Discrete Time Horizon Research Articles

Dynamic Mechanism Design with Budget-Constrained Buyers Under Limited Commitment

We study the dynamic mechanism design problem of a seller who repeatedly auctions independent items over a discrete time horizon to buyers who face a cumulative budget constraint. A driving motivation behind our model is the emergence of real-time bidding markets for online display advertising in which such budgets are prevalent. We assume the seller has a strong form of limited commitment: she commits to the rules of the current auction but cannot commit to those of future auctions. We show that the celebrated Myersonian approach that leverages the envelope theorem fails in this setting, and therefore, characterizing the dynamic optimal mechanism seems intractable. Despite these challenges, we derive and characterize a near-optimal dynamic mechanism. To do so, we show that the Myersonian approach is recovered in a corresponding fluid continuous time model in which the time interval between consecutive items becomes negligible. Then we leverage this approach to characterize the optimal dynamic direct-revelation mechanism, highlighting novel incentives at play in settings with buyers’ budget constraints and seller’s limited commitment. We show through a combination of theoretical and numerical results that the optimal mechanism arising from the fluid continuous time model approximately satisfies incentive compatibility for the buyers and is approximately sequentially rational for the seller in the original discrete time model.Supplemental material is available at https://doi.org/10.1287/opre.2018.1830 .

Effective matheuristics for the multi-item capacitated lot-sizing problem with remanufacturing

Effective Mixed Integer Programming (MIP) based matheuristics for the multi-item capacitated lot-sizing problem with remanufacturing (CLSP-RM) are proposed in this paper. This NP-hard problem consists of determining an optimal plan for the capacitated production and remanufacture of multiple items in order to satisfy their deterministic dynamic demands over a discrete time horizon. Two constructive approaches are proposed: a column generation rounding heuristic followed by a Linear Programming (LP)/MIP repairing mechanism, and a relax-and-fix method. A fix-and-optimize local search procedure is also applied to improve the quality of the solutions obtained by the constructive techniques. Computational results show that the new approach combining a column generation rounding heuristic with a fix-and-optimize local search procedure is very competitive with the state-of-the-art heuristic. Specifically, the obtained solutions are at least as good as the best known in the literature for 86.85% of the instances, with new best solutions encountered for 69.72% of them. Moreover, all obtained solutions are within 5.54% of optimality, showing the robustness of the newly proposed approach.

Complex-demand scheduling problem with application in smart grid

We consider the problem of scheduling complex-valued demands over a discretized time horizon. Given a set of users, each user is associated with a set of demands representing different power consumption preferences. A demand is represented by a complex number, a time interval, and a utility value obtained if it is satisfied. At each time slot, the magnitude of the total selected demands should not exceed a given generation capacity. This naturally captures the supply constraints in alternating current (AC) electric systems. In this paper, we consider maximizing the aggregate user utility subject to power supply limits over a time horizon. We present approximation algorithms characterized by the maximum angle ϕ between any two complex-valued demands. More precisely, a PTAS is presented for the case ϕ∈[0,π2], a bi-criteria FPTAS for ϕ∈[0,π-ε] for any polynomially small ε, assuming the number of time slots in the discretized time horizon is a constant. Furthermore, if the number of time slots is part of the input, we present a reduction to the real-valued unsplittable flow problem on a path with only a constant approximation ratio. Finally, we present a practical greedy algorithm for the single time slot case with an approximation ratio of 12cos⁡ϕ2 and a running time complexity of only O(Nlog⁡N), N standing for the aggregate number of user demands, which can be implemented efficiently in practice.

Optimally configuring a measurement system to detect diversions from a nuclear fuel cycle

The civilian nuclear fuel cycle is an industrial process that produces electrical power from the nuclear fission of uranium. Using a measurement system to accurately account for possibly dangerous nuclear material, such as uranium, in a fuel cycle is important because of its possible loss or diversion. A measurement system is defined by a set of measurement methods, or “devices,” used to account for material flows and inventory values at specific locations at facilities in the fuel cycle. We develop a simulation-optimization algorithm and an integer-programming model to find the best, or near-best, resource-limited measurement system with a high degree of confidence. The simulation-optimization algorithm minimizes a weighted sum of false positive and false negative diversion-detection probabilities while accounting for material quantities and measurement errors across a finite, discrete time horizon in hypothetical non-diversion and diversion contexts. In each time period, the estimated cumulative material unaccounted for is compared to a fixed or an optimized threshold value to assess if a “significant amount of material” is lost from a measurement system. The integer-programming model minimizes the population variance of the estimated material loss, i.e., material unaccounted for, in a measurement system. We analyze three potential problems in nuclear fuel cycle measurement systems: (i) given location-dependent device precisions, find the configuration of n devices at n locations ( $$n=3$$ ) that provides the lowest corresponding objective values using the simulation-optimization algorithm and integer-programming model, (ii) find the location at which improving device precision reduces objective values the most using the simulation-optimization algorithm (given the device accuracy is 100%), and (iii) determine the effect of measurement frequency on measurement system configurations and objective values using the simulation-optimization algorithm. We obtain comparable results for each problem at least an order of magnitude faster than existing methods do. Using an optimized, rather than a fixed, detection threshold in the simulation-optimization algorithm reduces the weighted sum of false positive and false negative probabilities.

Stackelberg Equilibrium in a Dynamic Stimulation Model with Complete Information

We consider a stimulation model with Markov dynamics and discounted optimality criteria in case of discrete time and infinite planning horizon. In this model, the regulator has an economic impact on the executor, choosing a stimulating function that depends on the system state and the actions of the executor, who employs positional control strategies. System dynamics, revenues of the regulator and costs of the executor depend on the system state and the executor’s actions. We show that finding an approximate solution of the (inverse) Stackelberg game reduces to solving the optimal control problem with criterion equal to the difference between the revenue of the regulator and the costs of the executor. Here the e-optimal strategy of the regulator is to economically motivate the executor to follow this optimal control strategy.

The min-up/min-down unit commitment polytope

The min-up/min-down unit commitment problem (MUCP) is to find a minimum-cost production plan on a discrete time horizon for a set of fossil-fuel units for electricity production. At each time period, the total production has to meet a forecast demand. Each unit must satisfy minimum up-time and down-time constraints besides featuring production and start-up costs. A full polyhedral characterization of the MUCP with only one production unit is provided by Rajan and Takriti (Minimum up/down polytopes of the unit commitment problem with start-up costs. IBM Research Report, 2005). In this article, we analyze polyhedral aspects of the MUCP with n production units. We first translate the classical extended cover inequalities of the knapsack polytope to obtain the so-called up-set inequalities for the MUCP polytope. We introduce the interval up-set inequalities as a new class of valid inequalities, which generalizes both up-set inequalities and minimum up-time inequalities. We provide a characterization of the cases when interval up-set inequalities are valid and not dominated by other inequalities. We devise an efficient Branch and Cut algorithm, using up-set and interval up-set inequalities.

Optimal Control, Dichotomy, and Closed Range

The paper deals with optimality conditions of variational type for infinite horizon discrete time optimal control problems. We show that for unconstrained problems standard first order conditions e...

On discrete time infinite horizon optimal growth problem

Optimal growth problem is an important optimization problem in the theory of economic dynamics. This paper provides an overview of the main approaches used in the existing literature in solving infinite horizon discrete time optimal growth problem and includes very recent developments.

Mean–variance optimization of discrete time discounted Markov decision processes

In this paper, we study a mean–variance optimization problem in an infinite horizon discrete time discounted Markov decision process (MDP). The objective is to minimize the variance of system rewards with the constraint of mean performance. Different from most of works in the literature which require the mean performance already achieve optimum, we can let the discounted performance equal any constant. The difficulty of this problem is caused by the quadratic form of the variance function which makes the variance minimization problem not a standard MDP. By proving the decomposable structure of the feasible policy space, we transform this constrained variance minimization problem to an equivalent unconstrained MDP under a new discounted criterion and a new reward function. The difference of the variances of Markov chains under any two feasible policies is quantified by a difference formula. Based on the variance difference formula, a policy iteration algorithm is developed to find the optimal policy. We also prove the optimality of deterministic policy over the randomized policy generated in the mean-constrained policy space. Numerical experiments demonstrate the effectiveness of our approach.

A decision support algorithm for assessing the engagement of a demand response program in the industrial sector of the smart grid

Abstract In the industrial sector of the smart grid (SG), a demand response program (DRP) is offered to consumers to motivate them to shift their demand for electricity to the off-peak period. DRP can cause a dilemma for industrial consumers when energy load is decreased since it may disrupt the production process and they may consequently incur losses. Hence, industrial units may choose to accept or reject a DRP. If they choose to engage in a DRP, they may use the available back-up on-site energy resources to access the required amount of energy. Hence, any decision about load curtailment requires a comprehensive assessment of all layers of production and operational management. This paper utilises several methodologies to evaluate the effects of DRP engagement on operational management. Firstly, the Delphi method is employed for extracting and identifying twenty-six criteria embedded in ten operational and production management factors. Secondly, based on these criteria, the production equipment is ranked using the TOPSIS method. This ranking shows which equipment will have less impact on the organisation’s profit as a result of participating in a DRP; but, it will not support production and energy planning which is affected by DRP engagement. So, thirdly, a linear programming (LP) model in a discrete scheduling time horizon is proposed which considers the TOPSIS method output and all the constraints imposed by the DRP and the production resources. Finally, based on the proposed methodology, a decision-making algorithm is designed to assist the operation and energy managers to decide whether to accept or reject the offer to engage in a DRP and if they decide to participate, how to best utilize the available distributed energy resources to regain the energy lost. The main contribution of this paper is the proposed methodology which combines the outcome of the Delphi and TOPSIS methods with a linear optimisation model, the effectiveness of which is clearly demonstrated by the sensitivity analysis.

A dynamic game theoretic framework for process plant competitive upgrade and production planning

A dynamic potential game theoretic production planning framework is presented in which production plants are treated as individual competing entities and competition occurs dynamically over a discrete finite time horizon. A modified Cournot oligopoly with sticky prices provides the basis for dynamic game theoretic competition in a multimarket nonlinear and nonconvex production planning model wherein market price adapts to a value that clears cumulative market supply. The framework is used to investigate a petrochemical refining scenario in which a single inefficient refiner faces elimination by its competitors; we demonstrate that there exist conditions under which the threatened refiner may upgrade itself to become competitive and escape the threat, or alternatively in which the threat of elimination is illegitimate and the refiner is effectively safe in the given market configuration. Globally optimal dynamic Nash equilibrium production trajectories are presented for each case. © 2017 American Institute of Chemical Engineers AIChE J, 2017

The impact of service level constraints in deterministic lot sizing with backlogging

This paper studies the impact of service level constraints in the context of the capacitated and uncapacitated lot sizing problems with deterministic demand and backlogging over a discrete and finite time horizon. Because the standard formulation of these problems cannot distinguish between a backlog and a backorder, we use a reformulation to explicitly make this distinction. We then formulate several service level constraints adapted from the stochastic inventory control literature and introduce new ones. We analyze the impact of these constraints on the structure of the optimal solutions in terms of the number of backorders and total backlog, and on the performance of these solutions in terms of various service level indicators (proportion of periods with no backlog, proportion of backorders over total demand, maximum delay, average unit waiting time for an item). We also consider the introduction of various types of backordering costs (both fixed and duration dependent) and study their impact. We find that they tend to stabilize the structure of solutions, regardless of the service level constraints imposed. Finally, we analyze the impact of a first-in first-out order management policy and show that it helps improving most solutions.

A Matheuristic for the Multivehicle Inventory Routing Problem

We consider the inventory routing problem, in which a supplier has to replenish a set of customers by means of a limited fleet of capacitated vehicles over a discrete time horizon. The goal is to minimize the total cost of the distribution that comprises the inventory cost at the supplier and at the customers and the routing cost. We present a matheuristic that combines a tabu search and mathematical programming formulations. When compared with two exact methods on 640 small instances, the matheuristic finds 192 (48%) optima over the 402 instances with known optima and improves 125 upper bounds. Tested on 240 large instances (with up to 200 customers) for which no optimal solutions are known, it improves the best solution for 220 (92%) of the 240 instances. The online supplement is available at https://doi.org/10.1287/ijoc.2016.0737 .

An algorithm for general infinite horizon lot sizing with deterministic demand

We present an algorithm for solving an infinite horizon discrete time lot sizing problem with deterministic non-stationary demand and discounting of future cost. Besides non-negativity and finite supremum over infinite horizon, no restrictions are placed on single period demands. (In particular, they need not follow any cyclical pattern). Variable procurement cost, fixed ordering cost, and holding cost can be different in different periods. The algorithm uses forward induction and its essence lies in the use of critical periods. Period j following t is the critical period of t if satisfying demands in any subset of the set of periods between t and j, including j and excluding t, from an order in t is not more expensive than satisfying it from an order in a later period and j is the last period with this property. When deciding whether to place an order in period t, all demands from t to its critical period are taken into account.

Distributed Scheduling of Network Connectivity Using Mobile Access Point Robots

In this paper, we consider scenarios where mobility can be exploited to enable reliable communications in wireless networks with scarce resources that are unable to concurrently service their nodes. Specifically, we consider cases where a team of robots operate as mobile access points (APs) that provide service, namely sufficient end-to-end communication routes, to a multihop network of static source nodes which generate data. We introduce the connectivity scheduling problem, a novel framework that combines motion planning of the APs with service scheduling of the source nodes and network routing control so that integrity of communications is guaranteed over time. We formulate the connectivity scheduling problem as a multistage mixed integer programming (MIP) problem, where path planning, service scheduling, and routing decisions are all jointly optimized over a discrete-time horizon. Since MIP problems can grow intractable quickly, we further consider a continuous convex reformulation of the problem and employ sparse optimization techniques, specifically the reweighted $\ell _1$ regularization scheme, to recover the desired integrality structure of the solution. We propose a decentralized method to solve the above relaxation that is based on the recently developed accelerated distributed augmented Lagrangians (ADAL) algorithm. Specifically, we modify ADAL by incorporating in the algorithm the reweighted $\ell _1$ scheme, which enables us to recover the desired sparsity structure of the original MIP at the final solution. Numerical results are presented that validate the effectiveness of the proposed framework.

A heuristic approach for in-pit crusher and conveyor system’s time and location problem in large open-pit mining

Cost efficiency and high reliability of In-Pit Crushing and Conveying (IPCC) system make it more appealing to be used in deeper open-pit mining activities. Determination of the optimum time and location (OT–OL) for applying the IPCC system were always a challenge. Application of mathematical programming approaches suffers from reduced computational efficiency due to a lot of decision variables. The situation will be worse; by adding the IPCC’s time and location variables to these models. Also, the problem involves a lot of variables that each one of them is a function of others. To tackle these, this paper provides a new heuristic approach to find a good quality solution hierarchically (here in two levels). The problem was broken into two parts (i.e. OT and OL). The model works in a discrete time horizon. It has been verified by the data gathered from Sungun copper mine of Iran. Our computational results show that the system is efficient for developed countries with a stable and low interest rate.

Stochastic regret minimization for revenue management problems with nonstationary demands

AbstractWe study an admission control model in revenue management with nonstationary and correlated demands over a finite discrete time horizon. The arrival probabilities are updated by current available information, that is, past customer arrivals and some other exogenous information. We develop a regret‐based framework, which measures the difference in revenue between a clairvoyant optimal policy that has access to all realizations of randomness a priori and a given feasible policy which does not have access to this future information. This regret minimization framework better spells out the trade‐offs of each accept/reject decision. We proceed using the lens of approximation algorithms to devise a conceptually simple regret‐parity policy. We show the proposed policy achieves 2‐approximation of the optimal policy in terms of total regret for a two‐class problem, and then extend our results to a multiclass problem with a fairness constraint. Our goal in this article is to make progress toward understanding the marriage between stochastic regret minimization and approximation algorithms in the realm of revenue management and dynamic resource allocation. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 433–448, 2016

Dynamic Mechanism Design with Budget Constrained Buyers Under Limited Commitment

We study the dynamic mechanism design problem of a seller that repeatedly auctions independent items over a discrete time horizon to buyers that face a cumulative budget constraint. A driving motivation behind our model is the emergence of real-time bidding markets for online display advertising in which such budgets are prevalent. We assume the seller has a strong form of limited commitment: she commits to the rules of the current auction but cannot commit to those of future auctions. We show that the celebrated Myersonian approach that leverages the envelope theorem fails in this setting, and therefore, characterizing the dynamic optimal mechanism appears intractable. In turn, we make three main contributions. First, we show that the Myersonian approach is recovered in a corresponding fluid continuous-time model in which the time interval between consecutive items becomes negligible. Second, we leverage this approach to characterize the optimal dynamic direct-revelation mechanism, highlighting novel incentives at play in settings with buyers' budget constraints and seller's limited commitment. Third, we provide theoretical results that provide support for using the prescription arising from the fluid continuous-time model in the original discrete-time model.

Improving the efficiency of decentralized supply chains with fixed ordering costs

In a decentralized two-stage supply chain where a supplier serves a retailer who, in turn, serves end customers, operations decisions based on local incentives often lead to suboptimal system performance. Operating decisions based on local incentives may in such cases lead to a lack of system coordination, wherein one party’s decisions put the other party and/or the system at a disadvantage. While models and mechanisms for such problem classes have been considered in the literature, little work to date has considered such problems under nonstationary demands and fixed replenishment order costs. This paper models such two-stage problems as a class of Stackelberg games where the supplier announces a set of time-phased ordering costs to the retailer over a discrete time horizon of finite length, and the retailer then creates an order plan, which then serves as the supplier’s demand. We provide metrics for characterizing the degree of efficiency (and coordination) associated with a solution, and provide a set of easily understood and implemented mechanisms that can increase this efficiency and reduce the negative impacts of uncoordinated decisions.

A multi-objective dynamic programming-based metaheuristic to solve a bi-objective unit commitment problem using a multi-objective decoder

The unit commitment problem UCP is a heavily constrained scheduling problem, where the on/off scheduling and production amounts of heterogeneous power production units have to be determined for a discrete time horizon. Due to environmental concerns, the traditional UCP to solely minimise the production cost is no longer adequate, and a second objective, to minimise the gas emissions has to be added to properly model reality. In this paper, we propose an efficient metaheuristic to solve this multi-objective version of the UCP. The proposed method, MO-DYNAMOP, is a generalisation of DYNAMOP DYNAmic programming-based Metaheuristic for Optimisation Problems, a state-of-the-art hybrid optimiser which was successfully applied to the single-objective unit commitment problem. The main difficulty in extending DYNAMOP to the multi-objective UCP is that it uses an indirect representation of solution that gives the on/off scheduling of each unit. The real production amounts are computed by an exact sub-optimiser which minimises the production cost assuming that the on/off scheduling is fixed. Since the sub-optimiser now has to solve a multi-objective problem, each on/off scheduling induces an entire Pareto optimal set of solutions. We handle this complication by assigning an approximation of the corresponding set to each on/off scheduling solution. A comparison study with methods previously proposed in the literature indicates that MO-DYNAMOP performs considerably better on many benchmark instances.