This paper has always been one of my favorite “children,” combining as it does elements of the duality of linear programming and combinatorial tools from graph theory. It may be of some interest to tell the story of its origin.

The Hungarian Method for the Assignment Problem.

We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branch-and-bound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. We then discuss computational issues and implementation of column generation, branch-and-bound algorithms, including special branching rules and efficient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality.

/pdf/branch-and-price-column-generation-for-solving-huge-integer-4jvlny6blr.pdf

Branch-And-Price: Column Generation for Solving Huge Integer Programs

https://hesso.tind.io/record/8224/files/Preprint.pdf

Benchmarks for basic scheduling problems

Perspectives in supply chain risk management

The importance of remanufacturing used products into new ones has been widely recognized in the literature and in practice. In this paper, we address the problem of choosing the appropriate reverse channel structure for the collection of used products from customers. Specifically, we consider a manufacturer who has three options for collecting such products: (1) she can collect them herself directly from the customers, (2) she can provide suitable incentives to an existing retailer (who already has a distribution channel) to induce the collection, or (3) she can subcontract the collection activity to a third party. Based on our observations in the industry, we model the three options described above as decentralized decision-making systems with the manufacturer being the Stackelberg leader. When considering decentralized channels, we find that ceteris paribus, the agent, who is closer to the customer (i.e., the retailer), is the most effective undertaker of product collection activity for the manufacturer. In addition, we show that simple coordination mechanisms can be designed such that the collection effort of the retailer and the supply chain profits are attained at the same level as in a centrally coordinated system.

https://ink.library.smu.edu.sg/cgi/viewcontent.cgi?article=6102&context=lkcsb_research

Closed-Loop Supply Chain Models with Product Remanufacturing

Handbooks in operations research and management science

History (A. Hoffman and P. Wolfe). Motivation and Modeling (R. Garfinkel). Computational Complexity (D. Johnson and C. Papadimitriou). Well-Solved Special Cases (P. Gilmore, et al.). Performance Guarantees for Heuristics (D. Johnson and C. Papadimitriou). Probabilistic Analysis of Heuristics (R. Karp and J. Steele). Empirical Analysis of Heuristics (B. Golden and W. Stewart). Polyhedral Theory (M. Grotschel and M. Padberg). Polyhedral Algorithms (M. Padberg and M. Grotschel). Branch and Bound Methods (E. Balas and P. Toth). Hamiltonian Cycles (V. Chvatal). Vehicle Routing (N. Christofides). Bibliography.

The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization

/pdf/sequencing-and-scheduling-algorithms-and-complexity-lnutyrr3kz.pdf

Sequencing and scheduling: algorithms and complexity

1. Introduction.- 2. Problem Formulation.- 2.1. Notations and representations.- 2.2. Restrictive assumptions.- 2.3. Optimality criteria.- 2.3.1. Regular measures.- 2.3.1.1. Criteria based on completion times.- 2.3.1.2. Criteria based on due dates.- 2.3.1.3. Criteria based on inventory cost and utilization.- 2.3.2. Relations between criteria.- 2.3.3. Analysis of scheduling costs.- 2.4. Classification of problems.- 3. Methods of Solution.- 3.1. Complete enumeration.- 3.2. Combinatorial analysis.- 3.3. Mixed integer and non-linear programming.- 3.3.1. [Bowman 1959].- 3.3.2. [Pritsker et al. 1969].- 3.3.3. [Wagner 1959].- 3.3.4. [Manne 1960].- 3.3.5. [Nepomiastchy 1973].- 3.4. Branch-and-bound.- 3.5. Dynamic programming.- 3.5.1. [Held and Karp 1962 Lawler 1964].- 3.5.2. [Lawler and Moore 1969].- 3.6. Complexity theory.- 3.7. Heuristic methods.- 3.7.1. Priority rules.- 3.7.2. Bayesian analysis.- 4. One-Machine Problems.- 4.1. n|1?cmax problems.- 4.1.1. The n|1?Cmax problem.- 4.1.2. The n|1?Lmax problem.- 4.1.3. The general n|1?cmax problem.- 4.2. n|1|i|Cmax problems.- 4.2.1. The n|1|ri0|cmax problem.- 4.2.1.1. Lower bound by job splitting.- 4.2.1.2. The algorithm of McMahon and Florian 62.- 4.2.1.3. Precedence constraints.- 4.2.2. The n|1|seq dep|cmax problem.- 4.2.3. The n|1|prec|cmax problem.- 4.3. n|1??ci problems.- 4.3.1. The n|1??wiCi problem.- 4.3.2. The n|1??wiTi problem.- 4.3.3. The general n|1??Ci problem.- 4.3.3.1. Elimination criteria.- 4.3.3.2. A branch-and-bound algorithm.- 4.4. n|1|?|?Ci problems.- 4.4.1. The n|1|ri ? 0|?Ci problem.- 4.4.2. The n|1|seq dep|?Ci problem.- 4.4.3. The n|1|prec|?Ci problem.- 5. Two-Machine and Three-Machine Problems.- 5.1. The n|2|?,?|cmax and n|3|?,?|cmax problem.- 5.2. The n|2|F|?Ci problem.- 5.3. The n|2|P|Cmax problem with time lags.- 6. General Flow-Shop and Job-Shop Problems.- 6.1. The n|m|P|? problem.- 6.1.1. Elimination criteria for the n|m|P|Cmax problem.- 6.1.2. Lower bounds for the n|m|P|cmax problem.- 6.2. The n|m|F|? problem.- 6.3. The n|m|G|? problem.- 6.3.1. Lower bounds.- 6.3.2. Branching rules.- 6.3.2.1. The procedure 'actsched'.- 6.3.2.2. Branching on disjunctive arcs.- 6.4. The n|m|?, no wait|? problem.- 7. Concluding Remarks.- 7.1. Complexity of scheduling problems.- 7.2. Practical scheduling problems.- 7.3. Conclusions.- List Of Notations.- References.- Author Index.

Machine Scheduling Problems: Classification, complexity and computations

In a capacitated routing problem, the objective is to minimize the total distance travelled by vehicles of limited capacity to serve a set of customers that are located in the Euclidean plane. We develop asymptotically optimal bounds and heuristics for this problem, under the assumption that the capacity of a vehicle is expressed in terms of an upper bound on the number of customers that it can serve. The analysis culminates in an algorithm that, for a given capacity and given ϵ, will find a solution with relative error at most ϵ in time polynomial in the number of customers.

A. H. G. Rinnooy Kan

Papers

Handbooks in operations research and management science

The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization

Sequencing and scheduling: algorithms and complexity

Machine Scheduling Problems: Classification, complexity and computations

Bounds and Heuristics for Capacitated Routing Problems