assignment problem r code

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Transportation and assignment problems with r.

In the previous post “ Linear Programming with R ” we examined the approach to solve general linear programming problems with “Rglpk” and “lpSolve” packages. Today, let’s explore “lpSolve” package in depth with two specific problems of linear programming: transportation and assignment.

1. Transportation problem

Code & Output:

The solution is shown as lptrans$solution and the total cost is 20500 as lptrans$objval.

2. Assignment problem

Similarly, the solution and the total cost are shown as lpassign$solution and lpassign$objval respectively.

This article was really helpful, but I am facing issue while solving unbalanced transportation problem when there is excess demand. Could you please guide me on what has to be done in this case.

Hello sir, this article was really helpful. But, I am facing issue while solving unbalanced transportation problem when there is excess demand, it gives solution as no feasible solution. works perfectly fine for balanced and excess supply problems. Could you please guide me on why this issue is occurring and a possible solution for the same. Thank you.

Assigning students to courses

Introduction.

In this article we will look at assignment problems .

As a real world example problem we would like to match a group of students to a set of courses with the following constraints:

• Each course has a capacity
• Every student needs to be assigned to exactly one course.
• All students have stated individual preferences on a scale from 1 to 3, where 3 is the most favorable course.

We have \(n\) students:

And \(m\) courses with equal capacity. The capacity can vary among courses though.

In addition, each student has three preferences. To model this we have a function that gives us three courses for each student. The first component has perference 1, second 2, and third 3:

The last component we need is a weight functions to make the model formulation easier. This function gives us the preference weighting for a course and student pair.

Some examples:

Let’s take a look at our random preferences. We plot the number of votes for each course grouped by the preference (1, 2, 3).

The idea is to introduce a binary variable \(x_{i, j}\) that is \(1\) if student \(i\) is matched to course \(j\) . As an objective we will try to satisfy preferences according to their weight. So assigning a student to a course with preference 3 gives 3 points and so forth. The model assumes, that the total capacity of the courses is enough for all students.

Here it is in mathematical notation:

\[ \begin{equation*} \begin{array}{ll@{}ll} \text{max} & \displaystyle\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}weight_{i,j} \cdot x_{i, j} & &\\ \text{subject to}& \displaystyle\sum\limits_{i=1}^{n} x_{i, j} \leq capacity_j, & j=1 ,\ldots, m&\\ & \displaystyle\sum\limits_{j=1}^{m} x_{i, j} = 1, & i=1 ,\ldots, n&\\ & x_{i,j} \in \{0,1\}, &i=1 ,\ldots, n, & j=1 ,\ldots, m \end{array} \end{equation*} \]

Or directly in R:

Solve the model

We will use glpk to solve the above model.

We solved the problem with an objective value of 118.

38 students got their top preference. 2 students were assigned to their second choice and 0 students got their least preferable course.

The course assignment now looks like this:

Using lpsolve from R

We will not discuss the specifics of R here but instead refer the reader to the R website. Also see An Introduction to R

R and lpsolve

lpsolve is callable from R via an extension or module. As such, it looks like lpsolve is fully integrated with R. Matrices can directly be transferred between R and lpsolve in both directions. The complete interface is written in C so it has maximum performance.

There are currently two R packages based on lp_solve. Both packages are available from CRAN .

The lpSolve R package is the first implementation of an interface of lpsolve to R. It provides high-level functions for solving general linear/integer problems, assignment problems and transportation problems. The following link contains the version of the driver: lpSolve: Interface to Lp_solve v. 5.5 to solve linear/integer programs . It does not contain the lpsolve API. Only the higher level calls. Documentation for this interface can be found on: Interface to Lp_solve v. 5.5 to solve linear/integer programs This driver is written and maintained by Sam Buttrey .

The lpSolveAPI R package is a second implementation of an interface of lpsolve to R. It provides an R API mirroring the lp_solve C API and hence provides a great deal more functionality but has a steeper learning curve. The R interface to lpsolve contains its own documentation. See An R interface to the lp_solve library for the driver. This driver is written and maintained by Kjell Konis .

Installing the lpsolve driver in R

How to install the driver depends on the environment.

In the RGui menu, there is a menu item 'Packages'. From there a package can be installed from a CRAN mirror or from a local zip file.

R command line

Packages can also be installed from the R command line. This is a more general approach that will work under all environments. Installing the package takes a single command:

> install.packages("lpSolve", repos = "http://r-forge.r-project.org") --> The lpSolve R package:

The > shown before each R command is the R prompt. Only the text after > must be entered.

Loading the lpsolve driver in R

Getting help.

Documentation is provided for each function in the lpSolve package using R's built-in help system. For example, the command

Building and Solving Linear Programs Using the lpSolve R Package

This implementation provides the functions lp , lp.assign , lp.object , lp.transport and print.lp . These functions allow a linear program (and transport and assignment problems) to be defined and solved using a single command.

For more information enter:

See also Interface to Lp_solve v. 5.5 to solve linear/integer programs

Building and Solving Linear Programs Using the lpSolveAPI R Package

This implementation provides an API for building and solving linear programs that mimics the lp_solve C API. This approach allows much greater flexibility but also has a few caveats. The most important is that the lpSolve linear program model objects created by make.lp and read.lp are not actually R objects but external pointers to lp_solve 'lprec' structures. R does not know how to deal with these structures. In particular, R cannot duplicate them. Thus one must never assign an existing lpSolve linear program model object in R code.

To load the library, enter:

Consider the following example. First we create an empty model x.

And finally, take a look at y.

The safest way to use the lpSolve API is inside an R function - do not return the lpSolve linear program model object.

Learning by Example

Note that there are some commands that return an answer. For the accessor functions (generally named get.*) the output should be clear. For other functions (e.g., solve ), the interpretation of the returned value is described in the documentation. Since solve is generic in R, use the command

Cleaning up

To free up resources and memory, the R command rm() must be used. For example:

See also Using lpsolve from MATLAB , Using lpsolve from O-Matrix , Using lpsolve from Sysquake , Using lpsolve from Octave , Using lpsolve from FreeMat , Using lpsolve from Euler , Using lpsolve from Python , Using lpsolve from Sage , Using lpsolve from PHP , Using lpsolve from Scilab Using lpsolve from Microsoft Solver Foundation

lp.assign: Integer Programming for the Assignment Problem

Description.

Interface to lp_solve linear/integer programming system specifically for solving assignment problems

An lp object. See documentation for details. The constraints are assumed (each row adds to 1, each column adds to 1, and no others) and are not returned.

Matrix of costs: the ij-th element is the cost of assigning source i to destination j.

Character vector, length 1, containing either "min" (the default) or "max"

Numeric: presolve? Default 0 (no); any non-zero value means "yes." Currently ignored.

Numeric: compute sensitivity? Default 0 (no); any non-zero value means "yes." In that case presolving is attempted.

Sam Buttrey, [email protected]

This is a particular integer programming problem. All the decision variables are assumed to be integers; each row has the constraint that its entries must add up to 1 (so that there is one 1 and the remaining entries are 0) and each column has the same constraint. This is assumed to be a minimization problem.

lp , lp.transport

• Linear programming

Solving linear transport problem with lp.transport in R, using lpSolve

• Linnart Felkl

The transportation problem is one of the classical problems teached in linear programming classes. The problem, put simply, states that a given set of customers with a specified demand must be satisfied by another set of supplier with certain capacities (“supply”). For a detailed explanation of the transportation problem you can, e.g. read this:  https://econweb.ucsd.edu/~jsobel/172aw02/notes8.pdf .

The lpSolve package available in R can be used for modelling and solving the transportation problem.

I will show how to do this. I define a problem in which 3 suppliers seek to satisfy 4 customers. The suppliers have capacities 100, 300, and 400, respectively. The customers have demand 100, 100, 200, and 400, respectively. Furthermore, the cost for supplying customer i by supplier j is defined for every possible combination and stated in a cost matrix.

Using this information we can model and solve the transportation problem (deciding which demand to fulfill by which supplier) with the lpSolve package in R.

First, I prepare the modeling-part:

Then, I solve the problem:

Let us review the “optimal” costs:

Let us review the optimal solution of the transportation problem (i.e., the optimal material flows to this problem):

The assignment problem is another classical problem, and you can see how I solved it in R here: Cost minimal production scheduling – solving the assignment problem with lpSolve, using lp.assign.

Data scientist focusing on simulation, optimization and modeling in R, SQL, VBA and Python

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hi. I am trying to “go to the next level”. I want to look deeper than this and say, “OK, truck A costs me $100 and takes 3 weeks, but flight A costs me $200 and takes just 1 week to move the good” For me, i am still trying to move things from A to B, but now i am expanding it to try and look at the total cost of holding the inventory, reduction in cash flow and the cost of delivery. I want to “optimise” my transportation so that maybe 90% of my forecast is shipped via truck, then i can “top up” with the flight if demand requires. Any suggestions how to approach this issue or where to turn for some guidance? Thanks

thanks for your comment.

You can try to formulate your problem as a linear optimization problem, from scratch. For example, the cost of holding inventory can be derived from the travel time, and the travel time depends on transportation distance (one variable) and transportation mode (binary or integer variable).

The objective of covering 90% of your forecast by truck could be formulated as a constraint, or maybe a better approach would be to define two objective functions, and to apply multi-goal linear programming.

If you provide some more info I might very well try to model it and post it, maybe in a simplified version!

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Introduction to probability and statistics, r: information, tutorials, and sample code.

R is an industrial strength open-source statistical package. We’ll use it for simulation, computation, and visualization. Don’t worry if you are not familiar with R, we will provide plenty of tutorials and guidance in its use.

R takes 2 minutes to install. Here are the steps:

• Go to the main page of the R project . Click the ‘CRAN’ link on the left. Then choose any of the mirror sites.
• On a Mac, download the latest version that works for your machine. You double-click to install.
• On Windows, just follow the instructions.
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We will also be using RStudio which provides an integrated environment for working in R. It is also a 2-minute install. Here are the steps.

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lpSolve Interface to 'Lp_solve' v. 5.5 to Solve Linear/Integer Programs

• lp: Linear and Integer Programming
• lp.assign: Integer Programming for the Assignment Problem
• lp.object: LP (linear programming) object
• lp.transport: Integer Programming for the Transportation Problem
• make.q8: Generate sparse constraint matrix for 8-queens problem
• print.lp: Print an lp object
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lp.assign : Integer Programming for the Assignment Problem In lpSolve: Interface to 'Lp_solve' v. 5.5 to Solve Linear/Integer Programs

Integer programming for the assignment problem, description.

Interface to lp_solve linear/integer programming system specifically for solving assignment problems

This is a particular integer programming problem. All the decision variables are assumed to be integers; each row has the constraint that its entries must add up to 1 (so that there is one 1 and the remaining entries are 0) and each column has the same constraint. This is assumed to be a minimization problem.

An lp object. See documentation for details. The constraints are assumed (each row adds to 1, each column adds to 1, and no others) and are not returned.

Sam Buttrey, [email protected]

lp , lp.transport

Related to lp.assign in lpSolve ...

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Algorithms for solving the Traffic Assignment Problem (TAP).

Description.

Estimation of the User Equilibrium (UE)

The most well-known assumptions in traffic assignment models are the ones following Wardrop's first principle. Traffic assignment models are used to estimate the traffic flows on a network. These models take as input a matrix of flows that indicate the volume of traffic between origin and destination (O-D) pairs. Unlike All-or-Nothing assignment (see get_aon ), edge congestion is modeled through the Volume Decay Function (VDF) . The Volume Decay Function used is the most popular in literature, from the Bureau of Public Roads :

t = t0 * (1 + a * (V/C)^b) with t = actual travel time (minutes), t0 = free-flow travel time (minutes), a = alpha parameter (unitless), b = beta parameter (unitless), V = volume or flow (veh/hour) C = edge capacity (veh/hour)

Traffic Assignment Problem is a convex problem and solving algorithms can be divided into two categories :

link-based : Method of Successive Average ( msa ) and Frank-Wolfe variants (normal : fw , conjugate : cfw and bi-conjugate : bfw ). These algorithms uses the descent direction given by AON assignment at each iteration, all links are updated at the same time.

bush-based : Algorithm-B ( dial ) The problem is decomposed into sub-problems, corresponding to each origin of the OD matrix, that operate on acyclic sub-networks of the original transportation network, called bushes. Link flows are shifted from the longest path to the shortest path recursively within each bush using Newton method.

Link-based algorithms are historically the first algorithms developed for solving the traffic assignment problem. It require low memory and are known to tail in the vicinity of the optimum and usually cannot be used to achieve highly precise solutions. Algorithm B is more recent, and is better suited for achieve the highest precise solution. However, it require more memory and can be time-consuming according the network size and OD matrix size. In cppRouting , the implementation of algorithm-B allow "batching", i.e. bushes are temporarily stored on disk if memory limit, defined by the user, is exceeded. Please see the package website for practical example and deeper explanations about algorithms. ( https://github.com/vlarmet/cppRouting/blob/master/README.md )

Convergence criterion can be set by the user using max_gap argument, it is the relative gap which can be written as : abs(TSTT/SPTT - 1) with TSTT (Total System Travel Time) = sum(flow * cost), SPTT (Shortest Path Travel Time) = sum(aon * cost)

Especially for link-based algorithms (msa, *fw), the larger part of computation time rely on AON assignment. So, choosing the right AON algorithm is crucial for fast execution time. Contracting the network on-the-fly before AON computing can be faster for large network and/or large OD matrix.

AON algorithms are :

bi : bidirectional Dijkstra algorithm

nba : bidirectional A* algorithm, nodes coordinates and constant parameter are needed

d : Dijkstra algorithm

cbi : contraction hierarchies + bidirectional search

cphast : contraction hierarchies + phast algorithm

These AON algorithm can be decomposed into two families, depending the sparsity of origin-destination matrix :

recursive pairwise : bi , nba and cbi . Optimal for high sparsity. One-to-one algorithm is called N times, with N being the length of from.

recursive one-to-many : d and cphast . Optimal for dense matrix. One-to-many algorithm is called N times, with N being the number of unique from (or to) nodes

For large instance, it may be appropriate to test different aon_method for few iterations and choose the fastest one for the final estimation.

Hyperparameters for algorithm-b are :

inneriter : number of time bushes are equilibrated within each iteration. Default to 20

max_tol : numerical tolerance. Flow is set to 0 if less than max_tol. Since flow shifting consist of iteratively adding or substracting double types, numerical error can occur and stop convergence. Default to 1e-11.

tmp_path : Path for storing bushes during algorithm-B execution. Default using tempdir()

max_mem : Maximum amount of RAM used by algorithm-B in gigabytes. Default to 8.

In New Bidirectional A star algorithm, euclidean distance is used as heuristic function. To understand the importance of constant parameter, see the package description : https://github.com/vlarmet/cppRouting/blob/master/README.md All algorithms are partly multithreaded (AON assignment).

A list containing :

The relative gap achieved

Number of iteration

A data.frame containing edges attributes, including equilibrated flows, new costs and free-flow travel times.

from , to and demand must be the same length. alpha , beta and capacity must be filled in during network construction. See makegraph .

Wardrop, J. G. (1952). "Some Theoretical Aspects of Road Traffic Research".

M. Fukushima (1984). "A modified Frank-Wolfe algorithm for solving the traffic assignment problem".

R. B. Dial (2006). "A path-based user-equilibrium traffic assignment algorithm that obviates path storage and enumeration".

M. Mitradjieva, P. O. Lindberg (2012). "The Stiff Is Moving — Conjugate Direction Frank-Wolfe Methods with Applications to Traffic Assignment".

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Job Assignment Problem using Branch And Bound

• N Queen Problem using Branch And Bound
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Let there be N workers and N jobs. Any worker can be assigned to perform any job, incurring some cost that may vary depending on the work-job assignment. It is required to perform all jobs by assigning exactly one worker to each job and exactly one job to each agent in such a way that the total cost of the assignment is minimized.

Let us explore all approaches for this problem.

Solution 1: Brute Force  

We generate n! possible job assignments and for each such assignment, we compute its total cost and return the less expensive assignment. Since the solution is a permutation of the n jobs, its complexity is O(n!).

Solution 2: Hungarian Algorithm  

The optimal assignment can be found using the Hungarian algorithm. The Hungarian algorithm has worst case run-time complexity of O(n^3).

Solution 3: DFS/BFS on state space tree  

A state space tree is a N-ary tree with property that any path from root to leaf node holds one of many solutions to given problem. We can perform depth-first search on state space tree and but successive moves can take us away from the goal rather than bringing closer. The search of state space tree follows leftmost path from the root regardless of initial state. An answer node may never be found in this approach. We can also perform a Breadth-first search on state space tree. But no matter what the initial state is, the algorithm attempts the same sequence of moves like DFS.

Solution 4: Finding Optimal Solution using Branch and Bound  

The selection rule for the next node in BFS and DFS is “blind”. i.e. the selection rule does not give any preference to a node that has a very good chance of getting the search to an answer node quickly. The search for an optimal solution can often be speeded by using an “intelligent” ranking function, also called an approximate cost function to avoid searching in sub-trees that do not contain an optimal solution. It is similar to BFS-like search but with one major optimization. Instead of following FIFO order, we choose a live node with least cost. We may not get optimal solution by following node with least promising cost, but it will provide very good chance of getting the search to an answer node quickly.

There are two approaches to calculate the cost function:  

• For each worker, we choose job with minimum cost from list of unassigned jobs (take minimum entry from each row).
• For each job, we choose a worker with lowest cost for that job from list of unassigned workers (take minimum entry from each column).

In this article, the first approach is followed.

Let’s take below example and try to calculate promising cost when Job 2 is assigned to worker A. 

Since Job 2 is assigned to worker A (marked in green), cost becomes 2 and Job 2 and worker A becomes unavailable (marked in red). 

Now we assign job 3 to worker B as it has minimum cost from list of unassigned jobs. Cost becomes 2 + 3 = 5 and Job 3 and worker B also becomes unavailable. 

Finally, job 1 gets assigned to worker C as it has minimum cost among unassigned jobs and job 4 gets assigned to worker D as it is only Job left. Total cost becomes 2 + 3 + 5 + 4 = 14. 

Below diagram shows complete search space diagram showing optimal solution path in green. 

Complete Algorithm:  

Below is the implementation of the above approach:

Time Complexity: O(M*N). This is because the algorithm uses a double for loop to iterate through the M x N matrix.  Auxiliary Space: O(M+N). This is because it uses two arrays of size M and N to track the applicants and jobs.

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How MSNBC’s Leftward Tilt Delivers Ratings, and Complications

NBC’s leaders have been forced to grapple with how to square its cable news network’s embrace of progressive politics with the company’s straight-news operation.

By Jim Rutenberg and Michael M. Grynbaum

MSNBC placed a big bet on becoming comfort TV for liberals. Then it doubled down.

Time slots on the cable network once devoted to news programming are now occupied by Trump-bashing opinion hosts. The channel has become a landing spot for high-profile alumni of President Biden’s administration like Jen Psaki, who went from hosting White House press briefings to hosting her own show. On Super Tuesday, when producers aired a portion of a live speech by former President Donald J. Trump, Rachel Maddow chastised her bosses on the air.

The moves have been a hit with viewers. MSNBC has leapfrogged past its erstwhile rival CNN in the ratings and has seen viewership rise over the past year, securing second place in cable news behind the perennial leader, Fox News.

But MSNBC’s success has had unintended consequences for its parent company, NBC, an original Big Three broadcaster that still strives to appeal to a mass American audience.

NBC’s traditional political journalists have cycled between rancor and resignation that the cable network’s partisanship — a regular target of Mr. Trump — will color perceptions of their straight news reporting. Local NBC stations between the coasts have demanded, again and again, that executives in New York do more to preserve NBC’s nonpartisan brand, lest MSNBC’s blue-state bent alienate their red-state viewers.

Even Comcast, NBC’s corporate owner, which is loath to intervene in news coverage, took the rare step of conveying its concern to MSNBC’s leaders when some hosts and guests criticized Israel as the Hamas attack was unfolding on Oct. 7, according to three people with knowledge of the discussions. An abrupt course correction to that coverage followed.

This account of the tensions roiling NBC and its corporate overseers is based on interviews with more than two dozen people with knowledge of the company’s inner workings, almost all of whom insisted on anonymity to share details of internal discussions.

NBC declined to make its top executives available for interviews. The chairman of the NBCUniversal News Group, Cesar Conde, has said he wants his division — which encompasses MSNBC, CNBC, a digital streaming service, Telemundo and journalistic stalwarts like “Nightly News,” “Meet the Press” and “Today” — to be a big tent.

Yet his recent efforts to include more conservative voices on the airwaves generated newsroom suspicion and ultimately led to an embarrassing rebellion over the hiring of Ronna McDaniel, a former Republican Party chair who aided Mr. Trump’s attempt to overturn his 2020 election loss.

MSNBC hosts, for their part, view their role in the political debate as more important than ever. They dismiss the accusation that MSNBC is a “Fox News for Democrats” and say their message — that Mr. Trump’s candidacy represents a unique and clear threat to democracy — is an urgent one for the electorate to hear.

And executives inside NBC’s corporate suites at Rockefeller Center say they are confident that viewers know the differences between the company’s various news brands. Any related challenges, they argue, are of a high-class sort — because their cable channels give NBC an advantage in relevance and revenue over its original Big Three competitors, ABC and CBS, which have no cable presence.

“Our strategy is built on our distinct, complementary brands including NBC News, CNBC, NBC News Now, MSNBC and Telemundo,” the NBCUniversal News Group said in a statement. “That has driven our performance as the nation’s leading news organization with the largest reach.” (Comcast does not disclose the news division’s earnings in its reports to Wall Street.)

The tensions inside NBC are, in some ways, a microcosm of the challenges facing many traditional news organizations as the country hurtles toward a tense presidential election: how to maintain trust and present neutral, fact-based reporting in a fractionalized era when partisanship carries vast financial and cultural rewards.

But the company’s challenge is also unique. It must juggle a broadcast news operation bound by traditional standards of impartiality and a cable channel increasingly bound by the partisan preferences of an intensely loyal viewership. How NBC navigates these dueling imperatives will have important implications for Comcast, a Philadelphia-based conglomerate known for its aversion to the political spotlight.

It will also have consequences for coverage of the presidential campaign. Where MSNBC’s cable news opinion-makers sustain and galvanize the Democratic faithful, the NBC broadcast network reaches millions of the potentially persuadable voters critical to both parties, which have sought to turn NBC’s internal tensions to their own advantage.

Left, Right, Left

MSNBC has caused corporate headaches since its inception.

NBC formed the channel as a joint venture with Microsoft in 1996 with the hope that it would thrust “all the value of NBC News into the cable world,” as Tom Rogers, a former NBC executive who helped found the cable network, described it in an interview.

But critics mocked the new 24-hour channel for its informal approach to news, mixing NBC’s biggest stars with younger personalities on a set reminiscent of Central Perk on “Friends.” It was almost immediately outflanked by Fox News, which followed MSNBC to market that same year and rose to the top of the cable news ratings as the first 24-hour TV channel with an overt political appeal.

MSNBC struggled with its identity. It moved to the left ahead of the Iraq war — and later moved right by hiring new hosts like the former Republican congressman Joe Scarborough. Soon it shifted leftward again, as the host Keith Olbermann hit a nerve with his strident anti-Bush — and often anti-Fox — commentary.

But when Andrew Lack, a veteran producer, took over NBC’s news division in 2015, he decided the channel needed to tone down its partisan image. Under Mr. Lack — who oversaw MSNBC’s creation in an earlier NBC stint — the cable network bumped the Rev. Al Sharpton from the weekday schedule, hired the former Fox anchor Greta Van Susteren and added more straightforward news programs, including a daily version of “Meet the Press,” NBC’s flagship political show, with Chuck Todd.

Mr. Todd was game — but would come to believe that his MSNBC duties ultimately hurt the “Meet the Press” franchise, several people at NBC said in interviews. The daily version of the show fell increasingly out of step with MSNBC’s partisan slant even as Republicans used its association with the liberal cable network to deny interview requests from the flagship Sunday edition of “Meet the Press.”

Then, Mr. Trump’s ascent shocked the Democratic base and spiked viewership of Ms. Maddow and other left-leaning hosts, whose programs became a kind of televised safe space. MSNBC’s ratings surged .

Conde Faces the Messiness

Mr. Conde succeeded Mr. Lack in spring 2020. A Wharton-trained business executive who sits on the boards of Walmart and PepsiCo, he came up through the corporate side of news, having led a turnaround at Telemundo after serving as the president of Univision Networks. Accordingly, Mr. Conde was expected to impose a more disciplined and neater corporate sensibility to the division.

He was almost immediately confronted by the messiness he had inherited.

Within a few weeks of Mr. Conde’s ascension, Mr. Trump attacked NBC when it announced the hiring of a new contributor: Lisa Page, a former F.B.I. lawyer who became a lightning rod on the right for her role in the investigation into his campaign ties to Russia. After an initial MSNBC appearance she did not show up again.

A few months later, NBC faced criticism from the other direction when it booked Mr. Trump for a prime-time interview on the night of a presidential debate that he had boycotted. (Mr. Biden was appearing at the same time on ABC.) Ms. Maddow chastised her bosses about it on the air.

That sort of partisan tumult has often riled another important constituency for Mr. Conde: NBC’s affiliated regional stations, which the company relies on to carry its major news programs to markets throughout the country.

The stations tend to be deeply embedded — and deeply trusted — in their communities. Many of them operate in red states or counties and chafed whenever MSNBC, which Mr. Trump regularly calls “MSDNC,” drew conservative ire.

Over the years the affiliates, many of which would have been thrilled to see MSNBC’s leftward tilt abandoned entirely, increasingly urged NBC executives to better distinguish its content from the NBC journalism like “Today” and “Nightly News” that they carried on their stations.

At one point after Mr. Conde took over, executives talked about the possibility of doubling down on partisanship and stripping MSNBC of news altogether, defining it as a pure opinion channel. The company would use the new NBC News Now streaming service, started under Noah Oppenheim when he was NBC News president, for 24-hour news, according to two people with knowledge of the conversations.

That idea fizzled. Mr. Conde was not prepared to entirely abandon news, but he began to better distinguish the various parts of his news division — which effectively moved MSNBC and NBC News further apart.

In the Lack era, Mr. Oppenheim of NBC News and Phil Griffin, the longtime chief of MSNBC, often worked closely as they managed a collection of stars who worked for both networks, like Mr. Todd, Craig Melvin and Hallie Jackson.

Creating more distance between the cable and broadcast outlets, Mr. Conde and Mr. Griffin’s successor, Rashida Jones, moved Mr. Todd, Ms. Jackson and Mr. Melvin off MSNBC to work exclusively at NBC News and NBC News Now. MSNBC’s daytime block of hard news shrank to six hours from eight, as the cable network extended by an hour each two opinion shows with loyal followings: “Morning Joe” featuring Mr. Scarborough and his wife Mika Brzezinski, and “Deadline: White House” with Nicolle Wallace as host.

Nothing did more to signal that MSNBC was more tightly embracing its partisan direction than Ms. Jones’s decision to hire Ms. Psaki and another Biden aide, Symone D. Sanders, straight from the White House.

It was the kind of revolving-door hiring that liberal pundits used to criticize when it happened with Fox News and the Trump administration.

It also created an awkward situation for the NBC News White House team, which was caught off guard when word that Ms. Psaki was in talks for the job leaked while she was still serving as White House press secretary.

A tense, televised confrontation followed in the White House briefing room when Kristen Welker, then NBC News’s co-chief White House correspondent, asked her future colleague: “How is it ethical to have these conversations with media outlets while you continue to have a job standing behind that podium?”

Chasing a Broad Appeal

At the same time, NBC News was going through its own changes.

Early last year, Mr. Oppenheim left his post running NBC News, and Mr. Conde split his job in three. In a jigsaw-like structure, one executive now oversaw “Today,” another “Nightly News” and NBC News Now, and a third “Meet the Press,” “Dateline” and news coverage across numerous shows and platforms.

Mr. Conde said the new setup would provide “growth opportunities,” with each show acting like its own megafranchise. “Today,” for instance, includes an e-commerce business and online sites dedicated to cooking, wellness and books.

He gave his deputies another brief: making additional efforts to ensure that news coverage reflected a wider range of political viewpoints.

Mr. Conde wanted to get Republicans back onto shows.

That was in line with an industrywide recalibration. After four years of combat between the press and Mr. Trump, media companies have sought better ways to reach Trump supporters who feel alienated from mainstream news. Television executives were also concerned that Republican elected officials were shunning their shows in favor of the congenial confines of right-wing media.

It was especially thorny for NBC, as Mr. Trump continued to yoke NBC News to MSNBC while accusing them, along with Comcast, of committing “Country Threatening Treason.”

A chance for a fresh start seemed to come last September when Ms. Welker succeeded Mr. Todd as the moderator of “Meet the Press.”

According to several people with knowledge of the internal discussions, Mr. Conde and Ms. Welker agreed that she should make booking both Mr. Trump and Mr. Biden for interviews a priority. Mr. Biden declined; Mr. Trump accepted.

But when Mr. Conde said she should schedule the Trump interview for her debut episode, Ms. Welker disagreed. Questioning the mendacious former president can be a high-wire act for even the most experienced TV interviewers, and Ms. Welker did not think it was a wise way to introduce herself to viewers. She acquiesced only after coaxing from Mr. Conde and several of his deputies.

Ms. Welker worked to fact-check Mr. Trump in real time while also eliciting an admission that he ignored his own campaign lawyers when they told him there was no evidence the 2020 presidential election results were rigged. Mr. Trump steamrolled ahead with a litany of lies nonetheless. The interview was panned on social media — complete with a “#boycottmeetthepress” campaign — but was deemed a success by Mr. Conde.

Mr. Conde and Rebecca Blumenstein, a former editor at The New York Times whom Mr. Conde hired as one of his top deputies, also worked aggressively to secure a Republican primary debate in fall 2023, pitching Ms. McDaniel and other Republican officials in person.

They succeeded, but only after accepting terms that unsettled some journalists within the company. NBC agreed to include a moderator from a right-wing media company, Salem Radio, and stream the debate live on Rumble, a video site that frequently hosts pro-Nazi and other extremist content. (NBC executives have defended the decision, noting that Rumble was already the party’s official streamer and had no editorial input.)

The debate received good marks in the press. And in general, red-state affiliates felt that Mr. Conde was doing a better job of bringing balance to NBC News, according to an executive at one company that owns affiliates.

Reverberations Continue

Each network was now set on its own distinct course: MSNBC toward more partisan and progressive opinion, and NBC News toward Mr. Conde’s commitment to “presenting our audiences with a widely diverse set of viewpoints and experiences,” as he put it.

But each tripped over the limits of its approach in an election landscape already littered with ideological tripwires.

When Hamas staged its terror attack against Israel on Oct. 7, MSNBC mixed breaking news of the attacks with discussions about the historical backdrop of Israel’s treatment of Palestinians. The coverage reflected views on the left — and presaged the pro-Palestinian demonstrations that would soon grow in number — but it struck many others as discordant, or even offensive, given that the violence was still coming into view.

“I love this network, but I’ve got to ask: Who’s writing your scripts? Hamas?” Jonathan Greenblatt, the Anti-Defamation League chief executive, asked two days later on “Morning Joe.”

Some of the blowback came from within.

In a call with Mr. Conde, Michael Cavanagh, the president of Comcast, who oversees NBC, shared concerns about that initial coverage, according to three people with knowledge of the discussions. Mr. Conde harbored the same concerns, according to a person briefed on their conversation, and he directed MSNBC to be more circumspect and to focus on facts, not opinions, in those initial days.

Five months later, Mr. Conde thought he had achieved a milestone at NBC News in his efforts to integrate right-wing perspectives into its programming. At the recommendation of Ms. Blumenstein and Carrie Budoff Brown, who oversees political coverage, Mr. Conde hired Ms. McDaniel, the former Republican Party chair, as a contributor who could offer on-air commentary.

If the hiring was in service of Mr. Conde’s goal of adding balance, it came as an unwelcome surprise to NBC’s ranks of correspondents, hosts and anchors. Ms. Welker had booked Ms. McDaniel for her next episode of “Meet the Press” — as a guest, not as a colleague. In the interview, she grilled Ms. McDaniel about her role in Mr. Trump’s effort to overturn the 2020 election result, actions that many at NBC and MSNBC viewed as disqualifying for a job there.

Mr. Todd, appearing as a guest on that day’s episode, unleashed a live, on-air denunciation of his bosses after the interview that left the control room in stunned silence. His rebellion carried over the next day on MSNBC, from “Morning Joe” up through “The Rachel Maddow Show.” Under pressure, Mr. Conde broke the deal with Ms. McDaniel, a move that only served to upset the Republicans he was trying to attract.

In the aftermath, NBC’s public stumble turned into a point of contention on the presidential campaign trail. The Republican Party said it was weighing an attempt to restrict NBC News at this summer’s convention, while Mr. Trump yet again bashed “Fake News NBC.”

Aides to Mr. Biden were also perturbed about the McDaniel hire, viewing it as part of a broader attempt by NBC News to overcompensate for MSNBC’s decidedly pro-Biden stance. In private conversations with NBC correspondents, Biden aides have argued that “Nightly News,” whose huge audience is of critical political importance to the campaign, was taking it easy on Mr. Trump and treating Mr. Biden too harshly.

Executives at NBC dismissed these complaints, saying the partisan brickbats simply come with the territory. They believe that each campaign will use anything at its disposal to pressure news organizations for more favorable coverage.

The company pointed to comments made by Mr. Conde after the McDaniel imbroglio: “We will redouble our efforts to seek voices that represent different parts of the political spectrum.” It also shared data intended to show strong performance across its cable, broadcast and online operations.

The message was clear. Regardless of any turbulence, NBC has no plans to change course.

Jim Rutenberg is a writer at large for The Times and The New York Times Magazine and writes most often about media and politics. More about Jim Rutenberg

Michael M. Grynbaum writes about the intersection of media, politics and culture. He has been a media correspondent at The Times since 2016. More about Michael M. Grynbaum