Dynamic journeying under uncertainty

We introduce a journey planning problem in multi-modal transportation networks under uncertainty. The goal is to find a journey, possibly involving transfers between different transport services, from a given origin to a given destination within a specified time horizon. Due to uncertainty in travel times, the arrival times of transport services at public transport stops are modeled as random variables. If a transfer between two services is rendered unsuccessful, the commuter has to reconsider the remaining path to the destination. The problem is modeled as a Markov decision process in which states are defined as paths in the transport network. The main contribution is a backward induction method that generates an optimal policy for traversing the public transport network in terms of maximizing the probability of reaching the destination in time. By assuming history independence and independence of successful transfers between services we obtain approximate methods for the same problem. Analysis and numerical experiments suggest that while solving the path dependent model requires the enumeration of all paths from the origin to the destination, the proposed approximations may be useful for practical purposes due to their computational simplicity. In addition to on-time arrival probability, we show how travel and overdue costs can be taken into account, making the model applicable to freight transportation problems.     

Online purchasing under uncertainty

Suppose there is a collection of independent uniform random variables, and a hypergraph of target structures on the vertex set . We would like to purchase a target structure at small cost, but we do not know all the costs x_i ahead of time. Instead, we inspect the random variables x_i one at a time, and after each inspection, choose to either keep the vertex i at cost x_i, or reject vertex i forever. In the present paper, we consider the case where is the edge‐set of a complete graph (or digraph), and the target structures are the spanning trees of a graph, spanning arborescences of a digraph, the paths between a fixed pair of vertices, perfect matchings, Hamilton cycles or the cliques of some fixed size.     

Forestry management under uncertainty

The forest harvest and road construction planning problem consists fundamentally of managing land designated for timber production and divided into harvest cells. For each time period the planner must decide which cells to cut and what access roads to build in order to maximize expected net profit. We have previously developed deterministic mixed integer linear programming models for this problem. The main contribution of the present work is the introduction of a multistage Stochastic Integer Programming model. This enables the planner to make more robust decisions based on a range of timber price scenarios over time, maximizing the expected value instead of merely analyzing a single average scenario. We use a specialization of the Branch-and-Fix Coordination algorithmic approach. Different price and associated probability scenarios are considered, allowing us to compare expected profits when uncertainties are taken into account and when only average prices are used. The stochastic approach as formulated in this work generates solutions that were always feasible and better than the average solution, while the latter in many scenarios proved to be infeasible.     

Optimisation under hybrid uncertainty

Fuzzy Stochastic Optimisation is emerging as a subfield of Mathematical programming, the disciplinary matrix of which consists of analysis of mathematical programs under fuzziness and randomness along with methods for solving them. The “primum movens” of this paper is to describe a unifying methodological approach, that is suitable for finding a satisfying solution of a mathematical program in the presence of fuzzy data and random variables. Properties of fuzzy random variables (FRVs) serve as the backdrop to this approach which also lends itself better to handling mathematical programs with fuzzy random coefficients. For the paper to be somewhat self-contained, the notion of FRV is briefly discussed and a synopsis of Fuzzy Stochastic Optimisation provided. A systematically solved example aimed at illustrating the proposed approach is also included.     

Cooperation under interval uncertainty

In this paper, the classical theory of two-person cooperative games is extended to two-person cooperative games with interval uncertainty. The core, balancedness, superadditivity and related topics are studied. Solutions called ψ ^α-values are introduced and characterizations are given.     

Responsive pricing under supply uncertainty

Consider a retailer orders a seasonal product from a supplier and sells the product over a selling season. While the product demand is known to be a linear function of price, the supply yield is uncertain and is distributed according to a general discrete probability distribution. This paper presents a two-stage stochastic model for analyzing two pricing policies: No Responsive Pricing and Responsive Pricing. Under the No Responsive Pricing policy, the retailer would determine the order quantity and the retail price before the supply yield is realized. Under the Responsive Pricing policy, the retailer would specify the order quantity first and then decide on the retail price after observing the realized supply yield. Therefore, the Responsive Pricing policy enables the retailer to use pricing as a response mechanism for managing uncertain supply. Our analysis suggests that the retailer would always obtain a higher expected profit under the Responsive Pricing policy. In addition to examining the impact of yield distribution and system parameters on the optimal order quantities, retail prices, and profits under these two pricing policies, we analyze two issues arising from responsive pricing. The first issue deals with the case in which the retailer can place an emergency order with an alternative source after observing the realized yield, while the second issue deals with a situation in which the retailer has to allocate his order among multiple suppliers.     

Comparative performance evaluation under uncertainty

In this article CCPR, a multidimensional framework for comparative performance evaluation is proposed, which is elaborated and illustrated through a real-life case. A particular feature of the approach is that it takes account of and corrects for the influence of risks, which are beyond the control of the decision maker. Here risk is seen as a multidimensional measure; it is expressed by means of sensitivities to unexpected changes of a multitude of risk factors. Furthermore, the approach corrects for differences in characteristics between the firms that are being compared. Some characteristics are fixed and thus uncontrollable for the decision maker, whereas others may be changed and controlled by the decision maker. The approach also answers whether the changes in firm characteristics made by management have been appropriate or not.     

An outranking method under uncertainty

The uncertainty of consequences and the imprecision of data often imply, in multicriteria decision problems, the use of probability distributions to characterize the evaluation of each action with respect to eacg criterion. To keep as much information as possible, the analysis should treat directly these probability distributions instead of reducing them to single values such as mean or median. In this context, the paper proposes a multicriteria procedure which transforms these distributive evaluations of actions, according to decisionmaker's preferences, in order to progress to a ranking of these actions. The procedure consists, for each couple of actions, to construct a distributive preference degree with respect to each criterion and a distributive outranking degree over all criteria. These distributive outranking degrees are then explored in order to rank the actions, totally or partially.     

Optimal cash management under uncertainty

We solve an agent’s optimization problem of meeting demands for cash over time with cash deposited in bank or invested in stock. The stock pays dividends and uncertain capital gains, and a commission is incurred in buying and selling of stock. We use a stochastic maximum principle to obtain explicitly the optimal transaction policy.     

Location choices under quality uncertainty

We examine a linear city duopoly where firms choose their locations to maximize expected profits, uncertain about how consumers will assess the relative quality of their products. Equilibrium locations depend on the ratio of the expected quality superiority and the strength of horizontal differentiation. When this ratio is small, firms locate at opposite endpoints. As it becomes larger, agglomeration also emerges as an equilibrium with both firms choosing the same location within an interval around the center. Eventually, when the ratio is large enough, agglomeration becomes the only equilibrium and can occur at any point of the linear city.     

Outranking under uncertainty using scenarios

This paper considers the use of scenarios to treat uncertain attribute evaluations in the outranking methods. The scenario-based approach allows the decision maker to think deterministically about the problem by attaching causal links to a small number of potential outcomes, instead of using probability distributions. The scenario approach can be expressed as a simplified version of the comprehensive but practically complex “distributive” outranking method of d’Avignon and Vincke. Using a scenario approach has distinct practical advantages, but also presents the inherent danger that meaningful information is ignored. The extent of this danger is assessed using a simulation experiment, where it is found to be of a magnitude that is non-trivial but still potentially acceptable for certain decision contexts.     

Flexible waste management under uncertainty

In this paper, we use stochastic dynamic programming to model the choice of a municipality which has to design an optimal waste management program under uncertainty about the price of recyclables in the secondary market. The municipality can, by undertaking an irreversible investment, adopt a flexible program which integrates the existing landfill strategy with recycling, keeping the option to switch back to landfilling, if profitable. We determine the optimal share of waste to be recycled and the optimal timing for the investment in such a flexible program. We find that adopting a flexible program rather than a non-flexible one, the municipality: (i) invests in recycling capacity under circumstances where it would not do so otherwise; (ii) invests earlier; and (iii) benefits from a higher expected net present value.     

Cooperative games under bubbly uncertainty

The allocation problem of rewards/costs is a basic question for players, namely, individuals and companies that are planning cooperation under uncertainty. The involvement of uncertainty in cooperative game theory is motivated by the real world in which noise in observation and experimental design, incomplete information and vagueness in preference structures and decision-making play an important role. In this study, a new class of cooperative games, namely, the cooperative bubbly games, where the worth of each coalition is a bubble instead of a real number, is presented. Furthermore, a new solution concept, the bubbly core, is defined. Finally, the properties and the conditions for the non-emptiness of the bubbly core are given. The paper ends with a conclusion and an outlook to related and future studies.     

Multicriteria decision making under uncertainty

Formalization for problems of multicriteria decision making under uncertainty is constructed in terms of guaranteed and weak estimates. A relevant definition of the vector maximinimax value is given. Parameterization and approximation of maximum, minimax, and maximinimax values based on the inverse logical convolution are suggested. An application for multicommodity networks is considered. Received: December 13, 2000 / Accepted: August 21, 2001?Published online May 8, 2002