A two-stage stochastic programming model for scheduling replacements in sow farms

This paper presents a formulation and resolution of a two-stage stochastic linear programming model with recourse for sow farms producing piglets. The proposed model considers a medium-term planning horizon and specifically allows optimal replacement and schedule of purchases to be obtained for the first stage. This model takes into account sow herd dynamics, housing facilities, reproduction management, herd size with initial and final inventory of sows and uncertain parameters such as litter size, mortality and fertility rates. These last parameters are explicitly incorporated via a finite set of scenarios. The proposed model is solved by using the algebraic modelling software OPL Studio from ILOG, in combination with the solver CPLEX to solve the linear models resulting from different instances considered. The article also presents results obtained with previous deterministic models assessing the suitability of the stochastic approach. Finally, the conclusions drawn from the study including an outlook are presented.      

Parking Capacity and Pricing in Park'n Ride Trips: A Continuous Equilibrium Network Design Problem

In this paper we consider the problem of designing parking facilities for park'n ride trips. We present a new continuous equilibrium network design problem to decide the capacity and fare of these parking lots at a tactical level. We assume that the parking facilities have already been located and other topological decisions have already been taken.The modeling approach proposed is mathematical programming with equilibrium constraints. In the outer optimization problem, a central Authority evaluates the performance of the transport network for each network design decision. In the inner problem a multimodal traffic assignment with combined modes, formulated as a variational inequality problem, generates the share demand for modes of transportation, and for parking facilities as a function of the design variables of the parking lots. The objective is to make optimal parking investment and pricing decisions in order to minimize the total travel cost in a subnetwork of the multimodal transportation system.We present a new development in model formulation based on the use of generalized parking link cost as a design variable.The bilevel model is solved by a simulated annealing algorithm applied to the continuous and non-negative design decision variables. Numerical tests are reported in order to illustrate the use of the model, and the ability of the approach to solve applications of moderate size.     

Facility location for market capture when users rank facilities by shorter travel and waiting times

A firm wants to locate several multi-server facilities in a region where there is already a competitor operating. We propose a model for locating these facilities in such a way as to maximize market capture by the entering firm, when customers choose the facilities they patronize, by the travel time to the facility and the waiting time at the facility. Each customer can obtain the service or goods from several (rather than only one) facilities, according to a probabilistic distribution. We show that in these conditions, there is demand equilibrium, and we design an ad hoc heuristic to solve the problem, since finding the solution to the model involves finding the demand equilibrium given by a nonlinear equation. We show that by using our heuristic, the locations are better than those obtained by utilizing several other methods, including MAXCAP, p-median and location on the nodes with the largest demand.     

Mathematical analysis of a model for the transmission dynamics of bovine tuberculosis

A deterministic model for studying the transmission dynamics of bovine tuberculosis in a single cattle herd is presented and qualitatively analyzed. A notable feature of the model is that it allows for the importation of asymptomatically infected cattle (into the herd) because re‐stocking from outside sources. Rigorous analysis of the model shows that the model has a globally‐asymptotically stable disease‐free equilibrium whenever a certain epidemiological threshold, known as the reproduction number, is less than unity. In the absence of importation of asymptomatically infected cattle, the model has a unique endemic equilibrium whenever the reproduction number exceeds unity (this equilibrium is globally asymptotically stable for a special case). It is further shown that, for the case where asymptomatically infected cattle are imported into the herd, the model has a unique endemic equilibrium. This equilibrium is also shown to be globally asymptotically stable for a special case. Copyright © 2011 John Wiley & Sons, Ltd.     

Multi‐level hierarchic Markov processes as a framework for herd management support

A general problem in relation to application of Markov decision processes to real world problems is the curse of dimensionality, since the size of the state space grows to prohibitive levels when information on all relevant traits of the system being modeled are included. In herd management, we face a hierarchy of decisions made at different levels with different time horizons, and the decisions made at different levels are mutually dependent. Furthermore, decisions have to be made without certainty about the future state of the system. These aspects contribute even further to the dimensionality problem. A new notion of a multilevel hierarchic Markov process specially designed to solve dynamic decision problems involving decisions with varying time horizon has been presented. The method contributes significantly to circumvent the curse of dimensionality, and it provides a framework for general herd management support instead of very specialized models only concerned with a single decision as, for instance, replacement. The applicational perspectives of the technique are illustrated by potential examples relating to the management of a sow herd and a dairy herd.     

The design of multiactivity multifacility systems

We consider a service/distribution system in which each of N activities is to be carried out at one or several facility locations. Each activity is to be assigned to one out of a specified set of configurations; each configuration is a specific subset of the set of L facilities being considered, along with a specific strategy for their use. We call such a system a multiactivity multifacility system and present a mathematical formulation for its optimal design that includes capacity restrictions at the facilities and the treatment of multiple criteria. The design problem is simply to choose an appropriate configuration for each of the N activities. We discuss various criteria, and we show that the multiactivity multifacility design problem includes many familiar discrete location problems as special cases. We introduce a 0–1 linear optimization model called the Team Generalized Assignment Problem (T-GAP) and show that parametric solution of a T-GAP will yield all efficient solutions of the multiactivity multifacility design problem with multiple criteria. Rather than attempting to find all efficient solutions, however, we advocate an interactive approach and describe an interactive branch-and-bound algorithm that solves the design problem as a finite sequence of T-GAP's.     

Multicriteria equilibrium programming: Extraproximal methods

Multicriteria equilibrium optimization is an efficient tool for mathematical modeling of various situations in operations research, design automation, control, etc. In this paper, a formal formulation of the problem of multicriteria equilibrium optimization is given, and an approach to solving this problem is examined.     

On the preliminary design of axial fans: extending the validity of the flow assumptions over the whole hydraulic characteristic and improvement of modeling

In order to carry out a preliminary axial fan design, established assumptions are made onto the flow structure within the fan at the design point. As an example the very common hypothesis of radial equilibrium, states that the radial velocity component downstream the fan is zero, and that the flow in this area is axisymmetric. With the help of this hypothesis, the flow is idealized in order to draw quantitative conclusions. The present communication aims at extending these assumptions on the whole flow regime in order to help fan designers to reduce the try-and-error cycle number. An analytical formulation for axial fan velocity distributions at design and off design conditions is proposed for an arbitrary work distribution. It is shown that the total pressure characteristics can be described by means of hyperbola and straight line. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Truss topology optimization including unilateral contact

This work extends the ground structure approach of truss topology optimization to include unilateral contact conditions. The traditional design objective of finding the stiffest truss among those of equal volume is combined with a second objective of achieving a uniform contact force distribution. Design variables are the volume of bars and the gaps between potential contact nodes and rigid obstacles. The problem can be viewed as that of finding a saddle point of the equilibrium potential energy function (a convex problem) or as that of minimizing the external work among all trusses that exhibit a uniform contact force distribution (a nonconvex problem). These two formulations are related, although not completely equivalent: they give the same design, but concerning the associated displacement states, the solutions of the first formulation are included among those of the second but the opposite does not necessarily hold.In the classical noncontact single-load case problem, it is known that an optimal truss can be found by solving a linear programming (LP) limit design problem, where compatibility conditions are not taken into account. This result is extended to include unilateral contact and the second objective of obtaining a uniform contact force distribution. The LP formulation is our vehicle for proving existence of an optimal design: by standard LP theory, we need only to show primal and dual feasibility; the primal one is obvious, and the dual one is shown by the Farkas lemma to be equivalent to a condition on the direction of the external load. This method of proof extends results in the classical noncontact case to structures that have a singular stiffness matrix for all designs, including a case with no prescribed nodal displacements.Numerical solutions are also obtained by using the LP formulation. It is applied to two bridge-type structures, and trusses that are optimal in the above sense are obtained.This work was supported by The Center for Industrial Information Technology (CENIIT) and the Swedish Research Council for Engineering Sciences (TFR).     

Heuristic algorithms for delivered price spatially competitive network facility location problems

We review previous formulations of models for locating a firm's production facilities while simultaneously determining production levels at those facilities so as to maximize the firm's profit. We enhance these formulations by adding explicit variables to represent the firm's shipping activities and discuss the implications of this revised approach. In these formulations, existing firms, as well as new entrants, are assumed to act in accordance with an appropriate model of spatial equilibrium. The firm locating new production facilities is assumed to be a large manufacturer entering an industry composed of a large number of small firms. Our previously reported proof of existence of a solution to the combined location-equilibrium problem is briefly reviewed. A heuristic algorithm based on sensitivity analysis methods which presume the existence of a solution and which locally approximate price changes as linear functions of production perturbations resulting from newly established facilities is presented. We provide several numerical tests to illustrate the contrasting locational solutions which this paper's revised delivered price formulation generates relative to those of previous formulations. An exact, although computationally burdensome, method is also presented and employed to check the reliability of the heuristic algorithm.     

Multi-period reverse logistics network design

The configuration of the reverse logistics network is a complex problem comprising the determination of the optimal sites and capacities of collection centers, inspection centers, remanufacturing facilities, and/or recycling plants. In this paper, we propose a profit maximization modeling framework for reverse logistics network design problems. We present a mixed-integer linear programming formulation that is flexible to incorporate most of the reverse network structures plausible in practice. In order to consider the possibility of making future adjustments in the network configuration to allow gradual changes in the network structure and in the capacities of the facilities, we consider a multi-period setting. We propose a multi-commodity formulation and use a reverse bill of materials in order to capture component commonality among different products and to have the flexibility to incorporate all plausible means in tackling product returns. The proposed general framework is justified by a case study in the context of reverse logistics network design for washing machines and tumble dryers in Germany. We conduct extensive parametric and scenario analysis to illustrate the potential benefits of using a dynamic model as opposed to its static counterpart, and also to derive a number of managerial insights.     

The Multi-Story Space Assignment Problem

The Multi-Story Space Assignment Problem (MSAP) is an innovative formulation of the multi-story facility assignment problem that allows one to model the location of departments of unequal size within multi-story facilities as a Generalized Quadratic 3-dimensional Assignment Problem (GQ3AP). Not only can the MSAP generate the design of the location of the departments in the facility, the MSAP also includes the evacuation planning for the facility. The formulation, background mathematical development, and computational experience with a branch and bound algorithm for the MSAP are also presented.     

Capacitated facility location/network design problems

We introduce a combined facility location/network design problem in which facilities have constraining capacities on the amount of demand they can serve. This model has a number of applications in regional planning, distribution, telecommunications, energy management, and other areas. Our model includes the classical capacitated facility location problem (CFLP) on a network as a special case. We present a mixed integer programming formulation of the problem, and several classes of valid inequalities are derived to strengthen its LP relaxation. Computational experience with problems with up to 40 nodes and 160 candidate links is reported, and a sensitivity analysis provides insight into the behavior of the model in response to changes in key problem parameters.     

高维Kramers系统离出点的分布问题

通过对高维Kramers系统与之对应稳态Fokker-Planck方程的渐近分析,仔细探讨了该系统在平衡点吸引域的边界上离出点的分布问题.运用变量替换、匹配原理、局部坐标变换、边界层展开等方法,对外解、远离鞍点处的边界层及鞍点处的边界层进行分析,得出离出点分布的渐近表达式.     

Mass transportation and the consistency of the empirical optimal conditional locations

We consider the problem of finding the optimal locations of new facilities given the locations of existing facilities and clients. We analyze the general situation where the locations of existing facilities are deterministic while the locations of clients are stochastic with the same unknown marginal distribution. We show how this conditional location-allocation problem can be modeled as a variation of the standard Monge-Kantorovich mass transference problem. We provide a probabilistic formulation of the optimal locations of the new facilities and derive consistent estimators of these theoretical locations from a sample of identically distributed random clients. The integrity of our method is illustrated through some simulation experiments and a real case study.     

A new formulation and an exact approach for the many-to-many hub location-routing problem

An important problem of the freight industry is the parcel delivery network design, where several facilities are responsible for assembling flows from several origins, re-routing them to other facilities where the flows are disassembled and the packages delivered to their final destinations. In order to provide this service, local tours are established for the vehicles assigned to each of the processing facilities, which are then responsible for the pickup and delivery tasks. This application gives rise to the many-to-many hub location routing problem that is the combination of two well known problems: the vehicle routing problem and the single assignment hub location problem. In this work, a new formulation for this important problem is proposed and solved by a specially tailored Benders decomposition algorithm. The proposed method is robust enough to solve instances up to 100 nodes having 4 million integer variables.     

After-sales services network design of a household appliances manufacturer

After-sales services have become high-margin businesses that account for larger portions of corporate profits. Delivering the after-sales services is challenging as after-sales services supply chains are significantly different than production–distribution supply chains. The literature provides little guidance on the use of quantitative methods for after-sales services network design. We present a mixed integer linear programming problem formulation to determine warehouse locations, assign repair vendors to facilities and choose mode of transportation while minimizing the total network cost. We transform the large-scale real-life problem of a household appliances manufacturer into a smaller scale to solve optimally in reasonable time. Through a scenario-based approach, we evaluate different configurations of a decentralized network with choices of transportation mode. We test the sensitivity of the solutions. The total cost decreases with additional choices of transportation mode and only slightly increases with the next-day delivery policy while the network solution may change significantly.     

Dynamic programming in minimum-weight design of axisymmetric plates

The paper deals with minimum-weight design of axisymmetric plates. The design is subject to restrictions imposed by static equilibrium conditions and various yield criteria governing the failure of the material. Herein, the optimization problem is formulated as a dynamic programming problem and is solved. Such a formulation unifies the design procedure. The paper also discusses the dynamic programming approach.     

A multi-objective integrated facility location-hardening model: Analyzing the pre- and post-disruption tradeoff

Two methods of reducing the risk of disruptions to distribution systems are (1) strategically locating facilities to mitigate against disruptions and (2) hardening facilities. These two activities have been treated separately in most of the academic literature. This article integrates facility location and facility hardening decisions by studying the minimax facility location and hardening problem (MFLHP), which seeks to minimize the maximum distance from a demand point to its closest located facility after facility disruptions. The formulation assumes that the decision maker is risk averse and thus interested in mitigating against the facility disruption scenario with the largest consequence, an objective that is appropriate for modeling facility interdiction. By taking advantage of the MFLHP’s structure, a natural three-stage formulation is reformulated as a single-stage mixed-integer program (MIP). Rather than solving the MIP directly, the MFLHP can be decomposed into sub-problems and solved using a binary search algorithm. This binary search algorithm is the basis for a multi-objective algorithm, which computes the Pareto-efficient set for the pre- and post-disruption maximum distance. The multi-objective algorithm is illustrated in a numerical example, and experimental results are presented that analyze the tradeoff between objectives.