MBA SEMESTER II MB0048 –Operation Research- 4 Credits (Book ID: B1137) Assignment Set- 1 (60 Marks)

Q1. a. Explain how and why Operation Research methods have been valuable inaiding executive decisions. b. Discuss the usefulness of Operation Research indecision making process and the role of computers in this field.Answer.Churchman, Aackoff and Aruoff defined Operations Research as:“the application of scientific methods, techniques and tools to operation of a systemwith optimum solutions to the problems”, where ‘optimum’ refers to the best possiblealternative.The objective of Operations Research is to provide a scientific basis to the decisionmakersfor solving problems involving interaction of various components of theorganization. You can achieve this by employing a team of scientists from differentdisciplines, to work together for finding the best possible solution in the interest of theorganization as a whole. The solution thus obtained is known as an optimal decision.You can also define Operations Research as“ The use of scientific methods to provide criteria for decisions regarding man,machine, and systems involving repetitive operations”.OR“Operation Techniques is a bunch of mathematical techniques.”b. “Operation Research is an aid for the executive in making his decisions based onscientific methods analysis”. Discuss the above statement in brief.Ans.“Operation Research is an aid for the executive in making his decisions based onscientific methods analysis”.Discussion:-Any problem, simple or complicated, can use OR techniques to find thebest possible solution. This section will explain the scope of OR by seeing its

application in various fields of everyday life.i) In Defense Operations:In modern warfare, the defense operations are carried out by three major independentcomponents namely Air Force, Army and Navy. The activities in each of thesecomponents can be further divided in four sub-components namely: administration,intelligence, operations and training and supply. The applications of modern warfaretechniques in each of the components of military organizations require expertiseknowledge in respective fields. Furthermore, each component works to drive maximumgains from its operations and there is always a possibility that the strategy beneficial toone component may be unfeasible for another component. Thus in defense operations,there is a requirement to co-ordinate the activities of various components, which givesmaximum benefit to the organization as a whole, having maximum use of the individualcomponents. A team of scientists from various disciplines come together to study thestrategies of different components. After appropriate analysis of the various courses ofactions, the team selects the best course of action, known as the ‘optimum strategy’.ii) In Industry: The system of modern industries is so complex that the optimum pointof operation in its various components cannot be intuitively judged by an individual.The business environment is always changing and any decision useful at one time maynot be so good sometime later. There is always a need to check the validity of decisionscontinuously against the situations. The industrial revolution with increased division oflabor and introduction of management responsibilities has made each component anindependent unit having their own goals. For example: production departmentminimizes the cost of production but maximise output. Marketing departmentmaximizes the output, but minimizes cost of unit sales. Finance department tries to

optimize the capital investment and personnel department appoints good people atminimum cost. Thus each department plans its own objectives and all these objectivesof various department or components come to conflict with one another and may notagree to the overall objectives of the organization. The application of OR techniqueshelps in overcoming this difficulty by integrating the diversified activities of variouscomponents to serve the interest of the organization as a whole efficiently. OR methodsin industry can be applied in the fields of production, inventory controls and marketing,purchasing, transportation and competitive strategies.iii)Planning:In modern times, it has become necessary for every government to havecareful planning, for economic development of the country. OR techniques can befruitfully applied to maximise the per capita income, with minimum sacrifice and time.A government can thus use OR for framing future economic and social policies.iv) Agriculture:With increase in population, there is a need to increase agriculture output. But thiscannot be done arbitrarily. There are several restrictions. Hence the need to determine acourse of action serving the best under the given restrictions. You can solve thisproblem by applying OR techniquesv) In Hospitals:OR methods can solve waiting problems in out-patient department of big hospitals andadministrative problems of the hospital organizations.vi) In Transport:You can apply different OR methods to regulate the arrival of trains and processingtimes minimize the passengers waiting time and reduce congestion, formulate suitabletransportation policy, thereby reducing the costs and time of trans-shipment.vii) Research and Development:You can apply OR methodologies in the field of R&D for several purposes, such as tocontrol and plan product introductions.

Q2. Explain how the linear programming technique can be helpful in decisionmakinginthe areas of Marketing and Finance.Ans. Linear programming problems are a special class of mathematical programmingproblems for which the objective functions and all constraints are linear. A classicexample of the application of linear programming is the maximization of profits givenvarious production or cost constraints. Linear programming can be applied to a varietyof business problems, such as marketing mix determination, financial decision making,production scheduling, workforce assignment, and resource blending. Such problemsare generally solved using the “simplex method.”. The local Chamber of Commerceperiodically sponsors public service seminars and programs. Promotional plans areunder way for this year’s program. Advertising alternatives include television, radio,and newspaper. Audience estimates, costs, and maximum media usage limitations areshown in Exhibit 1.If the promotional budget is limited to $18,200, how manycommercial messages should be run on each medium to maximize total audiencecontact? Linear programming can find the answer.Q3. a. How do you recognise optimality in the simplex method?b. Write the role of pivot element in simplex table?Ans.Simplex method is used for solving Linear programming problem especially when morethan two variables are involvedSIMPLEX METHOD1. Set up the problem.That is, write the objective function and the constraints.2. Convert the inequalities into equations.This is done by adding one slack variable for each inequality.3. Construct the initial simplex tableau.Write the objective function as the bottom row.4. The most negative entry in the bottom row identifies a column.5. Calculate thequotients. The smallest quotient identifies a row. The element in the intersection of thecolumn identified in step 4 and the row identified in this step is identified as the pivot

element.MaximizeZ = 40×1 + 30×2Subject to:x1 + x2 ≤ 122×1 + x2 ≤ 16x1 ≥ 0; x2 ≥ 02. Convert the inequalities into equations.This is done by adding one slack variable for eachi n equality.Forexample to convert the inequality x1 + x2 ≤ 12 into an equation, weadd a non-negativevariable y1, and we getx1 + x2 + y1 = 12Here thevariable y1 picks up the slack, and it represents the amount by whichx1 + x2 fallsshort of 12. In this problem, if Niki works fewer that 12hours, say 10, then y1 is 2. Later whenwe read off the final solutionfrom the simplex table, the values of the slack variables willidentifythe unused amounts.We can even rewrite the objective function Z =40×1 + 30×2 as – 40×1 – 30×2 + Z = 0.After adding the slackvariables, our problem readsObjective function: – 40×1 – 30×2 + Z = 0Subject to constraints: x1+ x2 + y1 = 122×1 + x2 + y2 = 16x1 ≥ 0; x2 ≥ 03. Construct the initial simplex tableau.Write the objective function as the bottom row. Now that theinequalities are converted into equations, we can represent theproblem into an augmented matrix called the initial simplex tableauas followsHere the vertical line separates the left hand side of the equationsfrom the right side. The horizontal line separates the constraints fromthe objective function. The right side of the equation is representedby the column CThe reader needs to observe that the last four columns of this matrixlook like the final matrix for the solution of a system of equations. Ifwe arbitrarily choosex1 = 0 and x2 = 0, we get Which reads y1 = 12 y2 = 16 Z =0The solution obtained by arbitrarily assigning values to somevariables and then solving for the remaining variables is called thebasic solution associated with the tableau. So the above solution is thebasic solution associated with the initial simplex tableau. We can labelthe basic solution variable in the right of the last column as shown inthe table below.The most negative entry in the bottom row identifies a column.The most negative entry in the bottom row is –40, therefore thecolumn 1 is identifiedQ.4 What is the significance of duality theory of linearprogramming?Describe the general rules for writing the dual of a linearprogramming problem.Ans. Linear programming

(LP) is a mathematical method for determining a way to achievethe best outcome (such as maximum profit or lowest cost) in a givenmathematical model for some list of requirements represented aslinear relationships. Linear programming is a specific caseof mathematical programming. More formally, linear programming isa technique for the optimization of a linear objective function, subjectto linear equality and linear inequality constraints. Given a polytopeand a real- value daffine function defined on this polytope, a linearprogramming method will find a point on the polytope where thisfunction has the smallest (or largest) value if such point exists, bysearching through the polytope vertices Linear programs areproblems that can be expressed in canonical form : where xrepresents the vector of variables (to be determined), c and bare vectors of (known) coefficients and A is a (known)matrix ofcoefficients. The expression to be maximized or minimized is calledthe objective function (c Tx in this case).The equations A x ≤ b are the constraints which specify a convexpolytope over which the objective function is to be optimized. (In thiscontext, two vectors are comparable when every entry in one is lessthanor equal-to the corresponding entry in the other. Otherwise, theyare incomparable.)Linear programming can be applied to variousfields of study. It is used most extensively in business and economics,but can also be utilized for some engineering problems. Industriesthat use linear programming models include transportation, energy,telecommunications, and manufacturing. It has proved useful inmodeling diverse types of problems in planning, routing, scheduling,assignment, and design.Duality:Every linear programming problem, referred to as aprimalproblem, can be convertedinto adual problem, which provides anupper bound to the optimal value of the primal problem.In matrixform, we can express theprimalproblem as:Maximizec T x subject to A x ≤ b , x≥ 0;with the correspondingsymmetric dual problem,Minimizeb T y subject to A T y ≥ c ,y ≥ 0.An alternative primal formulation is:Maximizec T x subject toA x ≤ b;with the correspondingasymmetricdual problem,

Minimizeb T y subject to A T y = c, y ≥ 0.There are two ideas fundamental to duality theory. One is the factthat (for the symmetric dual) the dual of a dual linear program is theoriginal primal linear program. Additionally, every feasible solution fora linear program gives a bound on the optimal value of the objectivefunction of its dual.The weak duality theorem states that the objective function value ofthe dual at any feasible solution is always greater than or equal to theobjective function value of the primal at any feasible solution. Thestrong duality theorem states that if the primal has an optimalsolution,x*, then the dual also has an optimal solution, y*, such thatcTx*=bTy* . A linear program can also be unbounded or infeasible.Duality theory tells us that if the primal isunbounded then the dual isinfeasible by the weak duality theorem. Likewise, if the dualisunbounded, then the primal must be infeasible. However, it ispossible for both the dual and the primal to be infeasible

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MBA SEMESTER II MB0048 –Operation Research- 4 Credits (Book ID: B1137) Assignment Set- 2 (60 Marks)

1. What are the essential characteristics of Operation Research? Mention different phases in an Operation Research study. Point out some limitations of O.R? [10 Marks]

Ans. Characteristics of Operations ResearchOperations research, an interdisciplinary division of mathematics andscience, uses statistics, algorithms and mathematical modelingtechniques to solve complex problems for the best possible solutions.This science is basically concerned with optimizing maxima andminima of the objective functions involved. Examples of maximacould be profit, performance and yield. Minima could be loss and risk.The management of various companies has benefited immenselyfrom operations research.Operations research is also known as OR. It has basic characteristicssuch as systems orientation, using interdisciplinary groups, applyingscientific methodology, providing quantitative answers, revelation ofnewer problems and the consideration of human factors in relation tothe state under which research is being conducted.Systems Orientationo This approach recognizes the fact that the behavior of any part ofthe system has an effect on the system as a whole. This stresses theidea that the interaction between parts of the system is whatdetermines the functioning of the system. No single part of thesystem can have a bearing effect on the whole. OR attempts appraisethe effect the changes of any single part would have on theperformance of the system as a whole. It then searches for the causesof the problem that has arisen either in one part of the system or inthe interrelation parts.

Interdisciplinary groupso The team performing the operational research is drawn fromdifferent disciplines. The disciplines could include mathematics,psychology, statistics, physics, economics and engineering. Theknowledge of all the people involved aids the research andpreparation of the scientific model.Application of Scientific Methodologyo OR extensively uses scientific means and methods to solveproblems. Most OR studies cannot be conducted in laboratories, andthe findings cannot be applied to natural environments. Therefore,

scientific and mathematical models are used for studies. Simulation ofthese models is carried out, and the findings are then studied withrespect to the real environment.New Problems Revealedo Finding a solution to a problem in OR uncovers additional problems.To obtain maximum benefits from the study, ongoing and continuousresearch is necessary. New problems must be pursued immediately tobe resolved. A company looking to reduce costs in manufacturingmight discover in the process that it needs to buy one morecomponent to manufacture the end product. Such a scenario wouldresult in unexpected costs and budget overruns. Ensuring flexibilityfor such contingencies is a key characteristic of OR.Provides Quantitative Answerso The solutions found by using operations research are alwaysquantitative. OR considers two or more options and emphasizes thebest one. The company must decide which option is the bestalternative for it.Human Factorso In other forms of quantitative research, human factors are notconsidered, but in OR, human factors are a prime consideration.People involved in the process may become sick, which would affectthe company’s output.PHASES OPERATIONS RESEARCH・Formulate the problem:This is the most important process, it is generally lengthy and timeconsuming. The activities that constitute this step are visits,observations, research, etc. With the help of such activities, the O.R.scientist gets sufficient information and support to proceed andis better prepared to formulate the problem. This process starts withunderstanding of the organizational climate, its objectives andexpectations. Further, the alternative courses of action are discoveredin this step.• Develop a model:Once a problem is formulated, the next step is to express the probleminto a mathematical model that represents systems, processes orenvironment in the form of equations, relationships or formulas. Wehave to identify both the static and dynamic structural elements, anddevice mathematical formulas to represent the interrelationshipsamong elements. The proposed model may be field tested andmodified in order to work under stated environmental constraints. Amodel may also be modified if the management is not satisfied withthe answer that it gives.• Select appropriate data input:Garbage in and garbage out is a famous saying. No model will workappropriately if data input is not appropriate. The purpose of this stepis to have sufficient input to operate and test the model.

• Solution of the model:After selecting the appropriate data input, the next step is to find asolution. If the model is not behaving properly, then updating andmodification is considered at this stage.• Validation of the model:A model is said to be valid if it can provide a reliable prediction of thesystem’s performance. A model must be applicable for a longer timeand can be updated from time to time taking into consideration thepast, present and future aspects of the problem.• Implement the solution:The implementation of the solution involves so many behaviouralissues and the implementing authority is responsible for resolvingthese issues. The gap between one who provides a solution and onewho wishes to use it should be eliminated. To achieve this, O.R.scientist as well as management should play a positive role. Aproperly implemented solution obtained through O.R. techniquesresults in improved working and wins the management support.Limitations• Dependence on an Electronic Computer:O.R. techniques try to find out an optimal solution taking into accountall the factors. In the modern society, these factors are enormous andexpressing them in quantity and establishing relationships amongthese require voluminous calculations that can only be handled bycomputers.• Non-Quantifiable Factors:O.R. techniques provide a solution only when all the elements relatedto a problem can be quantified. All relevant variables do not lendthemselves to quantification. Factors that cannot be quantified find noplace in O.R. models.• Distance between Manager and Operations Researcher:O.R. being specialist’s job requires a mathematician or a statistician,who might not be aware of the business problems. Similarly, amanager fails to understand the complex working of O.R. Thus, thereis a gap between the two.• Money and Time Costs:When the basic data are subjected to frequent changes, incorporatingthem into the O.R. models is a costly affair. Moreover, a fairly goodsolution at present may be more desirable than a perfect O.R.solution available after sometime.• Implementation:Implementation of decisions is a delicate task. It must take intoaccount the complexities of human relations and behaviour.Q2. What are the common methods to obtain an initial basicfeasible solution for a transportation problem whose cost andrequirement table is given? Give a stepwise procedure for oneof them?

Ans.Transportation Problem & its basic assumptionThis model studies the minimization of the cost of transporting acommodity from a number of sources to several destinations. Thesupply at each source and the demand at each destination are known.The transportation problem involves m sources, each of which hasavailable. i (i = 1, 2, …..,m) units of homogeneous product and ndestinations, each of which requires bj (j = 1, 2…., n) units ofproducts. Here a i and bj are positive integers. The cost cij oftransporting one unit of the product from the ith source to the jthdestination is given for each i and j. The objective is to develop anintegral transportation schedule that meets all demands from theinventory at a minimum total transportation cost. It is assumed thatthe total supply and the total demand are equal.i.e.Condition (1)Thecondition (1) is guaranteed by creating either a fictitious destinationwith ademand equal to the surplus if total demand is less than thetotal supply or a (dummy)source with a supply equal to the shortageif total demand exceeds total supply. The cost of transportation fromthe fictitious destination to all sources and from all destinations to thefictitious sources are assumed to be zero so that total cost oftransportation will remain the same.Formulation of Transportation ProblemThe standard mathematical model for the transportation problem isas follows. Let xij be number of units of the homogenous product tobe transported from source i to the destination j Then objective is toTheorem:A necessary and sufficient condition for the existence of a feasiblesolution to the transportation problem (2) is thatQ3. a. What are the properties of a game? Explain the “beststrategy” on the basis of minmax criterion of optimality.b. State the assumptions underlying game theory. Discuss itsimportance to business decisions.Ans. a) Minimax (sometimes minmax) is a decision rule usedindecision theory, game theory, statistics and philosophy forminimizing the possible loss while maximizing the potential gain.Alternatively, it can be thought of as maximizing the minimum gain(maxim in). Originally formulated for two-player zero-sum gametheory, covering both the cases where players take alternate movesand those where they make simultaneous moves, it has also beenextended to more complex games and to general decision making inthe presence of uncertainty.Game theoryIn the theory of simultaneous games, a minima strategy is a mixedstrategy which is part of the solution to a zero-sum game. In zero-sumgames, the minima solution is the same as theNashequilibrium.Minimax theorem The minimax theorem states: For

every two-person, zero-sum game with finitely many strategies, thereexists a value V and a mixed strategy for each player, such that (a)Given player 2′s strategy, the best payoff possible for player 1 is V,and (b) Given player 1′s strategy, the best payoff possible for player 2is −V.Equivalently, Player 1′s strategy guarantees him a payoff of Vregardless of Player 2′s strategy, and similarly Player 2 can guaranteehimself a payoff of −V. The name minimax arises because eachplayer minimizes the maximum payoff possible for the other—sincethe game is zero-sum, he also maximizes his own minimum payoff.This theorem was established by John von Neumann,[1]who is quoted as saying “As far as I can see, there could be notheory of games … without that theorem … I thought there wasnothing worth publishing until the Minimax Theorem was proved”.[2]See Scion’s minimax theorem and Parthasarathy’s theorem forgeneralizations; see also example of a game without a value. ExampleThe following example of a zero-sum game, whereAand Bmakesimultaneous moves,illustratesminimaxsolutions. Suppose eachplayer has three choices and consider the payoff matrixforAdisplayed at right.Assume the payoff matrix for Bis thesame matrixwith the signs reversed(i.e. if the choices are A1 and B1 thenBpays 3toA). Then, the minimaxchoice for Ais A2 since the worst possibleresult is then having to pay 1, while the simpleminimax choice for BisB2 since the worst possible result is then no payment. However, thissolution is not stable, since if BbelievesAwill choose A2 then willchoose B1 to gain 1; then if AbelievesBwill choose B1 then will chooseA1 to gain 3; and then will choose B2; and eventually both players willrealize the difficulty of making a choice. So a more stable strategyisneeded.Some choices are dominated by others and can beeliminated: will not choose A3 since either A1 or A2 will produce abetter result, no matter whatBchooses;Bwill not choose B3 sincesome mixtures of B1 and B2 will produce a better result, no matterwhatAchooses.Acan avoid having to make an expected payment ofmore than 1/3 by choosing A1 with probability 1/6 and A2 withprobability 5/6, no matter whatBchooses.Bcan ensure an expectedBchooses B1B chooses B2B chooses B3A chooses A1+3 −2 +2Achooses A2−1 0 +4A chooses A3−4 −3 +1 wouldn’t have to pay asmuch to license these characters. Changing the rules is another wayin which companies can benefit. The authors introduce the idea ofjudo economics, where a large company may be willing to allow asmaller company to capture a small market share rather thancompete by lowering its prices. As long as it does not become toopowerful or greedy, a small company can often participate in thesame market without having to compete with larger companies onunfavorable terms. Kiwi International Air Lines introduced services onits carriers that were of lower prices to get market share, but madesure that the competitors understood that they had no intention of

capturing more than 10% of any market. Companies can also changeperceptions to make themselves better off. This can be accomplishedeither by making things clearer or more uncertain. In 1994, the NewYork Post attempted to make radical price changes in order to get theDaily News to raise its price to regain subscribers. However, the DailyNews misunderstood and both newspapers were headed for a pricewar. The New York Post had to make its intentions clear, and bothpapers were able to raise their prices and not lose revenue. Theauthors also show an example of how investment banks can maintainambiguity to benefit themselves. If the client is more optimistic thanthe investment bank, the bank can try to charge a higher commissionas long as the client does not develop a more realistic appraisal of thecompany’s value. Finally, companies can change the boundarieswithin which they compete. For example, when Sega was unable togain market share from Nintendo’s 8-bit systems, it changed thegame by introducing a new 16-bit system. It took Nintendo 2 years torespond with its own 16-bit system, which gave Sega the opportunityto capture market share and build a strong brand image. Thisexample shows how companies can think outside the box to changethe way competition takes place in their industry.Brandenburger andNalebuff have illustrated how companies that recognize they canchange the rules of competition can vastly improve their odds ofsuccess, and sometimes respond in a way that benefits boththemselves and the competition. If companies are able to develop asystem where they can make both themselves and their competitorsbetter off, then they do not have to worry so much about theircompetitors trying to counter their moves. Also, because companiescan easily copy each other’s ideas, it is to a firm’s advantage if theycan benefit when their competitors copy their idea, which is notusually possible under the traditional win-losestructure.This articlehas some parallels with the article “Competing on Analytics” by ().The biggest factor that both of these articles have in common is howcrucial it is for managers to understand everything they can abouttheir business and the environment in which they work. In“Competingon Analytics”, the authors say that it is important to be familiar withthis information so that managers can change the way they competeto improve their chances of success. At the end of “The Right Game:Use Game Theory to Shape Strategy”, the authors discuss how inorder for companies to be able to change the environment or rulesunder which they compete they need to understand everything theycan about the constructs under which they are competing. Whether amanager intends to use analytics or game theory to be successful, heor she must first have all available information and use thatinformation to understand how to make the company better off.However, the work shown in “Competing on Analytics” tends to placean emphasis almost exclusively on the use of quantitative data to

improve efficiency or market share of the company. “The RightGame”, however focuses more on using information to find creativeways of changing the constructs or rules applied between companies,often yielding a much broader impact.

Q4. a. Compare CPM and PERT explaining similarities andmentioning where they mainly differ.b. What is meant by graphing in Network Analysis? [5 Marks]

Ans. The Major Differences and Similarities between CPM andPERTCPM (Critical Path Method) & PERT(Program Evaluation andReview Technique)1PERT is a probabilistic tool used with threeCPM is a deterministic tool, with only single Estimating the durationfor completion of estimate of duration.This tool is basically a tool for planningCPM also allows and explicit estimate of and control of time. costs inaddition to time, therefore CPM can control both time and cost.PERT is more suitable for R&D relatedCPM is best suited for routine and those projects where the project isperformed for projects where time and cost estimates can the firsttime and the estimate of duration be accurately calculated areuncertain.The probability factor I major in PERTThe deterministic factor is more so values or so outcomes may not beexact. outcomes are generally accurate and realistic. Extensions ofboth PERT and CPM allow the user to manage other resources inaddition to time and money, to trade off resources, to analyzedifferent types of schedules, and to balance the use of resources.Tensions of both PERT and CPM allow the user to manage otherresources in addition to time and money, to trade off resources, toanalyze different types of schedules, and to balance the use ofresources.Graphs _ In mathematics, networks are called graphs, the entities arenodes, and the links are edges _ Graph theory starts in the 18thcentury, with Leonhard Euler _ The problem of Konigsberg bridges _Since then graphs have been studied extensively. Graph Theory_ Graph G=(V,E) _ V = set of vertices _ E = set of edges 2 _ An edgeis defined by the two vertices which it connects _ optionally: 1 3Adirection and/or a weight _ Two vertices are adjacent if they areconnected by an edge 4 5 _ A vertex’s degree is the number of itsedgesGraph G=(V,E)2V = set of vertices E = set of edges Each edge is now an 1 3arrow,not just a line ->direction The in degree of a vertex is the number of5incoming edges 4The out degree of a vertex is the number of

outgoing edges.

b) Mathematica provides state-of-the-art functionality for analyzing and synthesizing graphs and networks. Building on Mathematica's powerful numerical and symbolic capabilities,Mathematica 8 brings numerous high-level functions for computing with graphs.

Modern extensible platform for graph computation and network analysis. » Support for directed, undirected, and weighted graphs. Hundreds of built-in Mathematica functions and standard graph algorithms. Direct support for random graph distributions. » Extensive collection of graph operations and modifications. » Support for set-theoretic and Boolean-based operations on graphs. » Selection of graph elements and subgraphs using Mathematica pattern language. Comprehensive collection of predicates for testing graph properties. » Efficient graph isomorphism testing. » Local and global structural properties, including components, covers, and matchings. 15+ metrics and centrality measures to characterize graphs and networks. » Efficient shortest path, cycle, and navigation functions. » Multi-paradigm approach to graph programming with matrix, optimization, and Boolean-based

frameworks. » Generic BFS and DFS algorithms with a flexible programmatic interface. » Support of arbitrary properties for graph elements. Full integration of graphs and networks into Mathematica.

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