assignment problem r

All The Dots Are Connected / It's Not Rocket Science

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

• 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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Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)

• Hungarian Algorithm for Assignment Problem | Set 2 (Implementation)
• Introduction to Exact Cover Problem and Algorithm X
• Greedy Approximate Algorithm for Set Cover Problem
• Job Assignment Problem using Branch And Bound
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• 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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Title: multi-auv kinematic task assignment based on self-organizing map neural network and dubins path generator.

Abstract: To deal with the task assignment problem of multi-AUV systems under kinematic constraints, which means steering capability constraints for underactuated AUVs or other vehicles likely, an improved task assignment algorithm is proposed combining the Dubins Path algorithm with improved SOM neural network algorithm. At first, the aimed tasks are assigned to the AUVs by improved SOM neural network method based on workload balance and neighborhood function. When there exists kinematic constraints or obstacles which may cause failure of trajectory planning, task re-assignment will be implemented by change the weights of SOM neurals, until the AUVs can have paths to reach all the targets. Then, the Dubins paths are generated in several limited cases. AUV's yaw angle is limited, which result in new assignments to the targets. Computation flow is designed so that the algorithm in MATLAB and Python can realizes the path planning to multiple targets. Finally, simulation results prove that the proposed algorithm can effectively accomplish the task assignment task for multi-AUV system.

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3 Astros takeaways: Selloff possibilities, Yordan Alvarez’s RISP problem and 2 homegrown DFAs

As he is prone to do, Justin Verlander strapped the Houston Astros to his back on Sunday before Houston’s bats finally backed up its ace during a 9-3 win against the Detroit Tigers at Comerica Park.

The victory secured a series win and salvaged a 3-3 road trip. A critical 10-game homestand looms for a Houston team still seeking some stability. Seven of the 10 games are against American League West opponents, offering the club its best chance at gaining ground from the cellar.

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Here are three takeaways from the road trip:

Dana Brown must consider all options

Most clubhouses have at least one television always tuned to MLB Network, so general manager Dana Brown had no choice but to back a ballclub that may be listening. Saying he “can’t predict any scenario” in which the Astros sell at the trade deadline is one of the only acceptable responses to a question that isn’t going away, even after a respectable road trip.

"No. No, I can't envision that. This ballclub is too good." – Astros GM Dana Brown on if he envisions a scenario where the team is a seller #MLBNow | #Relentless pic.twitter.com/0kEEULJUoW — MLB Now (@MLBNow) May 7, 2024

Since 1901, only seven teams have started a season 15-25 and finished with a winning record. Just three of them made the playoffs, though a third wild-card spot and an underperforming American League West give the Astros more hope than other teams mired in their situation.

Brown shouldn’t declare the season dead on May 7, but refusing to acknowledge the possibility of a selloff — even privately — is almost as misguided. Before the season, both The Athletic’s Keith Law and MLB Pipeline ranked Houston’s farm system 27th out of 30 teams. Infusing more talent is crucial, and part of the reason owner Jim Crane hired Brown in the first place.

Nothing Crane has done during his ownership tenure suggests he’s about to offer the type of extension it will take to retain Kyle Tucker , who launched his league-leading 13th home run on Sunday. Alex Bregman ’s brutal start also won’t stop agent Scott Boras from seeking the sort of deal Crane has never been willing to give.

If the Astros can’t engineer a turnaround and enter July with realistic playoff chances, it would be a dereliction of duty not to dangle one or both of those players during the trade deadline. Pitchers Verlander and Ryan Pressly are obvious candidates, too, but both have no-trade clauses in their contracts and would prefer to stay in Houston.

Since his franchise’s golden era began, Crane has reiterated: “While I’m here, the window is always going to be open.” His stance means much more than anything Brown will utter across the next three months.

Barring a total collapse, it’s difficult to envision Crane softening on such a strong statement and conceding, especially while carrying the largest payroll during his ownership tenure. He resides in a city and owns a franchise well-versed in this exact situation. The last team to start 15-25 and make the playoffs? The 2005 National League champion Houston Astros.

Yordan Alvarez ’s run production problem

Since April 10, Alvarez has taken 33 plate appearances with runners in scoring position. He has one hit: a game-tying single down the right-field line during Friday’s 5-2 win against the Tigers.

Among Astros, only Yainer Diaz has taken more at-bats than Alvarez with runners in scoring position across that 28-game span. Since it began, Alvarez’s OPS has plummeted 259 points from 1.038 to .779. It never dipped below .895 last season.

The situation epitomizes the Astros’ first 40 games. Houston has the sport’s second-highest batting average but has been outscored by 13 other teams. Getting the club’s best hitter a bevy of at-bats with runners in scoring position should be its foremost goal. The Astros are accomplishing it — and Alvarez isn’t coming through.

Entering Sunday, Alvarez had a minus-5 batter run value with runners in scoring position, according to Baseball Savant. Last season, it was 28. Alvarez is hitting .157 with runners in scoring position this season. A .231 expected batting average suggests there isn’t much bad luck involved, either.

Dissecting Alvarez’s issues with runners in scoring position is difficult and also arrives with the caveat of a small sample size. His 15.2 percent whiff rate with runners in scoring position is lower than his 22.9 percent career clip while his 92.8 mph average exit velocity is in line with his 93 mph season average. A 39.4 percent hard-hit rate, however, is far lower than his usual clip.

All season, manager Joe Espada has bemoaned a lack of plate discipline during run-scoring situations. Entering Sunday’s game, Alvarez had swung at 27 of the 87 pitches he saw out of the strike zone with runners in scoring position — a 31 percent clip almost identical to his overall 30.8 percent chase rate this season. That, it should be noted, is elevated from Alvarez’s career 26.6 percent rate.

Last season, Alvarez had a 28.4 chase rate with runners in scoring position. Perhaps he, like so many in this lineup, is feeling the pressure of a poor start and trying to compensate. All 176 of Alvarez’s plate appearances have arrived from either the second or third spots in the batting order. Espada isn’t about to move him, either, putting the onus on a preseason MVP candidate to discover a way to clutch up.

Farewell to two homegrown success stories?

The Astros drafted Corey Julks and Brandon Bielak three rounds apart in 2017, part of a 42-man draft class that’s already produced 13 major-league players. Neither signed for more than $150,000 and both were overshadowed by more noticeable names taken beforehand.

Both Julks and Bielak ascended to viable major-league players: a testament to Houston’s amateur scouting and player development. The team designated both players for assignment this weekend and each stands a decent chance of catching on with another club, be it via a waiver claim or small trade.

Julks grew up in The Woodlands, a Houston suburb, before playing three seasons at the University of Houston. His inclusion on the team’s Opening Day roster last season — and subsequent April success — represented a feel-good story for both the franchise and the city that houses it.

Still, Julks is 28 and sported a career .650 OPS against major-league pitching. Joey Loperfido passed him on the organization’s outfield hierarchy, Pedro León is threatening to do the same, and top prospect Jacob Melton may be in Triple A soon.

At full strength, Houston already has two right-handed hitting outfielders on its 26-man roster: Chas McCormick and Jake Meyers . Utilityman Mauricio Dubón ’s increased time in the outfield only furthered a logjam Julks could not crack.

Julks has outperformed at Triple A and has two minor-league option years remaining, which will increase his value to 29 other teams. Teams are always searching for pitching depth and Bielak’s respectable numbers at the major-league level — a 4.65 ERA in 191 2/3 innings — may entice some clubs.

However, Bielak is out of minor-league options, one of the primary reasons he made Houston’s Opening Day roster as a swingman. The team is already bereft of pitching depth and did not want to risk losing Bielak on waivers so early in the season.

Activating Cristian Javier on Saturday forced them to confront that scenario. The Astros could have optioned Shawn Dubin to Triple A and kept Bielak on the major-league roster, but Dubin’s ability to handle multiple innings with higher-upside stuff must have appealed to an Astros team that’s already put Dubin in a few high-leverage situations this season.

Bielak never got those chances, perhaps the first sign that his days were numbered. José Urquidy ’s impending return from the injured list could force either Spencer Arrighetti or Hunter Brown to the bullpen, too, taking the place Bielak once occupied.

(Photo of Yordan Alvarez: Paul Sancya / Associated Press)

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Chandler Rome is a Staff Writer for The Athletic covering the Houston Astros. Before joining The Athletic, he covered the Astros for five years at the Houston Chronicle. He is a graduate of Louisiana State University. Follow Chandler on Twitter @ Chandler_Rome

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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Robert F Kennedy Jr says health issue caused by dead worm in his brain

Third-party US presidential candidate says 2010 issue ‘caused by a worm that got into my brain and ate a portion of it and then died’

Robert F Kennedy Jr, the third-party candidate for US president, said a health problem he experienced in 2010 “was caused by a worm that got into my brain and ate a portion of it and then died”, the New York Times reported .

The startling words were contained in a divorce case deposition from 2012 the Times said it obtained.

Two years before the deposition, the paper said, Kennedy experienced “memory loss and mental fogginess so severe that a friend grew concerned he might have a brain tumour”.

Neurologists who treated Kennedy’s uncle, the Massachusetts senator Ted Kennedy, before his death aged 77 from brain cancer in 2009, told the younger man he had a dark spot on his brain scans, and concluded he too had a tumour.

But, Kennedy reportedly said, a doctor at New York-Presbyterian hospital posited another explanation: a parasite in Kennedy’s brain.

Speaking this winter, the paper said , Kennedy told the Times that at around the same time he learned of the parasite in his brain he was also found to have mercury poisoning , which can cause neurological problems, probably due to eating a lot of fish.

In the 2012 deposition, Kennedy reportedly said: “I have cognitive problems, clearly. I have short-term memory loss, and I have longer-term memory loss that affects me.”

In his recent interview, the Times said, Kennedy said he had recovered from such problems. The paper also said Kennedy’s spokesperson, Stefanie Spear, responded to a question about whether the candidate’s health problems could compromise his fitness to be president by saying: “That is a hilarious suggestion, given the competition.”

Now 70, Kennedy has suffered other issues including a heart problem for which he has been repeatedly hospitalised and spasmodic dysphonia, a neurological condition that affects his voice.

Nonetheless, the scion of a famous political clan – his father was the US attorney general and New York senator Robert F Kennedy, his uncle John F Kennedy, the 35th president – has focused on showing off his physical fitness in contrast to that of Joe Biden, the oldest president ever elected, now 81, and Donald Trump, the 77-year-old Republican challenger.

The Times said Kennedy declined to share his medical records. Biden and Trump have not released theirs in this election cycle.

Observers on the left and right of US politics fear Kennedy’s campaign, which leans into his prominence as a Covid vaccine conspiracy theorist and other outsider positions, could siphon key votes from both Biden and Trump – candidates with whom the public is largely dissatisfied .

Kennedy has gained ballot access or is pursuing it in key states , including by securing nominations from fringe parties. In April, a host of Kennedy family members appeared in Philadelphia to publicly back Biden.

The Times said the 2012 deposition it obtained was given during Kennedy’s divorce from his second wife, Mary Richardson Kennedy .

Kennedy, then an environmental attorney and campaigner , argued that his earning power was diminished by his neurological and cognitive problems, the Times said.

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Doctors eventually concurred that the spot on Kennedy’s brain was the result of a parasite, Kennedy said, according to the Times. Kennedy reportedly said he thought he might have contracted the parasite in southern Asia. The Times said experts who did not treat Kennedy thought the parasite “was likely a pork tapeworm larva”.

“Some tapeworm larvae can live in a human brain for years without causing problems,” the Times said. “Others can wreak havoc, often when they start to die, which causes inflammation. The most common symptoms are seizures, headaches and dizziness.”

But Scott Gardner, curator of the Manter Laboratory for Parasitology at the University of Nebraska-Lincoln, told the paper severe memory loss, as described by Kennedy (who told the Times he experienced “severe brain fog”), was more often associated with mercury poisoning .

In his 2012 deposition, the Times said, Kennedy also discussed his spasmodic dysphonia, which he said had affected his earnings from public speaking.

Kennedy recently told Vladislav “ DJ Vlad ” Lyubovny, “a Ukrainian-American interviewer, journalist and former DJ”, that last year he went to Kyoto, Japan, “to get a surgery, a procedure … that is not available here in the United States” and which involved putting “a titanium bridge between your vocal chords”.

Kennedy’s deposition also included discussion of his heart problems, which he said began in college, the Times reported .

Saying the condition was triggered by stress, caffeine and sleep deprivation, Kennedy reportedly said: “It feels like there’s a bag of worms in my chest.”

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'Gloves are off': Trump attacks Biden, sweeping criminal charges at raucous New Jersey rally

WILDWOOD, N.J. −Angered by his criminal indictments, former President Donald Trump declared Saturday during a campaign rally that “the gloves are off” and promised a brutal election against President Joe Biden, one centered on rooting out what he called America's “enemies from within.”

It was another sign the election is on a dark path, even as Trump was often lighthearted during a nearly 90-minute speech to a raucous Wildwood, New Jersey, crowd numbering in the tens of thousands.

Trump boasted about drawing bigger crowds than Bruce Springsteen, a favored son of the state, claimed he will win New Jersey despite it going for Biden by 16 points in 2020 and joked about eating a hot dog because it's among the foods least affected by inflation.

The big, boisterous rally was a show of force in a blue state, and at a time when Trump is on trial just a few hours away in New York City in a case stemming from hush money payments to an adult film actress.

Trump fans from across New Jersey, Pennsylvania and New York packed the beach in the popular Jersey Shore town, crowding onto a stretch of sand bordered by amusement rides on two sides, the ocean on one and a boardwalk with shops and carnival games on the other.

Prep for the polls: See who is running for president and compare where they stand on key issues in our Voter Guide

Trump spoke in front of a giant Ferris wheel, roller coaster, water slide and other rides. The former president injected his mix of increasingly apocalyptic rhetoric into the festival atmosphere.

He lamented what he described as the “plunder, rape, slaughter and destruction of" American communities, saying “our country’s in trouble, it’s in big trouble.”

Trump’s 2024 campaign has drawn attention for some of the harsh language he is using that echoes authoritarian leaders, calling his opponents “vermin” and saying immigrants are “poisoning the blood of the country.”

On Saturday he said America is less threatened by outside forces than the “enemies from within.”

“We have enemies on the outside and we have enemies from within,” he said. “The enemies from within are more dangerous to me than the enemies on the outside. Russia and China we can handle, but these lunatics within our government that are gonna destroy our country, and probably want to, we have to get it stopped.”

Trump said Biden is “surrounded by fascists,” using a term the former president’s critics have used to describe him after he sought to overturn the 2020 election and after a crowd of his supporters stormed the U.S. Capitol on Jan. 6, 2021 in an attempt to stop the election from being certified.

Trump's efforts to overturn the last race for the White House resulted in two criminal indictments. He also is facing charges for allegedly mishandling classified documents.

Trump said Saturday that he is going harder at Biden because he blames the president, without evidence, for his indictments.

“I talk about him differently now because now the gloves are off. He’s a bad guy, should’ve never done that because it’s so bad for our country,” Trump said in blaming his criminal cases on Biden.

Biden spokesperson James Singer in a statement called Trump "a danger to the Constitution and a threat to our democracy, promising to rule as a dictator on ‘day one,’ punish those who stand against him, condone violence done on his behalf, and put his own revenge and retribution ahead of what is best for America."

New Jersey is a deep blue state that hasn’t voted for a Republican for president since 1988, raising questions about the wisdom of a Trump rally in the area. Trump claimed the state is in play, though it's widely expected to favor Biden again this year.

Wildwood is a Republican enclave within New Jersey, close to where Trump is on trial in New York City and near the former president’s estate in Bedminster.

It also is close to Philadelphia, a crucial source of votes in swing state Pennsylvania. The rally drew coverage in the Philadelphia media market, and attendees from the city and surrounding suburbs.

Trump mentioned Pennsylvania repeatedly in his speech.

“Whose from Pennsylvania?” He asked. “I went to school in Pennsylvania.”

John Llewellyn, 57, drove from the Philadelphia area with his wife and daughter to attend the rally. Wearing a Philadelphia Eagles jacket, the postal worker said he voted for Trump the last two elections and plans to again.

Supporting Trump means being “part of a movement. It’s part of a family,” Llewellyn said.

The southern Jersey Shore includes Atlantic City, where Trump operated casinos , and Llewellyn said he views it as “kind of like his home turf.”

Llewellyn slammed Trump’s criminal cases as “trumped up charges… just to throw him off his game” and said he believes unfounded claims that the 2020 election was stolen.

Home builder Mark Bass, 64, also drove from the Philadelphia area for the rally to “support America, first and foremost, and support probably the best president we’ve had in several decades.”

If Trump goes to jail, Bass predicted “he’ll gain more popularity and gain more votes.”

Many rallygoers wore shirts with Trump’s mug shot from his indictment in Georgia and the words “Never Surrender.” Others wore shirts supporting those arrested for rioting in the Capitol building.

Attendees cheered as Trump’s plane – dubbed “Trump Force One” – flew over two hours before the rally began.

Planes trolling Trump − and one praising him − also crisscrossed the sky trailing banners.

One read “SCOTUS justice can’t wait” while another said “Jack Ciat said don’t vote Trump he’s embarrassing,” an apparent reference to New Jersey Republican candidate for governor Jack Ciattarelli , who criticized Trump during his first campaign before backing him in 2020.

Another banner flying behind a plane read “God bless D Trump.”

North Dakota Gov. Doug Burgum warmed up the crowd. Burgum is on Trump’s short list of potential running mates, and the former president teased a potential Trump/Burgum ticket.

“You won’t find anybody better than this gentlemen in terms of his knowledge… so get ready for something, OK. Just get ready," Trump said.

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Biden’s Warning Over Rafah Sharpens a Problem for Netanyahu

“If we need to stand alone, we will stand alone,” the Israeli prime minister said, as the U.S. threatened to withhold more weapons.

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By Adam Rasgon

Reporting from Jerusalem

• May 9, 2024

President Biden’s warning over halting weapons supplies has tightened the bind that Prime Minister Benjamin Netanyahu of Israel faces, as he is increasingly caught between international calls for a cease-fire and right-wing Israeli demands to proceed with a wide-scale invasion of Rafah, in southern Gaza.

Mr. Netanyahu, who has insisted over American objections that invading Rafah is necessary, now finds the U.S.-Israel relationship at a moment of crisis that could affect how he conducts the next phase of the war against Hamas.

On Thursday, the Israeli leader, alluding to Mr. Biden’s remarks, said in a statement: “If we need to stand alone, we will stand alone. I have said that, if necessary, we will fight with our fingernails. But we have much more than fingernails and with that same strength of spirit, with God’s help, together we will win.”

With Mr. Biden threatening for the first time to withhold more American weapons, including heavy bombs and artillery shells, if Israel carries out a major operation in Rafah, a city crammed with about a million Palestinians, analysts say that the Israeli military risks losing the support of its most important supplier of foreign arms.

“The United States provides Israel with a steel dome — it’s not only military support; it’s strategic and political; it’s at the United Nations, the international court, and so on,” said Amos Gilead, a former senior Israeli defense official who worked closely with American security officials for decades.

“If we lose the United States with the unbelievable friendship of President Biden, it won’t be forgiven,” he added.

But Rear Adm. Daniel Hagari, spokesman for the Israeli military, said on Thursday that the military had sufficient “munitions for its planned operations, including operations in Rafah.”

While Israel has enough weapons in its stockpiles to conduct a wide-scale invasion of the Gazan city, U.S. restrictions could force the Israeli military to cut back on deploying specific munitions, experts said.

“It’s possible we’ll have to economize the way we use our arms and hit more targets without precision bombs,” said Jacob Nagel, a former national security adviser.

Avi Dadon, a former leader of procurement at Israel’s Defense Ministry, told Kan, the Israeli public broadcaster, that he “could be worried” if American arms were withheld. But outwardly, at least, key members of Mr. Netanyahu’s government said the war effort would not be affected.

“I turn to Israel’s enemies as well as to our best of friends and say: The state of Israel cannot be subdued,” Defense Minister Yoav Gallant said at a memorial ceremony, adding that the country would do “whatever is necessary” to defend its citizens and “to stand up to those who attempt to destroy us.”

Bezalel Smotrich, the far-right finance minister, declared that Israel would achieve “complete victory” despite what he described as Mr. Biden’s “pushback and arms embargo.”

American-made weapons, including heavy bombs, have been essential to Israel’s war effort since the country was attacked by Hamas and other militant groups on Oct. 7. But Mr. Biden has been under growing domestic pressure to rein in Israel’s military as the death toll has risen in Gaza. It is now more than 34,000, according to local health authorities.

And in his comments on Wednesday in an interview with CNN, Mr. Biden acknowledged for the first time that U.S. bombs had killed innocent civilians in the conflict.

The American concerns have only grown since the Israeli army sent tanks and troops into the eastern part of Rafah on Monday night, taking over the main border crossing between Gaza and Egypt. Israeli forces have stopped short of entering built-up parts of the city, but Mr. Netanyahu and others have signaled that such an operation is necessary to eliminate Hamas battalions there.

On Tuesday, American officials said Mr. Biden had withheld 1,800 2,000-pound bombs and 1,700 500-pound bombs that he feared could be dropped on Rafah. The administration was reviewing whether to hold back future transfers, including guidance kits that convert so-called dumb bombs into precision-guided munitions, the officials said.

In addition to the bombs, Mr. Biden said the United States would not supply artillery shells if Israel invaded population centers in Rafah.

Gilad Erdan, the Israeli ambassador to the United Nations, described the Biden administration’s decision as “very disappointing” and “frustrating.”

“We have a cruel enemy here,” he said. “Is this the time to put restrictions on Israel’s weapons?”

Nadav Eyal, a prominent columnist for a centrist Israeli newspaper, said Mr. Biden had essentially decided to declare an end to the war. Writing on the social media platform X, he called it “the most serious clash between an American administration and the government of Israel since the first Lebanon war.”

The Israeli ambassador to Washington, Michael Herzog, said on Thursday that Mr. Biden’s decision “sends the wrong message to Hamas and to our enemies in the region.”

“It puts us in a corner,” he said in a public conversation hosted by the Carnegie Endowment for International Peace in Washington. He added, “Nobody presented to me or to us a strategy of defeating Hamas without dealing with Rafah.”

Some analysts, however, downplayed the significance of the crisis, arguing it wasn’t as bad as past fissures between the United States and Israel. The rupture in relations over the Iran nuclear deal in 2015 was “much worse,” said Mr. Nagel.

Amid the tense state of affairs, Israel’s president, Isaac Herzog, thanked the United States for supporting Israel and appeared to lash out at Itamar Ben-Gvir, the national security minister, who had posted on X, “Hamas ♥ Biden.”

“Even when there are disagreements and moments of disappointment between friends and allies, there is a way to clarify the disputes,” Mr. Herzog said.

Myra Noveck , Michael Crowley and Johnatan Reiss contributed reporting.

Adam Rasgon reports from Israel for The Times's Jerusalem bureau. More about Adam Rasgon

Our Coverage of the Israel-Hamas War

News and Analysis

As the Israeli military stepped up pressure on what it calls Hamas’s last stronghold in Gaza, fighting elsewhere in the Palestinian enclave  led to warnings that the militants might remain a force for a long time to come.

On Israel’s Memorial Day, many were drawn to the site of the music festival  that was attacked on Oct. 7 by Hamas, while peace activists broadcast a joint Israeli-Palestinian ceremony .

Around 300,000 Palestinians in southern and northern Gaza were being forced to flee once again , the U.N. said, just as Israel issued new and expanded evacuation orders.

A Key Weapon: When President Biden threatened to pause some weapons shipments to Israel if it invaded Rafah, the devastating effects of the 2,000-pound Mark 84 bomb  were of particular concern to him.

A Presidential Move: Ronald Reagan also used the power of American arms to influence  Israeli war policy. The comparison underscores how much the politics of Israel have changed in the United States since the 1980s.

Netanyahu’s Concerns: Prime Minister Benjamin Netanyahu of Israel, under pressure from all sides, is trying to reassure his many domestic, military and diplomatic critics. Here’s a look at what he is confronting .

Al Jazeera Shutdown: The influential Arab news network says it will continue reporting from Gaza and the West Bank, but its departure from Israel is a new low in its long-strained history with the country .