The document discusses Gomory's Cutting Plane Method for solving integer programming problems (IPPs). It begins by introducing all-integer linear programs (AILPs) and mixed-integer linear programs (MILPs). It then describes how Gomory's method works by taking the linear programming (LP) relaxation of an IPP, obtaining the fractional solution, deriving a cutting plane constraint, and adding it to strengthen the LP relaxation until an optimal integer solution is found. The key steps are to decompose the LP into basic and non-basic variables, derive cutting plane coefficients from the LP tableau, and add constraints of the form [yij]xj + xBi - yi0 ≤ 0 to eliminate fractional solutions.
A problem is provided which is solved by using graphical and analytical method of linear programming method and then it is solved by using geometrical concept and algebraic concept of simplex method.
this ppt is helpful for BBA/B.tech//MBA/M.tech students.
the ppt is on simulation topic...its covers -
Meaning
Advantages & Disadvantages
Uses
Process
Monte Carlo SImulation
Advantages & Disadvantages
Its example
The transportation problem is a special type of linear programming problem where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations.
Because of its special structure, the usual simplex method is not suitable for solving transportation problems. These problems require a special method of solution.
A problem is provided which is solved by using graphical and analytical method of linear programming method and then it is solved by using geometrical concept and algebraic concept of simplex method.
this ppt is helpful for BBA/B.tech//MBA/M.tech students.
the ppt is on simulation topic...its covers -
Meaning
Advantages & Disadvantages
Uses
Process
Monte Carlo SImulation
Advantages & Disadvantages
Its example
The transportation problem is a special type of linear programming problem where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations.
Because of its special structure, the usual simplex method is not suitable for solving transportation problems. These problems require a special method of solution.
Medical Conferences, Pharma Conferences, Engineering Conferences, Science Conferences, Manufacturing Conferences, Social Science Conferences, Business Conferences, Scientific Conferences Malaysia, Thailand, Singapore, Hong Kong, Dubai, Turkey 2014 2015 2016
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"SSumM: Sparse Summarization of Massive Graphs", KDD 2020KyuhanLee4
A presentation slides of Kyuhan Lee, Hyeonsoo Jo, Jihoon Ko, Sungsu Lim, Kijung Shin, "SSumM: Sparse Summarization of Massive Graphs", KDD 2020.
Given a graph G and the desired size k in bits, how can we summarize G within k bits, while minimizing the information loss?
Large-scale graphs have become omnipresent, posing considerable computational challenges. Analyzing such large graphs can be fast and easy if they are compressed sufficiently to fit in main memory or even cache. Graph summarization, which yields a coarse-grained summary graph with merged nodes, stands out with several advantages among graph compression techniques. Thus, a number of algorithms have been developed for obtaining a concise summary graph with little information loss or equivalently small reconstruction error. However, the existing methods focus solely on reducing the number of nodes, and they often yield dense summary graphs, failing to achieve better compression rates. Moreover, due to their limited scalability, they can be applied only to moderate-size graphs.
In this work, we propose SSumM, a scalable and effective graph-summarization algorithm that yields a sparse summary graph. SSumM not only merges nodes together but also sparsifies the summary graph, and the two strategies are carefully balanced based on the minimum description length principle. Compared with state-of-the-art competitors, SSumM is (a) Concise: yields up to 11.2X smaller summary graphs with similar reconstruction error, (b) Accurate: achieves up to 4.2X smaller reconstruction error with similarly concise outputs, and (c) Scalable: summarizes 26X larger graphs while exhibiting linear scalability. We validate these advantages through extensive experiments on 10 real-world graphs.
Directed Optimization on Pareto FrontiereArtius, Inc.
New multi-objective optimization technology is presented which considers Pareto frontier as
a search space for finding Pareto optimal solutions that meet the user’s preferences.
Typically, 80-90% of points evaluated by new optimization algorithms are Pareto optimal,
and the majority of them are located in the user’s area of interest on the Pareto frontier. In
contrast, conventional optimization techniques search for Pareto optimal solutions in the
entire domain, which increases computational effort by orders of magnitude. New
optimization technology is represented by two new algorithms: Multi-Gradient Pathfinder
(MGP), and Hybrid Multi-Gradient Pathfinder (HMGP) (patent pending). MGP is a pure
gradient-based algorithm; it starts from a Pareto-optimal point, and steps along the Pareto
surface in the direction that allows improving a subset of objective functions with higher
priority. HMGP is a hybrid of a gradient-based technique and genetic algorithms (GA); it
works similarly to MGP, but in addition, searches for dominating Pareto frontiers. HMGP is
designed to find the global Pareto frontier and the best Pareto optimal points on this frontier
with respect to preferable objectives. Both algorithms are designed for optimizing very
expensive models, and are able to optimize models ranging from a few to thousands of design
variables.
Main obstacles of Bayesian statistics or Bayesian machine learning is computing posterior distribution. In many contexts, computing posterior distribution is intractable. Today, there are two main stream to detour directly computing posterior distribution. One is using sampling method(ex. MCMC) and another is Variational inference. Compared to Variational inference, MCMC takes more time and vulnerable to high-dimensional parameters. However, MCMC has strength in simplicity and guarantees of convergence. I'll briefly introduce several methods people using in application.
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Crude-Oil Scheduling Technology: moving from simulation to optimizationBrenno Menezes
Scheduling technology either commercial or homegrown in today’s crude-oil refining industries relies on a complex simulation of scenarios where the user is solely responsible for making many different decisions manually in the search for feasible solutions over some limited time-horizon i.e., trial-and-error heuristics. As a normal outcome, schedulers abandon these solutions and then return to their simpler spreadsheet simulators due to: (i) time-consuming efforts to configure and manage numerous scheduling scenarios, and (ii) requirements of updating premises and situations that are constantly changing. Moving to solutions based in optimization rather than simulation, the lecture describes the future steps in the refactoring of the scheduling technology in PETROBRAS considering in separate the graphic user interface (GUI) and data communication developments (non-modeling related), and the modeling and process engineering related in an automated decision-making with built-in problem representation facilities and integrated data handling features among other techniques in a smart scheduling frontline.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
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Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
2. Outline
1. Why Integer Programming
2. Introduction to All Integer Linear Programming Problem (AILP) and Mixed Integer Linear
Programming Problem (MILP)
3. Common Approach for solving AILP
4. Introduction to Gomory’s Cutting Plane Method
5. Derivation of Gomory’s Cutting Plane Method
6. Gomory’s Cutting Plane Method Algorithms
7. Explaination of Gomory’s Cutting Plane Method Algorithm with Example
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3. Why Integer Programming
Production Problem
◦ Items being produced may be in complete units
◦ E.g. TV Sets of 21” and 29”
◦ Therefore fractional number of item have no meaning
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4. IPP Expression
PROBLEM DEFINITION
𝑀𝑎𝑥 𝑧 = 𝑗=1
𝑛
𝑐𝑗 𝑥𝑗
subject to
𝑗=1
𝑛
𝑎𝑖𝑗 𝑥𝑗 = 𝑏𝑖 (𝑖 = 1, … , 𝑚)
𝑥𝑗 ≥ 0 (j=1,…,n)
and
𝑥𝑗 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑓𝑜𝑟 𝑗1∁ 𝑗
where j={1,2, … ,n}
DEFINITION
All Integer LPP (AILP):- If all variable take
integer values only. (if 𝒋 𝟏 = 𝒋)
(slack & surplus variable take integer value)
Mixed Integer LPP (MILP):- If some but not
all variable of the problem are constrained
Integer values.
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5. IPP Example
EXAMPLE OF AILP
𝑀𝑎𝑥 𝑧 = 4𝑥1 + 3𝑥2
subject to
𝑥1 + 𝑥2 ≤ 8
2𝑥1 + 𝑥2 ≤ 10
𝑥1, 𝑥2 ≥ 0
and
𝑥1 𝑎𝑛𝑑 𝑥2 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
𝑥1 and 𝑥2 are non-negative integer
slack variable
𝑥3 = 8 − 𝑥1 − 𝑥2 &
𝑥4 = 10 − 2𝑥1 − 𝑥2
are also non-negative integer
if we consider 2nd constraints is given as:
2𝑥1 + 𝑥2 ≤ 10;
𝑥4 = 10 − 2𝑥1 − 𝑥2
Then this problem no more AILP. But MILP
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6. Common Approach (Rounding off)
PROBLEM
𝑀𝑎𝑥 𝑧 = 21𝑥1 + 11𝑥2
subject to
7𝑥1 + 4𝑥2 ≤ 13
𝑥1, 𝑥2 ≥ 0
and
𝑥1 𝑎𝑛𝑑 𝑥2 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
The Feasible Set of discrete points
0,0 , 0,1 , 1,0 , 1,1 , 0,2 , 0,3 .
Lies inside feasible region, can be visualize in
figure
Optimal Soln. of ILP 𝑥1
∗
= 0, 𝑥2
∗
= 3, 𝑧∗
= 33
Optimal Soln. of LLP (𝑥1
∗
= 13/7, 𝑥2
∗
= 0, 𝑧∗
= 39)
Rounding LLP Soln. (𝑥1
∗
= 2, 𝑥2
∗
= 0, 𝑧∗
= 42),
two obj. fn are not close in any meaningful sense
Rounding off is not correct approach to solve ILP’s5/8/2015 HTTPS://SITES.GOOGLE.COM/SITE/PIRYANIRAJESH/ 6
7. Common Approach (Convex Hull)
PROBLEM
𝑀𝑎𝑥 𝑧 = 21𝑥1 + 11𝑥2
subject to
7𝑥1 + 4𝑥2 ≤ 13
𝑥1, 𝑥2 ≥ 0
and
𝑥1 𝑎𝑛𝑑 𝑥2 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
The Feasible Set of the given ILP is non convex, its convex hull is a
polytope whose corner points meet the integer requirements.
0,0 , 0,1 , 1,0 , 1,1 , 0,2 , 0,3 .
Lies inside feasible region, can be visualize in figure
Optimal Soln. of ILP 𝑥1
∗
= 0, 𝑥2
∗
= 3, 𝑧∗
= 33
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8. Common Approach (Convex Hull)
PROBLEM (ILP EQUIVALENT TO SOLVING LPP)
𝑀𝑎𝑥 𝑧 = 𝑗=1
𝑛
𝑐𝑗 𝑥𝑗
subject to
(𝑥1, … , 𝑥 𝑛) ∈ 𝑆,
Where S is the polytope
𝑗=1
𝑛
𝑎𝑖𝑗 𝑥𝑗 = 𝑏𝑖(𝑖 = 1, … , 𝑚)
𝑥𝑗 ≥ 0 (j=1,…,n)
and
𝑥𝑗 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑓𝑜𝑟 𝑗 ∈ 𝐽1∁ 𝐽 = {1, … , 𝑛}.
Optimal Soln. of ILP 𝑥1
∗
= 0, 𝑥2
∗
= 3, 𝑧∗
= 33
This method is perfectly valid except that there are
certain practical difficulties in getting the convex
hull. When Euclidean space is more than two or
three
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9. Gomory’s Cutting Plane method for AILP
PROBLEM (ILP EQUIVALENT TO SOLVING LPP)
𝑀𝑎𝑥 𝑧 = 𝑐 𝑇
𝑥
subject to
𝐴𝑥 = 𝑏,
𝑥 ≥ 0
𝑥 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
𝐴, 𝑏 𝑎𝑛𝑑 𝑐 are integer,
The objective function is automatically constrained to be integer.
Let
(𝑳𝑷) 𝟏→ 𝑨𝒔𝒔𝒐𝒄𝒊𝒂𝒕𝒆𝒅 𝑳𝑷𝑷 𝒇𝒐𝒓 𝑨𝑰𝑳𝑷
𝒙(𝟏)
→ 𝑶𝒑𝒕𝒊𝒎𝒂𝒍 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏
(if all constrained are integer then it is optimal
solution.
Else according to Gomory,
A new constrained 𝒑 𝑻
𝒙 ≤ 𝒅 append to new
(𝑳𝑷) 𝟏 to get a new (𝑳𝑷) 𝟐
The basic purpose of the cut constrained
◦ Delete a part of the feasible region 𝑺 𝟏
◦ Don’t delete the points which have integer
coordinates
Finitely many cut constrained will be needed to
solve the given AILP.
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11. Gomory’s Cutting Plane method for AILP
DERIVATION OF THE GOMORY’S CUT
CONSTRAINT
𝑨𝑰𝑳𝑷 𝑹𝒆𝒑𝒓𝒔𝒆𝒏𝒕𝒂𝒕𝒊𝒐𝒏 𝑀𝑎𝑥 𝑧 = 𝑐 𝑇
𝑥
subject to
𝐴𝑥 = 𝑏,
𝑥 ≥ 0, 𝑥 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 (Eq. 0)
𝑨𝒔𝒔𝒐𝒄𝒊𝒂𝒕𝒆𝒅 𝑳𝑷𝑷
𝑀𝑎𝑥 𝑧 = 𝑐 𝑇
𝑥
subject to
𝐴𝑥 = 𝑏, 𝑥 ≥ 0 (Eq. 1)
𝒙 𝑩 𝒊
= 𝒚𝒊𝟎 −
𝒋∈𝑹
𝒚𝒊𝒋 𝒙𝒋 𝒇𝒐𝒓 𝒊 = 𝟎, 𝟏, … , 𝒎 (𝑬𝒒. 𝟒)
This holds for any feasible solution of LPP (Eq. 1) and (Eq. 0)
If for any real number a
Fractional part 𝒇 𝒂 = 𝒂 − 𝒂
[𝒂] → 𝒈𝒓𝒆𝒂𝒕𝒆𝒔𝒕 𝒊𝒏𝒕𝒆𝒈𝒆𝒓 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏
𝟎 ≤ 𝒇 𝒂 < 𝟏
For 𝒂 = −𝟏 𝒇 𝒂 = 𝟎
But 𝒂 = −𝟏. 𝟔,
𝒇 𝒂 = −𝟏. 𝟔 − −𝟏. 𝟔 = −𝟏. 𝟔 − −𝟐 = 𝟎. 𝟒
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12. Gomory’s Cutting Plane method for AILP
𝒙 𝑩 𝒊
= 𝒚𝒊𝟎 −
𝒋∈𝑹
𝒚𝒊𝒋 𝒙𝒋 𝒇𝒐𝒓 𝒊 = 𝟎, 𝟏, … , 𝒎 (𝑬𝒒. 𝟒)
𝒋∈𝑹
𝒚𝒊𝒋 𝒙𝒋 +
𝒋∈𝑹
𝒚𝒊𝒋 − 𝒚𝒊𝒋 𝒙𝒋 + 𝒙 𝑩 𝒊
= 𝒚𝒊𝟎 + (𝒚𝒊𝟎 − [𝒚𝒊𝟎])
i.e.
𝒋∈𝑹
[𝒚𝒊𝒋] 𝒙𝒋 + 𝒙 𝑩 𝒊
− 𝒚𝒊𝟎 = 𝒚𝒊𝟎 − 𝒚𝒊𝟎 −
𝒋∈𝑹
𝒚𝒊𝒋 − 𝒚𝒊𝒋 𝒙𝒋
i.e.
𝒋∈𝑹
[𝒚𝒊𝒋] 𝒙𝒋 + 𝒙 𝑩 𝒊
− 𝒚𝒊𝟎 = 𝒇𝒊𝟎 −
𝒋∈𝑹
𝒇𝒊𝒋 𝒙𝒋 (𝑬𝒒. 𝟓)
(Eq. 5) holds for all feasible points of points LPP (Eq. 0) and for the given AILP.
Therefore the R.H.S must also be integer.
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13. Gomory’s Cutting Plane method for AILP
𝑓𝑖0 −
𝑗∈𝑅
𝑓𝑖𝑗 𝑥𝑗
𝑖 = 0 included because for the AILP, the objective function is also constrained to be integer.
𝑁𝑜𝑤 𝑓𝑖𝑗 ≥ 0 𝑎𝑛𝑑 𝑥𝑗 ≥ 0 𝑓𝑜𝑟 𝑗 ∈ 𝑅. 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒
𝑗∈𝑅 𝑓𝑖𝑗 𝑥𝑗 ≥ 0 (Eq. 6)
𝑓𝑖0 < 1 𝑎𝑛𝑑 𝐸𝑞. 6 𝑔𝑖𝑣𝑒𝑠
𝑓𝑖0 − 𝑗∈𝑅 𝑓𝑖𝑗 𝑥𝑗 < 1 is an integer
𝒇𝒊𝟎 − 𝒋∈𝑹 𝒇𝒊𝒋 𝒙𝒋 ≤ 𝟎 (Eq. 7)
The inequality (Eq. 7) is satisfied by every integer feasible point of the given AILP.
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14. Gomory’s Cutting Plane method for AILP
If current b.f.s. 𝒙 𝑩 is not an integer. It doesn’t meet the requirement of AILP.
In that case, inequality is not satisfied.
“ It certainly deletes a part of the feasible region of the associated LLP ( at least the current
b.f.s. 𝒙 𝑩 and may be more points) but does not delete any feasible point with integer co-
ordinates. Hence it is valid cut constraint and it is called Gomory’s cut constraint”
−𝑓𝑖0= 𝑠𝑖 −
𝑗∈𝑅
𝑓𝑖𝑗 𝑥𝑗
Append this to associated LPP, (𝐿𝑃)1 to get the new LPP (𝐿𝑃)2 Therefore we solve
(𝐿𝑃)2 𝑎𝑛𝑑 𝑟𝑒𝑝𝑒𝑎𝑡 𝑡ℎ𝑒 𝑝𝑟𝑜𝑐𝑒𝑑𝑢𝑟𝑒.
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15. Stepwise Description
Step 1: Solve the associated LLP, say (𝑳𝑷) 𝟏,by the simplex method. Set 𝒌 = 𝟏
Step 2:
◦ If the optimal solution obtained at Step 1 is integer
◦ Stop
◦ Otherwise go to Step3
Step 3: For any updated constraint 𝒊 whose 𝒚𝒊𝟎 value is fractional (including 𝒊 = 𝟎, i.e. obj. fun.)
◦ Generate Gomory’s cut constraint as given at (6.13).
◦ Select the value of 𝒊, 𝟎 ≤ 𝒊 ≤ 𝒎 for which 𝒇𝒊𝟎 value is maximum.
◦ Theoretically we can choose any i for which 𝒇𝒊𝟎 > 𝟎 but the maximum of 𝒇𝒊𝟎 is chosen with the hope that it may give a
deeper cut
Step 4: Append the Gomory’s cut constraint derived at Step 3 above the (𝑳𝑷) 𝒌 to get the new LPP
(𝑳𝑷) 𝒌+𝟏 .
◦ Solve by the dual simplex method and return to Step2
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16. Theorem
“The number of Gomory’s cut constraints needed to solve any
instance of all integer linear programming (AILP) problem is
always finite”
As the no. of cut constraints needed is always finite, we are solving
only finitely many LPP to get an optimal solution of the given AILP.
But unfortunately, even for a problem of “average” size, the no. of
cut constraints needed may be ‘too many’ as AILP belongs to the
class of Hard Problem.
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17. Example
CONSIDER THE INTEGER LPP
𝑀𝑎𝑥 𝑧 = 5𝑥1 + 2𝑥2
subject to
2𝑥1 + 2𝑥2 ≤ 9
3𝑥1 + 𝑥2 ≤ 11
𝑥1, 𝑥2 ≥ 0
𝑥1, 𝑥2integer
THE GIVEN ILP IS EQUIVALENT TO
𝑀𝑎𝑥 𝑧 = 5𝑥1 + 2𝑥2 + 0𝑥3 + 0𝑥4
subject to
2𝑥1 + 2𝑥2 + 𝑥3 = 9
3𝑥1 + 𝑥2 + 𝑥4 = 11
𝑥1, 𝑥2, 𝑥3, 𝑥4 ≥ 0
all integer
𝑥3 = 9 − 2𝑥1 − 2𝑥2 and
𝑥4 = 11 − 3𝑥1 − 𝑥2
𝑥1, 𝑥2 𝑎𝑟𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟, 𝑠𝑜 𝑥3, 𝑥4 𝑎𝑟𝑒 𝑎𝑙𝑠𝑜 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
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