Idempotent convexity and algebras for the capacity monad and its submonads

Idempotent analogues of convexity are introduced. It is proved that the category of algebras for the capacity monad in the category of compacta is isomorphic to the category of $(\max,\min)$-idempotent biconvex compacta and their biaffine maps. It is also shown that the category of algebras for the monad of sup-measures ($(\max,\min)$-idempotent measures) is isomorphic to the category of $(\max,\min)$-idempotent convex compacta and their affine maps

Approximation algorithms for the joint replenishment problem with deadlines

The Joint Replenishment Problem ($${\hbox {JRP}}$$JRP) is a fundamental optimization problem in supply-chain management, concerned with optimizing the flow of goods from a supplier to retailers. Over time, in response to demands at the retailers, the supplier ships orders, via a warehouse, to the retailers. The objective is to schedule these orders to minimize the sum of ordering costs and retailers’ waiting costs. We study the approximability of $${\hbox {JRP-D}}$$JRP-D, the version of $${\hbox {JRP}}$$JRP with deadlines, where instead of waiting costs the retailers impose strict deadlines. We study the integrality gap of the standard linear-program (LP) relaxation, giving a lower bound of $$1.207$$1.207, a stronger, computer-assisted lower bound of $$1.245$$1.245, as well as an upper bound and approximation ratio of $$1.574$$1.574. The best previous upper bound and approximation ratio was $$1.667$$1.667; no lower bound was previously published. For the special case when all demand periods are of equal length, we give an upper bound of $$1.5$$1.5, a lower bound of $$1.2$$1.2, and show APX-hardness

Approximation algorithms for the joint replenishment problem with deadlines

The Joint Replenishment Problem ((Formula presented.)) is a fundamental optimization problem in supply-chain management, concerned with optimizing the flow of goods from a supplier to retailers. Over time, in response to demands at the retailers, the supplier ships orders, via a warehouse, to the retailers. The objective is to schedule these orders to minimize the sum of ordering costs and retailers' waiting costs. We study the approximability of (Formula presented.), the version of (Formula presented.) with deadlines, where instead of waiting costs the retailers impose strict deadlines. We study the integrality gap of the standard linear-program (LP) relaxation, giving a lower bound of (Formula presented.), a stronger, computer-assisted lower bound of (Formula presented.), as well as an upper bound and approximation ratio of (Formula presented.). The best previous upper bound and approximation ratio was (Formula presented.); no lower bound was previously published. For the special case when all demand periods are of equal length, we give an upper bound of (Formula presented.), a lower bound of (Formula presented.), and show APX-hardness. © 2014 Springer Science+Business Media New York

Nash equilibria for games in capacities

This paper provides a formal generalization of Nash equilibrium for games under Knightian uncertainty. The paper is devoted to counterparts of the results of Glycopantis and Muir (Econ Theory 13:743-751, 1999, Econ Theory 16:239-244, 2000) for capacities. We prove that the expected payoff defined as the integral of a payoff function with respect to the tensor product of capacities on compact Hausdorff spaces of pure strategies is continuous if so is the payoff function. We prove also an approximation theorem for Nash equilibria when the expected utility payoff functions are defined on the space of capacities