operations research question paper 2019

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TYBMS Semester 6 Operation Research (Q.P. April 2019 with Solution)

 paper/ subject code 86001/ operation research, tybms semester 6 operation research , (q.p. april 2019 with solution), 1) april 2019 q.p. with   solution (pdf) , 2) november 2019 q.p. with  solution (pdf), 4 ) april 2023 q.p. with  solution (pdf).

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___________________________________________________________________________

Q1.A) Multiple choice questions (Attempt Any 8)                                                  (8)

I) ln linear programming, unbounded solution means ____________. .

a) infeasible solution   b) Degenerate solution

c) Infinite solutions     d) Unique solution

  Ans:  c) Infinite solutions

Ii) if m + n-1 = number of allocations in transportation, it means _________..

(Where 'M' is number of rows and N' is number of columns)

a) There is no degeneracy       b) Problem is unbalanced

c) Problem is degenerate        d) Solution is optimal

 Ans:  a) There is no degeneracy   

Ijü) floats for critical activities will be always _________________..

a) One             b) Zero

c) Highest        d) Same ás duration of the activity

 Ans:  b) Zero

Iv) the total time required to complete all the jobs-in a job sequencing problem is known as __________,.

a) Idle time                                 b) Processing time

c)Elapsed time                        d) Processing order

 Ans:  c)Elapsed time

V) in linear programming, _____________ represents mathematical equation of the limitations imposed by the problem..

a) Objective function              b) Decision variable

c)Redundancy                         d) Constraints

Ans:    d) Constraints

Vi) if in an assignment problem, number of rows is not equal to number of columns then ____________..

a) Problem is degenerate                                    b) Problem is ‘unbalanced

c) It is a maximization problem                      d) Optimal solution is not possible

Ans:    b ) Problem is ‘unbalanced

 vii) the maximum time in which an activity will be completed assuming all possible delays and postponements is termed as __________..

a) Optimistic time                   b) Most likely time

c) Pessimistic time                  d) Expected time

Ans:  c) Pessimistic time

Viii) the various alternatives or courses of actions available to each player in a game are called as __________..

a) Saddle points                      b) Strategies

c) Pay off                                d) 'n' player game

Ans:  b) Strategies

Ix) in simplex, a maximization problem is optimal when all delta j, i.e., cj-zj values are.

a) Either zero or positive                       b) Either zero or negative

c) Only positive                                  d) Only negative

Ans:     b) Either zero or negative

X) which of the following considers difference between two least costs for each row and column while finding initial basic feasible solution in transportation.

a) North west corner rule                    b) Least cost method

c) Vogel's Approximation method       d) Row minima method

Ans:    b) Least cost method

Q1.B) True or false (Attempt Any 7)                                               (7)

i) Probability of a project completing in its expected time (Te) will be always 100%

Ii) lf saddle point is available in a game, it is called as pure strategy game., iii) slack represents unutilized resources., iv) if in a transportation problem, number of rows is not equal to number of columns, then the problem is unbalanced., v) lf we introduce an unnecessary dummy activity, the error is termed as redundancy., vi) job sequencing problems are solved to ensure that, both, the total time to complete all jobs and idle time of each machine are maximum., vii) when more than one optimal solution is possible in a linear programming problem, it is termed as 'unique solution'., viii) regret matrix is made to convert a maximization problem into minimization problem in assignment., ix) critical path method (cpm) considers the three-time estimates: most likely, optimistic and pessimistic time estimates., x) in solving a job sequencing problem, it is assumed that all jobs require the same sequence of operations., q.2 a) a company produces 2 products a and b: x 1 and x 2 , are the quantities manufactured of products a and b respectively. the following objective function along with constraints is given to you: .

Max Z = 8x 1 + 16x 2 ,

Subject to constraints:

x 1 + x 2 < 200

x 2 < 125

x 1 + 2x 2 < 300

x 1 , x 2 > 0

Find how many units of Product A and Product B should be produced by the company so that the profit is maximized. Ls it a case of multiple optimal solutions? Use graphical method to solve the LPP.

Video solution.

x 1 + x 2 < 200   ---- (i)

x 2 < 125           ---- (ii)

x 1 + 2x 2 < 300 ---- (iii)

x 1 , x 2 > 0  

x 1 + x 2 = 200

Let X 1 = 0

0 + x 2 = 200

Therefore x 2 = 200                  (x 1 , x 2 ) = (0, 200)

Let X 2 = 0

 x 1 + 0 = 200

Therefore x 1 = 200                      (x 1 , x 2 ) = (200, 0)

  x 2 < 125    ---- (ii)

Let x 2 = 0

x 2 = 125                 (x 1 , x 2 ) = (0, 125)

x 1 + 2x 2 < 300     ---- (iii)

x 1 + 2x 2 = 300

Let x 1 = 0

 0 + 2x 2 = 300

            x 2 = 300 / 2                             Therefore x 2 = 150             (x 1 , x 2 ) = (0, 150)

x 1 + 2x 2 = 300 Let x 2 = 0 x 1 + 2(0) = 300 x 1 = 300                     (x 1 , x 2 ) = (300, 0)

Corner Point Method:

Optimal Profit Maximization is Rs. 2,400

Q2.b) you are given the per unit cost of transporting goods from 3 factories to 4 customers. the 3 factories a b and c have capacity to supply 500,300 and 200 units respectively. the 4 customers p, q, r and s require 180,150,350 and 320 units respectively., i) you are required to find the initial basic feasible solution using vogel's approximation method                                                                                                                 (5) ii) find the total cost of transportation schedule obtained using vam          (2).

Total Demand = 1000

Total Supply = 1000

Total Demand = Total Supply

Cost data is given

Minimization Problem

RIM Condition:

M + N – 1 = No. of Allocation

3 + 4 – 1 = 6

Therefore, RIM Condition is satisfied.

Optimal Solution is Non-Degeneracy

(ii) Total Cost of Transportation :

            = (150 x 10) + (230 x 12) + (120 x 13) + (180 x 7) + (120 x 8) + (200 x 7)

            = Rs. 9, 440

Q.2 C) You are given the following details for a project consisting of 8 activities:

(i) Draw the network diagram and identify the critical path.                    (3) (ii) Find earliest start time, earliest finish time, latest start time and latest finish time for each activity.                                                                                                      (4) (iii) Find free float for activity B.                                                              (1)

  Solution:    [Video Explanation ]

(i)                  Network Diagram

Paths in Network:

(1) A-E-F = 4 + 20 + 10 = 34 days

(2) A-D-G = 4 + 5 + 6 = 15 days

(3) B-G = 6 + 6 = 12 days

(4) C- H = 13 + 16 = 29

Projection Completion Time : 34 days

(ii)               Free Float for Activity B:

= Ej – Ei – t

= 9 – 0 – 6

= 15  

Q.2 D) There are 6 jobs to be performed in a factory and each would go through 2 machines A and B in the order AB. The processing time (in hours) is given for each job in each machine.

(i) determine the sequence of performing the jobs that would minimize the total time of completing all the jobs:                                                                                 (2) (ii) find the elapsed time.                                                                               (3) (iii) find idle time for both the machines.                                                    (2).

Solution :    [Video Explanation ]

(i)                  Optimal Sequence

A                                                                                                                           B

(ii)                 Elapsed Time Calculation :

  Total Min. Elapsed Time = 36 Hrs.

(iii)            Idle time for Machinery A    = 36 – 35 = 1 Hrs

Idle time for Machinery A    = 36 – 28 = 8 Hrs

Q.3. B) You are given the following details for a project with 8 activities.

(i) Draw the network diagram.                                                                                     (3)

(ii) Find the expected time of project completion along with standard deviation.       (2)

(iii) What is the probability of the project completing in 55 days?                               (2)

Ans:    [Video Explanation]

Network Diagram:

Paths in the Network:

1)       1-2-3-5-7-8  = 47 days

2)       1-2-4-6-7-8 = 33 days

Expected Project Completion Time : 47 days

Variance of critical path = 16/36 + 100/36 + 256/36 + 64/36 + 4/36 = 440/36 =12.22

SD of critical path = √variance

SD = √ 12.22   = 3.495                        = 3.5 days

Time = 55 days = te

Z = t – CP / SD

= 55 – 47 / 3.5 = 2.29 

Table value = 0.4890

P(55 days) = 0.5 + 0.4890      = 0.9890                      = 98.90% 

Q.3. C) You are given information about the cost (in Rs. Thousands) of performing different jobs by different persons. PI cannot perform J3. P3 cannot perform J4

(i)Obtain optimal assignmnt and find cost of such assignment.

(i)Is it a case of alternative optimal solution?

Ans:      [Video Explanation]

Prohibited Assignment = X (infinity)

The problem is unbalanced.

Taking P5 as Dummy

Cost is given.

Hence Minimization Problem.

Row Minimization

Column Minimization Not Required.

Min. No. of Lines = 4

Size of Matrix = 5 x 5

Therefore Not optimal

Change = Min. open Value = 4

Optimal Cost = 66,000

Q3.D) Two firms, Lacko textiles and Rayon textiles have 3 strategies each to select from. The 3 strategies are no advertisement, using moderate advertising and using heavy advertising. You are given the pay off matrix from view point of Lacko textiles, showing its market share under several combinations of strategies:

(    (i) Find the saddle point and value of game.      (ii) Comment on the strategy to be selected by both the companies.

Value of the Game = 50 x 10000 =     5,00,000

Q. 4 a) you are given a solution for a transportation cost problem. figures in each cell represent per unit transportation cost. figures in circle within each cell represent number of units allocated for transportation. x, y and z. are the 3 factories and a, b, c and d are the 4 customers..

(i) You are required to check the above solution for optimality.

(ii) If it is not optimal, use modified distribution method to obtain optimal solution.

(iii) Find optimal transportation cost.

Ans:  [Video Explanation]

No negative ∆, Hence Optimal Solution.

(iii) Optimal transportation Cost:

= (200 x 7) + (120 x 18) + (100 x 15) + (280 x 7) + (180 x 11) + (12 x 5)

= RS. 9,600

B) You are given the following information, for a project with 8 activities:

The cost of completing the eight activities in normal time is Rs.6,500.

Indirect cost is Rs 160 per day. The contract includes a penalty of Rs. 100 per day for every day of delay more than 17 days.

  (i)   Draw the network diagram and find critical path.

(i     (ii)  Crash the projèct duration to find the total 'cost of completing the project in 17 days (4)

  Ans:  [Video Explanation]

Q.4. C) A company produces 2 products x and xX) using three resources S1, S2 and S. Product x1 gives profit of Rs.30 per unit and product x2 gives profít of Rs.40 per unit. The 3 resources Si. S and S; are available to the extent of 200 units, 600 units and 500 units respectiveiy The following objective function and constraints are given to you:

Max Z = 30x1 + 40x2

Subject to constraints;

x 1 + 2x 2 < 200

8x 1 + 5x 2 < 600

3x 1 + 4x 2 < 500

X1 >0; x2 >0

You are given the following simplex solution to the above problem:

  A) With reference to the above table answer the following:

i) Check if the above solution is optimal or not.

ii) lf it is not optimal, find optimal solution.

B) With reference to the optimal simplex table in the above problem obtained by you, answer the following:

iii) Find the optimal product mix and optimal profit

iv) Which resources are scarce and which are, unutilized?

v) Is it a case of alternative solution? Justify your answer

vi) What are the shadow prices of the resources? Justify

Ans:       [Video Explanation]

Testing the given solution for optimality ;

Positive ∆

Hence, solution not optimal.

(ii) Revising the Solution for Optimality:

No positive ∆

Therefore, Optimal Solution

(iii) Optimal Product Mix;

No. of units of X1 = 200/11        = 18.18

No. of units of X1 = 1000/11      = 90.90

Optimal Profit:

Max Z = [30 x 200/11] + [40 x 1000/11]

          = Rs. 46,000 /11

          = Rs. 4,181.81

(iv) Abundant Resource:

S3 = 900/11 ----- in optimal solution

Therefore Unutilized Capacity of S3 = 900 / 11

S3 is Abundant resource.

S1 and S2 are non-basis variable in optimal solution.

 Unutilized Capacity of S1 and S2 are 0

(i)                 ∆ of non basis variable :

∆ of S1 = - 170 / 11

∆ of S2 = -20/11

No zero ∆ value.

Hence, this is a case of Unique Solution.

There is no alternate optimal solution.

(ii)              Zj value of slack variables:

Q.5 A) Explain the concepts : Total float, free float, Independent float and Interfering float.           (8) 

(a) Total float:

lt is the amount of time by which the completion of an activity can be delayed beyond the earliest possible finishing time without affecting total project completion time.

 Total Float = (LST - EST) or (LFT - EFT)

.:: Total float = LST - EST -[;-t]-E Total Float = [Lj- E -t

(b) Free float:  It is the amount of time by which the completion of an activity can be delayed beyond the earliest finishing time without affecting the earliest possible start of a subsequent activity.

Free float = Earliest starting time of subsequent activity - Earliest finishing time of activity ij = Ej- [E, + t;] . Free Float = Ej- E-tj

(c) Independent float:  It is the amount of time by which the start of an activity can be delayed without affecting the earliest start of a subsequent activity, assuming that the preceding activity has finished at its latest finishing time. .

:. Independent float = Earliest starting time of subsequent activity - Latest finishing time of preceding activity - ij

:. Independent Float = [Ej-Li]-tij;

(d) Interfering float:  It is that part of total float which reduces the float of a subsequent activity. It is the amount of time by which the earliest possible start of a subsequent activity will be delayed if activity ÷ finishes on latest finishing time.

:. Interfering float = Latest finishing time of activity ij - Earliest starting time of subsequent activity. :. Interfering Float= Lj-Ej

B) Discuss any 5 areas where techniques of operation Research can be applied.    (7) 

OR techniques are applied to a variety of business problems. Some examples are:

(A) Production management:

(1) To calculate loss of time due to waiting time, queuing time etc.

(2) To decide optimum allocation of jobs and optimum sequence in which jobs should be sequenced.

(B) Personnel management:

(1) To study labour turnover. (2) To do human resource planning. (3) To decide number of personnel required to be kept on standby in case of demand for higher manpower.

(C) Inventory management: (1) To study economic lot size to be ordered.

(D) Marketing management: (1) To decide optimal product mix for maximum profit. (2) Media selection for advertising for maximum reach. (3) Sales forecasting.

(E) Transportation management: (1) To determine transportation schedule for minimum cost or minimum time.

(F) Project management (1) To identify critical and non-critical activities of a project (2) To determine minimum project completion time. (3) To determine optimal project cost.

(G) Financial management: project. (1) To decide investment portfolio to maximize return on investment. of such nature that it is impossible to find a feasible solution that can satis1y a the constraints. 

c) Answer any 3 of the following :            (15) 

I) explain the terms: redundant constraints and infeasible solution in linear programming. .

Ans:  A redundant constraint is the one which does not affect the optimal solution. Even if that constraint is not considered, we can still obtain the solution to the problem. Since the constraint is not required for obtaining the optimal solution, it is called redundant constraint cost.

Following sketch explains redundancy (redundant constraint):

There is no common feasible region for line AB and line CD

Hence, solution is infeasible.

A redundarnt constraint is the one which does not affect the optimal solution. Even if that constraint is not considered, we can still obtain the solution to the problem. Since the constraint is not required for obtaining the optimal solution, it is called redundant constraint. cost.

The feasible region for the above problem is OABC. The 3rd constraint does not affect the feasible region. Hence, the constraint [X1 S 16] is redundant constraint.

ii) what do you means by alternative optimal solution in transportation? How do you find that alternative solution? 

Alternate optimal solution or multiple optimal solutions mean there are two sets of solutions which provide the same optimal cost or optimal profit:

(a) In the optimal solution to a transportation problem, if there is an empty cell with zero A value, it means there is alternate optimal solution.

(b) To find alternate optimal solution, we should construct a closed loop from the empty cell with zero A. The new table we obtain after looping is the alternate optimal solution.

iii) Explain the time coat trade off in project crashing. 

The execution of a project involves two types of costs, direct cost and indirect cost.

(a) Direct costs: These include costs of materials, machinery, tools, manhours etc. When we estimate the duration of various activities in the project, it is assumed that normal amount of labour (manhours) & machines will be required to complete these activities. The estimation of direct costs is done based on the normal amount of resource required. This estimation gives us minimum possible direct cost required to complete the project. When we want to complete the project in a shorter period than critical path, we will need to employ more resources. Hence, direct cost will increase.

(b) Indirect costs: These include rent, overheads, administrative costs etc. Indirect costs vary with time. They are expressed on per day (or per week etc.) basis. Hence, when we shorten the project completion time, total indirect cost reduces.

Total P'roject Cost = Direct Cost + Indirect Cost

iV) Discuss the signification of theory of game. Briefly discuss the terms. Player and pay off.

Ans: Every one business party. The situation (also called competitive situation) involves more parties involved could be corporate. individuals, small firms or A number of outcome alternatives are available to each party for decision making. The (i.e. profit or loss) of the business decision does not depend on the alternative selected by one party alone, but on the interaction between the decisions of all the parties involved in the situation. Each party will try to select the alternative which maximizes gains for it, at the cost of its competitors. "Games Theory" deals with such business situations or problems where multiple parties are involved in the conflict through interaction of their alternatives or decisions.

Players: The various participants or decision markers in the game are called "Players". A game must have minimum two players (or competitors). A game having two competitors opposing each other is called 'two person game' and a game having more than 2 players is called 'n-person game'.

Payoff: Payoff is the outcome of the interaction of selected strategies of opponents in the game. Positive payoff is gain and negative payoff is loss. In a 'two person game', when one party gains, the other party loses.

v) Distinguished between PERT and CPM. 

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Operations Research Proceedings 2019

Selected Papers of the Annual International Conference of the German Operations Research Society (GOR), Dresden, Germany, September 4-6, 2019

• Conference proceedings
• © 2020
• Janis S. Neufeld 0 ,
• Udo Buscher 1 ,
• Rainer Lasch 2 ,
• Dominik Möst 3 ,
• Jörn Schönberger 4

Faculty of Business and Economics, TU Dresden, Dresden, Germany

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Faculty of Transportation and Traffic, TU Dresden, Dresden, Germany

• Presents latest findings from operations research (OR)
• Gathers selected papers from a leading OR conference
• Summarizes the perspectives of OR researchers on recent and upcoming challenges

Part of the book series: Operations Research Proceedings (ORP)

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Table of contents (99 papers)

Front matter, analysis and optimization of urban energy systems.

• Kai Mainzer

Optimization in Outbound Logistics—An Overview

• Stefan Schwerdfeger

Incorporating Differential Equations into Mixed-Integer Programming for Gas Transport Optimization

• Mathias Sirvent

Scheduling a Proportionate Flow Shop of Batching Machines

• Christoph Hertrich

Vehicle Scheduling and Location Planning of the Charging Infrastructure for Electric Buses Under the Consideration of Partial Charging of Vehicle Batteries

• Luisa Karzel

Data-Driven Integrated Production and Maintenance Optimization

• Anita Regler

Business Analytics, Artificial Intelligence and Forecasting

Multivariate extrapolation: a tensor-based approach.

• Josef Schosser

Business Track

Heuristic search for a real-world 3d stock cutting problem.

• Katerina Klimova, Una Benlic

Control Theory and Continuous Optimization

Model-based optimal feedback control for microgrids with multi-level iterations.

• Robert Scholz, Armin Nurkanovic, Amer Mesanovic, Jürgen Gutekunst, Andreas Potschka, Hans Georg Bock et al.

Mixed-Integer Nonlinear PDE-Constrained Optimization for Multi-Modal Chromatography

• Dominik H. Cebulla, Christian Kirches, Andreas Potschka

Sparse Switching Times Optimization and a Sweeping Hessian Proximal Method

• Alberto De Marchi, Matthias Gerdts

Toward Global Search for Local Optima

• Jens Deussen, Jonathan Hüser, Uwe Naumann

First Experiments with Structure-Aware Presolving for a Parallel Interior-Point Method

• Ambros Gleixner, Nils-Christian Kempke, Thorsten Koch, Daniel Rehfeldt, Svenja Uslu

A Steepest Feasible Direction Extension of the Simplex Method

• Biressaw C. Wolde, Torbjörn Larsson

Convex Quadratic Mixed-Integer Problems with Quadratic Constraints

• Simone Göttlich, Kathinka Hameister, Michael Herty
• business analytics
• decision analytics
• digital economy
• mathematical optimization
• multicriteria decision making
• optimization and simulation
• logistics scheduling
• transportation
• OR applications in production
• OR applications in supply chains
• OR applications using big data

About this book

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Janis S. Neufeld, Udo Buscher, Rainer Lasch, Dominik Möst

Jörn Schönberger

About the editors

Bibliographic information.

Book Title : Operations Research Proceedings 2019

Book Subtitle : Selected Papers of the Annual International Conference of the German Operations Research Society (GOR), Dresden, Germany, September 4-6, 2019

Editors : Janis S. Neufeld, Udo Buscher, Rainer Lasch, Dominik Möst, Jörn Schönberger

Series Title : Operations Research Proceedings

DOI : https://doi.org/10.1007/978-3-030-48439-2

Publisher : Springer Cham

eBook Packages : Business and Management , Business and Management (R0)

Copyright Information : The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

Softcover ISBN : 978-3-030-48438-5 Published: 25 September 2020

eBook ISBN : 978-3-030-48439-2 Published: 24 September 2020

Series ISSN : 0721-5924

Series E-ISSN : 2197-9294

Edition Number : 1

Number of Pages : XV, 819

Number of Illustrations : 118 b/w illustrations, 30 illustrations in colour

Topics : Operations Research/Decision Theory , Operations Research, Management Science , Optimization , Operations Management

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Tuesday, March 2, 2021

Anna university ba 5201-applied operations research november/december 2019 question paper.

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Question Paper Code: 70163

M.B.A.  DEGREE EXAMINATION, NOVEMBER/DECEMBER 2019

Second Semester

BA 5201-APPLIED OPERATIONS RESEARCH 

(Regulation 2017)

Time: Three hours

Maximum: 100 marks

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Operations Research–T.Y.B.M.S.–Semester–VI (Vikram Shrotri)

1. Introduction to Operations Research

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Operations Research: Evolution, methodology and role in decision making; Linear programming: Meaning, assumptions, advantages, scope and limitations: Formulation of Problem and its solution by graphical and simplex methods (Including Big M Method and Two Phase Simplex Method); special cases in simplex method; infeasibility, degeneracy, unboundedness and multiple optimal solutions; duality. Dual Simplex Method.

Transportation problems including transshipment problems; Special cases in transportation problems; unbalanced problems, degeneracy; maximization objective and multiple optimal solutions; assignment problems including travelling salesman’s problem. Special cases in assignment problems; unbalanced problems, maximization objective and multiple optimal solutions.

PERT/CPM: Difference between PERT and CPM, network construction, calculating EST, EFT, LST, LFT and floats, probability considerations in PERT, time cost trade-off. Decision theory: decision making under uncertainty and risk, Bayesian analysis, decision trees. Replacement problem (Individual and Group replacement problems both).

Game theory, pure and mixed strategy games; a principle of dominance; two people zero sum game; Queuing theory: concept, assumptions and applications; analysis of queue system, Poisson distributed arrivals and exponentially distributed service time model (MMI and MMK); simulation; meaning, process, advantages, limitations and applications.

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What the data says about abortion in the U.S.

Pew Research Center has conducted many surveys about abortion over the years, providing a lens into Americans’ views on whether the procedure should be legal, among a host of other questions.

In a  Center survey  conducted nearly a year after the Supreme Court’s June 2022 decision that  ended the constitutional right to abortion , 62% of U.S. adults said the practice should be legal in all or most cases, while 36% said it should be illegal in all or most cases. Another survey conducted a few months before the decision showed that relatively few Americans take an absolutist view on the issue .

Find answers to common questions about abortion in America, based on data from the Centers for Disease Control and Prevention (CDC) and the Guttmacher Institute, which have tracked these patterns for several decades:

How many abortions are there in the U.S. each year?

How has the number of abortions in the u.s. changed over time, what is the abortion rate among women in the u.s. how has it changed over time, what are the most common types of abortion, how many abortion providers are there in the u.s., and how has that number changed, what percentage of abortions are for women who live in a different state from the abortion provider, what are the demographics of women who have had abortions, when during pregnancy do most abortions occur, how often are there medical complications from abortion.

This compilation of data on abortion in the United States draws mainly from two sources: the Centers for Disease Control and Prevention (CDC) and the Guttmacher Institute, both of which have regularly compiled national abortion data for approximately half a century, and which collect their data in different ways.

The CDC data that is highlighted in this post comes from the agency’s “abortion surveillance” reports, which have been published annually since 1974 (and which have included data from 1969). Its figures from 1973 through 1996 include data from all 50 states, the District of Columbia and New York City – 52 “reporting areas” in all. Since 1997, the CDC’s totals have lacked data from some states (most notably California) for the years that those states did not report data to the agency. The four reporting areas that did not submit data to the CDC in 2021 – California, Maryland, New Hampshire and New Jersey – accounted for approximately 25% of all legal induced abortions in the U.S. in 2020, according to Guttmacher’s data. Most states, though,  do  have data in the reports, and the figures for the vast majority of them came from each state’s central health agency, while for some states, the figures came from hospitals and other medical facilities.

Discussion of CDC abortion data involving women’s state of residence, marital status, race, ethnicity, age, abortion history and the number of previous live births excludes the low share of abortions where that information was not supplied. Read the methodology for the CDC’s latest abortion surveillance report , which includes data from 2021, for more details. Previous reports can be found at  stacks.cdc.gov  by entering “abortion surveillance” into the search box.

For the numbers of deaths caused by induced abortions in 1963 and 1965, this analysis looks at reports by the then-U.S. Department of Health, Education and Welfare, a precursor to the Department of Health and Human Services. In computing those figures, we excluded abortions listed in the report under the categories “spontaneous or unspecified” or as “other.” (“Spontaneous abortion” is another way of referring to miscarriages.)

Guttmacher data in this post comes from national surveys of abortion providers that Guttmacher has conducted 19 times since 1973. Guttmacher compiles its figures after contacting every known provider of abortions – clinics, hospitals and physicians’ offices – in the country. It uses questionnaires and health department data, and it provides estimates for abortion providers that don’t respond to its inquiries. (In 2020, the last year for which it has released data on the number of abortions in the U.S., it used estimates for 12% of abortions.) For most of the 2000s, Guttmacher has conducted these national surveys every three years, each time getting abortion data for the prior two years. For each interim year, Guttmacher has calculated estimates based on trends from its own figures and from other data.

The latest full summary of Guttmacher data came in the institute’s report titled “Abortion Incidence and Service Availability in the United States, 2020.” It includes figures for 2020 and 2019 and estimates for 2018. The report includes a methods section.

In addition, this post uses data from StatPearls, an online health care resource, on complications from abortion.

An exact answer is hard to come by. The CDC and the Guttmacher Institute have each tried to measure this for around half a century, but they use different methods and publish different figures.

The last year for which the CDC reported a yearly national total for abortions is 2021. It found there were 625,978 abortions in the District of Columbia and the 46 states with available data that year, up from 597,355 in those states and D.C. in 2020. The corresponding figure for 2019 was 607,720.

The last year for which Guttmacher reported a yearly national total was 2020. It said there were 930,160 abortions that year in all 50 states and the District of Columbia, compared with 916,460 in 2019.

• How the CDC gets its data: It compiles figures that are voluntarily reported by states’ central health agencies, including separate figures for New York City and the District of Columbia. Its latest totals do not include figures from California, Maryland, New Hampshire or New Jersey, which did not report data to the CDC. ( Read the methodology from the latest CDC report .)
• How Guttmacher gets its data: It compiles its figures after contacting every known abortion provider – clinics, hospitals and physicians’ offices – in the country. It uses questionnaires and health department data, then provides estimates for abortion providers that don’t respond. Guttmacher’s figures are higher than the CDC’s in part because they include data (and in some instances, estimates) from all 50 states. ( Read the institute’s latest full report and methodology .)

While the Guttmacher Institute supports abortion rights, its empirical data on abortions in the U.S. has been widely cited by  groups  and  publications  across the political spectrum, including by a  number of those  that  disagree with its positions .

These estimates from Guttmacher and the CDC are results of multiyear efforts to collect data on abortion across the U.S. Last year, Guttmacher also began publishing less precise estimates every few months , based on a much smaller sample of providers.

The figures reported by these organizations include only legal induced abortions conducted by clinics, hospitals or physicians’ offices, or those that make use of abortion pills dispensed from certified facilities such as clinics or physicians’ offices. They do not account for the use of abortion pills that were obtained  outside of clinical settings .

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The annual number of U.S. abortions rose for years after Roe v. Wade legalized the procedure in 1973, reaching its highest levels around the late 1980s and early 1990s, according to both the CDC and Guttmacher. Since then, abortions have generally decreased at what a CDC analysis called  “a slow yet steady pace.”

Guttmacher says the number of abortions occurring in the U.S. in 2020 was 40% lower than it was in 1991. According to the CDC, the number was 36% lower in 2021 than in 1991, looking just at the District of Columbia and the 46 states that reported both of those years.

(The corresponding line graph shows the long-term trend in the number of legal abortions reported by both organizations. To allow for consistent comparisons over time, the CDC figures in the chart have been adjusted to ensure that the same states are counted from one year to the next. Using that approach, the CDC figure for 2021 is 622,108 legal abortions.)

There have been occasional breaks in this long-term pattern of decline – during the middle of the first decade of the 2000s, and then again in the late 2010s. The CDC reported modest 1% and 2% increases in abortions in 2018 and 2019, and then, after a 2% decrease in 2020, a 5% increase in 2021. Guttmacher reported an 8% increase over the three-year period from 2017 to 2020.

As noted above, these figures do not include abortions that use pills obtained outside of clinical settings.

Guttmacher says that in 2020 there were 14.4 abortions in the U.S. per 1,000 women ages 15 to 44. Its data shows that the rate of abortions among women has generally been declining in the U.S. since 1981, when it reported there were 29.3 abortions per 1,000 women in that age range.

The CDC says that in 2021, there were 11.6 abortions in the U.S. per 1,000 women ages 15 to 44. (That figure excludes data from California, the District of Columbia, Maryland, New Hampshire and New Jersey.) Like Guttmacher’s data, the CDC’s figures also suggest a general decline in the abortion rate over time. In 1980, when the CDC reported on all 50 states and D.C., it said there were 25 abortions per 1,000 women ages 15 to 44.

That said, both Guttmacher and the CDC say there were slight increases in the rate of abortions during the late 2010s and early 2020s. Guttmacher says the abortion rate per 1,000 women ages 15 to 44 rose from 13.5 in 2017 to 14.4 in 2020. The CDC says it rose from 11.2 per 1,000 in 2017 to 11.4 in 2019, before falling back to 11.1 in 2020 and then rising again to 11.6 in 2021. (The CDC’s figures for those years exclude data from California, D.C., Maryland, New Hampshire and New Jersey.)

The CDC broadly divides abortions into two categories: surgical abortions and medication abortions, which involve pills. Since the Food and Drug Administration first approved abortion pills in 2000, their use has increased over time as a share of abortions nationally, according to both the CDC and Guttmacher.

The majority of abortions in the U.S. now involve pills, according to both the CDC and Guttmacher. The CDC says 56% of U.S. abortions in 2021 involved pills, up from 53% in 2020 and 44% in 2019. Its figures for 2021 include the District of Columbia and 44 states that provided this data; its figures for 2020 include D.C. and 44 states (though not all of the same states as in 2021), and its figures for 2019 include D.C. and 45 states.

Guttmacher, which measures this every three years, says 53% of U.S. abortions involved pills in 2020, up from 39% in 2017.

Two pills commonly used together for medication abortions are mifepristone, which, taken first, blocks hormones that support a pregnancy, and misoprostol, which then causes the uterus to empty. According to the FDA, medication abortions are safe  until 10 weeks into pregnancy.

Surgical abortions conducted  during the first trimester  of pregnancy typically use a suction process, while the relatively few surgical abortions that occur  during the second trimester  of a pregnancy typically use a process called dilation and evacuation, according to the UCLA School of Medicine.

In 2020, there were 1,603 facilities in the U.S. that provided abortions,  according to Guttmacher . This included 807 clinics, 530 hospitals and 266 physicians’ offices.

While clinics make up half of the facilities that provide abortions, they are the sites where the vast majority (96%) of abortions are administered, either through procedures or the distribution of pills, according to Guttmacher’s 2020 data. (This includes 54% of abortions that are administered at specialized abortion clinics and 43% at nonspecialized clinics.) Hospitals made up 33% of the facilities that provided abortions in 2020 but accounted for only 3% of abortions that year, while just 1% of abortions were conducted by physicians’ offices.

Looking just at clinics – that is, the total number of specialized abortion clinics and nonspecialized clinics in the U.S. – Guttmacher found the total virtually unchanged between 2017 (808 clinics) and 2020 (807 clinics). However, there were regional differences. In the Midwest, the number of clinics that provide abortions increased by 11% during those years, and in the West by 6%. The number of clinics  decreased  during those years by 9% in the Northeast and 3% in the South.

The total number of abortion providers has declined dramatically since the 1980s. In 1982, according to Guttmacher, there were 2,908 facilities providing abortions in the U.S., including 789 clinics, 1,405 hospitals and 714 physicians’ offices.

The CDC does not track the number of abortion providers.

In the District of Columbia and the 46 states that provided abortion and residency information to the CDC in 2021, 10.9% of all abortions were performed on women known to live outside the state where the abortion occurred – slightly higher than the percentage in 2020 (9.7%). That year, D.C. and 46 states (though not the same ones as in 2021) reported abortion and residency data. (The total number of abortions used in these calculations included figures for women with both known and unknown residential status.)

The share of reported abortions performed on women outside their state of residence was much higher before the 1973 Roe decision that stopped states from banning abortion. In 1972, 41% of all abortions in D.C. and the 20 states that provided this information to the CDC that year were performed on women outside their state of residence. In 1973, the corresponding figure was 21% in the District of Columbia and the 41 states that provided this information, and in 1974 it was 11% in D.C. and the 43 states that provided data.

In the District of Columbia and the 46 states that reported age data to  the CDC in 2021, the majority of women who had abortions (57%) were in their 20s, while about three-in-ten (31%) were in their 30s. Teens ages 13 to 19 accounted for 8% of those who had abortions, while women ages 40 to 44 accounted for about 4%.

The vast majority of women who had abortions in 2021 were unmarried (87%), while married women accounted for 13%, according to  the CDC , which had data on this from 37 states.

In the District of Columbia, New York City (but not the rest of New York) and the 31 states that reported racial and ethnic data on abortion to  the CDC , 42% of all women who had abortions in 2021 were non-Hispanic Black, while 30% were non-Hispanic White, 22% were Hispanic and 6% were of other races.

Looking at abortion rates among those ages 15 to 44, there were 28.6 abortions per 1,000 non-Hispanic Black women in 2021; 12.3 abortions per 1,000 Hispanic women; 6.4 abortions per 1,000 non-Hispanic White women; and 9.2 abortions per 1,000 women of other races, the  CDC reported  from those same 31 states, D.C. and New York City.

For 57% of U.S. women who had induced abortions in 2021, it was the first time they had ever had one,  according to the CDC.  For nearly a quarter (24%), it was their second abortion. For 11% of women who had an abortion that year, it was their third, and for 8% it was their fourth or more. These CDC figures include data from 41 states and New York City, but not the rest of New York.

Nearly four-in-ten women who had abortions in 2021 (39%) had no previous live births at the time they had an abortion,  according to the CDC . Almost a quarter (24%) of women who had abortions in 2021 had one previous live birth, 20% had two previous live births, 10% had three, and 7% had four or more previous live births. These CDC figures include data from 41 states and New York City, but not the rest of New York.

The vast majority of abortions occur during the first trimester of a pregnancy. In 2021, 93% of abortions occurred during the first trimester – that is, at or before 13 weeks of gestation,  according to the CDC . An additional 6% occurred between 14 and 20 weeks of pregnancy, and about 1% were performed at 21 weeks or more of gestation. These CDC figures include data from 40 states and New York City, but not the rest of New York.

About 2% of all abortions in the U.S. involve some type of complication for the woman , according to an article in StatPearls, an online health care resource. “Most complications are considered minor such as pain, bleeding, infection and post-anesthesia complications,” according to the article.

The CDC calculates  case-fatality rates for women from induced abortions – that is, how many women die from abortion-related complications, for every 100,000 legal abortions that occur in the U.S .  The rate was lowest during the most recent period examined by the agency (2013 to 2020), when there were 0.45 deaths to women per 100,000 legal induced abortions. The case-fatality rate reported by the CDC was highest during the first period examined by the agency (1973 to 1977), when it was 2.09 deaths to women per 100,000 legal induced abortions. During the five-year periods in between, the figure ranged from 0.52 (from 1993 to 1997) to 0.78 (from 1978 to 1982).

The CDC calculates death rates by five-year and seven-year periods because of year-to-year fluctuation in the numbers and due to the relatively low number of women who die from legal induced abortions.

In 2020, the last year for which the CDC has information , six women in the U.S. died due to complications from induced abortions. Four women died in this way in 2019, two in 2018, and three in 2017. (These deaths all followed legal abortions.) Since 1990, the annual number of deaths among women due to legal induced abortion has ranged from two to 12.

The annual number of reported deaths from induced abortions (legal and illegal) tended to be higher in the 1980s, when it ranged from nine to 16, and from 1972 to 1979, when it ranged from 13 to 63. One driver of the decline was the drop in deaths from illegal abortions. There were 39 deaths from illegal abortions in 1972, the last full year before Roe v. Wade. The total fell to 19 in 1973 and to single digits or zero every year after that. (The number of deaths from legal abortions has also declined since then, though with some slight variation over time.)

The number of deaths from induced abortions was considerably higher in the 1960s than afterward. For instance, there were 119 deaths from induced abortions in  1963  and 99 in  1965 , according to reports by the then-U.S. Department of Health, Education and Welfare, a precursor to the Department of Health and Human Services. The CDC is a division of Health and Human Services.

Note: This is an update of a post originally published May 27, 2022, and first updated June 24, 2022.

Support for legal abortion is widespread in many countries, especially in Europe

Nearly a year after roe’s demise, americans’ views of abortion access increasingly vary by where they live, by more than two-to-one, americans say medication abortion should be legal in their state, most latinos say democrats care about them and work hard for their vote, far fewer say so of gop, positive views of supreme court decline sharply following abortion ruling, most popular.

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