Linear Programming Solution Research Articles

Reconfiguration Aware Orchestration for Network Function Virtualization With Time-Varied Workload in Virtualized Datacenters

Network function virtualization (NFV) is an emerging paradigm that has revolutionized the deployment and provisioning of telecommunication network services (NS). In NFV, a network service is implemented as a chain of network functions (NFs). NFs are virtualized and packaged into virtual network functions (VNF), so that they can be decoupled from the physical middleboxes and dynamically deployed on standard virtualization platforms in cloud or datacenters. Therefore, NFV has the potential to provide significant cost reductions in operating expenses (OPEX) and capital expenses (CAPEX) and increased agility for network service deployments. However, to build and manage a network function virtualization infrastructure (NFVI) is challenging. One of the key problems is to determine the mapping between underlying physical resources and VNFs in order to minimize energy consumption without violating service-level agreements (SLA). Many previous studies use embedding solutions to solve the problem by only considering the resource demand of NS and the resource capacity of physical network at a given time instance. But in practice, the resource demands of NS can vary over time, hence the embedding decision must be reconfigured accordingly. Many techniques can be used for reconfiguration, including resource scaling, instance migration and traffic re-direction. But each of these operation is associated with some costs, such as service interruptions, and additional resource or energy consumption. In this work, we investigate various reconfiguration strategies, formulate their costs by designing a model in the NFV placement problem, and propose Integer Linear Programming (ILP) and heuristic solutions, called REAP. In our evaluation, we observed 60% and 25% cost reduction for energy and reconfiguration costs, and we analyze the cost reduction from each of the reconfiguration methods to validate the design of our heuristic algorithm.

Computing the Rearrangement Distance of Natural Genomes.

The computation of genomic distances has been a very active field of computational comparative genomics over the past 25 years. Substantial results include the polynomial-time computability of the inversion distance by Hannenhalli and Pevzner in 1995 and the introduction of the double cut and join distance by Yancopoulos et al. in 2005. Both results, however, rely on the assumption that the genomes under comparison contain the same set of unique markers (syntenic genomic regions, sometimes also referred to as genes). In 2015, Shao et al. relax this condition by allowing for duplicate markers in the analysis. This generalized version of the genomic distance problem is NP-hard, and they give an integer linear programming (ILP) solution that is efficient enough to be applied to real-world datasets. A restriction of their approach is that it can be applied only to balanced genomes that have equal numbers of duplicates of any marker. Therefore, it still needs a delicate preprocessing of the input data in which excessive copies of unbalanced markers have to be removed. In this article, we present an algorithm solving the genomic distance problem for natural genomes, in which any marker may occur an arbitrary number of times. Our method is based on a new graph data structure, the multi-relational diagram, that allows an elegant extension of the ILP by Shao et al. to count runs of markers that are under- or over-represented in one genome with respect to the other and need to be inserted or deleted, respectively. With this extension, previous restrictions on the genome configurations are lifted, for the first time enabling an uncompromising rearrangement analysis. Any marker sequence can directly be used for the distance calculation. The evaluation of our approach shows that it can be used to analyze genomes with up to a few 10,000 markers, which we demonstrate on simulated and real data.

Virtual Network Function Service Provisioning in MEC Via Trading Off the Usages Between Computing and Communication Resources

Mobile edge computing (MEC) has emerged as a promising technology that offers resource-intensive yet delay-sensitive applications from the edge of mobile networks. With the emergence of complicated and resource-hungry mobile applications, offloading user tasks to cloudlets of nearby mobile edge-cloud networks is becoming an important approach to leverage the processing capability of mobile devices, reduce mobile device energy consumptions, and improve experiences of mobile users. In this article we first study the provisioning of virtualized network function (VNF) services for user requests in an MEC network, where each user request has a demanded data packet rate with a specified network function service requirement, and different user requests need different services that are represented by virtualized network functions instantiated in cloudlets. We aim to maximize the number of user request admissions while minimizing their admission cost, where the request admission cost consists of the computing cost on instantiations of requested VNF instances and the data packet traffic processing of requests in their VNF instances, and the communication cost of routing data packet traffic of requests between users and the cloudlets hosting their requested VNF instances. We study the joint VNF instance deployment and user requests assignment in MEC, by explicitly exploring a non-trivial usage tradeoff between different types of resources. To this end, we first formulate the cost minimization problem that admits all requests by assuming that there is sufficient computing resource in MEC to accommodate the requested VNF instances of all requests, for which we formulate an Integer Linear Programming solution and two efficient heuristic algorithms. We then deal with the problem under the computing resource constraint. We term this problem as the throughput maximization problem by admitting as many as requests, subject to computing resource capacity on each cloudlet, for which we formulate an ILP solution when the problem size is small; otherwise, we devise efficient algorithms for it. We finally evaluate the performance of the proposed algorithms through experimental simulations. Experimental results demonstrate that the proposed algorithms are promising. To the best of our knowledge, we are the first to explicitly explore the usage tradeoff between computing and communication resources in the admissions of user requests in MEC through introducing a novel load factor concept to minimize the request admission cost and maximize the network throughput.

Coverage Path Planning Techniques for Inspection of Disjoint Regions With Precedence Provision

Recent times are witnessing an emergence of sites that are hazardous for human access. This has created a global demand to equip agents with the ability to autonomously inspect such environments by computing a coverage path effectively and efficiently. However, inspection of such sites requires agents to consider the correlation of work, providing precedence provision in visiting regions. The current approaches to compute coverage path in the hazardous sites, however, do not consider precedence provision. To this end, coverage path planning strategies are proposed, which provide precedence provision. To meet the challenges, the problem is divided into two phases: inter-region and intra-region path planning. In the `inter-region' path planning of the approach, the site comprising of multiple disjoint regions is modelled as connectivity graph. Two novel approaches, Mixed Integer Linear Programming (MILP) solution and heuristic based techniques, are proposed to generate the ordered sequence of regions to be traversed. In the `intra-region' path planning of the approach, each region is decomposed into a grid and Boustrophedon Motion is planned over each region. The ability of combined approach to provide complete coverage is proved under minor assumption. An investigative study has been conducted to elucidate the efficiency of the proposed approach in different scenarios using simulation experiments. The proposed approach is evaluated against baseline approaches. The results manifest a significant reduction in cost and execution time, which caters to inspection of target sites comprising of multiple disjoint regions with precedence provision.

Maximum likelihood reconstruction of ancestral networks by integer linear programming.

The study of the evolutionary history of biological networks enables deep functional understanding of various bio-molecular processes. Network growth models, such as the Duplication-Mutation with Complementarity (DMC) model, provide a principled approach to characterizing the evolution of protein-protein interactions (PPIs) based on duplication and divergence. Current methods for model-based ancestral network reconstruction primarily use greedy heuristics and yield sub-optimal solutions. We present a new Integer Linear Programming (ILP) solution for maximum likelihood reconstruction of ancestral PPI networks using the DMC model. We prove the correctness of our solution that is designed to find the optimal solution. It can also use efficient heuristics from general-purpose ILP solvers to obtain multiple optimal and near-optimal solutions that may be useful in many applications. Experiments on synthetic data show that our ILP obtains solutions with higher likelihood than those from previous methods, and is robust to noise and model mismatch. We evaluate our algorithm on two real PPI networks, with proteins from the families of bZIP transcription factors and the Commander complex. On both the networks, solutions from our ILP have higher likelihood and are in better agreement with independent biological evidence from other studies. A Python implementation is available at https://bitbucket.org/cdal/network-reconstruction. Supplementary data are available at Bioinformatics online.

A column generation-based diving heuristic to solve the multi-project personnel staffing problem with calendar constraints and resource sharing

Project managers are often responsible for the management of multiple projects and associated personnel budgeting decisions. In order to determine the workforce size and mix accurately, we integrate the multi-project scheduling problem and personnel staffing problem. We construct a baseline personnel roster that takes the personnel scheduling constraints into account and model the workload stemming from the project activities as an endogenous variable in the model. In order to decrease the overall staffing budget, we consider the sharing of resources, i.e. personnel resources can be transferred between projects. In the problem under study, we consider unproductive resource transfer times and different restrictions related to the sharing of resources imposed on the schedule of individual workers, which is often neglected in the literature. We propose a multi-stage solution procedure that exploits the optimal linear programming solution, obtained via column generation, to find a high-quality integer solution based on a diving heuristic. In order to improve the computational performance, different speed-up mechanisms are included relying on a state space reduction. Detailed computational experiments are presented to evaluate and benchmark different alternative optimisation strategies. Managerial insights are provided which give insight in the benefits of resource sharing.

An optimal rounding for half-integral weighted minimum strongly connected spanning subgraph

In the weighted minimum strongly connected spanning subgraph (WMSCSS ) problem we must purchase a minimum-cost strongly connected spanning subgraph of a digraph. We show that half-integral linear program (LP) solutions for WMSCSS can be efficiently rounded to integral solutions at a multiplicative 1.5 cost. This rounding matches a known 1.5 integrality gap lower bound for a half-integral instance. More generally, we show that LP solutions whose non-zero entries are at least a value f>0 can be rounded at a multiplicative cost of 2−f.

A receding horizon approach to peak power minimization for EV charging stations in the presence of uncertainty

The increasing penetration of plug-in electric vehicles in recent years asks for specific solutions concerning the charging policies to be used in parking lots equipped with charging stations. In fact, simple policies based on uncoordinated charge of vehicles lead, in general, to high peak power demand, which may cause high costs to the car park owner. In this paper, the problem of minimizing the daily peak power of a charging station is addressed. Three sources of uncertainty affect the incoming vehicles: the arrival time, the departure time and the demanded energy to be charged. To assess the quality of the charging service under such uncertainties, a suitable customer satisfaction policy is employed. Depending on the information available on the uncertain variables, two algorithms based on a receding horizon approach are designed. Such algorithms require the solution of linear programs and provide the charging power for each plugged-in vehicle. Numerical simulations are provided to assess performance and computational burden of the algorithms, showing the effectiveness and feasibility of the proposed techniques.

Reliability-Aware Virtualized Network Function Services Provisioning in Mobile Edge Computing

Along with Network Function Virtualization (NFV), Mobile Edge Computing (MEC) is becoming a new computing paradigm that enables accommodating innovative applications and services with stringent response delay and resource requirements, including autonomous vehicles and augmented reality. Provisioning reliable network services for users is the top priority of most network service providers, as unreliable services or severe service failures can result in tremendous losses of users, particularly for their mission-critical applications. In this paper, we study reliability-aware VNF instances provisioning in an MEC, where different users request different network services with different reliability requirements through paying their requested services with the aim to maximize the network throughput. To this end, we first formulate a novel reliability-aware VNF instance placement problem by provisioning primary and secondary VNF instances at different cloudlets in MEC for each user while meeting the specified reliability requirement of the user request. We then show that the problem is NP-hard and formulate an Integer Linear Programming (ILP) solution. Due to the NP-hardness of the problem, we instead devise an approximation algorithm with a logarithmic approximation ratio for the problem. Moreover, we also consider two special cases of the problem. For one special case where each request only requests one primary and one secondary VNF instances, the problem is still NP-hard, and we devise a constant approximation algorithm for it. For another special case where different VNFs have the same amounts of computing resource demands, we show that it is polynomial-time solvable by developing a dynamic programming solution for it. We finally evaluate the performance of the proposed algorithms through experimental simulations. Experimental results demonstrate that the proposed algorithms are promising, and the empirical results of the algorithms outperform their analytical counterparts as theoretical estimations usually are very conservative.

MIP model-based heuristics for the minimum weighted tree reconstruction problem

We consider the minimum weighted tree reconstruction (MWTR) problem and two matheuristic methods to obtain optimal or near-optimal solutions: the Feasibility Pump heuristic and the Local Branching heuristic. These matheuristics are based on a Mixed Integer Programming model used to find feasible solutions. We discuss the applicability and effectiveness of the matheuristics to obtain solutions to the MWTR problem. The purpose of the MWTR problem is to find a minimum weighted tree connecting a set of leaves in such a way that the length of the path between each pair of leaves is greater than or equal to a given distance between the considered pair of leaves. The Feasibility Pump matheuristic starts with the Linear Programming solution, iteratively fixes the values of some variables and solves the corresponding problem until a feasible solution is achieved. The Local Branching matheuristic, in its turn, improves a feasible solution by using a local search. Computational results using two different sets of instances, one from the phylogenetic area and another from the telecommunications area, show that these matheuristics are quite effective in finding feasible solutions and present small gap values. Each matheuristic can be used independently; however, the best results are obtained when used together. For instances of the problem having up to 17 leaves, the feasible solution obtained by the Feasibility Pump heuristic is improved by the Local Branching heuristic. Noticeably, when comparing with existing based models processes that solve instances having up to 15 leaves, this achievement of the matheuristic increases the size of solved instances.

A Cost-Effective Electric Vehicle Intelligent Charge Scheduling Method for Commercial Smart Parking Lots Using a Simplified Convex Relaxation Technique.

Deployment of efficient and cost-effective parking lots is a known bottleneck for the electric vehicles (EVs) sector. A comprehensive solution incorporating the requirements of all key stakeholders is required. Taking up the challenge, we propose a real-time EV smart parking lot model to attain the following objectives: (a) maximize the smart parking lot revenue by accommodating maximum number of EVs and (b) minimize the cost of power consumption by participating in a demand response (DR) program offered by the utility since it is a tool to answer and handle the electric power usage requirements for charging the EV in the smart parking lot. With a view to achieving these objectives, a linear programming-based binary/cyclic (0/1) optimization technique is developed for the EV charge scheduling process. It is difficult to solve the problems of binary optimization in real-time given that the complexity of the problem increases with the increase in number of EV. We deploy a simplified convex relaxation technique integrated with the linear programming solution to overcome this problem. The algorithm achieves: minimum power consumption cost of the EV smart parking lot; efficient utilization of available power; maximization of the number of the EV to be charged; and minimum impact on the EV battery lifecycle. DR participation provide benefits by offering time-based and incentive-based hourly intelligent charging schedules for the EV. A thorough comparison is drawn with existing variable charging rate-based techniques in order to demonstrate the comparative validity of our proposed technique. The simulation results show that even under no DR event, the proposed scheme results in 2.9% decrease in overall power consumption cost for a 500 EV scenario when compared to variable charging rate method. Moreover, in similar conditions, such as no DR event and for 500 EV arrived per day, there is a 2.8% increase in number of EV charged per day, 3.2% improvement in the average state-of-charge (SoC) of the EV, 12.47% reduction in the average time intervals required to achieve final SoC.

Preference heterogeneity, neighbourhood effects and basic services: logit kernel models for farmers’ climate adaptation in Ethiopia

Climate change adaptations by farmers are usually analysed with multinomial choice models, which ignore heterogeneity and ‘mixing’ across alternatives and individuals. Moreover, not enough attention is devoted to basic utilities other than extension services. Aimed at a reassessment that accounts for these issues, this paper formulates conditional and mixed logit (logit kernel) models with/without neighbourhood effects. In particular, a spatial logit kernel allows capturing behavioural differences and interdependence between neighbours. The analysis was based on a survey covering 162 villages of the Ethiopian Nile basin, with 50 farm households—multistage stratified randomly sampled—in each of 20 districts (woredas). Supplementary information concerned linear programming solutions on adaptation choice attributes in smallholder farming areas, under two scenarios. Lastly, cross-district aid emergency allocation priorities for food security and agriculture, among others, served as yardsticks for comparison with prevailing adjustment decisions and location characteristics. Closeness to farms with more educated farmers and larger plot sizes turned out to increase the likelihood of a household opting to sell livestock and land along with combined measures of farmland enhancement. Logit kernel outperformed conditional logit in explanatory power, and random parameters highlighted individual and group heterogeneity in preferences. Access to electricity, healthcare, and reliable and safe water supply strengthen farmers’ ability to respond to climate change. Specific adaptation measures can contribute in turn to improve the capacity of rural communities to mitigate the severity of health and nutrition crises. By contrast, in woredas hit by recurrent epidemics and droughts, many farmers appear to be unable to resort even to basic adaptation measures, such as planting of new trees. This prompts an additional need for institutional support and infrastructure development.

Computationally Efficient Pricing and Benefit Distribution Mechanisms for Incentivizing Stable Peer-to-Peer Energy Trading

Peer-to-peer (P2P) energy trading has emerged as a promising market paradigm toward maximizing the value of distributed energy resources (DERs) for electricity prosumers, by enabling direct energy trading among them. However, state-of-the-art P2P mechanisms either fail to adequately incentivize prosumers to participate, prevent prosumers from accessing the highest achievable monetary benefits, or suffer severely from the curse of dimensionality. This article proposes two computationally efficient mechanisms to construct a stable grand coalition of prosumers participating in P2P trading, founded on cooperative game-theoretic principles. The first one involves a benefit distribution scheme inspired by the core tâtonnement process while the second involves a novel pricing mechanism based on the solution of a single linear program. The performance of the proposed mechanisms is validated against state-of-the-art mechanisms through numerous case studies using real-world data. The results demonstrate that the proposed mechanisms exhibit superior computational performance than the nucleolus and are superior to the rest of the examined mechanisms in incentivizing prosumers to remain in the grand coalition.

QoS-Aware VNF Placement and Service Chaining for IoT Applications in Multi-Tier Mobile Edge Networks

Mobile edge computing and network function virtualization (NFV) paradigms enable new flexibility and possibilities of the deployment of extreme low-latency services for Internet-of-Things (IoT) applications within the proximity of their users. However, this poses great challenges to find optimal placements of virtualized network functions (VNFs) for data processing requests of IoT applications in a multi-tier cloud network, which consists of many small- or medium-scale servers, clusters, or cloudlets deployed within the proximity of IoT nodes and a few large-scale remote data centers with abundant computing and storage resources. In particular, it is challenging to jointly consider VNF instance placement and routing traffic path planning for user requests, as they are not only delay sensitive but also resource hungry. In this article, we consider admissions of NFV-enabled requests of IoT applications in a multi-tier cloud network, where users request network services by issuing service requests with service chain requirements, and the service chain enforces the data traffic of the request to pass through the VNFs in the chain one by one until it reaches its destination. To this end, we first formulate the throughput maximization problem with the aim to maximize the system throughput. We then propose an integer linear program solution if the problem size is small; otherwise, we devise an efficient heuristic that jointly takes into account VNF placements to both cloudlets and data centers and routing path finding for each request. For a special case of the problem with a set of service chains, we propose an approximation algorithm with a provable approximation ratio. Next, we also devise efficient learning-based heuristics for VNF provisioning for IoT applications by incorporating the mobility and energy conservation features of IoT devices. We finally evaluate the performance of the proposed algorithms by simulations. The simulation results show that the performance of the proposed algorithms is promising.

Robust estimation and economic predictive control for dynamic metabolic flux systems under probabilistic uncertainty

A novel formulation is proposed for propagating parametric uncertainty for Dynamic Metabolic Flux models involving the solution of Linear Programming (LP) problems. The propagation approach is based on the calculation of all possible active sets of constraints at each time interval. Then, a tree-based approach is used to propagate the uncertainty onto the worst case of each active set where each branch of the tree corresponds to an active set of constraints. Each branch is assigned a relative probability according to the relative hypervolume occupied by the active set solutions in the parameter space. The approach is applied for the robust estimation of the Ecoli fermentation process and for the design of a robust economic predictive controller for this system. The proposed method is found to be significantly more efficient as compared with other uncertainty propagation approaches such as Monte Carlo (MC) simulations.

A multi-objective optimization model for dairy feeding management

The allocation of feedstuff to intensively managed dairy cows to achieve different objectives is challenging due to the inherent complexity of the system and the combinatorial problem that has to be solved. Pareto-based multi-objective optimization approaches using evolutionary algorithms can help to address these challenges and show the trade-offs and synergies among various objectives. Here we present a framework for multi-objective optimization with the Differential Evolution (DE) algorithm applied to dairy feeding systems with grazing and concentrate supply to generate an approximation of the Pareto front. The available feed resources are located in different feeding areas, and the number of animals and groups of animals with similar feeding requirements are distributed across these areas for feeding purposes. To evaluate the DE algorithm, we performed two in-silico experiments to: (1) compare the solutions quality of single-objective DE with exact Linear Programming (LP) solutions, and (2) assess the influence of different stocking rates (number of cows/ha) on milk production, feed allocation and economic performance indicators. The DE solutions that minimize the feeding costs for different stocking rates (1.1–2.6 cows/ha) closely approached the solutions derived with LP, confirming the quality of the heuristic algorithm. The multi-objective model scenarios demonstrated that increasing stocking density would enhance milk production and gross margin per unit of area at largely unchanged productivity per animal by shifting the feed ration from roughage to a large proportion of supplementary concentrate feed. At low stocking rates solutions with high productivity and gross margin and a large proportion of roughage in the ration and limited supplementary feeding were identified. We conclude that the multi-objective optimization with a Pareto-based DE algorithm is highly effective to explore the interrelations among conflicting objectives and to find suitable solutions.

Separation routine and extended formulations for the stable set problem in claw-free graphs

The maximum weighted stable set problem in claw-free graphs is a well-known generalization of the maximum weighted matching problem, and a classical problem in combinatorial optimization. In spite of the recent development of fast(er) combinatorial algorithms and some progresses in the characterization of the corresponding stable set polytope, the problem of “providing a decent linear description” for this polytope (Grotschel et al. in Geometric algorithms and combinatorial optimization, Springer, New York, 1988) is still open. The main contribution of this paper is to propose an algorithmic answer to that question by providing a polynomial-time and computationally attractive separation routine for the stable set polytope of claw-free graphs, that only requires a combinatorial decomposition algorithm, the solution of (moderate sized) compact linear programs, and Padberg and Rao’s algorithm for separating over the matching polytope. In particular, it is a generalization of the latter and avoids the heavy computational burden of resorting to the ellipsoid method, on which the only poly-time separation routine known so far relied. Besides, our separation routine comes with a ‘small’ (but not polynomial) extended linear programming formulation and a procedure to derive a linear description of the stable set polytope of claw-free graphs in the original space.

Hybrid risk‐averse energy management optimizer for large‐scale industrial building microgrids

Energy system has been facing problems such as soaring energy cost and environmental concerns. Among different types of customers, large-sized industrial building microgrids (μGs) with heavy load, that can contribute significantly to demand response and greenhouse gas (GHG) emissions reduction. Therefore, an optimal risk-averse energy management strategy is required for this class of customers. The objective of this article is to devise an energy management system (EMS) for large-scale industrial μG to reduce energy consumption cost and GHG emissions. Framework has been solved in MATLAB using conventional flower pollination algorithm (FPA). However, metaheuristic techniques such as FPA take large execution time and trap in local optimum. On the other hand, deterministic techniques cannot handle large problems, however, reach an optimum solution in a shorter time span. To address these issues of both classes of algorithms, article devises and validates hybrid modified FPA-mixed-integer linear programming solution algorithm. Simulations show that the proposed technique produces improved results with low execution time, providing a justification for the practical implementation of the concept in the smart energy distribution system.

Canadian Regional Agriculture Model: PMP Applied to the Beef Sector in Canada

This paper describes the Positive Mathematical Programming (PMP), the method for calibrating models of agricultural livestock production and resource use by a nonlinear total cost function. The PMP method is applied to agricultural sectoral models to study changes in policy and market signals. The Canadian Regional Agriculture Model (CRAM) is a regional, multi-sectoral, comparative static, partial equilibrium, mathematical programming model developed and maintained by Agriculture and Agri-Food Canada (AAFC) since mid-eighties. The PMP process converts a linear model using flexibility constraints into a nonlinear model in the absence of the flexibility constraints. A component of CRAM is the beef sector. The elements of the set of total cost curves are defined as quadratic function in terms of the number of cows and calves in the beef production activities. The marginal cost curves were then approximated using the shadow values from linear programming solution with linear curves. Once the flexibility constraints were removed, the model automatically calibrates to the base year production levels. The results from four scenarios indicated the beef sector of CRAM could predict the impact of the scenarios on the size of beef herd. In Scenario 1 where cash costs were increased by 10 percent, the breeding herd size decreased from 3.73 percent in New Brunswick to 0.0 percent in Ontario and Quebec. In Scenario 2 where barley costs were decreased by 10 percent, the breeding herd size increased from 0 percent for British Columbia, Alberta, Ontario, Quebec, Prince Edward Island and Nova Scotia to 1.93 percent for New Brunswick. In Scenario 3 where carcass weight per beef cow could be increased by 10 percent, the increase in beef herd size ranged from 0 percent for Ontario and Quebec to 2.56 percent for New Brunswick. In Scenario 4 where world beef prices were increased by 10 percent increase in beef herd size ranged from 4.48 percent for Manitoba to 25.78 percent for New Brunswick.

Canadian Regional Agriculture Model: Positive Mathematical Programming Applied to the Beef Sector in Canada

This paper describes the Positive Mathematical Programming (PMP), the method for calibrating models of agricultural livestock production and resource use using a nonlinear total cost function. The PMP method is applied to agricultural sectoral models to study changes in policy and market signals. The Canadian Regional Agriculture Model (CRAM) is a regional, multi-sectoral, comparative static, partial equilibrium, mathematical programming model developed and maintained by Agriculture and Agri-Food Canada (AAFC) since mid-eighties. The PMP process converts a linear model using flexibility constraints into a nonlinear model in the absence of the flexibility constraints. A component of CRAM is the beef sector. The elements of the set of total cost curves are defined as quadratic function in terms of the number of cows and calves in the beef production activities. The marginal cost curves are then approximated using the shadow values from linear programming solution with linear curves. Once the flexibility constraints were removed, the model automatically calibrates to the base year production levels. The results from four scenarios indicated the beef sector of CRAM could predict the impact of the scenarios on the size of beef herd. In Scenario 1 where cash costs were increased by 10 percent, the breeding herd size decreased from 3.73 percent in New Brunswick to 0.0 percent in Ontario and Quebec. In Scenario 2 where barley costs were decreased by 10 percent, the breeding herd size increased from 0 percent for British Columbia, Alberta, Ontario, Quebec, Prince Edward Island and Nova Scotia to 1.93 percent for New Brunswick. In Scenario 3 where carcass weight per beef cow could be increased by 10 percent, the increase in beef herd size ranged from 0 percent for Ontario and Quebec to 2.56 percent for New Brunswick. In Scenario 4 where world beef prices were increased by 10 percent increase in beef herd size ranged from 4.48 percent for Manitoba to 25.78 percent for New Brunswick.