OPTIMAL ORDERING IN A DUAL‐SUPPLIER SYSTEM WITH DEMAND FORECAST UPDATES

We study how an updated demand forecast affects a manufacturer's choice in ordering raw materials. With demand forecast updates, we develop a model where raw materials are ordered from two suppliers—one fast but expensive and the other cheap but slow—and further provide an explicit solution to the resulting dynamic optimization problem. Under some mild conditions, we demonstrate that the cost function is convex and twice‐differentiable with respect to order quantity. With this model, we are able to evaluate the benefit of demand information updating which leads to the identification of directions for further improvement. We further demonstrate that the model applies to multiple‐period problems provided that some demand regularity conditions are satisfied. Data collected from a manufacturer support the structure and conclusion of the model. Although the model is described in the context of in‐bound logistics, it can be applied to production and out‐bound logistics decisions as well.     

Quantitative Comparison of Approximate Solution Sets for Bi‐criteria Optimization Problems*

We present the Integrated Preference Functional (IPF) for comparing the quality of proposed sets of near‐pareto‐optimal solutions to bi‐criteria optimization problems. Evaluating the quality of such solution sets is one of the key issues in developing and comparing heuristics for multiple objective combinatorial optimization problems. The IPF is a set functional that, given a weight density function provided by a decision maker and a discrete set of solutions for a particular problem, assigns a numerical value to that solution set. This value can be used to compare the quality of different sets of solutions, and therefore provides a robust, quantitative approach for comparing different heuristic, a posteriori solution procedures for difficult multiple objective optimization problems. We provide specific examples of decision maker preference functions and illustrate the calculation of the resulting IPF for specific solution sets and a simple family of combined objectives.     

Improving the Numerical Performance of Static and Dynamic Aggregate Discrete Choice Random Coefficients Demand Estimation

The widely used estimator of Berry, Levinsohn, and Pakes (1995) produces estimates of consumer preferences from a discrete‐choice demand model with random coefficients, market‐level demand shocks, and endogenous prices. We derive numerical theory results characterizing the properties of the nested fixed point algorithm used to evaluate the objective function of BLP's estimator. We discuss problems with typical implementations, including cases that can lead to incorrect parameter estimates. As a solution, we recast estimation as a mathematical program with equilibrium constraints, which can be faster and which avoids the numerical issues associated with nested inner loops. The advantages are even more pronounced for forward‐looking demand models where the Bellman equation must also be solved repeatedly. Several Monte Carlo and real‐data experiments support our numerical concerns about the nested fixed point approach and the advantages of constrained optimization. For static BLP, the constrained optimization approach can be as much as ten to forty times faster for large‐dimensional problems with many markets.     

Tractable Consideration Set Structures for Assortment Optimization and Network Revenue Management

Discrete‐choice models are widely used to model consumer purchase behavior in assortment optimization and revenue management. In many applications, each customer segment is associated with a consideration set that represents the set of products that customers in this segment consider for purchase. The firm has to make a decision on what assortment to offer at each point in time without the ability to identify the customer's segment. A linear program called the Choice‐based Deterministic Linear Program (CDLP) has been proposed to determine these offer sets. Unfortunately, its size grows exponentially in the number of products and it is NP‐hard to solve when the consideration sets of the segments overlap. The Segment‐based Deterministic Concave Program with some additional consistency equalities (SDCP+) is an approximation of CDLP that provides an upper bound on CDLP's optimal objective value. SDCP+ can be solved in a fraction of the time required to solve CDLP and often achieves the same optimal objective value. This raises the question under what conditions can one guarantee equivalence of CDLP and SDCP+. In this study, we obtain a structural result to this end, namely that if the segment consideration sets overlap with a certain tree structure or if they are fully nested, CDLP can be equivalently replaced with SDCP+. We give a number of examples from the literature where this tree structure arises naturally in modeling customer behavior.     

Rationalizing Policy Functions by Dynamic Optimization

We derive necessary and sufficient conditions for a pair of functions to be the optimal policy function and the optimal value function of a dynamic maximization problem with convex constraints and concave objective functional. It is shown that every Lipschitz continuous function can be the solution of such a problem. If the maintained assumptions include free disposal and monotonicity, then we obtain a complete characterization of all optimal policy and optimal value functions. This is the case, e.g., in the standard aggregative optimal growth model.     

Coordinating pricing and production decisions for multiple products

We consider a dynamic problem of joint pricing and production decisions for a profit-maximizing firm that produces multiple products. We model the problem as a mixed integer nonlinear program, incorporating capacity constraints, setup costs, and dynamic demand. We assume demand functions to be convex, continuous, differentiable, and strictly decreasing in price. We present a solution approach which is more general than previous approaches that require the assumption of a specific demand function. Using real-world data from a manufacturer, we study problem instances for different demand scenarios and capacities and solve for optimal prices and production plans. We present analytical results that provide managerial insights on how the optimal prices change for different production plans and capacities. We extend some of the earlier works that consider single product problems to the case of multiple products and time variant production capacities. We also benchmark performance of proposed algorithm with a commercial solver and show that it outperforms the solver both in terms of solution quality and computational times.     

Inventory Models with Continuous and Poisson Demands and Discounted and Average Costs

We develop a new, unified approach to treating continuous‐time stochastic inventory problems with both the average and discounted cost criteria. The approach involves the development of an adjusted discounted cycle cost formula, which has an appealing intuitive interpretation. We show for the first time that an (s, S) policy is optimal in the case of demand having a compound Poisson component as well as a constant rate component. Our demand structure simultaneously generalizes the classical EOQ model and the inventory models with Poisson demand, and we indicate the reasons why this task has been a difficult one. We do not require the surplus cost function to be convex or quasi‐convex as has been assumed in the literature. Finally, we show that the optimal s is unique, but we do not know if optimal S is unique.     

Conditional Choice Probability Estimation of Dynamic Discrete Choice Models With Unobserved Heterogeneity

We adapt the expectation–maximization algorithm to incorporate unobserved heterogeneity into conditional choice probability (CCP) estimators of dynamic discrete choice problems. The unobserved heterogeneity can be time‐invariant or follow a Markov chain. By developing a class of problems where the difference in future value terms depends on a few conditional choice probabilities, we extend the class of dynamic optimization problems where CCP estimators provide a computationally cheap alternative to full solution methods. Monte Carlo results confirm that our algorithms perform quite well, both in terms of computational time and in the precision of the parameter estimates.     

Commitment vs. Flexibility

We study the optimal trade‐off between commitment and flexibility in a consump‐ tion–savings model. Individuals expect to receive relevant information regarding tastes and thus they value the flexibility provided by larger choice sets. On the other hand, they also expect to suffer from temptation, with or without self‐control, and thus they value the commitment afforded by smaller choice sets. The optimal commitment problem we study is to find the best subset of the individual's budget set. This problem leads to a principal–agent formulation. We find that imposing a minimum level of savings is always a feature of the solution. Necessary and sufficient conditions are derived for minimum‐savings policies to completely characterize the solution. We also discuss other applications, such as the design of fiscal constitutions, the problem faced by a paternalist, and externalities.     

Employment Behaviour of Marginal Workers: The Roles of Preferences and Opportunities

Abstract. We use structural estimation techniques to analyse labour supply effects of changes in economic incentives for individuals who have just finished vocational rehabilitation in Norway. The complicated and sometimes non‐convex budget sets for this group of marginal workers are accounted for. We also focus on the limitation in the choice sets this group faces. Parametric bootstrap and simulation techniques are applied to construct confidence intervals for the predicted impacts of changes in the economic environment. The results show that there is a small to moderate effect of changes in economic incentives on the chance of vocational rehabilitation bringing individuals back to employment. We also find that individual health status and local labour market conditions are the most important factors affecting the transition from rehabilitation to work.     

Physicians’ Labour Supply: The Wage Impact on Hours and Practice Combinations

Abstract. It is a stated aim to improve physician services in underserved sectors and areas. Increased wages is one instrument for boosting the hours provided by the personnel to the prioritized sub‐markets. This study applies an econometric framework that allows for non‐convex budget sets, non‐linear labour supply curves and imperfect markets with institutional constraints. The labour supply decision is viewed as a choice from a set of discrete alternatives (job packages) in a structural labour supply model estimated on Norwegian micro data. An out‐of‐sample prediction is also presented and evaluated by means of a natural experiment.     

Social Optimal Location of Facilities with Fixed Servers,Stochastic Demand,and Congestion

We consider two capacity choice scenarios for the optimal location of facilities with fixed servers, stochastic demand, and congestion. Motivating applications include virtual call centers, consisting of geographically dispersed centers, walk‐in health clinics, motor vehicle inspection stations, automobile emissions testing stations, and internal service systems. The choice of locations for such facilities influences both the travel cost and waiting times of users. In contrast to most previous research, we explicitly embed both customer travel/connection and delay costs in the objective function and solve the location–allocation problem and choose facility capacities simultaneously. The choice of capacity for a facility that is viewed as a queueing system with Poisson arrivals and exponential service times could mean choosing a service rate for the servers (Scenario 1) or choosing the number of servers (Scenario 2). We express the optimal service rate in closed form in Scenario 1 and the (asymptotically) optimal number of servers in closed form in Scenario 2. This allows us to eliminate both the number of servers and the service rates from the optimization problems, leading to tractable mixed‐integer nonlinear programs. Our computational results show that both problems can be solved efficiently using a Lagrangian relaxation optimization procedure.     

Self‐Selective Social Choice Functions Verify Arrow and Gibbard‐Satterthwaite Theorems

This paper introduces a new notion of consistency for social choice functions, called self‐selectivity, which requires that a social choice function employed by a society to make a choice from a given alternative set it faces should choose itself from among other rival such functions when it is employed by the society to make this latter choice as well. A unanimous neutral social choice function turns out to be universally self‐selective if and only if it is Paretian and satisfies independence of irrelevant alternatives. The neutralunanimous social choice functions whose domains consist of linear order profiles on nonempty sets of any finite cardinality induce a class of social welfare functions that inherit Paretianism and independence of irrelevant alternatives in case the social choice function with which one starts is universally self‐selective. Thus, a unanimous and neutral social choice function is universally self‐selective if and only if it is dictatorial. Moreover, universal self‐selectivity for such functions is equivalent to the conjunction of strategy‐proofness and independence of irrelevant alternatives or the conjunction of monotonicity and independence of irrelevant alternatives again.     

Data‐Driven Classification Using Boundary Observations

Classification is often a critical task for business managers in their decision‐making processes. It is generally more difficult for a classification scheme to produce accurate results when the input domains of the various output classes coincide, to some degree, with one another. In an attempt to address this issue, this article discusses a data‐driven algorithm that identifies the region of coincidence, or overlap, for two‐group classification problems by empirically determining the convex boundary for each group. The results are extendable to multigroup classification. The class membership of a new observation is then determined by its relative position with respect to each of these boundaries. Due to minimal data storage requirements, this boundary‐point classification technique can adapt to changing conditions far more easily than other approaches. Test results demonstrate that the new classification technique has similar performance to a back‐propagation neural network under static conditions and significantly outperforms a back‐propagation neural network under dynamic conditions.     

Simultaneous Search

We introduce and solve a new class of “downward‐recursive” static portfolio choice problems. An individual simultaneously chooses among ranked stochastic options, and each choice is costly. In the motivational application, just one may be exercised from those that succeed. This often emerges in practice, such as when a student applies to many colleges or when a firm simultaneously tries several technologies. We show that such portfolio choice problems quite generally entail maximizing a submodular function of finite sets—which is NP‐hard in general. Still, we show that a greedy algorithm finds the optimal set, finding first the best singleton, then the best single addition to it, and so on. We show that the optimal choices are “less aggressive” than the sequentially optimal ones, but “more aggressive” than the best singletons. Also, the optimal set in general contains gaps. We provide some comparative statics results on the chosen set.     

Existence and Uniqueness of Solutions to the Bellman Equation in the Unbounded Case

We study the problem of the existence and uniqueness of solutions to the Bellman equation in the presence of unbounded returns. We introduce a new approach based both on consideration of a metric on the space of all continuous functions over the state space, and on the application of some metric fixed point theorems. With appropriate conditions we prove uniqueness of solutions with respect to the whole space of continuous functions. Furthermore, the paper provides new sufficient conditions for the existence of solutions that can be applied to fairly general models. It is also proven that the fixed point coincides with the value function and that it can be approached by successive iterations of the Bellman operator.     

Competing Mechanisms in a Common Value Environment

Consider strategic risk‐neutral traders competing in schedules to supply liquidity to a risk‐averse agent who is privately informed about the value of the asset and his hedging needs. Imperfect competition in this common value environment is analyzed as a multi‐principal game in which liquidity suppliers offer trading mechanisms in a decentralized way. Each liquidity supplier behaves as a monopolist facing a residual demand curve resulting from the maximizing behavior of the informed agent and the trading mechanisms offered by his competitors. There exists a unique equilibrium in convex schedules. It is symmetric and differentiable and exhibits typical features of market‐power: Equilibrium trading volume is lower than ex ante efficiency would require. Liquidity suppliers charge positive mark‐ups and make positive expected profits, but these profits decrease with the number of competitors. In the limit, as this number goes to infinity, ask (resp. bid) prices converge towards the upper (resp. lower) tail expectations obtained in Glosten (1994) and expected profits are zero.     

Distributionally robust maximum probability shortest path problem

In this study, we discuss and develop a distributionally robust joint chance-constrained optimization model and apply it for the shortest path problem under resource uncertainty. In sch a case, robust chance constraints are approximated by constraints that can be reformulated using convex programming. Since the issue we are discussing here is of the multi-resource type, the resource related to cost is deterministic; however, we consider a robust set for other resources where covariance and mean are known. Thus, the chance-constrained problem can be expressed in terms of a cone constraint. In addition, since our problem is joint chance-constrained optimization, we can use Bonferroni approximation to divide the problem into L separate problems in order to build convex approximations of distributionally robust joint chance constraints. Finally, numerical results are presented to illustrate the rigidity of the bounds and the value of the distributionally robust approach.

     

Heuristics for Base‐Stock Levels in Multi‐Echelon Distribution Networks

We study inventory optimization for locally controlled, continuous‐review distribution systems with stochastic customer demands. Each node follows a base‐stock policy and a first‐come, first‐served allocation policy. We develop two heuristics, the recursive optimization (RO) heuristic and the decomposition‐aggregation (DA) heuristic, to approximate the optimal base‐stock levels of all the locations in the system. The RO heuristic applies a bottom‐up approach that sequentially solves single‐variable, convex problems for each location. The DA heuristic decomposes the distribution system into multiple serial systems, solves for the base‐stock levels of these systems using the newsvendor heuristic of Shang and Song (2003), and then aggregates the serial systems back into the distribution system using a procedure we call “backorder matching.” A key advantage of the DA heuristic is that it does not require any evaluation of the cost function (a computationally costly operation that requires numerical convolution). We show that, for both RO and DA, changing some of the parameters, such as leadtime, unit backordering cost, and demand rate, of a location has an impact only on its own local base‐stock level and its upstream locations’ local base‐stock levels. An extensive numerical study shows that both heuristics perform well, with the RO heuristic providing more accurate results and the DA heuristic consuming less computation time. We show that both RO and DA are asymptotically optimal along multiple dimensions for two‐echelon distribution systems. Finally, we show that, with minor changes, both RO and DA are applicable to the balanced allocation policy.     

A Continuous‐Review Inventory Model with Disruptions at Both Supplier and Retailer

We consider a continuous‐review inventory problem for a retailer who faces random disruptions both internally and externally (from its supplier). We formulate the expected inventory cost at this retailer and analyze the properties of the cost function. In particular, we show that the cost function is quasi‐convex and therefore can be efficiently optimized to numerically find the optimal order size from the retailer to the supplier. Computational experiments provide additional insight into the problem. In addition, we introduce an effective approximation of the cost function. Our approximation can be solved in closed form, which is useful when the model is embedded into more complicated supply chain design or management models.