term assignment problem

Assignment Problem: Meaning, Methods and Variations | Operations Research

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.

Meaning of Assignment Problem:

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

Definition of Assignment Problem:

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Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:

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Assignment problem

The problem of optimally assigning $ m $ individuals to $ m $ jobs. It can be formulated as a linear programming problem that is a special case of the transport problem :

maximize $ \sum _ {i,j } c _ {ij } x _ {ij } $

$$ \sum _ { j } x _ {ij } = a _ {i} , i = 1 \dots m $$

(origins or supply),

$$ \sum _ { i } x _ {ij } = b _ {j} , j = 1 \dots n $$

(destinations or demand), where $ x _ {ij } \geq 0 $ and $ \sum a _ {i} = \sum b _ {j} $, which is called the balance condition. The assignment problem arises when $ m = n $ and all $ a _ {i} $ and $ b _ {j} $ are $ 1 $.

If all $ a _ {i} $ and $ b _ {j} $ in the transposed problem are integers, then there is an optimal solution for which all $ x _ {ij } $ are integers (Dantzig's theorem on integral solutions of the transport problem).

In the assignment problem, for such a solution $ x _ {ij } $ is either zero or one; $ x _ {ij } = 1 $ means that person $ i $ is assigned to job $ j $; the weight $ c _ {ij } $ is the utility of person $ i $ assigned to job $ j $.

The special structure of the transport problem and the assignment problem makes it possible to use algorithms that are more efficient than the simplex method . Some of these use the Hungarian method (see, e.g., [a5] , [a1] , Chapt. 7), which is based on the König–Egervary theorem (see König theorem ), the method of potentials (see [a1] , [a2] ), the out-of-kilter algorithm (see, e.g., [a3] ) or the transportation simplex method.

In turn, the transportation problem is a special case of the network optimization problem.

A totally different assignment problem is the pole assignment problem in control theory.

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Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)

• For each row of the matrix, find the smallest element and subtract it from every element in its row.
• Do the same (as step 1) for all columns.
• Cover all zeros in the matrix using minimum number of horizontal and vertical lines.
• Test for Optimality: If the minimum number of covering lines is n, an optimal assignment is possible and we are finished. Else if lines are lesser than n, we haven’t found the optimal assignment, and must proceed to step 5.
• Determine the smallest entry not covered by any line. Subtract this entry from each uncovered row, and then add it to each covered column. Return to step 3.

Try it before moving to see the solution

Explanation for above simple example:

  An example that doesn’t lead to optimal value in first attempt: In the above example, the first check for optimality did give us solution. What if we the number covering lines is less than n.

Time complexity : O(n^3), where n is the number of workers and jobs. This is because the algorithm implements the Hungarian algorithm, which is known to have a time complexity of O(n^3).

Space complexity :   O(n^2), where n is the number of workers and jobs. This is because the algorithm uses a 2D cost matrix of size n x n to store the costs of assigning each worker to a job, and additional arrays of size n to store the labels, matches, and auxiliary information needed for the algorithm.

In the next post, we will be discussing implementation of the above algorithm. The implementation requires more steps as we need to find minimum number of lines to cover all 0’s using a program. References: http://www.math.harvard.edu/archive/20_spring_05/handouts/assignment_overheads.pdf https://www.youtube.com/watch?v=dQDZNHwuuOY

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One of the most well-known combinatorial optimization problems is the assignment problem . Here's an example: suppose a group of workers needs to perform a set of tasks, and for each worker and task, there is a cost for assigning the worker to the task. The problem is to assign each worker to at most one task, with no two workers performing the same task, while minimizing the total cost.

You can visualize this problem by the graph below, in which there are four workers and four tasks. The edges represent all possible ways to assign workers to tasks. The labels on the edges are the costs of assigning workers to tasks.

An assignment corresponds to a subset of the edges, in which each worker has at most one edge leading out, and no two workers have edges leading to the same task. One possible assignment is shown below.

The total cost of the assignment is 70 + 55 + 95 + 45 = 265 .

The next section shows how solve an assignment problem, using both the MIP solver and the CP-SAT solver.

Other tools for solving assignment problems

OR-Tools also provides a couple of other tools for solving assignment problems, which can be faster than the MIP or CP solvers:

• Linear sum assignment solver
• Minimum cost flow solver

However, these tools can only solve simple types of assignment problems. So for general solvers that can handle a wide variety of problems (and are fast enough for most applications), we recommend the MIP and CP-SAT solvers.

Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4.0 License , and code samples are licensed under the Apache 2.0 License . For details, see the Google Developers Site Policies . Java is a registered trademark of Oracle and/or its affiliates.

Last updated 2023-01-02 UTC.

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Unbalanced Assignment Problem: Definition, Formulation, and Solution Methods

Table of Contents

Are you familiar with the assignment problem in Operations Research (OR)? This problem deals with assigning tasks to workers in a way that minimizes the total cost or time needed to complete the tasks. But what if the number of tasks and workers is not equal? In this case, we face the Unbalanced Assignment Problem (UAP). This blog will help you understand what the UAP is, how to formulate it, and how to solve it.

What is the Unbalanced Assignment Problem?

The Unbalanced Assignment Problem is an extension of the Assignment Problem in OR, where the number of tasks and workers is not equal. In the UAP, some tasks may remain unassigned, while some workers may not be assigned any task. The objective is still to minimize the total cost or time required to complete the assigned tasks, but the UAP has additional constraints that make it more complex than the traditional assignment problem.

Formulation of the Unbalanced Assignment Problem

To formulate the UAP, we start with a matrix that represents the cost or time required to assign each task to each worker. If the matrix is square, we can use the Hungarian algorithm to solve the problem. But when the matrix is not square, we need to add dummy tasks or workers to balance the matrix. These dummy tasks or workers have zero costs and are used to make the matrix square.

Once we have a square matrix, we can apply the Hungarian algorithm to find the optimal assignment. However, we need to be careful in interpreting the results, as the assignment may include dummy tasks or workers that are not actually assigned to anything.

Solutions for the Unbalanced Assignment Problem

Besides the Hungarian algorithm, there are other methods to solve the UAP, such as the transportation algorithm and the auction algorithm. The transportation algorithm is based on transforming the UAP into a transportation problem, which can be solved with the transportation simplex method. The auction algorithm is an iterative method that simulates a bidding process between the tasks and workers to find the optimal assignment.

In summary, the Unbalanced Assignment Problem is a variant of the traditional Assignment Problem in OR that deals with assigning tasks to workers when the number of tasks and workers is not equal. To solve the UAP, we need to balance the matrix by adding dummy tasks or workers and then apply algorithms such as the Hungarian algorithm, the transportation algorithm, or the auction algorithm. Understanding the UAP can help businesses and organizations optimize their resource allocation and improve their operational efficiency.

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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
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8 Integer Programming

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9 Dynamic Programming

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10 Non-Linear Programming

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11 Introduction to game theory and its Applications

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12 Monte Carlo Simulation

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• Some practical applications of simulation
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13 Queueing Models

• Characteristics of a queueing model
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Encyclopedia of Optimization pp 1153–1162 Cite as

Generalized Assignment Problem

• O. Erhun Kundakcioglu 3 &
• Saed Alizamir 3  
• Reference work entry

2822 Accesses

15 Citations

Article Outline

Introduction

  Multiple-Resource Generalized Assignment Problem

  Multilevel Generalized Assignment Problem

  Dynamic Generalized Assignment Problem

  Bottleneck Generalized Assignment Problem

  Generalized Assignment Problem with Special Ordered Set

  Stochastic Generalized Assignment Problem

  Bi-Objective Generalized Assignment Problem

  Generalized Multi-Assignment Problem

  Exact Algorithms

  Heuristics

Conclusions

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Kundakcioglu, O.E., Alizamir, S. (2008). Generalized Assignment Problem . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_200

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Hungarian Method

The Hungarian method is a computational optimization technique that addresses the assignment problem in polynomial time and foreshadows following primal-dual alternatives. In 1955, Harold Kuhn used the term “Hungarian method” to honour two Hungarian mathematicians, Dénes Kőnig and Jenő Egerváry. Let’s go through the steps of the Hungarian method with the help of a solved example.

Hungarian Method to Solve Assignment Problems

The Hungarian method is a simple way to solve assignment problems. Let us first discuss the assignment problems before moving on to learning the Hungarian method.

What is an Assignment Problem?

A transportation problem is a type of assignment problem. The goal is to allocate an equal amount of resources to the same number of activities. As a result, the overall cost of allocation is minimised or the total profit is maximised.

Because available resources such as workers, machines, and other resources have varying degrees of efficiency for executing different activities, and hence the cost, profit, or loss of conducting such activities varies.

Assume we have ‘n’ jobs to do on ‘m’ machines (i.e., one job to one machine). Our goal is to assign jobs to machines for the least amount of money possible (or maximum profit). Based on the notion that each machine can accomplish each task, but at variable levels of efficiency.

Hungarian Method Steps

Check to see if the number of rows and columns are equal; if they are, the assignment problem is considered to be balanced. Then go to step 1. If it is not balanced, it should be balanced before the algorithm is applied.

Step 1 – In the given cost matrix, subtract the least cost element of each row from all the entries in that row. Make sure that each row has at least one zero.

Step 2 – In the resultant cost matrix produced in step 1, subtract the least cost element in each column from all the components in that column, ensuring that each column contains at least one zero.

Step 3 – Assign zeros

• Analyse the rows one by one until you find a row with precisely one unmarked zero. Encircle this lonely unmarked zero and assign it a task. All other zeros in the column of this circular zero should be crossed out because they will not be used in any future assignments. Continue in this manner until you’ve gone through all of the rows.
• Examine the columns one by one until you find one with precisely one unmarked zero. Encircle this single unmarked zero and cross any other zero in its row to make an assignment to it. Continue until you’ve gone through all of the columns.

Step 4 – Perform the Optimal Test

• The present assignment is optimal if each row and column has exactly one encircled zero.
• The present assignment is not optimal if at least one row or column is missing an assignment (i.e., if at least one row or column is missing one encircled zero). Continue to step 5. Subtract the least cost element from all the entries in each column of the final cost matrix created in step 1 and ensure that each column has at least one zero.

Step 5 – Draw the least number of straight lines to cover all of the zeros as follows:

(a) Highlight the rows that aren’t assigned.

(b) Label the columns with zeros in marked rows (if they haven’t already been marked).

(c) Highlight the rows that have assignments in indicated columns (if they haven’t previously been marked).

(d) Continue with (b) and (c) until no further marking is needed.

(f) Simply draw the lines through all rows and columns that are not marked. If the number of these lines equals the order of the matrix, then the solution is optimal; otherwise, it is not.

Step 6 – Find the lowest cost factor that is not covered by the straight lines. Subtract this least-cost component from all the uncovered elements and add it to all the elements that are at the intersection of these straight lines, but leave the rest of the elements alone.

Step 7 – Continue with steps 1 – 6 until you’ve found the highest suitable assignment.

Hungarian Method Example

Use the Hungarian method to solve the given assignment problem stated in the table. The entries in the matrix represent each man’s processing time in hours.

\(\begin{array}{l}\begin{bmatrix} & I & II & III & IV & V \\1 & 20 & 15 & 18 & 20 & 25 \\2 & 18 & 20 & 12 & 14 & 15 \\3 & 21 & 23 & 25 & 27 & 25 \\4 & 17 & 18 & 21 & 23 & 20 \\5 & 18 & 18 & 16 & 19 & 20 \\\end{bmatrix}\end{array} \)

With 5 jobs and 5 men, the stated problem is balanced.

\(\begin{array}{l}A = \begin{bmatrix}20 & 15 & 18 & 20 & 25 \\18 & 20 & 12 & 14 & 15 \\21 & 23 & 25 & 27 & 25 \\17 & 18 & 21 & 23 & 20 \\18 & 18 & 16 & 19 & 20 \\\end{bmatrix}\end{array} \)

Subtract the lowest cost element in each row from all of the elements in the given cost matrix’s row. Make sure that each row has at least one zero.

\(\begin{array}{l}A = \begin{bmatrix}5 & 0 & 3 & 5 & 10 \\6 & 8 & 0 & 2 & 3 \\0 & 2 & 4 & 6 & 4 \\0 & 1 & 4 & 6 & 3 \\2 & 2 & 0 & 3 & 4 \\\end{bmatrix}\end{array} \)

Subtract the least cost element in each Column from all of the components in the given cost matrix’s Column. Check to see if each column has at least one zero.

\(\begin{array}{l}A = \begin{bmatrix}5 & 0 & 3 & 3 & 7 \\6 & 8 & 0 & 0 & 0 \\0 & 2 & 4 & 4 & 1 \\0 & 1 & 4 & 4 & 0 \\2 & 2 & 0 & 1 & 1 \\\end{bmatrix}\end{array} \)

When the zeros are assigned, we get the following:

The present assignment is optimal because each row and column contain precisely one encircled zero.

Where 1 to II, 2 to IV, 3 to I, 4 to V, and 5 to III are the best assignments.

Hence, z = 15 + 14 + 21 + 20 + 16 = 86 hours is the optimal time.

Practice Question on Hungarian Method

Use the Hungarian method to solve the following assignment problem shown in table. The matrix entries represent the time it takes for each job to be processed by each machine in hours.

\(\begin{array}{l}\begin{bmatrix}J/M & I & II & III & IV & V \\1 & 9 & 22 & 58 & 11 & 19 \\2 & 43 & 78 & 72 & 50 & 63 \\3 & 41 & 28 & 91 & 37 & 45 \\4 & 74 & 42 & 27 & 49 & 39 \\5 & 36 & 11 & 57 & 22 & 25 \\\end{bmatrix}\end{array} \)

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Frequently Asked Questions on Hungarian Method

What is hungarian method.

The Hungarian method is defined as a combinatorial optimization technique that solves the assignment problems in polynomial time and foreshadowed subsequent primal–dual approaches.

What are the steps involved in Hungarian method?

The following is a quick overview of the Hungarian method: Step 1: Subtract the row minima. Step 2: Subtract the column minimums. Step 3: Use a limited number of lines to cover all zeros. Step 4: Add some more zeros to the equation.

What is the purpose of the Hungarian method?

When workers are assigned to certain activities based on cost, the Hungarian method is beneficial for identifying minimum costs.

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Quantitative Techniques: Theory and Problems by P. C. Tulsian, Vishal Pandey

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WHAT IS ASSIGNMENT PROBLEM

Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons.

The assignment problem in the general form can be stated as follows:

“Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to assign each facility to one and only one job in such a way that the measure of effectiveness is optimised (Maximised or Minimised).”

Several problems of management has a structure identical with the assignment problem.

Example I A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost of assigning (i.e. effectiveness) each job to each ...

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Assignment Problems

Definition of the assignment problem, mathematical formulation of the assignment problem, hungarian method for solving assignment problem, flow chart of hungarian method.

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

Suppose there are \(n\) jobs to be performed and \(n\) persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let \(c_{ij}\) be the cost if the \(i-th\) person is assigned to the \(j-th\) job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

An assignment problem can be mathematically formulated as follows:

Minimise the total cost

\(x_{ij} =1\), if \(i^{th}\) person is assigned to the \(j^{th}\) job \(x_{ij}=0\), if \(i^{th}\) person is that assigned to the \(j^{th}\) job

subject to the constraints

i) \(\sum_{i=1}^n x_{ij} = 1, j=1, 2, \cdots n\)

which means that only one job is done by the \(i^{th}\) person, \(i= 1, 2, \cdots, n\)

ii) \(\sum_{i=1}^n x_{ij} = 1, j=1, 2, \cdots n\)

which means that only one person should be assigned to the \(j^{th}\) person, \(j= 1, 2, \cdots, n\)

The Hungarian method of assignment provides us with an efficient method of finding the optimal solution without having to make a-direct comparison of every solution. It works on the principle of reducing the given cost matrix to a matrix of opportunity costs.

Opportunity cost show the relative penalties associated with assigning resources to an activity as opposed to making the best or least cost assignment. If we can reduce the cost matrix to the extent of having at least one zero in each row and column, it will be possible to make optimal assignment.

The Hungarian method can be summarized in the following steps:

Step 1: Develop the Cost Table from the given Problem

If the no of rows are not equal to the no of columns and vice versa, a dummy row or dummy column must be added. The assignment cost for dummy cells are always zero.

Step 2: Find the Opportunity Cost Table

(a) Locate the smallest element in each row of the given cost table and then subtract that from each element of that row, and

(b) In the reduced matrix obtained from 2 (a) locate the smallest element in each column and then subtract that from each element. Each row and column now have at least one zero value.

Step 3: Make Assignment in the Opportunity Cost Matrix

The procedure of making assignment is as follows:

(a) Examine rows successively until a row with exactly one unmarked zero is obtained. Make an assignment single zero by making a square around it.

(b) For each zero value that becomes assigned, eliminate (Strike off) all other zeros in the same row and/ or column

(c) Repeat step 3 (a) and 3 (b) for each column also with exactly single zero value all that has not been assigned.

(d) If a row and/or column has two or more unmarked zeros and one cannot be chosen by inspection, then choose the assigned zero cell arbitrarily.

(e) Continue this process until all zeros in row column are either enclosed (Assigned) or struck off (x)

Step 4: Optimality Criterion

If the member of assigned cells is equal to the numbers of rows column then it is optimal solution. The total cost associated with this solution is obtained by adding original cost figures in the occupied cells.

If a zero cell was chosen arbitrarily in step (3), there exists an alternative optimal solution. But if no optimal solution is found, then go to step (5).

Step 5: Revise the Opportunity Cost Table

Draw a set of horizontal and vertical lines to cover all the zeros in the revised cost table obtained from step (3), by using the following procedure:

(a) For each row in which no assignment was made, mark a tick (√)

(b) Examine the marked rows. If any zero occurs in those columns, tick the respective rows that contain those assigned zeros.

(c) Repeat this process until no more rows or columns can be marked.

If a no of lines drawn is equal to the no of (or columns) the current solution is the optimal solution, otherwise go to step 6.

Step 6: Develop the New Revised Opportunity Cost Table

(a) From among the cells not covered by any line, choose the smallest element, call this value K

(b) Subtract K from every element in the cell not covered by line.

(c) Add K to very element in the cell covered by the two lines, i.e., intersection of two lines.

(d) Elements in cells covered by one line remain unchanged.

Step 7: Repeat Step 3 to 6 Unlit an Optimal Solution is Obtained

The flow chart of steps in the Hungarian method for solving an assignment problem is shown in following figures:

In a factory there are five operator \(O_1\), \(O_2\), \(O_3\), \(O_4\), \(O_5\) and five machine \(M_1\), \(M_2\), \(M_3\), \(M_4\), \(M_5\). The operating costs are given when the \(O_i\) operator operates the \(M_j\) machine \((i,j=1,2,..,5)\). But there is a restriction that \(O_3\) cannot be allowed to operate the third machine \(M_3\) and \(O_2\) cannot be allowed to operate the fifth machine \(M_5\). The cost matrix is given above. Find the optional assignment and the optimal assignment cost also.

[2018, 15M]

2) Solve the following assignment problem to maximize the sales: \(\begin{array}{|c|c|c|c|c|c|} \hline {} & {I} & {II} & {III} & {IV} & {V} \\ \hline {A} & {3} & {4} & {5} & {6} & {7} \\ \hline {B} & {4} & {15} & {13} & {7} & {6} \\ \hline {C} & {6} & {13} & {12} & {5} & {11} \\ \hline {D} & {7} & {12} & {15} & {8} & {5} \\ \hline {E} & {8} & {13} & {10} & {6} & {9} \\ \hline \end{array}\)

where \(I\), \(II\), \(III\), \(IV\) and \(V\) are Territories; \(A\), \(B\), \(C\), \(D\), \(E\) are Salesmen.

[2015, 10M]

3) Solve the minimum time assignment problem: \(\begin{array}{|c|c|c|c|c|} \hline { } & {I} & {II} & {III} & {IV} \\ \hline {A} & {3} & {12} & {5} & {4} \\ \hline {B} & {7} & {9} & {8} & {12} \\ \hline {C} & {5} & {11} & {10} & {12} \\ \hline {D} & {6} & {14} & {4} & {11} \\ \hline \end{array}\)

where \(I\), \(II\), \(III\) and \(IV\) are Machines; \(A\), \(B\), \(C\) and \(D\) are Jobs.

[2013, 15M]

4) A travelling salesman has to visit 5 cities. He wishes to start from a particular city, visit each city once and then return to his starting point. Cost of going from one city to another is given below:

You are required to find the least cost route.

[2004, 15M]

5) Find the optimal solution for the assignment problem with the following cost matrix: \(\begin{bmatrix}{6} & {1} & {9} & {11} & {12} \\ {2} & {8} & {17} & {2} & {5} \\ {11} & {8} & {3} & {3} & {3} \\ {4} & {10} & {8} & {6} & {11} \\ {8} & {10} & {11} & {5} & {13}\end{bmatrix}\)

Indicate clearly the rule you apply to arrive at the complete assignment.

[2003, 15M]

Table of Contents

Collaboration, information literacy, writing process, problem definition assignment.

A Problem Definition is a genre of writing that can also function as a mode of discourse

Assignment Guidelines

• Review the the deliverables for The Consulting Simulation. Skim the Problem Definition overview.
• Brainstorm about potential problems you would like to learn more about. Try to think about this for a bit of time, a day or two. Here you may find it useful to talk over ideas with friends. As always, it’s helpful to read what you can about topics of interest. Do a bit of preliminary research to develop more robust ideas.
• Post your problem definition to your new team members.

Discussion Post Guidelines

• Project Manager or Product Manager
• Research Director
• Senior Editor
• Design Director
• Quality Assurance
• Who experiences this problem?
• How do they experience the problem?
• Who are the stakeholders?
• What is the history of this problem?
• What unsatisfactory situation do you propose to investigate?
• What specifically is unsatisfactory about it? 
• Supply any data needed to prove that this is indeed a significant problem. Cite your sources.
• Note that for this discussion post you are not expected to have solutions to the problem. For now it’s enough that you have identified a real problem–a problem for specific people in specific contexts.

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Guest Essay

The Problem With Saying ‘Sex Assigned at Birth’

By Alex Byrne and Carole K. Hooven

Mr. Byrne is a philosopher and the author of “Trouble With Gender: Sex Facts, Gender Fictions.” Ms. Hooven is an evolutionary biologist and the author of “T: The Story of Testosterone, the Hormone That Dominates and Divides Us.”

As you may have noticed, “sex” is out, and “sex assigned at birth” is in. Instead of asking for a person’s sex, some medical and camp forms these days ask for “sex assigned at birth” or “assigned sex” (often in addition to gender identity). The American Medical Association and the American Psychological Association endorse this terminology; its use has also exploded in academic articles. The Cleveland Clinic’s online glossary of diseases and conditions tells us that the “inability to achieve or maintain an erection” is a symptom of sexual dysfunction, not in “males,” but in “people assigned male at birth.”

This trend began around a decade ago, part of an increasing emphasis in society on emotional comfort and insulation from offense — what some have called “ safetyism .” “Sex” is now often seen as a biased or insensitive word because it may fail to reflect how people identify themselves. One reason for the adoption of “assigned sex,” therefore, is that it supplies respectful euphemisms, softening what to some nonbinary and transgender people, among others, can feel like a harsh biological reality. Saying that someone was “assigned female at birth” is taken to be an indirect and more polite way of communicating that the person is biologically female. The terminology can also function to signal solidarity with trans and nonbinary people, as well as convey the radical idea that our traditional understanding of sex is outdated.

The shift to “sex assigned at birth” may be well intentioned, but it is not progress. We are not against politeness or expressions of solidarity, but “sex assigned at birth” can confuse people and creates doubt about a biological fact when there shouldn’t be any. Nor is the phrase called for because our traditional understanding of sex needs correcting — it doesn’t.

This matters because sex matters. Sex is a fundamental biological feature with significant consequences for our species, so there are costs to encouraging misconceptions about it.

Sex matters for health, safety and social policy and interacts in complicated ways with culture. Women are nearly twice as likely as men to experience harmful side effects from drugs, a problem that may be ameliorated by reducing drug doses for females. Males, meanwhile, are more likely to die from Covid-19 and cancer, and commit the vast majority of homicides and sexual assaults . We aren’t suggesting that “assigned sex” will increase the death toll. However, terminology about important matters should be as clear as possible.

More generally, the interaction between sex and human culture is crucial to understanding psychological and physical differences between boys and girls, men and women. We cannot have such understanding unless we know what sex is, which means having the linguistic tools necessary to discuss it. The Associated Press cautions journalists that describing women as “female” may be objectionable because “it can be seen as emphasizing biology,” but sometimes biology is highly relevant. The heated debate about transgender women participating in female sports is an example ; whatever view one takes on the matter, biologically driven athletic differences between the sexes are real.

When influential organizations and individuals promote “sex assigned at birth,” they are encouraging a culture in which citizens can be shamed for using words like “sex,” “male” and “female” that are familiar to everyone in society, as well as necessary to discuss the implications of sex. This is not the usual kind of censoriousness, which discourages the public endorsement of certain opinions. It is more subtle, repressing the very vocabulary needed to discuss the opinions in the first place.

A proponent of the new language may object, arguing that sex is not being avoided, but merely addressed and described with greater empathy. The introduction of euphemisms to ease uncomfortable associations with old words happens all the time — for instance “plus sized” as a replacement for “overweight.” Admittedly, the effects may be short-lived , because euphemisms themselves often become offensive, and indeed “larger-bodied” is now often preferred to “plus sized.” But what’s the harm? No one gets confused, and the euphemisms allow us to express extra sensitivity. Some see “sex assigned at birth” in the same positive light: It’s a way of talking about sex that is gender-affirming and inclusive .

The problem is that “sex assigned at birth”— unlike “larger-bodied”— is very misleading. Saying that someone was “assigned female at birth” suggests that the person’s sex is at best a matter of educated guesswork. “Assigned” can connote arbitrariness — as in “assigned classroom seating” — and so “sex assigned at birth” can also suggest that there is no objective reality behind “male” and “female,” no biological categories to which the words refer.

Contrary to what we might assume, avoiding “sex” doesn’t serve the cause of inclusivity: not speaking plainly about males and females is patronizing. We sometimes sugarcoat the biological facts for children, but competent adults deserve straight talk. Nor are circumlocutions needed to secure personal protections and rights, including transgender rights. In the Supreme Court’s Bostock v. Clayton County decision in 2020, which outlawed workplace discrimination against gay and transgender people, Justice Neil Gorsuch used “sex,” not “sex assigned at birth.”

A more radical proponent of “assigned sex” will object that the very idea of sex as a biological fact is suspect. According to this view — associated with the French philosopher Michel Foucault and, more recently, the American philosopher Judith Butler — sex is somehow a cultural production, the result of labeling babies male or female. “Sex assigned at birth” should therefore be preferred over “sex,” not because it is more polite, but because it is more accurate.

This position tacitly assumes that humans are exempt from the natural order. If only! Alas, we are animals. Sexed organisms were present on Earth at least a billion years ago, and males and females would have been around even if humans had never evolved. Sex is not in any sense the result of linguistic ceremonies in the delivery room or other cultural practices. Lonesome George, the long-lived Galápagos giant tortoise , was male. He was not assigned male at birth — or rather, in George’s case, at hatching. A baby abandoned at birth may not have been assigned male or female by anyone, yet the baby still has a sex. Despite the confusion sown by some scholars, we can be confident that the sex binary is not a human invention.

Another downside of “assigned sex” is that it biases the conversation away from established biological facts and infuses it with a sociopolitical agenda, which only serves to intensify social and political divisions. We need shared language that can help us clearly state opinions and develop the best policies on medical, social and legal issues. That shared language is the starting point for mutual understanding and democratic deliberation, even if strong disagreement remains.

What can be done? The ascendance of “sex assigned at birth” is not an example of unhurried and organic linguistic change. As recently as 2012 The New York Times reported on the new fashion for gender-reveal parties, “during which expectant parents share the moment they discover their baby’s sex.” In the intervening decade, sex has gone from being “discovered” to “assigned” because so many authorities insisted on the new usage. In the face of organic change, resistance is usually futile. Fortunately, a trend that is imposed top-down is often easier to reverse.

Admittedly, no one individual, or even a small group, can turn the lumbering ship of English around. But if professional organizations change their style guides and glossaries, we can expect that their members will largely follow suit. And organizations in turn respond to lobbying from their members. Journalists, medical professionals, academics and others have the collective power to restore language that more faithfully reflects reality. We will have to wait for them to do that.

Meanwhile, we can each apply Strunk and White’s famous advice in “The Elements of Style” to “sex assigned at birth”: omit needless words.

Alex Byrne is a professor of philosophy at M.I.T. and the author of “Trouble With Gender: Sex Facts, Gender Fictions.” Carole K. Hooven is an evolutionary biologist, a nonresident senior fellow at the American Enterprise Institute, an associate in the Harvard psychology department, and the author of “T: The Story of Testosterone, the Hormone That Dominates and Divides Us.”

The Times is committed to publishing a diversity of letters to the editor. We’d like to hear what you think about this or any of our articles. Here are some tips . And here’s our email: [email protected] .

Follow The New York Times Opinion section on Facebook , Instagram , TikTok , WhatsApp , X and Threads .

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For Success with AI, Bring Everyone On Board

• David De Cremer

AI is intimidating employees. As machines perform intellectually demanding tasks that were previously reserved for human workers, people feel more excluded and less necessary than ever. The problem is only getting worse. Eighty percent of organizations say their main technological goal is hyperautomation—or the complete end-to-end automation of as many business processes as possible. Executives often pursue that goal without feedback from employees—the people whose jobs, and lives, will feel the greatest impact from automation.

In this article the author examines what keeps leaders from involving rank-and-file employees in AI projects, how they should model inclusive behavior, and what organizations must do to develop employee-inclusive AI practices. Those practices will make companies more likely to improve long-term performance—and to keep their employees happy, productive, and engaged.

When you include rank-and-file employees, you’ll improve your overall performance.

Idea in Brief

The problem.

When employees are excluded from the adoption process, they become averse to working with AI, never develop trust in its capabilities, and resist even the positive changes that come from using it.

Eighty percent of organizations say their main technological goal is hyperautomation—the end-to-end automation of as many business processes as possible. Executives often pursue that goal without feedback from employees—the people whose jobs and lives will be most affected by achieving it.

The Solution

AI transformation requires constant human-to-human connection across business disciplines. Including rank-and-file employees in AI projects will make your long-term performance more likely to improve—and your employees more likely to be happy, productive, and engaged.

AI is intimidating your employees. As machines increasingly perform intellectually demanding tasks that were previously reserved for humans, your people feel more excluded and less necessary than ever. And the problem is getting worse. According to the market research company Vanson Bourne, 80% of organizations say that their main technological goal is hyperautomation—the end-to-end automation of as many business processes as possible. Executives have a tendency to pursue that goal without any feedback from their employees—the people whose jobs, and lives, will be most affected by achieving it. But my decades of research into the enterprise adoption of emerging technologies has proved one thing time and again: The savviest leaders prioritize participation by the rank and file throughout the adoption process.

• David De Cremer is a professor of management and technology at Northeastern University and the Dunton Family Dean of its D’Amore-McKim School of Business. His website is daviddecremer.com .

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Taking On the Short-Term Rental Problem in Santa Barbara

Enforcement Program Finds 1,147 Illegal Vacation Rentals in City, 151 Cases in First Year

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Last year, the City of Santa Barbara decided to seriously crack down on short-term vacation rentals, setting aside $1.175 million toward a one-year short-term rental enforcement program aimed at finding illegal operators in the city and collecting back taxes, fees, interest, and penalties. Since launching in August 2023, the enforcement program has handled 151 cases, and collected more than a half a million dollars, and now the city is asking for an extension on the temporary program to continue to chip away at the estimated 1,147 unlawful vacation rentals still operating in the city.

On Tuesday, April 16, the Santa Barbara City Council will hear the first update on the pilot program, and according to the report prepared by Assistant City Attorney Denny Wei and City Attorney’s Office Investigator William Alva, the city has followed up on 106 citizen complaints and found 46 more properties through proactive enforcement.

In total, there are 57 cases still in progress, while 45 were brought into compliance voluntarily, 18 were determined to be unfounded, and six were found to be outside city limits. Twenty-five cases are currently going through the court system. So far, the city has collected $269,272 from properties in the coastal zone and $233,607 for properties in the inland area.

According to the staff report, the program has been a success in its first year, spending only $99,083 out of the $1.175 million budget by “being mindful of spending taxpayer money” and taking “all possible steps to use the allocated funds judiciously.”

But the pilot program is set to expire at the end of July, and with more than a thousand illegal vacation rentals still operating in Santa Barbara, the City Attorney’s Office is asking for an extension to keep the program going forward. No additional funds are being requested.

The report also details how the cases were handled. The City Attorney’s Office employed three special investigators who worked to follow up on complaints, gather evidence, interview witnesses, and visit the properties, with the intention of getting operators to voluntarily come into compliance and pay any applicable back taxes, fees, and penalties. Property owners that did not come into compliance were then referred to the city prosecutor to begin the court process. Later, the city brought in a financial analyst to assist with the program.

While the program has brought in more than $500,000 in penalties and fees, city staff said that current enforcement of unpermitted short-term rentals in the coastal zone is limited to tax violations unless there were more serious issues regarding “tenant behavior or other nuisance-like complaints.”

In order to allow the city to go after the estimated 1,147 illegal short-term rentals that are still out there, Councilmembers Mike Jordan and Eric Friedman also requested to agendize a second item on the same day to allow the council to discuss potential changes to the city’s short-term rental regulations.

“Short-term vacation rentals have grown in popularity and numbers across the city,” the two councilmembers wrote in a memo sent out earlier this week. “Both regulated (zoned and registered) and unregulated (not zoned and/or not registered) continue to re-purpose the city’s housing stock away from permanent housing as they are converted to short term visitor uses only.”

Not only do short-term rentals ”intrude into existing single family residential” areas, but they can cause “significant tensions,” the memo continues, “as residents have observed neighborhood housing turned into transient visitor lodging,” along with reports of weekly turnover, lack of local oversight or management, increased parking and noise, and other impacts.

The main goal of the discussion will be to assess what types of changes could be made to the current regulations to ensure that the city continues “to protect and develop all types of housing stock and take steps to remedy the loss of such,” according to Jordan and Friedman’s statement.

This could include addressing zoning codes, which currently allow for single family residential, hotels, and short-term rentals to exist in the same areas.

“In order to clarify short-term vacation rental regulations and address conflicts with residential uses, while protecting erosion to city housing stock,” the memo states, “we believe that revisions to existing ordinances is warranted and appropriate.”

City Council will receive the update on the pilot program and discuss potential changes to short-term rental regulations at City Hall on Tuesday.

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David Chang’s Momofuku draws heat over its ‘chile crunch’ trademark

Celebrity chef david chang’s company is sending cease-and-desist letters to other companies about a term some critics say momofuku shouldn’t own.

In March, when Momofuku’s lawyers sent cease-and-desist letters to stop manufacturers from using the name of celebrity chef David Chang’s popular chili crunch condiment, they were doing what trademark lawyers always do: trying to protect a company’s investment from competitors.

But the letters soon became public in a ballooning number of news stories, opinion pieces and social media posts, many of which criticized Momofuku and Chang, its Korean American founder, for trying to seize control of a burgeoning, multimillion-dollar market.

Momofuku and Chang were accused of bullying mom-and-pop manufacturers that have ancestral connections to a spicy-oily-crispy condiment, often known as chili crisp or chili oil and with varying spellings of chili/chile, which is popular in China and other Asian countries. The company and its founder were denounced for trying to stifle competition with a trademark that many viewed as not distinctive enough to earn legal protection. Momofuku’s trademark has been repeatedly criticized as “merely descriptive,” but one trademark lawyer also told The Washington Post that it’s too late to challenge the trademark on those grounds.

James Park, a recipe developer and author who wrote “ Chili Crisp: 50+ Recipes to Satisfy Your Spicy, Crunchy, Garlicky Cravings ,” pointed out in an interview that Jing Gao, founder of Fly by Jing , tried to trademark “Sichuan chili crisp” in 2019. The application was denied because it was merely descriptive, which, according to the U.S. Patent and Trademark Office , “describes an ingredient, quality, characteristic, function, feature, purpose, or use of the specified goods or services.” That makes Park wonder why Momofuku owns the trademark for a term that strikes him as having the same issue.

“This feels like the same thing as if they were going after the term ‘hot sauce’ or ‘ketchup’ or ‘mustard,’” Park said.

For days as the controversy unfolded, no one with Momofuku — including Chang and chief executive Marguerite Mariscal — responded to reporters seeking comment. Neither Chang nor Mariscal would comment on the record for this story, but the company provided a statement and background information to tell its side.

On a July 2020 episode of his podcast , just ahead of Momofuku’s chili crunch debut, Chang talked about the long, painstaking process for creating the condiment. The recipe pulled from many sources, he said: not just Laoganma, the beloved Chinese company behind a variety of chili sauces and oils , but also Mexican salsa macha and salsa seca. Chang even name-checked restaurants from his youth in Northern Virginia.

“We’re not going to do anything that everyone else is doing,” Chang said on the podcast. “It’s got to be our story, and our story is not our story. Our story is going to be a blending of all these other stories that we’re trying to do differently.”

As such, Momofuku argues in its statement, the company wanted to create a name that “we could own and intentionally picked ‘Chili Crunch’ to further differentiate it from the broader chili crisp category, reflecting the uniqueness of Chili Crunch.” To the company, the name was an attempt to create a brand as unique as, say, the cereal brands Cap’n Crunch and Catalina Crunch.

Momofuku ran into trouble soon after the rollout. A company in Denver, Chile Colonial, already owned the trademark to “chile crunch,” which some experts say gave the company common-law protections for the use of the alternative spelling using “chili.” Colonial sent a cease-and-desist letter to Momofuku, but rather than fight it, the company worked with Colonial to purchase the trademark. Momofuku attained the trademark last year, according to the patent office.

As Momofuku was racking up sales with its chili crunch, moving from what the company said was a niche product to a leader in the chili crisp category, at least two other manufacturers were working to change the name of their condiments to “chili crunch” or a variation of it.

Based in Bellevue, Wash., MìLà had been selling a chili crisp for years, even when the direct-to-consumer company was operating under its previous name, Xiao Chi Jie, said Caleb Wang, a second-generation Chinese American who spent part of his childhood in Shanghai. But about two years ago, Wang and his wife and MìLà co-founder, Jennifer Liao, needed to find a solution for their leaky chili crisp jars. While working on that, the couple decided to reformulate their recipe.

“For chili crisp, we decided to put in more garlic, crunchiness,” Wang said. “And we’re like: ‘Oh, this is great. It’s no longer crispy. It actually feels crunchy.’ We had to call it chili crunch.”

Michelle Tew, a Malaysian native who moved to New York to study mathematics and philosophy at Columbia University, launched her own company, Homiah, in 2022. Her sambal chili crunch, based on a family recipe that dates back generations, made its debut last year and is now poised to appear on shelves in Whole Foods and Target later this year. (Whole Foods is owned by Amazon, whose founder, Jeff Bezos, owns The Post.) Tew struggled to come up with a name that would connect with American consumers. She explored a number of possibilities — sambal, crispy sambal, crispy sambal chili — before settling on sambal chili crunch.

“I could have maybe chosen ‘chili crisp,’” Tew told The Post. “But to me, ‘crunch’ was more descriptive of what the product is, because it didn’t have a lot of oil in it.”

The growing popularity of Momofuku’s product was not why either MìLà or Homiah landed on the term “chili crunch,” say their founders, who both also say that the term is not distinctive enough for a trademark.

“If you spend a lot of time developing ... something that’s very differentiated, where people have really good recognition of it, I think you should have the right to protect it,” Wang said. “And my personal opinion is just the use of ‘chili crunch’ is not that.”

Tew with Homiah was even more pointed: When a company trademarks a name that’s “generic or descriptive, ... the only thing that can result is monopoly power, which is directly anti-competitive,” she said.

In Homiah’s letter to Momofuku, its lawyer wrote: “This isn’t Momofuku’s first attempt to register generic and descriptive terms for Asian foods. Your client’s attempt to own SSÄM SAUCE (U.S. Trademark Application Serial No. 88881122) in 2021 was ruled generic, and demonstrates a pattern to attempt to own generic Asian cultural products to anticompetitive effect.”

The fight between Momofuku and the companies targeted by its cease-and-desist letters has, more or less, devolved into an argument over whether the patent office should have granted the trademark in the first place. (Last month, Momofuku also applied for a trademark for “chili crunch ” even though the company believes that its current trademark using the “chile” spelling covers both terms.) Before Momofuku acquired the “chile crunch” trademark last year, Chile Colonial had owned it since 2015, according to patent office documents. Colonial even once successfully defended its trademark.

In 2020, after Trader Joe’s debuted its chili onion crunch , the grocery chain sought to strip Chile Colonial’s trademark, arguing to the patent office that the phrase was merely descriptive and that it “is or has become generic.” But in 2021, the office suspended their dispute pending the outcome of a lawsuit the grocer had filed against the Denver condiment maker. The companies settled that case in 2021, and although the terms were not made public, Trader Joe’s now sells “ crunchy chili onion .”

Copyright and trademark attorney Nicholas Wells said he thinks Chang shouldn’t have been awarded the exclusive right in the first place because of the “merely descriptive” issue. The cease-and-desist letters, he said, are a way of keeping smaller companies who might attempt to challenge it in check by raising the specter of litigation they probably can’t afford.

A trademark can be challenged through the trademark office or via lawsuit, Wells says, with the latter route being particularly expensive. For scrappy companies without big budgets for legal fees, it might not be worth it. “They’re more likely to say, ‘We’re going to just fold, we can’t deal with this,’” he said. “And so, by virtue of the sledgehammer approach, he ends up owning it.”

Duke University law professor Jennifer Jenkins was skeptical that, on its own, Momofuku’s more recent trademark application for “chili crunch” would succeed. “There is no secondary meaning connecting it to a single producer,” Jenkins said. “When I see ‘chili crunch’ on a jar, it tells me what’s in the jar, not who produced it.” But, she added, because the company’s existing “chile crunch” trademark has been federally registered for more than five years and the owner has filed the necessary paperwork, it “can no longer be challenged on the ground that it’s merely descriptive.” According to the patent office, five years’ use is one of the pieces of evidence that can be used to prove a descriptive mark has “acquired distinctiveness.”

Before the Momofuku product debuted in 2020, the term “chili crunch” — at least relating to a condiment — did not often appear in news stories, according to a search in the database Nexis. Later that year, the phrase began popping up, in reference to the Momofuku product and more generally.

A 2020 Associated Press story headlined, “A guide to all the new condiments lining grocery shelves,” included “chili crunch or chili oil,” which it said was great for drizzling. “Many world cuisines have condiments that are essentially some sort of hot chili paste with a crunchy texture, and lately they’ve been having a moment,” the story read.

By 2021, food stories often used the phrase. A story in the Washington City Paper described a mandu served at Korean hot spot Anju topped with “pickled long chilies and chili crunch powered by gochugaru.” The Florida Times-Union enthused about a pasta dish at Jacksonville’s Town Hall restaurant “with cauliflower puree, pistachios and chili crunch.” A Seattle Times tofu stir-fry recipe called for chili crunch as an ingredient. That year, the Houston Chronicle conducted a test of three published recipes, so home cooks could craft their own version of the condiment it described as “chile crisp, or sometimes called chili crunch.”

As the term “chili crunch” enters into the mainstream, Momofuku noted that it’s not necessarily concerned with the true mom-and-pop shops that enter the chili crisp business. Yet it’s not always easy to tell the Davids from the Goliaths: Momofuku Goods, the consumer goods side of the company, reportedly had $50 million in sales last year and raised around $29 million from two separate funding rounds, while Wang said that MìLà has raised around $30 million in capital in 2022 and 2023. What’s more, actor Simu Liu is the chief content officer for MìLà.

Momofuku has indicated that it’s more concerned about the larger companies that may try to create a product to capitalize on the growing marketplace, the way Trader Joe’s did with its chili onion crunch. Which is why, Momofuku said, it had to defend its trademark, not to pick on small immigrant businesses.

“Setting this precedent is important to defend brands making innovative strides in new categories from having their work copied by much larger players,” the company said in its statement. “Failure to defend our trademark against any size company would leave us without recourse against these larger players who often try to enter categories on the rise. Our intent has never been to stifle innovation in a category that we care deeply about.”

“As we’ve said in our engagement with these companies,” the company added in its statement, “our goal is and has been to find an amicable resolution — not to harm the competition that makes this category so vibrant. And that is what we’re trying to do.”

History is littered with examples of terms that were once trademarked, including aspirin, escalator, cellophane and laundromat. But some companies have defended their rights even as their products became shorthand for an entire category, including Xerox, Band-Aid and Velcro.

Momofuku’s trademark moves have had at least one unexpected response. Fly by Jing decided to reopen its application for Sichuan chili crisp, which had been rejected years earlier, and also applied for Chengdu crunch, the name of another one of its condiments. The company filed the applications on April 3 .

“Our lawyers feared that since there was now a precedent set with the ‘chile crunch’ trademark, [Momofuku] or another party could also apply for ‘chili crisp’ as well in order to corner the market and eliminate competition,” said Gao, the founder, in a direct message on Instagram.

But on Saturday, when a reporter contacted Jing, the company changed course. It said it now believes “there’s been enough awareness raised about the descriptive nature of the term, that the USPTO will reconsider the chili/chile crunch trademarks, and we felt comfortable with filing a request to abandon the application for our product’s name, which we have already done as of today.”