## Procedure, Example Solved Problem | Operations Research - Solution of assignment problems (Hungarian Method) | 12th Business Maths and Statistics : Chapter 10 : Operations Research

Chapter: 12th business maths and statistics : chapter 10 : operations research.

Solution of assignment problems (Hungarian Method)

First check whether the number of rows is equal to the numbers of columns, if it is so, the assignment problem is said to be balanced.

Step :1 Choose the least element in each row and subtract it from all the elements of that row.

Step :2 Choose the least element in each column and subtract it from all the elements of that column. Step 2 has to be performed from the table obtained in step 1.

Step:3 Check whether there is atleast one zero in each row and each column and make an assignment as follows.

Step :4 If each row and each column contains exactly one assignment, then the solution is optimal.

Example 10.7

Solve the following assignment problem. Cell values represent cost of assigning job A, B, C and D to the machines I, II, III and IV.

Here the number of rows and columns are equal.

∴ The given assignment problem is balanced. Now let us find the solution.

Step 1: Select a smallest element in each row and subtract this from all the elements in its row.

Look for atleast one zero in each row and each column.Otherwise go to step 2.

Step 2: Select the smallest element in each column and subtract this from all the elements in its column.

Since each row and column contains atleast one zero, assignments can be made.

Step 3 (Assignment):

Thus all the four assignments have been made. The optimal assignment schedule and total cost is

The optimal assignment (minimum) cost

Example 10.8

Consider the problem of assigning five jobs to five persons. The assignment costs are given as follows. Determine the optimum assignment schedule.

∴ The given assignment problem is balanced.

Now let us find the solution.

The cost matrix of the given assignment problem is

Column 3 contains no zero. Go to Step 2.

Thus all the five assignments have been made. The Optimal assignment schedule and total cost is

The optimal assignment (minimum) cost = ` 9

Example 10.9

Solve the following assignment problem.

Since the number of columns is less than the number of rows, given assignment problem is unbalanced one. To balance it , introduce a dummy column with all the entries zero. The revised assignment problem is

Here only 3 tasks can be assigned to 3 men.

Step 1: is not necessary, since each row contains zero entry. Go to Step 2.

Step 3 (Assignment) :

Since each row and each columncontains exactly one assignment,all the three men have been assigned a task. But task S is not assigned to any Man. The optimal assignment schedule and total cost is

The optimal assignment (minimum) cost = ₹ 35

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## OPERATIONS RESEARCH

Lesson 9. solution of assignment problem.

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## EngineersTutor

Solved assignment problems – algorithms and flowcharts.

An algorithm is defined as sequence of steps to solve a problem (task) . The steps must be finite, well defined and unambiguous. Writing algorithm requires some thinking. Algorithm can also be defined as a plan to solve a problem and represents its logic. Note that an algorithm is of no use if it does not help us arrive at the desired solution

## Algorithm characteristics

- It should have finite number of steps . No one can be expected to execute infinite number of steps.
- The steps must be in order and simple
- Each step should be defined clearly i.e. without un-ambiguity (without doubtfulness)
- Must include all required information
- Should exhibit at least one output

## A flowchart is a pictorial (graphical) representation of an algorithm . A flowchart is drawn using different kinds of symbols. A symbol is used for a specific purpose. Each symbol has name.

Different algorithms have different performance characteristics to solve the same problem. some algorithms are fast. some are slow. some occupy more memory space. some occupy less memory space. some are complex and some algorithms are simple..

Logically algorithm, flowchart and program are the same.

Q1 . Create a program to compute the volume of a sphere. Use the formula: V = (4/3) *pi*r 3 where pi is equal to 3.1416 approximately. The r is the radius of sphere. Display the result.

Q2 . Write a program the converts the input Celsius degree into its equivalent Fahrenheit degree. Use the formula: F = (9/5) *C+32.

Q3 . Write a program that converts the input dollar to its peso exchange rate equivalent. Assume that the present exchange rate is 51.50 pesos against the dollar. Then display the peso equivalent exchange rate.

Q4 . Write a program that converts an input inch(es) into its equivalent centimeters. Take note that one inch is equivalent to 2.54cms.

Q5 . Write a program that exchanges the value of two variables: x and y. The output must be: the value of variable y will become the value of variable x, and vice versa.

Q6 . Design a program to find the circumference of a circle. Use the formula: C=2πr, where π is approximately equivalent 3.1416.

Q7 . Write a program that takes as input the purchase price of an item (P), its expected number of years of service (Y) and its expected salvage value (S). Then outputs the yearly depreciation for the item (D). Use the formula: D = (P – S) Y.

Q8 . Swapping of 2 variables without using temporary (or 3 rd variable).

Q9 . Determine the most economical quantity to be stocked for each product that a manufacturing company has in its inventory: This quantity, called economic order quantity (EOQ) is calculated as follows: EOQ=2rs/1 where: R= total yearly production requirement S=set up cost per order I=inventory carrying cost per unit.

Q10 . Write a program to compute the radius of a circle. Derive your formula from the given equation: A=πr², then display the output.

- ← Solved Assignment Problems in Java (with Algorithm and Flowchart)
- Simple if statement in C →

## Gopal Krishna

Hey Engineers, welcome to the award-winning blog,Engineers Tutor. I'm Gopal Krishna. a professional engineer & blogger from Andhra Pradesh, India. Notes and Video Materials for Engineering in Electronics, Communications and Computer Science subjects are added. "A blog to support Electronics, Electrical communication and computer students".

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## Designing Multiple-Choice Questions

A multiple-choice question (MCQ) is composed of two parts: a stem that identifies the question or problem, and a set of alternatives or possible answers that contain a key that is the best answer to the question, and a number of distractors that are plausible but incorrect answers to the question. Students respond to MCQs by indicating the alternative that they believe best answers or completes the stem. There are many advantages to using MCQs for assessment. One key advantage is that the questions are easy to mark and can even be scored by a computer, which makes them an attractive assessment approach for large classes. Well designed MCQs allow testing for a wide breadth of content and objectives and provide an objective measurement of student ability.

The following suggestions for designing MCQs are organized into three sections: 1) general strategies, 2) designing stems, and 3) designing alternatives.

## General strategies

Write questions throughout the term. Multiple-choice question exams are challenging and time-consuming to create. You will find it easier if you write a few questions each week, perhaps after a lecture when the course material is still fresh in your mind.

Instruct students to select the “best answer” rather than the “correct answer”. By doing this, you acknowledge the fact that the distractors may have an element of truth to them and discourage arguments from students who may argue that their answer is correct as well.

Use familiar language. The question should use the same terminology that was used in the course. Avoid using unfamiliar expressions or foreign language terms, unless measuring knowledge of such language is one of the goals of the question. Students are likely to dismiss distractors with unfamiliar terms as incorrect.

Avoid giving verbal association clues from the stem in the key. If the key uses words that are very similar to words found in the stem, students are more likely to pick it as the correct answer.

Avoid trick questions. Questions should be designed so that students who know the material can find the correct answer. Questions designed to lead students to an incorrect answer, through misleading phrasing or by emphasizing an otherwise unimportant detail of the solution, violate this principle.

Avoid negative wording. Students often fail to observe negative wording and it can confuse them. As a result, students who are familiar with the material often make mistakes on negatively worded questions. In general, avoid having any negatives in the stem or the options. Should you choose to use negatives, be sure to emphasize the key words by putting them in upper case, and bolding or underlining them. For example:

The University of Waterloo does NOT have a building of this name?

a.) B.C. Matthews Hall b.) Carl A. Pollock Hall c.) I.L. Neilson Hall d.) Douglas Wright Engineering Building

## Designing stems

Express the full problem in the stem. When creating the item, ask yourself if the students would be able to answer the question without looking at the options. This makes the purpose of the question clear.

Put all relevant material in the stem. Do not repeat in each of the alternatives information that can be included in the stem. This makes options easier to read and understand, and makes it easier for students to answer the question quickly.

Eliminate excessive wording and irrelevant information from the stem. Irrelevant information in the stem confuses students and leads them to waste time. For example:

A number of books have been published about the University of Waterloo. These books fall into various genres such as photographic histories, biographies of prominent people involved with the University, and accounts of the history of individual departments. Among them was a book whose author is known as "Simon the Troll". What is the title of this book?

a.) Dreaming in Technicolor b.) Water Under the Bridge c.) Of Mud and Dreams d.) Images of Waterloo

Most of the stem is not necessary to answer the question. A better question would be:

What is the title of the book about Waterloo written by “Simon the Troll”?

e.) Dreaming in Technicolor f.) *Water Under the Bridge g.) Of Mud and Dreams h.) Images of Waterloo

## Designing alternatives

Limit the number of alternatives . Use between three and five alternatives per question. Research shows that three-choice items are about as effective as four or five-choice items, mainly because it is difficult to come up with plausible distractors.

Make sure there is only one best answer. Avoid having two or more options that are correct, but where one is “more” correct than the others. The distractors should be incorrect answers to the question posed in the stem.

Make the distractors appealing and plausible. If the distractors are farfetched, students will too easily locate the correct answer, even if they have little knowledge. When testing for recognition of key terms and ideas keep the distractors similar in length and type of language as the correct solution. When testing conceptual understanding, distractors should represent common mistakes made by students.

Waterloo Counselling Services provides workshops about:

a.) cooking skills b.) hockey refereeing c.) *study skills d.) fire safety and prevention

It is unlikely that students would choose options a, b, or d, even if they didn’t know the answer. A better question would have plausible links between the stem and the distractors:

a.) preparing for marriage b.) presentation skills c.) * study skills d.) psychotherapy research

Make the choices gramatically consistent with the stem . Read the stem and each of the choices aloud to make sure that they are grammatically correct.

Place the choices in some meaningful order. When possible, place the choices in numerical, chronological or conceptual order. A better structured question is easier to read and respond to:

During what period was James Downey the president of Waterloo?

a.) 1990-1996 b.) 1991-1997 c.) 1992-1998 d.) *1993-1999

Randomly distribute the correct response. The exam should have roughly the same number of correct answers that are a's, b's, c's, and d's (assuming there are four choices per question).

Avoid using “all of the above”. If “all of the above” is an option and students know two of the options are correct, the answer must be “all of the above”. If they know one is incorrect, the answer must not be “all of the above”. A student may also read the first option, determine that it is correct, and be misled into choosing it without reading all of the options.

Avoid using “none of the above”. The option “none of the above” does not test whether the student knows the correct answer, but only that he/she knows the distractors aren’t correct.

Refrain from using words such as always, never, all, or none. Most students know that few things are universally true or false, so distractors with these words in them can often be easily dismissed.

Avoid overlapping choices. Make the alternatives mutually exclusive. It should never be the case that if one of the distractors is true, another distractor must be true as well.

Avoid questions of the form “Which of the following statements is correct?” There is no clear question being asked, and the choices are often heterogeneous. Such questions are better presented in the form of True/False questions.

If you would like support applying these tips to your own teaching, CTE staff members are here to help. View the CTE Support page to find the most relevant staff member to contact.

## Other Resources

CTE Teaching Tip “ Exam Questions: Types, Characteristics, and Suggestions ”

Cheser-Jacobs, L., & Chase, C.L. (1992). Developing and Using Tests Effectively: A Guide for Faculty . 1st ed. Jossey-Bass Publishers; San Francisco, CA.

Dirks, C., Wenderoth, M.P., & Withers, Michelle. (2014). Assessment in the College Classroom. 1st ed. W.H. Freeman and Company; New York, NY.

Kar, S.S., Lakshminarayanan, S., & Mahalakshmy, T. (2015). Basic principles of constructing multiple choice questions. Indian Journal of Community and Family Medicine, 1(2):65-69. doi:10.4103/2395-2113.251640

Towns, M.H. (2014). Guide to developing high-quality, reliable, and valid multiple-choice assessments. Journal of Chemical Education, 91(9):1426-1431. doi:10.1021/ed500076x

This Creative Commons license lets others remix, tweak, and build upon our work non-commercially, as long as they credit us and indicate if changes were made. Use this citation format: Designing multiple-choice questions . Centre for Teaching Excellence, University of Waterloo .

## Catalog search

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- Career Development
- Course Design
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## How to Solve the Assignment Problem: A Complete Guide

Table of Contents

Assignment problem is a special type of linear programming problem that deals with assigning a number of resources to an equal number of tasks in the most efficient way. The goal is to minimize the total cost of assignments while ensuring that each task is assigned to only one resource and each resource is assigned to only one task. In this blog, we will discuss the solution of the assignment problem using the Hungarian method, which is a popular algorithm for solving the problem.

## Understanding the Assignment Problem

Before we dive into the solution, it is important to understand the problem itself. In the assignment problem, we have a matrix of costs, where each row represents a resource and each column represents a task. The objective is to assign each resource to a task in such a way that the total cost of assignments is minimized. However, there are certain constraints that need to be satisfied – each resource can be assigned to only one task and each task can be assigned to only one resource.

## Solving the Assignment Problem

There are various methods for solving the assignment problem, including the Hungarian method, the brute force method, and the auction algorithm. Here, we will focus on the steps involved in solving the assignment problem using the Hungarian method, which is the most commonly used and efficient method.

## Step 1: Set up the cost matrix

The first step in solving the assignment problem is to set up the cost matrix, which represents the cost of assigning a task to an agent. The matrix should be square and have the same number of rows and columns as the number of tasks and agents, respectively.

## Step 2: Subtract the smallest element from each row and column

To simplify the calculations, we need to reduce the size of the cost matrix by subtracting the smallest element from each row and column. This step is called matrix reduction.

## Step 3: Cover all zeros with the minimum number of lines

The next step is to cover all zeros in the matrix with the minimum number of horizontal and vertical lines. This step is called matrix covering.

## Step 4: Test for optimality and adjust the matrix

To test for optimality, we need to calculate the minimum number of lines required to cover all zeros in the matrix. If the number of lines equals the number of rows or columns, the solution is optimal. If not, we need to adjust the matrix and repeat steps 3 and 4 until we get an optimal solution.

## Step 5: Assign the tasks to the agents

The final step is to assign the tasks to the agents based on the optimal solution obtained in step 4. This will give us the most cost-effective or profit-maximizing assignment.

## Solution of the Assignment Problem using the Hungarian Method

The Hungarian method is an algorithm that uses a step-by-step approach to find the optimal assignment. The algorithm consists of the following steps:

- Subtract the smallest entry in each row from all the entries of the row.
- Subtract the smallest entry in each column from all the entries of the column.
- Draw the minimum number of lines to cover all zeros in the matrix. If the number of lines drawn is equal to the number of rows, we have an optimal solution. If not, go to step 4.
- Determine the smallest entry not covered by any line. Subtract it from all uncovered entries and add it to all entries covered by two lines. Go to step 3.

The above steps are repeated until an optimal solution is obtained. The optimal solution will have all zeros covered by the minimum number of lines. The assignments can be made by selecting the rows and columns with a single zero in the final matrix.

## Applications of the Assignment Problem

The assignment problem has various applications in different fields, including computer science, economics, logistics, and management. In this section, we will provide some examples of how the assignment problem is used in real-life situations.

## Applications in Computer Science

The assignment problem can be used in computer science to allocate resources to different tasks, such as allocating memory to processes or assigning threads to processors.

## Applications in Economics

The assignment problem can be used in economics to allocate resources to different agents, such as allocating workers to jobs or assigning projects to contractors.

## Applications in Logistics

The assignment problem can be used in logistics to allocate resources to different activities, such as allocating vehicles to routes or assigning warehouses to customers.

## Applications in Management

The assignment problem can be used in management to allocate resources to different projects, such as allocating employees to tasks or assigning budgets to departments.

Let’s consider the following scenario: a manager needs to assign three employees to three different tasks. Each employee has different skills, and each task requires specific skills. The manager wants to minimize the total time it takes to complete all the tasks. The skills and the time required for each task are given in the table below:

The assignment problem is to determine which employee should be assigned to which task to minimize the total time required. To solve this problem, we can use the Hungarian method, which we discussed in the previous blog.

Using the Hungarian method, we first subtract the smallest entry in each row from all the entries of the row:

Next, we subtract the smallest entry in each column from all the entries of the column:

We draw the minimum number of lines to cover all the zeros in the matrix, which in this case is three:

Since the number of lines is equal to the number of rows, we have an optimal solution. The assignments can be made by selecting the rows and columns with a single zero in the final matrix. In this case, the optimal assignments are:

- Emp 1 to Task 3
- Emp 2 to Task 2
- Emp 3 to Task 1

This assignment results in a total time of 9 units.

I hope this example helps you better understand the assignment problem and how to solve it using the Hungarian method.

Solving the assignment problem may seem daunting, but with the right approach, it can be a straightforward process. By following the steps outlined in this guide, you can confidently tackle any assignment problem that comes your way.

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## Operations Research

1 Operations Research-An Overview

- History of O.R.
- Approach, Techniques and Tools
- Phases and Processes of O.R. Study
- Typical Applications of O.R
- Limitations of Operations Research
- Models in Operations Research
- O.R. in real world

2 Linear Programming: Formulation and Graphical Method

- General formulation of Linear Programming Problem
- Optimisation Models
- Basics of Graphic Method
- Important steps to draw graph
- Multiple, Unbounded Solution and Infeasible Problems
- Solving Linear Programming Graphically Using Computer
- Application of Linear Programming in Business and Industry

3 Linear Programming-Simplex Method

- Principle of Simplex Method
- Computational aspect of Simplex Method
- Simplex Method with several Decision Variables
- Two Phase and M-method
- Multiple Solution, Unbounded Solution and Infeasible Problem
- Sensitivity Analysis
- Dual Linear Programming Problem

4 Transportation Problem

- Basic Feasible Solution of a Transportation Problem
- Modified Distribution Method
- Stepping Stone Method
- Unbalanced Transportation Problem
- Degenerate Transportation Problem
- Transhipment Problem
- Maximisation in a Transportation Problem

5 Assignment Problem

- Solution of the Assignment Problem
- Unbalanced Assignment Problem
- Problem with some Infeasible Assignments
- Maximisation in an Assignment Problem
- Crew Assignment Problem

6 Application of Excel Solver to Solve LPP

- Building Excel model for solving LP: An Illustrative Example

7 Goal Programming

- Concepts of goal programming
- Goal programming model formulation
- Graphical method of goal programming
- The simplex method of goal programming
- Using Excel Solver to Solve Goal Programming Models
- Application areas of goal programming

8 Integer Programming

- Some Integer Programming Formulation Techniques
- Binary Representation of General Integer Variables
- Unimodularity
- Cutting Plane Method
- Branch and Bound Method
- Solver Solution

9 Dynamic Programming

- Dynamic Programming Methodology: An Example
- Definitions and Notations
- Dynamic Programming Applications

10 Non-Linear Programming

- Solution of a Non-linear Programming Problem
- Convex and Concave Functions
- Kuhn-Tucker Conditions for Constrained Optimisation
- Quadratic Programming
- Separable Programming
- NLP Models with Solver

11 Introduction to game theory and its Applications

- Important terms in Game Theory
- Saddle points
- Mixed strategies: Games without saddle points
- 2 x n games
- Exploiting an opponent’s mistakes

12 Monte Carlo Simulation

- Reasons for using simulation
- Monte Carlo simulation
- Limitations of simulation
- Steps in the simulation process
- Some practical applications of simulation
- Two typical examples of hand-computed simulation
- Computer simulation

13 Queueing Models

- Characteristics of a queueing model
- Notations and Symbols
- Statistical methods in queueing
- The M/M/I System
- The M/M/C System
- The M/Ek/I System
- Decision problems in queueing

## Assessing by Multiple Choice Questions

Multiple choice question (MCQ) tests can be useful for formative assessment and to stimulate students' active and self-managed learning. They improve students' learning performance and their perceptions of the quality of their learning experience ( Velan et al., 2008 ). When using MCQ tests for formative learning, you might still want to assign a small weight to them in the overall assessment plan for the course, to indicate to students that their grasp of the material tested is important.

MCQ tests are strongly associated with assessing lower-order cognition such as the recall of discrete facts. Because of this, assessors have questioned their use in higher education. It is possible to design MCQ tests to assess higher-order cognition (such as synthesis, creative thinking and problem-solving), but questions must be drafted with considerable skill if such tests are to be valid and reliable. This takes time and entails significant subjective judgement.

When determining whether an MCQ test is at the appropriate cognitive level, you may want to compare it with the levels as set out in Krathwohl's (2002) revision to Bloom's Taxonomy of Educational Objectives. If an MCQ test is, in fact, appropriate to the learning outcomes you want to assess and the level of cognition they involve, the next question is whether it would be best used for formative assessment (to support students' self-management of their learning), or for summative assessment (to determine the extent of students' learning at a particular point). When MCQ tests are being used for summative assessment, it's important to ask whether this is because of the ease of making them or for solid educational reasons. Where MCQ tests are appropriate, ensure that you integrate them effectively into assessment design. MCQ tests should never constitute the only or major form of summative assessment in university-level courses.

## When to use

MCQ tests are useful for assessing lower-order cognitive processes, such as the recall of factual information, although this is often at the expense of higher-level critical and creative reasoning processes. Use MCQ tests when, for example, you want to:

- assess lower-order cognition such as the recall of discrete facts, particularly if they will be essential to higher-order learning later in the course
- gather information about students' pre-course understanding, knowledge gaps and misconceptions, to help plan learning and teaching approaches.
- provide students with an accessible way to review course material, check that they understand key concepts and obtain timely feedback to help them manage their own learning
- test students' broad knowledge of the curriculum and learning outcomes.

It should be noted that the closed-ended nature of MCQ tests makes them particularly inappropriate for assessing originality and creativity in thinking.

MCQ tests are readily automated, with numerous systems available to compile, administer, mark and provide feedback on tests according to a wide range of parameters. Marking can be done by human markers with little increase in marking load; or automatically, with no additional demands on teachers or tutors.

## Objectivity

Although MCQ tests can lack nuance in that the answers are either right or wrong (provided the questions are well-designed), this has the advantage of reducing markers' bias and increasing overall objectivity. Moreover, because the tests do not require the students to formulate and write their own answers, the students' writing ability (which can vary widely even within a homogeneous cohort) becomes much less of a subjective factor in determining their grasp of the material. It should be noted, however, that they are highly prone to cultural bias (Bazemore-James et al., 2016).

## The development of question banks

MCQ tests can be refined and enhanced over time to incorporate new questions into an ever-growing question pool that can be used in different settings.

While MCQ tests are quick and straighforward to administer and mark, they require a large amount of up-front design effort to ensure that they are valid, relevant, fair and as free as possible from cultural, gender or racial bias. It's also challenging to write MCQs that resist mere guesswork. This is particularly the case if the questions are intended to test higher-order cognition such as synthesis, creative thinking and problem-solving.

The use of MCQ tests as summative assessments can encourage students to adopt superficial approaches to learning. This superficiality is exacerbated by the lack of in-depth, critical feedback inherent in a highly standardised assessment.

MCQ tests can disadvantage students with reading or English-language difficulties, regardless of how well they understand the content being assessed.

## Plan a summative MCQ test

If you decide that an MCQ test is appropriate for summative assessment according to the objectives and outcomes for a course, let students know in the course outline that you'll be using it.

Table 1 sets out a number of factors to consider in the test planning stage.

Table 1: Factors in planning MCQ test design

## Construct an MCQ test

Constructing effective MCQ tests and items takes considerable time and requires scrupulous care in the design, review and validation stages. Constructing MCQ tests for high-stakes summative assessment is a specialist task.

For this reason, rather than constructing a test from scratch, it may be more efficient for you to see what other validated tests already exist, and incorporate one into any course for which numerous decisions need to be made.

In some circumstances it may be worth the effort to create a new test. If you can undertake test development collaboratively within your department or discipline group, or as a larger project across institutional boundaries, you will increase the test's potential longevity and sustainability.

By progressively developing a multiple-choice question bank or pool, you can support benchmarking processes and establish assessment standards that have long-term effects on assuring course quality.

Use a design framework to see how individual MCQ questions will assess particular topic areas and types of learning objectives, across a spectrum of cognitive demand, to contribute to the test's overall balance. As an example, the "design blueprint" in Table 2 provides a structural framework for planning.

Table 2: Design blueprint for multiple choice test design (from the Instructional Assessment Resources at the University of Texas at Austin)

Use the most appropriate format for each question posed. Ask yourself, is it best to use:

- a single correct answer
- more than one correct answer
- a true/false choice (with single or multiple correct answers)
- matching (e.g. a term with the appropriate definition, or a cause with the most likely effect)
- sentence completion, or
- questions relating to some given prompt material?

To assess higher-order thinking and reasoning, consider basing a cluster of MCQ items on some prompt material, such as:

- a brief outline of a problem, case or scenario
- a visual representation (picture, diagram or table) of the interrelationships among pieces of information or concepts
- an excerpt from published material.

You can present the associated MCQ items in a sequence from basic understanding through to higher-order reasoning, including:

- identifying the effect of changing a parameter
- selecting the solution to a given problem
- nominating the optimum application of a principle.

You may wish to add some short-answer questions to a substantially MCQ test to minimise the effect of guessing by requiring students to express in their own words their understanding and analysis of problems.

Well in advance of an MCQ test, explain to students:

- the purposes of the test (and whether it is formative or summative)
- the topics being covered
- the structure of the test
- whether aids can be taken into the test (for example, calculators, notes, textbooks, dictionaries)
- how it will be marked
- how the mark will contribute to their overall grade.

Compose clear instructions on the test itself, explaining:

- the components of the test
- their relative weighting
- how much time you expect students to spend on each section, so that they can optimise their time.

## Quality assurance of MCQ tests

Whether you use MCQ tests to support learning in a formative way or for summative assessment, ensure that the overall test and each of its individual items are well aligned with the course learning objectives. When using MCQ tests for summative assessment, it's all the more critical that you assure their validity.

The following strategies will help you assure quality:

- Use a basic quality checklist ( such as this one from Knapp & Associates International ) when designing and reviewing the test.
- Take the test yourself. Calculate student completion time as being four times longer than your completion time.
- Work collaboratively across your discipline to develop an MCQ item bank as a dynamic (and growing) repository that everyone can use for formative or summative assessments, and that enables peer review, evaluation and validation.

Use peer review to:

- consider whether MCQ tests are educationally justified in your discipline
- critically evaluate MCQ test and item design
- examine the effects of using MCQs in the context of the learning setting
- record and disseminate the peer review outcomes to students and colleagues.

## Engage students in active learning with MCQ tests

Used formatively, MCQ tests can:

- engage students in actively reviewing their own learning progress, identifying gaps and weaknesses in their understanding, and consolidating their learning through rehearsal
- provide a trigger for collaborative learning activities, such as discussion and debate about the answers to questions.
- become collaborative through the use of technologies such as electronic voting systems
- through peer assessment, help students identify common areas of misconception within the class.

You can also create activities that disrupt the traditional agency of assessment. You might, for example, require students to indicate the learning outcomes with which individual questions are aligned, or to construct their own MCQ questions and prepare explanatory feedback on the right and wrong answers (Fellenz, 2010).

## Ensure fairness

Construct MCQ tests according to inclusive-design principles to ensure equal chances of success for all students. Take into account any diversity of ability, cultural background or learning styles and needs.

- Avoid sexual, racial, cultural or linguistic stereotyping in individual MCQ test items, to ensure that no groups of students are unfairly advantaged or disadvantaged.
- Provide alternative formats for MCQ-type exams for any students with disabilities. They may, for example, need more time to take a test, or to be provided with assistive technology or readers.
- Set up contingency plans for timed online MCQ tests, as computers can malfunction or system outages occur at any time. Also address any issues arising from students being in different time zones.
- Develop processes in advance so that you have time to inform students about the objectives, formats and delivery of MCQ tests and, for a summative test, the marking scheme being used and the test's effect on overall marks.
- To reduce the opportunity for plagiarism, specify a randomised question presentation for MCQ, so that different students will be presented with the same content in a different order.

Writing Good Multiple Choice Test Questions (Vanderbilt University).

Bazemore-James, C. M., Shinaprayoon, T., & Martin, J. (2016). Understanding and supporting students who experience cultural bias in standardized tests . Trends and Issues in Academic Support: 2016-2017 .

Fellenz, M.R. (2004). Using assessment to support higher level learning: the multiple choice item development assignment. Assessment and Evaluation in Higher Education 29 (6), 703-719.

Krathwohl, D. R. (2002). A revision of Bloom's Taxonomy: An overview . Theory into Practice , 41 (4), 212-218.

LeJeune, J. (2023). A multiple-choice study: The impact of transparent question design on student performance . Perspectives In Learning , 20 (1), 75-89.

Velan, G.M., Jones, P., McNeil, H.P. and Kumar, R.K. (2008). Integrated online formative assessments in the biomedical sciences for medical students: benefits for learning . BMC Medical Education 8(52).

## Events & news

## Designing multiple-choice questions

Designing multiple-choice questions.

A multiple-choice question (MCQ) is composed of two parts: a stem that identifies the question or problem, and a set of alternatives or possible answers that contain a key that is the best answer to the question, and several distractors that are plausible but incorrect answers to the question. Students respond to MCQs by indicating the alternative that they believe best answers or completes the stem. There are many advantages to using MCQs for assessment. One key benefit is that the questions are easy to mark and can even be scored by a computer, making them an attractive assessment approach for large classes. Well-designed MCQs allow testing for a wide breadth of content and objectives and provide an objective measurement of student ability.

We organized the following suggestions into three sections:

- General strategies,
- Designing stems (the question or unfinished statement)
- Developing alternatives (options)

## General strategies

- Write questions throughout the term. Multiple-choice question exams are challenging and time-consuming to create. You will find it easier to write a few questions each week, perhaps after a lecture when the course material is still fresh in your mind.
- Instruct students to select the “best answer” rather than the “correct answer.” By doing this, you acknowledge that the distractors may have an element of truth to them and discourage arguments from students who may argue that their answer is correct.
- Use familiar language. The question should use the same terminology used in the course. Avoid using unfamiliar expressions or foreign language terms unless measuring such language knowledge is one of the question’s goals. Students are likely to dismiss distractors with unfamiliar terms as incorrect.
- Avoid giving verbal association clues from the stem in the key. If the key uses words similar to the stem’s words, students are more likely to pick it as the correct answer.
- Avoid trick questions. Design questions so that students who know the material can find the correct answer. Questions designed to lead students to an incorrect answer, through misleading phrasing or by emphasizing an otherwise unimportant detail of the solution, violate this principle.
- Avoid negative wording. Students often fail to observe negative wording, and it can confuse them. As a result, students who are familiar with the material often make mistakes on negatively worded questions. In general, avoid having any negatives in the stem or the options. In the rare cases where you use negatives, be sure to emphasize the keywords by putting them in uppercase and bolding or underlining them. For example:

Iowa State University does NOT have a building of this name?

- a.) Beardshear Hall
- b.) Marston Hall
- c.) Agronomy Hall
- d.) Ames Hall

## Designing stems

- Express the full problem in the stem. When creating the item, ask yourself if the students would answer the question without considering the options. This step makes the purpose of the question clear.
- Put all relevant material in the stem. Do not repeat each of the alternative information included in the stem. This step makes options more straightforward to read and understand and makes it easier for students to answer the question quickly.
- Eliminate excessive wording and irrelevant information from the stem. Irrelevant information in the stem confuses students and leads them to waste time:

## Poor example

From this prairie beginning, Iowa State University’s campus has grown to encompass approximately 140 buildings and renowned national landscaping. However, throughout its history of growth, the campus also lost numerous buildings to fire and the wrecking ball, such as the Chemistry Building, Margaret Hall, Old Main, the original Veterinary Hospital, Clyde Williams Stadium, numerous barns, cottages, and temporary buildings. What building was designated a National Historic Landmark in 1965?

- Morrill Hall
- Beardshear Hall
- Farm House
- Agronomy Hall

## A better example

What is the oldest structure designated as a National Landmark on Campus?

- *Farm House

## Developing alternatives

- Limit the number of alternatives . Use between three and five options per question. Research shows that three-choice items are about as effective as four or five-choice items, mainly because it is challenging to develop plausible distractors.
- Make sure there is only one best answer. Avoid having two or more correct options, but one is “more” accurate than the others. The distractors should be incorrect answers to the question posed in the stem.
- Make the distractors appealing and plausible. If the distractors are farfetched, students will too quickly locate the correct answer. Even if they have little knowledge when testing to recognize key terms and ideas, keep the distractors similar in length and type of language as the correct solution. When testing conceptual understanding, distractors should represent common mistakes made by students.

Poor example: Which of the following artists is known for painting the ceiling of the Sistine Chapel?

- Homer Simpson.
- Michelangelo.

It is unlikely that students would choose a, b, or d, even if they didn’t know the answer.

A better question would have plausible links between the stem and the distractors:

Better example: Which of the following artists is known for painting the ceiling of the Sistine Chapel?

- Botticelli.
- Make the choices grammatically consistent with the stem . Read the stem and each of the options aloud to make sure that they are grammatically correct.
- Place the choices in some meaningful order. When possible, place the options in numerical, chronological, or conceptual order. A better-structured question is easier to read and respond to: During what period was George Washington Carver a student/faculty at Iowa State?
- Randomly distribute the correct response. The exam should have roughly the same number of correct answers: a’s, b’s, c’s, and d’s (assuming there are four choices per question).
- Avoid using “all of the above.” If “all of the above” is an option and students know two of the options are correct, the answer must be “all of the above.” If they know one is incorrect; the solution must not be “all of the above.” A student may also read the first option, determine that it is correct, and misled into choosing it without reading all of the possibilities.
- Avoid using “none of the above.” The option “none of the above” does not test whether the student knows the correct answer, but only that he/she knows the distractors aren’t correct.
- Refrain from using words such as always, never, all, or none. Most students know that few things are universally true or false, so distractors with these words in them can often be easily dismissed.
- Avoid overlapping choices. Make the alternatives mutually exclusive. It should never be the case that if one of the distractors is true, another distractor must be true as well.
- Avoid questions of the form “Which of the following statements is correct?” There is no clear question asked, and the choices are often different. A better practice is to present these questions in True/False format.

Designing Multiple-Choice Questions developed by the Centre for Teaching Excellence at the University of Waterloo (retrieved on February 22, 2021) from https://uwaterloo.ca/centre-for-teaching-excellence/teaching-resources/teaching-tips/developing-assignments/assignment-design/designing-multiple-choice-questions, and Multiple choice exam construction Center for Teaching Excellence at Duquesne University retrieved (February 23, 2021) from https://www.duq.edu/about/centers-and-institutes/center-for-teaching-excellence/teaching-and-learning-at-duquesne/multiple-choice-exam-construction

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