Assortment planning at a retailer entails both selecting the set of products to be carried and setting inventory levels for each product. We study an assortment planning model in which consumers might accept substitutes when their favorite product is unavailable. We develop an algorithmic process to help retailers compute the best assortment for each store. First, we present a procedure for estimating the parameters of substitution behavior and demand for products in each store, including the products that have not been previously carried in that store. Second, we propose an iterative optimization heuristic for solving the assortment planning problem. In a computational study, we find that its solutions, on average, are within 0.5% of the optimal solution. Third, we establish new structural properties (based on the heuristic solution) that relate the products included in the assortment and their inventory levels to product characteristics such as gross margin, case-pack sizes, and demand variability. We applied our method at Albert Heijn, a supermarket chain in The Netherlands. Comparing the recommendations of our system with the existing assortments suggests a more than 50% increase in profits.

http://home.ku.edu.tr/~gkok/papers/mrc.pdf

Demand Estimation and Assortment Optimization Under Substitution: Methodology and Application

The nonlinear knapsack problem – algorithms and applications

http://www.apmath.spbu.ru/cnsa/pdf/obzor/PatrikssonM2008.pdf

A survey on the continuous nonlinear resource allocation problem

Quantifying the notion of fairness is under-explored when users request different ratios of multiple distinct resource types. A typical example is datacenters processing jobs with heterogeneous resource requirements on CPU, memory, etc. A generalization of max-min fairness to multiple resources was recently proposed in [1], but may suffer from significant loss of efficiency. This paper develops a unifying framework addressing this fairness-efficiency tradeoff with multiple resource types. We develop two families of fairness functions which provide different tradeoffs, characterize the effect of user requests' heterogeneity, and prove conditions under which these fairness measures satisfy the Pareto efficiency, sharing incentive, and envy-free properties. Intuitions behind the analysis are explained in two visualizations of multi-resource allocation.

/pdf/multi-resource-allocation-fairness-efficiency-tradeoffs-in-a-21m844lqvg.pdf

Multi-resource allocation: Fairness-efficiency tradeoffs in a unifying framework

Competitive facility location model with concave demand

In this paper we propose an iterative algorithm for solving a convex quadratic program with one equality constraint and bounded variables. At each iteration, a separable convex quadratic program with the same constraint set is solved. Two variants are analyzed: one that uses an exact line search, and the other a unit step size. Preliminary testing suggests that this approach is efficient for problems with diagonally dominant matrices.

Convex quadratic programming with one constraint and bounded variables

In this paper, we consider the extension of the notion of well-posedness by perturbations, introduced by Zolezzi for optimization problems, to other related variational problems like inclusion problems and fixed-point problems. Then, we study the conditions under which there is equivalence of the well-posedness in the above sense between different problems. Relations with the so-called diagonal well-posedness are also given. Finally, an application to staircase iteration methods is presented.

/pdf/well-posedness-by-perturbations-of-variational-problems-2kvzf7zxrp.pdf

Well-Posedness by Perturbations of Variational Problems

In this Note we extend the notion of well-posedness by perturbations, introduced by Zolezzi for optimization problems, for inclusion and fixed-point problems. Then we investigate conditions under which we have equivalence for this concept between the different problems. An application to methods coupling iteration and approximation is presented.

Well-posedness of variational problems with applications to staircase methods

Extending the Tikhonov regularization method to the fixed point problem for a nonexpansive self mapping P on a real Hilbert space H, generates a family of fixed points u r of strongly nonexpansive self mappings P r on H with positive parameter r tending to 0. If the fixed point set C of P is nonempty, then u r converges strongly to u * the unique solution to some monotone variational inequality on the (closed convex) subset C. The iteration method suitably combined with this regularization generates a sequence that converges strongly to u *. When C is a priori defined by finitely many convex inequality constraints, expressing C as the fixed point set of a suitable nonexpansive mapping and applying the above method lead to an iterative scheme in which each step is decomposed in finitely many successive or parallel projections or proximal computations.

Bernard Lemaire

Papers

Convex quadratic programming with one constraint and bounded variables

Well-Posedness by Perturbations of Variational Problems

Well-posedness of variational problems with applications to staircase methods

From Fixed Point Regularization to Constraint Decomposition in Variational Inequalities