuristics, and the latter facilitates cutting algorithm research. The random cut concept of problem generation is first discussed in general and then details are provided on its implementation for symmetric traveling salesman problems.