Lower Bound Approach Research Articles

A comparison of different lower bounding procedures for the routing of automated guided vehicles in an urban context

We propose different lower bounding procedures to solve the problem of routing a set of automated guided vehicles in an urban context. This problem consists of designing a set of low-cost roads starting and ending at a depot while satisfying a set of transportation demands in an urban area. The problem is treated under a set of different constraints, such as battery constraints and time window constraints. Our lower bounding approach consists of a decomposition method in which we first solve a relaxed problem. We then deal with the set of infeasible routes obtained from the relaxed problem. The different procedures developed in this paper allow us to find a set of good-quality lower bounds.

Failure mechanism and shear strength of steel–concrete–steel sandwich deep beams

In steel–concrete–steel (SCS11SCS: steel–concrete–steel.) sandwich deep structural members the shear performance becomes rather critical. This paper describes an experimental study on the shear behavior of SCS continuous deep beams. Three beams with different shear spans were tested under anti-symmetric concentrate loads, and the failure pattern is found to be different from reinforced concrete members. Tests show that the shear capacity depends strongly on the steel plates and the shear connectors, and the membrane action of the outer steel plates provides the beams with excellent strength and ductile performance. On the basis of the experimental observations (both the continuous beam tests in this paper and the simple beam tests conducted earlier), a plastic limit analytical model is developed to explain the force transfer mechanism and predict the shear strength of both simple and continuous members. Considering equilibrium and boundary conditions, the lower bound approach shows satisfactory predictions for the shear performance of SCS deep beams.

On the Lower Predictive Bound Approach for Noninferiority Clinical Trials with Binary Data

Clinical trials are mostly designed to provide data that allow direct inference of the underlying hypotheses. This, however, is not the case for noninferiority trials where placebo controls are absent due to ethical considerations, resulting in the need for indirect inference about the underlying hypotheses of the treatment effect of a new drug against a putative placebo control. In particular, the European Committee for Medicinal Products for Human Use (CHMP) guideline for noninferiority trials requires indirect confidence bounds for the “putative” effect of the new drug. To meet this requirement, we develop a new approach by which the lower predictive bound of a given confidence level is used to determine the noninferiority margin. For this approach, we provide methods for calculating the sample size and for making indirect statistical inference (i.e., p-values, confidence intervals, and point estimates) on the putative effect of the new test drug. For an actual trial example, we assess designs by the lower predictive bound approach as well as other approaches for determining noninferiority margins.

Experimental Study and Theoretical Analysis of Resistance of Steel-Concrete-Steel Sandwich Beams

This paper investigates the structural performance of steel-concrete-steel (SCS) sandwich beams interconnected by channel connectors with large interval. Nine simply supported SCS beams with shear span/depth ratio between 1.0 and 5.5 were tested under static load. The contribution of the channel connectors was also studied. All the beams failed because of either tension plate failure in midspan (flexural failure) or tension plate failure in shear span (vertical shear failure). As shear span decreased, failure tended to be governed by shear, whereas for larger shear span, failure was governed by bending. The channel connectors were confirmed to directly affect the vertical shear resistance and ductile performance after critical diagonal cracking. On the basis of the observed failure mode, a mechanical model was developed to predict the ultimate shear resistance of SCS deep beams after critical diagonal cracking. The accuracy of this lower-bound approach was ascertained by comparing with the test results.

Using dual feasible functions to construct fast lower bounds for routing and location problems

In cutting and packing problems, Dual Feasible Functions (DFFs) represent a well established tool for deriving high quality lower bounds in very short times. A well-known DFF lower bounding approach consists of using DFFs to rapidly generate feasible values for the dual variables of a Column Generation (CG) program (i.e., of an extended formulation commonly associated to CG). This paper presents a method for extending this approach to problems such as Capacitated p-Median, Distance Constrained General Routing and related variants (e.g., Capacitated Arc-Routing). In contrast to classical DFF bounds for cutting and packing, our extended DFF bounds deal with non-equal column costs, i.e., the column costs (primal objective function coefficients) have a more complex structure depending on some sums of distances. The general idea is to use a (reformulated) classical CG model in which a feasible dual solution is expressed as a linear combination of both DFF and non-DFF terms. In fact, one of the proposed approaches (the mixed CG-DFF bound for Capacitatedp-Median in Section 2.2) still requires optimizing a restricted CG program, but with fewer variables and easier pricing sub-problems. The most refined bound version is the “DFF warm-started CG” from Section 2.2.2: it takes dual constraints generated while solving the above restricted CG program are re-uses them to warm-start a full CG phase. In the best cases, this can yield a speed-up between 2 and 3 relative to the pure CG. We present numerical experiments on Capacitated p-Median, Capacitated Arc-Routing (with fixed costs) and Distance Constrained Arc Routing; the comparisons with the CG bound concern both the quality and the running time.

Brownian Inventory Models with Convex Holding Cost, Part 1: Average-Optimal Controls

We consider an inventory system in which inventory level fluctuates as a Brownian motion in the absence of control. The inventory continuously accumulates cost at a rate that is a general convex function of the inventory level, which can be negative when there is a backlog. At any time, the inventory level can be adjusted by a positive or negative amount, which incurs a fixed cost and a proportional cost. The challenge is to find an adjustment policy that balances the holding cost and adjustment cost to minimize the long-run average cost. When both upward and downward fixed costs are positive, our problem is an impulse control problem. When both fixed costs are zero, our problem is a singular or instantaneous control problem. For the impulse control problem, we prove that a four-parameter control band policy is optimal among all feasible policies. For the singular control problem, we prove that a two-parameter control band policy is optimal. We use a lower-bound approach, widely known as the “verification theorem”, to prove the optimality of a control band policy for both the impulse and singular control problems. Our major contribution is to prove the existence of a “smooth” solution to the free boundary problem under some mild assumptions on the holding cost function. The existence proof leads naturally to a numerical algorithm to compute the optimal control band parameters. We demonstrate that the lower-bound approach also works for a Brownian inventory model which prohibits an inventory backlog. A companion paper (“Brownian inventory models with convex holding cost, part 2: discount-optimal controls”, Stochastic Systems, Vol. 3, No. 2, pp. 500–573) explains how to adapt the lower-bound approach to study a Brownian inventory model under a discounted cost criterion.

Brownian Inventory Models with Convex Holding Cost, Part 2: Discount-Optimal Controls

We consider an inventory system in which inventory level fluctuates as a Brownian motion in the absence of control. The inventory continuously accumulates cost at a rate that is a general convex function of the inventory level, which can be negative when there is a backlog. At any time, the inventory level can be adjusted by a positive or negative amount, which incurs a fixed positive cost and a proportional cost. The challenge is to find an adjustment policy that balances the inventory cost and adjustment cost to minimize the expected total discounted cost. We provide a tutorial on using a three-step lower-bound approach to solving the optimal control problem under a discounted cost criterion. In addition, we prove that a four-parameter control band policy is optimal among all feasible policies. A key step is the constructive proof of the existence of a unique solution to the free boundary problem. The proof leads naturally to an algorithm to compute the four parameters of the optimal control band policy.

Lower Bound Limit Analysis of Masonry Arches with CFRP Reinforcements: A Numerical Method

This paper presents a numerical method to predict the ultimate load of masonry arches strengthened with carbon-fiber-reinforced polymer (CFRP) strips bonded to the intrados. The voussoirs of the arch and the CFRP strip, ideally divided into the same number of parts as the voussoirs, are modeled as rigid blocks. A finite set of stress resultants represents the stress state acting on interfaces of the rigid blocks. The local failure modes at the block interfaces are defined according to experimental evidence. The model is developed within an associated framework in such a way that the normality rule is satisfied: the upper- and lower-bound theorems of classical limit analysis apply. The ultimate load is predicted by a lower bound approach. The feasible domain is defined by the equilibrium equations and by the linear constraints imposed on the stress resultants. All the relations defining the model are linear, so that a linear programming problem is imposed. The predictions of the numerical model compare well with experimental results.

Lower-Bound Analysis of Fiber-Reinforced Polymeric Laminated Cylindrical Shells

The imperfection-sensitive buckling of fiber-reinforced polymeric laminated shells is complicated by the large number of additional material and geometric parameters compared to the buckling of conventional metallic shells. As a basis for conceptual design, the traditional lower-bound approach of relying upon scatter of test results is no longer feasible when it comes to the buckling design of fiber-reinforced polymeric laminated shells. This study aimed to extend an analytical approach, the so-called reduced-stiffness method, to the lower-bound buckling analysis of fiber-reinforced polymeric laminated cylindrical shells under uniform axial compression. For a range of isotropic shells, the reduced-stiffness method has been shown capable of predicting safe lower bounds to experimental and numerical studies of imperfection-sensitive buckling loads. In this paper, the validity of the reduced-stiffness method applied to fiber-reinforced polymeric laminated cylindrical shells is verified by the nonlinear-buckling analysis performed using the finite element software ABAQUS/Standard. Through several case studies, it is suggested that the reduced-stiffness method offers an important alternative for improved prediction of lower-bound buckling loads in the design of fiber-reinforced polymeric laminated shells.

A scenario generation-based lower bounding approach for stochastic scheduling problems

In this paper, we investigate scenario generation methods to establish lower bounds on the optimal objective value for stochastic scheduling problems that contain random parameters with continuous distributions. In contrast to the Sample Average Approximation (SAA) approach, which yields probabilistic bound values, we use an alternative bounding method that relies on the ideas of discrete bounding and recursive stratified sampling. Theoretical support is provided for deriving exact lower bounds for both expectation and conditional value-at-risk objectives. We illustrate the use of our method on the single machine total weighted tardiness problem. The results of our numerical investigation demonstrate good properties of our bounding method, compared with the SAA method and an earlier discrete bounding method.

Variation of Weak Polyelectrolyte Persistence Length through an Electrostatic Contour Length

There is currently no consensus that the classical perspective of electrostatic persistence length (the so-called Odijk–Skolnick–Fixman, or OSF, formulation), derived by equating changes in electrostatic energy with bending rigidity, is applicable to weak polyelectrolytes. Here, with a focus on polyelectrolyte chains of finite length at a single scale, we formulate a simple and general theoretical model featuring the electrostatic persistence length, Pe, through the introduction of an electrostatic contour length, Le, such that the electrostatic energy balance is applicable to highly flexible charged polyelectrolytes (where the intrinsic stiffness, P0 is on the order of Pe). To isolate the effect of ionization, only the salt-free regime is considered. At the upper limit (relatively rigid molecules), the new formulation converges to the classical OSF form, while the lower bound (highly flexible molecules) approaches proportionality to the Debye screening length, κ–1. In general, the electrostatic persistence can be described retroactively by the Debye screening, κ–1, and the ratio of electrostatic to intrinsic contour lengths, A/a, complementary to predictive theoretical formulations. The theory is validated via full atomistic molecular dynamics simulations of single, isolated model weak polyelectrolyte chains in explicit solvent—specifically poly(acrylic acid), PAA, and poly(allylamine hydrochloride), PAH, both of which undergo significant increases in persistence length under ionization. An ensemble of equilibrated polymer states is obtained via temperature assisted sampling, implementing molecular dynamics to drive the polymers into physically accessible conformations via cyclical temperature fluctuations and equilibration.

Plastic deformation and development of antigorite crystal preferred orientation in high-pressure serpentinites

We have inferred the deformation mechanisms of antigorite by high-resolution EBSD mapping of samples from Cerro del Almirez ultramafic massif (Betic Cordillera, SE Spain). Textural relations and phase diagram calculations constrain the foliation development conditions to the subduction prograde path at up to 600–630°C and 1.6–1.9GPa. Deformation was followed by static annealing of antigorite at ca. 680°C. The Crystal Preferred Orientation (CPO) of antigorite is characterised by a strong alignment of (001) poles normal to the foliation plane and a weaker, but clear, parallelism between [100] axes and the macroscopic lineation defined by the elongation of magnetite aggregates. Analysis of misorientations across subgrains shows predominance of [010] rotation axis, consistent with activation of the [100](001) slip system. However, tilt subgrain boundaries subparallel to (100), probably formed by edge dislocations of this system, are subsidiary. Most subgrain boundaries are subparallel to (001) planes. They are interpreted as (001) twins wherein continuing viscoplastic deformation resulted in a slight increase of the misorientation between the twins. Modelling of the evolution of the antigorite CPO using a lower bound approach and considering different deformation regimes and sets of basal and non-basal slip systems has shown that intensities of [100] and [010] maxima reflect the relative strength of the antigorite [100](001) and [010](001) systems. Activation of other basal or non-basal slip systems does not change significantly the CPO patterns, but results in less concentrated CPO. 3D transpression models better reproduce the CPO of natural antigorite serpentinites. We propose that the widespread occurrence of strong CPO in high-pressure antigorite serpentinite is consistent with deformation by dislocation creep with dominant glide on [hk0](001), together with the activation of twinning, implying that a power law rheology would better account for the mechanical behaviour of antigorite serpentinite deep in the subduction channel and mantle wedge.

Brownian inventory models with convex holding cost, Part 2: Discount-optimal controls

We consider an inventory system in which inventory level uctuates as a Brownian motion in the absence of control. The inventory continuously accumulates cost at a rate that is a general convex function of the inventory level, which can be negative when there is a backlog. At any time, the inventory level can be adjusted by a positive or negative amount, which incurs a xed positive cost and a proportional cost. The challenge is to nd an adjustment policy that balances the inventory cost and adjustment cost to minimize the expected total discounted cost. We provide a tutorial on using a three-step lower-bound approach to solving the optimal control problem under a discounted cost criterion. In addition, we prove that a four-parameter control band policy is optimal among all feasible policies. A key step is the constructive proof of the existence of a unique solution to the free boundary problem. The proof leads naturally to an algorithm to compute the four parameters of the optimal control band policy. AMS classications: 60J70, 90B05, 93E20

Brownian inventory models with convex holding cost, Part 1: Average-optimal controls

We consider an inventory system in which inventory level fluctuates as a Brownian motion in absence of control. The inventory continuously accumulates cost at a rate that is a general convex function of inventory level, which can be negative when there is a backlog. At any time, inventory level can be adjusted by a positive or negative amount, which incurs a fixed cost and a proportional cost. The challenge is to find an adjustment policy that balances holding cost and adjustment cost to minimize long-run average cost. When both upward and downward fixed costs are positive, our model is an impulse control problem. When both fixed costs are zero, our model is a singular or instantaneous control problem. For impulse control problem, we prove that a four-parameter control band policy is optimal among all feasible policies. For singular control problem, we prove that a two-parameter control band policy is optimal. We use a lower-bound approach, widely known as the verification theorem, to prove optimality of a control band policy for both impulse and singular control problems. Our major contribution is to prove existence of a smooth solution to free boundary problem under some mild assumptions on holding cost function. The existence proof leads naturally to numerical algorithms to compute optimal control band parameters. We demonstrate that lower-bound approach also works for Brownian inventory model in which no inventory backlog is allowed. In a companion paper, we will show how lower-bound approach can be adapted to study a Brownian inventory model under a discounted cost criterion.

A coupled stream function-finite element analysis for wire drawing processes

A numerical approach has been developed based on the stream function technique and the finite element analysis to predict required power and temperature rise in wire drawing processes. An admissible velocity field is first proposed using a stream function and then power consumption in the wire drawing is optimized to achieve sensible and unique deformation geometry. In addition, the finite element method together with axi-symmetric Petrov–Galerkin algorithm is coupled with the deformation model to assess the temperature distribution in both the deforming wire and the die during the process. The work hardening effects are also considered in the model both in the deformation zone and on the velocity discontinuity surfaces. The model can estimate the effects of various process parameters such as drawing velocity and die geometry. In order to evaluate the results of the model, the predictions are compared with the established models including force equilibrium as a lower-bound approach and an upper-bound solution based on the spherical velocity field.

Simple lower bound limit analysis homogenization model for in- and out-of-plane loaded masonry walls

A simple homogenized equilibrated and admissible model for the limit analysis of masonry walls in- and out-of-plane loaded is presented. A rectangular running bond elementary cell (RVE) is discretized by means of a few constant stress triangular elements (CST) and interfaces. Non linearity is concentrated on brick–brick interfaces and joints reduced to interface, exhibiting a frictional behavior with limited tensile and compressive strength. When dealing with the flexural behavior, a Kirchhoff–Love homogenized model is derived, subdividing the REV into several layers along the thickness and in discretizing each layer analogously to the in-plane case. When dealing with the in-plane behavior, a linear programming problem with very few variables is obtained, which can be handled even manually for some cases of technical interest. The lower bound model proposed is validated at a cell level through literature kinematic models, to show that the lower bound approach provides almost identical results with respect to upper bound approaches, meaning that the actual solution is well approximated. Homogenized masonry behavior is then implemented at a structural level in consolidated FE limit analysis codes for the evaluation of collapse loads and failure mechanisms of real scale masonry structures.

Three-Dimensional Analysis of Masonry Vaults Using Limit State Analysis with Finite Friction

Within the past 20 years, a growing number of methods for the analysis of masonry vaults have been developed. However, most methods idealize the vaults as a system of many arches. This oversimplification could be admissible for barrel vaults and spherical domes under simple gravitational loads, but it cannot do justice to the three-dimensional effects developing in other types of vaults, especially in complex vaults without smooth and continuous surface. Moreover, although the results could be conservative for uniform load distributions, this model limits substantially the set of loading conditions that can be analyzed and hence provide an accurate assessment of vaults performance. To clear this limitation, this article proposes a tool for three-dimensional analysis of masonry vaults. Using the lower bound approach and taking into account the friction among block interfaces, the proposed analytical method may give a good indication of the actual surface of thrust within the framework of limit state analysis. Therefore, limit state analysis with finite friction is able to provide the crack pattern, the stress resultants and the horizontal thrust at the supports, important elements with regard to strengthening interventions. To show its simplicity and rigor, the procedure has been applied to pavilion vaults. The results obtained are validated by comparison with nonlinear finite element analysis and an application to a real case study is presented.

Challenges in the computation of lower-bound buckling loads for tanks under wind pressures

This paper reports on the implementation of a lower-bound approach for the buckling of imperfection-sensitive shells using general purpose finite element codes. The stability of cylindrical steel tanks under wind pressure is evaluated for two tank configurations: conical roof tanks and open top tanks. For both tank configurations, several geometric relations are considered in order to find the variation of the knock-down factor as the geometry changes. The reduced energy method is implemented to compute a lower-bound for critical wind pressures and the results are compared with the static non-linear analysis carried out on the same models. An alternative way to implement the reduced energy method is presented to improve the results obtained with the proposed methodology.

Evaluation of Interslice Force Function and Discussion on Convergence in Slope Stability Analysis by the Lower Bound Method

The interslice force function f(x) is a major assumption of the limit equilibrium method, which is important but has not been adequately considered in the past. In this paper, f(x) is taken as the control variable, and the upper and lower limits of the factor of safety for a general slope will be determined by a global optimization analysis. Based on this approach, f(x) will be determined and investigated. We demonstrate that f(x) cannot be arbitrarily assigned if a set of acceptable internal forces is required. The approach can be presented practically as a lower bound approach with the advantage that failure to converge is virtually eliminated, which is not possible with all other existing rigorous methods. The present attempts to answer several important questions in the basic theory of slope stability analysis, and provides a f(x) based on the lower bound approach statically admissible forces throughout the whole failure zone. Currently, different assumptions will give different factors of safety to the same problem, and this situation will be overcome by the use of the proposal. The proposal is also proven to give a result equal to the slip line solution for a simple footing on clay which is not possible for other classical slope stability methods, which has demonstrated that the applicability of the present for general difficult problems.

An Improved Approach for Estimating Product Performance Based on the Capability Index C pmk

In this article, we construct an improved procedure for estimating the process capability index C pmk . We propose a new C pmk lower-bound approach based on the GCI concept, and compare it with other existing methods. Based on the comparison results, we conclude with a recommendation, and construct a step-by-step procedure for the recommended approach to estimate the actual process capability C pmk for various sample sizes. The lower bound attended by our recommended approach, indeed, improves other existing lower bound methods. We also investigate a real-world application to illustrate how we could apply the recommended approach to the actual manufacturing processes.