Concave Cost Research Articles

A Lagrange barrier approach for the minimum concave cost supply problem via a logarithmic descent direction algorithm

The minimisation of concave costs in the supply chain presents a challenging non-deterministic polynomial (NP) optimisation problem, widely applicable in industrial and management engineering. To approximate solutions to this problem, we propose a logarithmic descent direction algorithm (LDDA) that utilises the Lagrange logarithmic barrier function. As the barrier variable decreases from a high positive value to zero, the algorithm is capable of tracking the minimal track of the logarithmic barrier function, thereby obtaining top-quality solutions. The Lagrange function is utilised to handle linear equality constraints, whilst the logarithmic barrier function compels the solution towards the global or near-global optimum. Within this concave cost supply model, a logarithmic descent direction is constructed, and an iterative optimisation process for the algorithm is proposed. A corresponding Lyapunov function naturally emerges from this descent direction, thus ensuring convergence of the proposed algorithm. Numerical results demonstrate the effectiveness of the algorithm.

Compensatory traits can explain the concave cost function of purely sexual traits.

The cost of ornamentation is often measured experimentally to study the relative importance of sexual and viability selection for ornamentation, but these experiments can lead to a misleading conclusion when compensatory trait is ignored. For example, a classic experiment on the outermost tail feathers in the barn swallow Hirundo rustica explains that the concave (or U-shaped) aerodynamic performance cost of the outermost tail feathers would be the evolutionary outcome through viability selection for optimal tail length, but this conclusion depends on the assumption that compensatory traits do not cause reduced performance. Using a simple "toy model" experiment, I demonstrated that ornamentation evolved purely though sexual selection can produce a concave cost function under the presence of compensatory traits, which was further reinforced by a simple mathematical model. Therefore, concave cost function (and the low performance of individuals with reduced ornaments) cannot be used to infer the evolutionary force favoring ornamentation, due to a previously overlooked concept, "overcompensation," which can worsen the whole body performance.

Smoother Entropy for Active State Trajectory Estimation and Obfuscation in POMDPs

We study the problem of controlling a partially observed Markov decision process (POMDP) to either aid or hinder the estimation of its state trajectory. We encode the estimation objectives via the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">smoother entropy</i> , which is the conditional entropy of the state trajectory given measurements and controls. Consideration of the smoother entropy contrasts with previous approaches that instead resort to marginal (or instantaneous) state entropies due to tractability concerns. By establishing novel expressions for the smoother entropy in terms of the POMDP belief state, we show that both the problems of minimising and maximising the smoother entropy in POMDPs can surprisingly be reformulated as belief-state Markov decision processes with concave cost and value functions. The significance of these reformulations is that they render the smoother entropy a tractable optimisation objective, with structural properties amenable to the use of standard POMDP solution techniques for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">both</i> active estimation and obfuscation. Simulations illustrate that optimisation of the smoother entropy leads to superior trajectory estimation and obfuscation compared to alternative approaches.

The Generalized c/μ Rule for Queues with Heterogeneous Server Pools

Dynamic Routing of Queues with Heterogeneous Server Pools In “The Generalized c/μ Rule for Queues with Heterogeneous Server Pools,” Long, Zhang, Zhang, and Zhang study the optimal control of queueing systems with heterogeneous server pools and a single customer class. The goal is to balance the holding cost of the queue with the operating costs of the server pools. They introduce the target-allocation policy, the Gc/μ rule, and the fixed priority policy for systems with general, convex, and concave cost functions, respectively. They also consider an extension to minimize operating costs and maintain a service-level target for customers waiting in the queue. Moreover, they show that their asymptotically optimal routing policies coincide with several classic policies in the literature in special cases.

Generic catastrophic poverty when selfish investors exploit a degradable common resource.

The productivity of a common pool of resources may degrade when overly exploited by a number of selfish investors, a situation known as the tragedy of the commons. Without regulations, agents optimize the size of their individual investments into the commons by balancing incurring costs with the returns received. The resulting Nash equilibrium involves a self-consistency loop between individual investment decisions and the state of the commons. As a consequence, several non-trivial properties emerge. For N investing actors we prove rigorously that typical payoffs do not scale as 1/N, the expected result for cooperating agents, but as (1/N)2. Payoffs are hence reduced with regard to the functional dependence on N, a situation denoted catastrophic poverty. We show that catastrophic poverty results from a fine-tuned balance between returns and costs. Additionally, a finite number of oligarchs may be present. Oligarchs are characterized by payoffs that are finite and not decreasing when N increases. Our results hold for generic classes of models, including convex and moderately concave cost functions. For strongly concave cost functions the Nash equilibrium undergoes a collective reorganization, being characterized instead by entry barriers and sudden death forced market exits.

The Cognitive Load of Financing Constraints: Evidence from Large-Scale Wage Surveys

In this paper, we take advantage of the implicit cognitive exercise available in standard Labor Force Surveys to propose a new indicator of financing constraints which is based on the cognitive load they generate (Mullainathan and Shafir, 2013). Survey respondents are requested to report their monthly wages, which we compare to their administrative, fiscal counterparts. We propose a well-defined index of worker-level uncertainty, which filters out their potential rounding behavior and reporting biases. We estimate it using unsupervised ML/EM techniques and find that workers tend to perceive their own wages with a degree of uncertainty of around 10%. Through the lens of a simple rational signal extraction model, this amounts to estimates of workers' attention ranging from 30% to 84% depending on their wage, education, tenure and gender. Most importantly, we show that the attention of the lowest paid 30% of workers is cyclical and increases steadily by 17 percentage points in the ten days preceding payday, before immediately dropping on that day, which, through the lens of a simple model, is indicative of end-of-month financing liquidity constraints. Furthermore, this pattern reveals that the cognitive cost induced by these financing constraints arises from the not too concave (or convex) costs of achieving high levels of attention, and the convex costs of maintaining it over time.

Distribution of decarbonization costs and externality regulation

The transformation of the economy to a low-carbon level is constrained by a high level of costs and the problem of balancing interests in the distribution of these costs among participants. The paper proposes to use the corporate game theory, in particular the Shapley value, for cost allocation. In contrast to the classical division of additional utility for cooperative games, in this case the costs are divided, and the concave cost function is minimized. The Shapley value coordinates determine the center of gravity of the multidimensional figure of possible cost distributions and are associated with a formal representation of fairness without taking into account financial opportunities and additional, historically established conditions. A comparative analysis of two approaches (carbon tax and carbon credits) to managing negative externalities shows that it is preferable to use the Shapley value for the base allocation of carbon credits. Both approaches to emissions regulation are constrained by institutional barriers to the transformation of the economy: the level of development of national institutions, the achievement of international agreements in the face of economic competition and political confrontation, and the lack of objective information. The authors come to the conclusion that the use of the Shapley value can contribute to the objective formation of quotas and reduce barriers to decarbonization.

Computer Programming Language Solution to a Transportation Problem Involving a Concave Cost Function

The work is on computer programming language solution to a transportation problem involving a concave cost function. Two computer programming languages; Wolfram Mathematica and Anaconda Python programming tools were employed in this study to effectively solve four real life examples from published works. The results from the programming languages yielded an optimal value of N253,000 with an optimal solution as z12 = 13, z22 = 5, z23 = 8, z31 = 11 and z33 = 4 in the first example and the remaining three examples were successfully solved with optimal values of N377,000, GH¢ 236,000 and N509,000 respectively, and the results agreed with the results of existing Karush-Kuhn-Tucker (KKT) procedure of Modified Distribution Method.

Simulating the Impact of the Sustained Melting Arctic on the Global Container Sea–Rail Intermodal Shipping

Global warming trends and the rapid reduction of summer Arctic sea ice extent have increased the feasibility of transarctic transport. How the process of glacier melting affects the existing containerized sea–rail shipping network and container flow assignment has become a challenging economic and policy issue. This paper first examines the meteorological influences on glacier melting and the assignment of container flow over the existing sea–rail network. Then, a three-layer simulation framework is constructed, with the upper layer simulating glacier melting based on the raster grid, the middle layer combining a grid and topology analysis to simulate the evolution of the global sea–rail network and the lower layer establishing a concave cost network flow model to simulate the container flow assignment. Finally, we use MicroCity to achieve the dynamic optimization and simulation of global container flow assignment, solving the large-scale sea–rail shipping network traffic assignment problem. The simulation results show that the proposed model and solution algorithm are feasible and effective, revealing the variation of container flow assignment in the global sea–rail shipping network under different Arctic ice melting scenarios. For instance, in the summer of 2050, the Arctic routes will share the global container flows, resulting in a significant reduction of container flows in the Malacca Strait, Suez Canal and Panama Canal.

Measurement-outlier robust Kalman filter for discrete-time dynamic systems

This paper proposes a recursive filter for discrete-time linear dynamic systems subject to output outliers or heavy-tailed noises. First, we introduce a weight matrix in the conventional MAP estimation. It is shown that this matrix plays an influential role in the innovation whitening and asymptotic variance of the modified MAP estimation and, consequently, can be used in outlier detection. Then, we propose two different constrained optimization problems to obtain this weight. These constraints, stemming from environmental noise characteristics, help to obtain the weight matrix more precisely, which increases the filtering performance significantly. In the first approach, we introduce a convex optimization problem to minimize the estimation upper bound of the error covariance matrix. The second approach converts the modified MAP estimation to a min–min optimization problem with a concave cost function. Consequently, to reduce the effect of outliers in estimation, a semidefinite program (SDP) is proposed for outlier detection. At last, simulation results show the effectiveness and verify the performance of the proposed filter for dynamic systems in the presence of measurement outliers.

A dynamic programming approach for the two-product capacitated lot-sizing problem with concave costs

In this paper, we analyze a two-product multi-period dynamic lot-sizing problem with a fixed capacity constraint in each period. Each product has a known demand in each period that must be satisfied over a finite planning horizon. The aim of this problem is to minimize the overall cost of placing orders and carrying inventory across all periods. The structure of an optimal solution is analyzed with respect to a type of period called regeneration period, which is a period where the inventory of one or both products reach zero. We show that there is an optimal arrangement of placing orders between consecutive regeneration periods. We propose a pseudo-polynomial algorithm to solve the two-product problem. First, we show how the optimal ordering pattern between two consecutive regeneration periods can be solved using a shortest path problem. Then, we explain how the optimal locations for regeneration periods can be found by solving a shortest path problem on a different network, where each arc corresponds to the shortest path in a subproblem network. We then show how this approach can be scaled up to a three-product problem and generalize this technique to any number of products, as long as it is small.

Two-Level Capacitated Discrete Location with Concave Costs

In this paper, we study a general class of two-level capacitated discrete location problems with concave costs. The concavity arises from the economies of scale in production, inventory, or handling at the facilities and from the consolidation of flows for transportation and transshipment on the links connecting the facilities. Given the discrete nature of the problem, it is naturally formulated as a mixed-integer nonlinear program that uses binary variables for locational decisions and continuous variables for routing flows. We present an alternative formulation that only uses continuous variables and discontinuous functions, resulting in a nonlinear program with a concave objective function. Our main goal is to computationally compare these two modeling approaches under the same solution framework. In particular, we present an exact branch-and-bound algorithm that uses (integer) linear relaxations of the proposed formulations to optimally solve large-scale instances. The algorithm is enhanced with a cost-dependent spatial branching strategy and preprocessing step to improve its convergence. Extensive computational experiments are performed to assess the performance of the exact algorithm. Based on real location data from 3,109 counties in the contiguous United States, we also present a sensitivity analysis to showcase the impact of considering concave costs in location and assignment decisions.

Formulation of branched transport as geometry optimization

The branched transport problem, a popular recent variant of optimal transport, is a non-convex and non-smooth variational problem on Radon measures. The so-called urban planning problem, on the contrary, is a shape optimization problem that seeks the optimal geometry of a street or pipe network. We show that the branched transport problem with concave cost function is equivalent to a generalized version of the urban planning problem. Apart from unifying these two different models used in the literature, another advantage of the urban planning formulation for branched transport is that it provides a more transparent interpretation of the overall cost by separation into a transport (Wasserstein-1-distance) and a network maintenance term, and it splits the problem into the actual transportation task and a geometry optimization.

Solving the non-linear multi-index transportation problems with genetic algorithms

In this paper we study the non-linear multi-index transporta\-tion problem with concave cost functions. We solved the non-linear transportation problem on a network with 5 indices (NTPN5I) described by sources, destinations, intermediate nodes, types of products, and types of transport, that is formulated as a non-linear transportation problem on a network with 3 indices (NTPN3I) described by arcs, types of products, and types of transport. We propose a genetic algorithm for solving the large-scale problems in reasonable amount of time, which was proven by the various tests shown in this paper. The convergence theorem of the algorithm is formulated and proved. The algorithm was implemented in Wolfram Language and tested in Wolfram Mathematica.

Non-Confrontational Extremists

Preference falsification, the act of misrepresenting one’s beliefs under social pressure, is widespread but not ubiquitous. We show that when individuals perceive a concave cost of deviating from their principles, ideological extremists are more likely to falsify preferences. Being in ideological minority, they cave-in on decisions they disagree with ideologically because once they deviate slightly, further deviations entail relatively little additional cost. To this result—ideological perfectionism—which is supported by recent lab experiments, we add the first evidence in a high-stakes field setting regarding which individuals, on an ideological scale, conform under social pressure and which stand their ground.

Cost-aware scheduling on uniform parallel machines

We consider scheduling problems with uniform parallel machines to minimize the sum of the total (weighted) completion time and the total cost for usage of machines. A cost density function is given for each machine in advance. Integrating the cost density function for the time period(s) when the machine is busy, one obtains the cost of using this machine in a schedule. We suggest the classification of the problems under consideration according to the type of the cost density functions. The notions of a simple cost density function and an almost concave cost density function are introduced. For these types of cost density functions, we investigate the properties of optimal schedules. Based on these properties, we develop two families of exact pseudo-polynomial DP algorithms. For the case of two uniform machines, we present an FPTAS. Besides, a polynomial-solvable case of the problem is considered.

The emergence of adaptive diversification from plant's light competition

Light competition, the main force in shaping plant height, is ubiquitous in the plant kingdom. Plant height was commonly treated as a competitive strategy and investigated under the game-theoretic framework. However, how plant heights are shaped by light environment under natural selection has not yet been fully understood. Aiming at revealing the potential mechanism causing the height diversification and complex evolutionary dynamics, we here propose an eco-evolutionary model of the plant competing for light, in which the benefit and cost due to light competition as well as density-dependent selection are considered. We provide the criteria for producing different evolutionary consequences. The results show that the moderate levels of competitive costs boost the diversity of plant height, and that the eco-evolutionary feedback results in complex nature of the height evolution. In the power-type cost, the evolutionary consequence of plant height highly depends on the shape of benefit and cost functions, and the height evolution under convex benefit or concave cost displays a trend of evident diversification, which is most pronounced when the two factors are added together, whereas a homogenization tendency of the height evolution occurs in the opposite cases. Meantime, the classification of evolutionary states under different benefit-cost combinations exhibits distinct structure patterns. Generally, more complex evolutionary dynamics are caused if we consider the Hill-type cost: the multiple evolutionary equilibria and phase transitions between these evolutionary equilibria could occur. These results help us in part understand why plant heights in the plant kingdom are highly diverse.

Estimating Monotone Concave Stochastic Production Frontiers

Recent research shows that the search for Bayesian estimation of concave production functions is a fruitful area of investigation. In this article, we use a flexible cost function that satisfies globally the monotonicity and curvature properties to estimate features of the production function. Specification of a globally monotone concave production function is a difficult task which is avoided here by using the first-order conditions for cost minimization from a globally monotone concave cost function. The problem of unavailable factor prices is bypassed by assuming structure for relative prices in the first-order conditions. The new technique is shown to perform well in a Monte Carlo experiment as well as in an empirical application to rice farming in India.

A new Lagrangian-Benders approach for a concave cost supply chain network design problem

This article presents an extended facility location model for multiproduct supply chain network design, that accounts for concave capacity, transportation, and inventory costs (induced by economies of scale, quantity discounts and risk pooling, respectively). The problem is formulated as a mixed-integer nonlinear program (MINLP) with linear constraints and a large number of separable concave terms in the objective function. We propose a solution approach that combines stabilized Lagrangian relaxation with a novel Benders decomposition. Lagrangian relaxation is used first to decompose the problem by potential warehouse location into low-rank concave minimization subproblems. Benders decomposition is then applied to solve the subproblems by shifting the concave terms to a low-dimensional master problem that is solved effectively through implicit enumeration. Finally, a feasible solution for the original problem is constructed from the partial subproblems solutions by solving a restricted set covering problem. The proposed approach is tested on several instances from the literature and compared against a state-of-the-art solver. Furthermore, a realistic case study is used to demonstrate the impact of concavities in the cost components with extensive sensitivity analyses.

Location and two-echelon inventory network design with economies and diseconomies of scale in facility operating costs

Most existing facility location and network design models assume the facility cost as either a fixed cost term or a linear/concave function of the volume of demand assigned to a facility. In particular, a concave cost function implies decreasing marginal cost and exhibits economies of scale. However, when the assigned demand volume exceeds a certain level, diseconomies of scale may occur due to facility congestions, overuse of resources, increasing operational complexity, and so on, leading to a convex cost function with increasing marginal cost. This paper studies a warehouse–retailer network design problem, which simultaneously optimizes the warehouse locations, the warehouse–retailer assignment, and the multi-echelon inventory replenishment policy to minimize the total cost, including the warehouse location and operating costs, the inventory replenishment cost, and the transportation cost. In this paper, the warehouse operating cost is defined as a general function of the volume of demand assigned to a warehouse, which could be concave first due to economies of scale and then convex due to diseconomies of scale. We formulate this problem as a set-covering model and propose a column generation algorithm to solve its linear relaxation. The pricing problem of the column generation process, although formulated as a nonlinear mixed integer program, has nice structural properties and can be solved efficiently by a branch-and-bound method. To validate the effectiveness and efficiency of the proposed column generation algorithm, we conduct numerical studies based on randomly generated instances. Numerical results also demonstrate the impact of economies and diseconomies of scale on network design decisions and warehouse operating cost.