An O(n^2 log^2 n) Time Algorithm for Minmax Regret Minsum Sink on Path Networks

We model evacuation in emergency situations by dynamic flow in a network. We want to minimize the aggregate evacuation time to an evacuation center (called a sink) on a path network with uniform edge capacities. The evacuees are initially located at the vertices, but their precise numbers are unknown, and are given by upper and lower bounds. Under this assumption, we compute a sink location that minimizes the maximum "regret." We present the first sub-cubic time algorithm in n to solve this problem, where n is the number of vertices. Although we cast our problem as evacuation, our result is accurate if the "evacuees" are fluid-like continuous material, but is a good approximation for discrete evacuees

Multi-objective decision-making for dietary assessment and advice

Unhealthy diets contribute substantially to the worldwide burden of non-communicable diseases, such as cardiovascular diseases, cancers, and diabetes. Globally, non-communicable diseases are the leading cause of death, and numbers are still rising, which makes healthy diets a global priority. In Nutrition Research, two fields are particularly relevant for formulating healthier diets: dietary assessment, which assesses food and nutrient intake in order to investigate the relation between diet and disease, and dietary advice, which translates food and nutrient recommendations into realistic food choices. Both fields face complex decision problems: which foods to include in dietary assessment or advice in order to pursue the multiple objectives of the researcher or fulfil the requirements of the consumer. This thesis connects the disciplines of Nutrition Research and Operations Research in order to contribute to formulating healthier diets. In the context of dietary assessment, the thesis proposes a MILP model for the selection of food items for food frequency questionnaires (a crucial tool in dietary assessment) that speeds up the selection process and increases standardisation, transparency, and reproducibility. An extension of this model gives rise to a 0-1 fractional programming problem with more than 200 fractional terms, of which in every feasible solution only a subset is actually defined. The thesis shows how this problem can be reformulated in order to eliminate the undefined fractional terms. The resulting MILP model can solved with standard software. In the context of dietary advice, the thesis proposes a diet model in which food and nutrient requirements are formulated via fuzzy sets. With this model, the impact of various achievement functions is demonstrated. The preference structures modelled via these achievement functions represent various ways in which multiple nutritional characteristics of a diet can be aggregated into an overall indicator for diet quality. Furthermore, for Operations Research the thesis provides new insights into a novel preference structure from literature, that combines equity and utilitarianism in a single model. Finally, the thesis presents conclusions of the research and a general discussion, which discusses, amongst others, the main modelling choices encountered when using MODM methods for optimising diet quality. Summarising, this thesis explores the use of MODM approaches to improve decision-making for dietary assessment and advice. It provides opportunities for better decision-making in research on dietary assessment and advice, and it contributes to model building and solving in Operations Research. Considering the added value for Nutrition Research and the new models and solutions generated, we conclude that the combination of both fields has resulted in synergy between Nutrition Research and Operations Research.</p

Essays in Retail Operations and Humanitarian Logistics

This dissertation introduces and analyzes research problems related to Retail Operations and Humanitarian Logistics. In Retail Operations, the inventory that ends up as unsaleable at primary markets can be significant (up to 20% of the retail product). Thus retailers look for strategies like selling in secondary markets at a discounted price. In such a setting, the decisions of how much to order for a product of limited shelf life and when (if at all) to start selling the product in the secondary market become critical because these decisions not only affect the retailer's cost of procurement and sales revenues obtained from the product but also affect utilization of shelf space, product rollover and assortment decisions of the retailer. Apart from using secondary markets, retailers that sell seasonal products or products with sales horizons shorter than the typical production/procurement lead time also enter into contractual agreements with suppliers. These contracts are in place to share risks associated with unknown or uncertain demand for the product. Presence of such contracts does affect a retailer's order quantity as well as the time to start selling in the secondary market. In our two essays on retail operations, we analyze a retailer's optimal order quantity and when he/she starts selling in the secondary market. We refer to the former as the 'ordering decision' and the latter as the 'timing decision.' These two decisions are studied first without risk sharing contracts in Essay 1, and then in the presence of contracts in Essay 2. In Essay 1, we build a two-stage model with demand uncertainty. The ordering decision is made in the first stage considering cost of procurement and expected sales revenue. The timing decision is made in the second stage and is conditional on the order quantity determined in the first stage. We introduce a new class of aggregate demand model for this model. We study the structural properties of the retailer's timing and ordering problem and identify optimality conditions for the timing decision. Finally, we complement our analytical results with computational experiments and show how retailer's optimal decisions change when problem parameters are varied. In Essay 2, we extend the work in first essay to include the contracts between the retailer and a supplier. In this essay, we introduce a time-based Poisson demand model. We define three di®erent types of contracts and investigate the effect of each of these contracts on the retailer's ordering and timing decisions. We investigate how the analytical structure of the retailer's decision changes in the presence of these contracts. For a given order quantity, we show that the timing decision depends on the type of contract. Our analytical results on the timing decision are complemented with computational experiments where we investigate the impact of contract type on the optimal order quantity of the retailer. In Humanitarian Logistics, non-profit organizations receive several-billion-dollars-worth of donations every year but lack a sophisticated system to handle their complex logistics operations; the absence of expertly-designed systems is one of the significant reasons why there has been a weak link in the distribution of relief aid. The distribution of relief aid is a complex problem as the goal is humanitarian yet at the same time, due to limited resources, the operations have to be efficient. In the two essays on humanitarian logistics, we study the distribution of aid using homogeneous fleet, with and without capacity restrictions. In Essay 3, we discuss routing for relief operations using one vehicle without capacity restrictions. Contrary to the existing vehicle routing models, the key property of our routing models is that the nodes have priorities along with humanitarian needs. We formulate this model with d-Relaxed Priority rule that captures distance and response time. We formulate routing models with strict and relaxed forms of priority restrictions as Mixed Integer Programs (MIP). We derive bounds for this problem and show that this bound is attained in limiting condition for a worst-case example. Finally, we evaluate the optimal solutions on test problems for response time and distance and show that our vehicle routing model with priorities captures the trade-off between distance and response time unlike existing Vehicle Routing Problem (VRP) models without priorities. In Essay 4, we extend the problem dealt in third essay to consider fleet consisting of multiple vehicles (homogeneous) with capacity and route length restrictions. First, we show that the humanitarian aspect imposes additional challenges and develop routing models that capture performance metrics like fill rate, distance traversed, response time and number of victims satisfied. Proposed routing models are formulated as Mixed Integer Programs and are solved to optimality for small test problems. We conduct computational experiment and show that our models perform well on these performance metrics

Weak coverage of a rectangular barrier

Assume n wireless mobile sensors are initially dispersed in an ad hoc manner in a rectangular region. They are required to move to final locations so that they can detect any intruder crossing the region in a direction parallel to the sides of the rectangle, and thus provide weak bar-rier coverage of the region. We study three optimization problems related to the movement of sensors to achieve weak barrier coverage: minimizing the number of sensors moved (MinNum), minimizing the average distance moved by the sensors (MinSum), and minimizing the maximum distance moved by the sensors (

Auction algorithm sensitivity for multi-robot task allocation

We consider the problem of finding a low-cost allocation and ordering of tasks between a team of robots in a d-dimensional, uncertain, landscape, and the sensitivity of this solution to changes in the cost function. Various algorithms have been shown to give a 2-approximation to the MinSum allocation problem. By analysing such an auction algorithm, we obtain intervals on each cost, such that any fluctuation of the costs within these intervals will result in the auction algorithm outputting the same solution

Counting and enumerating optimum cut sets for hypergraph kk-partitioning problems for fixed kk

We consider the problem of enumerating optimal solutions for two hypergraph $k$-partitioning problems -- namely, Hypergraph-$k$-Cut and Minmax-Hypergraph-$k$-Partition. The input in hypergraph $k$-partitioning problems is a hypergraph $G=(V, E)$ with positive hyperedge costs along with a fixed positive integer $k$. The goal is to find a partition of $V$ into $k$ non-empty parts $(V_1, V_2, \ldots, V_k)$ -- known as a $k$-partition -- so as to minimize an objective of interest. 1. If the objective of interest is the maximum cut value of the parts, then the problem is known as Minmax-Hypergraph-$k$-Partition. A subset of hyperedges is a minmax-$k$-cut-set if it is the subset of hyperedges crossing an optimum $k$-partition for Minmax-Hypergraph-$k$-Partition. 2. If the objective of interest is the total cost of hyperedges crossing the $k$-partition, then the problem is known as Hypergraph-$k$-Cut. A subset of hyperedges is a min-$k$-cut-set if it is the subset of hyperedges crossing an optimum $k$-partition for Hypergraph-$k$-Cut. We give the first polynomial bound on the number of minmax-$k$-cut-sets and a polynomial-time algorithm to enumerate all of them in hypergraphs for every fixed $k$. Our technique is strong enough to also enable an $n^{O(k)}p$-time deterministic algorithm to enumerate all min-$k$-cut-sets in hypergraphs, thus improving on the previously known $n^{O(k^2)}p$-time deterministic algorithm, where $n$ is the number of vertices and $p$ is the size of the hypergraph. The correctness analysis of our enumeration approach relies on a structural result that is a strong and unifying generalization of known structural results for Hypergraph-$k$-Cut and Minmax-Hypergraph-$k$-Partition. We believe that our structural result is likely to be of independent interest in the theory of hypergraphs (and graphs).Comment: Accepted to ICALP'22. arXiv admin note: text overlap with arXiv:2110.1481