Compact Mixed-integer Programming Formulations Research Articles

Two extended formulations for the virtual network function placement and routing problem

AbstractGiven a bi‐directed graph modeling a telecommunication network, and a set of origin‐destination pairs representing traffic requests (commodities) along with their associated Service Function Chains (SFCs), the Virtual Network Function Placement and Routing Problem (VNFPRP) aims to find, for each commodity, one latency‐constrained routing path that visits the required Virtual Network Functions in a specific order. The function installation costs together with the node activation costs have to be minimized. In this paper, we present two extended Mixed Integer Programming (MIP) formulations to model the VNFPRP. For each formulation we define the master problem, the pricing problem, the associated Lagrangian bound and a specific branching scheme, in order to derive an efficient Branch‐and‐Price algorithm. We also provide several families of valid inequalities to strengthen the LP‐relaxation bounds. Computational results are reported comparing the performance of the two Branch‐and‐Price algorithms with a compact MIP formulation and its Branch‐and‐Benders‐cut implementation on a set of SNDlib instances representing telecommunication networks.

Compact mixed-integer programming formulations in quadratic optimization

Compact mixed-integer programming formulations in quadratic optimization

The two-echelon vehicle routing problem with covering options: City logistics with cargo bikes and parcel lockers

We introduce the two-echelon vehicle routing problem with covering options (2E-VRP-CO). This problem arises in sustainable applications for e-commerce and city distribution. In the first echelon, trucks depart from a single depot and transport goods to two types of locations. At covering locations, such as parcel lockers, customers can pick up goods themselves. At satellite locations, goods are transferred to zero-emission vehicles (such as cargo bikes) that deliver to customers. If desired, customers can indicate their choice for delivery. The 2E-VRP-CO aims at finding cost-minimizing solutions by selecting locations and routes to serve all customers. We present a compact mixed integer programming formulation and an efficient and tailored adaptive large neighborhood search heuristic that provides high-quality, and often optimal, solutions to the 2E-VRP-CO. The 2E-VRP-CO has as special cases the two-echelon vehicle routing problem, and the simultaneous facility location and vehicle routing problem without duration constraints. On these special cases, for which our heuristic predominantly solves the established benchmark instances either to optimality or to the best-known solution, our heuristic finds three new best-known solutions. Moreover, we introduce a new set of benchmark instances for the 2E-VRP-CO and provide managerial insights when distribution via both satellite and covering locations is most beneficial. Our results indicate that customers in the same area are best-served either via cargo-bikes or parcel lockers (i.e., not both), and that the use of parcel lockers has a great potential to reduce driving distance.

In This Issue

In This Issue

Runway sequencing with holding patterns

We study a scheduling problem, motivated by air-traffic control. When aircraft reach the final descent in the “Terminal Radar Approach CONontrol” area ( tracon), a set of disjoint time windows in which the landing is possible, can be automatically assigned to each aircraft. The objective is then to determine landing times, within these time windows, which maximize the minimum time elapsed between consecutive landings. We study the complexity of the problem and describe several special cases that can be solved in polynomial time. We also provide a compact Mixed Integer Programming formulation that allows us to solve large instances of the general problem when all time windows have the same size. Finally, we introduce a general hybrid branch and cut framework to solve the problem with arbitrary time windows. Experimental results show that our approach outperforms earlier formulation of the problem.