assignment problem in transportation problem

Operations Research/Transportation and Assignment Problem

The Transportation and Assignment problems deal with assigning sources and jobs to destinations and machines. We will discuss the transportation problem first.

Suppose a company has m factories where it manufactures its product and n outlets from where the product is sold. Transporting the product from a factory to an outlet costs some money which depends on several factors and varies for each choice of factory and outlet. The total amount of the product a particular factory makes is fixed and so is the total amount a particular outlet can store. The problem is to decide how much of the product should be supplied from each factory to each outlet so that the total cost is minimum.

Let us consider an example.

Suppose an auto company has three plants in cities A, B and C and two major distribution centers in D and E. The capacities of the three plants during the next quarter are 1000, 1500 and 1200 cars. The quarterly demands of the two distribution centers are 2300 and 1400 cars. The transportation costs (which depend on the mileage, transport company etc) between the plants and the distribution centers is as follows:

Which plant should supply how many cars to which outlet so that the total cost is minimum?

The problem can be formulated as a LP model:

The whole model is:

subject to,

The problem can now be solved using the simplex method. A convenient procedure is discussed in the next section.

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Linear Programming and Its Applications pp 140–184 Cite as

Transportation and Assignment Problems

• James K. Strayer 2  

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Part of the Undergraduate Texts in Mathematics book series (UTM)

Transportation and assignment problems are traditional examples of linear programming problems. Although these problems are solvable by using the techniques of Chapters 2–4 directly, the solution procedure is cumbersome; hence, we develop much more efficient algorithms for handling these problems. In the case of transportation problems, the algorithm is essentially a disguised form of the dual simplex algorithm of 4§2. Assignment problems, which are special cases of transportation problems, pose difficulties for the transportation algorithm and require the development of an algorithm which takes advantage of the simpler nature of these problems.

• Assignment Problem
• Transportation Problem
• Basic Feasible Solution
• Unique Optimal Solution
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Strayer, J.K. (1989). Transportation and Assignment Problems. In: Linear Programming and Its Applications. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1009-2_7

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Solving Transshipment and Assignment Problems

In a previous blog, we provided an overview of the transportation problem – the problem of finding minimum cost shipping routes for products or resources between supply locations and demand locations. We also provided a little background on the algorithm for solving such problems along with several examples using an implementation in IMSL.

In this blog, we look at two special cases of the transportation problem – the transshipment problem and assignment problem.

What Is the Transshipment Problem?

The transshipment problem is a special case of the transportation problem in which shipping paths can include intermediate points. For example, shipping from Los Angeles to New York via Denver may be less expensive than shipping directly (non-stop) to New York. Intermediate points also arise when different modes of transportation are available at different costs (for example, shipping via truck versus rail) or when products must be stored for a time at intermediate points for special processing or packaging.

A table display helps illustrate the transshipment problem. In the standard table used for a transportation problem, the transshipment or intermediate points are appended to both the demand center columns and the supply center rows. For each of the transshipment points, the supply and demand are set equal to the total supply and the total demand for the problem (assuming the problem is balanced or has been balanced with dummy variables). In the cost matrix, the point-to-point costs are all still required, except that there will be zero values for the cost where the same label exists in both the supply and demand. Arbitrarily large values are used for any paths that aren’t logistically valid, for example, shipping between transshipment points. This table format is illustrated in the following example.

Transshipment Problem Example

Consider an example where wheat is harvested in Nebraska and Colorado and then must be shipped to grain elevators in Kansas City, Omaha, and Des Moines. These transshipment points then send the product along to Chicago, St. Louis, and Cincinnati.

The farms have outputs of 300 units each, while the demand is 200, 100, and 300 respectively at each destination. In this problem statement, the farms cannot ship to the endpoints directly, and the transshipment points cannot ship to each other.

Each of the point-to-point shipping costs appear in the table below. Wherever a path is not valid, a cost of 1000 is applied. In this case, the farms cannot ship directly to the destinations, and the transshipment points are not allowed to transfer between each other. For the transshipment points, which appear as both sources and destinations, there is zero cost to ship to itself.

Using the table format to formulate the problem and provide the point-to-point shipping costs gives:

Table 1: Cost Matrix for Transshipment of Wheat

The problem is balanced, noting that the transshipment points each have a supply and demand equal to the sum of the actual values, 600. The source code to solve this problem using IMSL is given in the next subsection. The solution matrix is shown below for a total cost of $12,400:

Table 2: Solution Matrix for Transshipment of Wheat

The solution indicates that the 300 units from Nebraska are shipped to Des Moines, while all 300 units from Colorado are shipped to Kansas City. Once in Kansas City, all 300 units are sent off to Cincinnati, but at Des Moines, the shipment is split with 200 heading to Chicago and 100 to St. Louis. Notice the 600 units in Omaha indicate that this point is not part of the solution as the site is meeting its (artificial) demand with its own (artificial) supply.

Solving the Transshipment Problem With IMSL

While this example is small enough to solve manually, most problems are too large to do this practically. The C code to solve this problem using IMSL is given below. Note how the inputs (the supply and demand arrays and the cost matrix) are drawn directly from Table 1. Conceptually, larger sized problems are set up in the same way.

What Is the Assignment Problem?

The assignment problem is another special case of the transportation problem. This type of problem arises when assigning workers to different tasks or, as illustrated below, assigning athletes to different legs of a relay.

Assignment Problem Example

Consider the example of a swimming relay team in the Summer Olympics. The objective is to build the best (fastest) swimming medley relay team given the four events and the times of five swimmers for each event. That is, the problem is to assign one and only one swimmer to one and only one leg of the medley relay that gives a minimum overall time. Each supply item (a swimmer) provides only one unit of supply, and each destination item (a leg of the race) demands only one unit, as shown in the following table:

Table 3 Cost Matrix for the Assignment of Swimmers

Note that swim times are measured to the tenths of a second (for example 48.1 seconds); to keep everything integer, the times have been multiplied by ten. Due to the linearity of the problem, scaling the cost matrix has no effect on the solution. With unit supply and demand across the board, the resulting solution will be mostly zeros with a value of one located for the best choice of swimmer for each leg of the medley relay.

No special formulation is required when using the IMSL transport function to solve the assignment problem. The source code is shown in the next subsection. The solution matrix is below, giving an optimal best time of 190.1 seconds.

Table 4 Solution Matrix for the Assignment of Swimmers

With this small example it’s easy to pick out a few characteristics of the solution. First, we see that swimmer E, while not the slowest in any event, is also not the fastest in any event and is not selected for the medley. Swimmer D, who is the slowest swimmer in some events, is nevertheless selected for the backstroke event because other, faster swimmers in that event are needed for other legs. In real life, the coach might average a swimmer’s last 5 times and might also consider the variation in those times. How consistently fast is a swimmer in any given event?

The idea of incorporating variation or additional uncertainties extends to any problem with respect to costs. If there is known or estimable variation in costs, how is it accounted for in the optimal solution? Similarly, how would we account for variation or uncertainties in the supplies and/or demands? These are ideas we will be exploring in future blogs on the transportation problem.

Solving the Assignment Problem With IMSL

The C code to solve this problem definition using the IMSL transport algorithm is as follows:

Final Thoughts

In this blog, we looked at the transshipment and assignment problems and described how they can be understood and solved as transportation problems.

If you missed our first blog on this topic, you can read about it here: Solving the Transportation Problem .

Easily Solve the Transshipment and Assignment Problems With IMSL

Transshipment and assignment problems along with traditional transportation problems are easily solved using the transportation algorithm included in IMSL.

Whether you’re working in C/C++, Fortran, Java, or Python, you can evaluate the IMSL library for your application free. Try it today via the link below.

TRY IMSL FREE

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Transportation Problem | Set 1 (Introduction)

Transportation problem is a special kind of Linear Programming Problem (LPP) in which goods are transported from a set of sources to a set of destinations subject to the supply and demand of the sources and destination respectively such that the total cost of transportation is minimized. It is also sometimes called as Hitchcock problem.

Types of Transportation problems: Balanced: When both supplies and demands are equal then the problem is said to be a balanced transportation problem.

Unbalanced: When the supply and demand are not equal then it is said to be an unbalanced transportation problem. In this type of problem, either a dummy row or a dummy column is added according to the requirement to make it a balanced problem. Then it can be solved similar to the balanced problem.

Methods to Solve: To find the initial basic feasible solution there are three methods:

• NorthWest Corner Cell Method.
• Least Call Cell Method.
• Vogel’s Approximation Method (VAM).

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Transportation Problem Explained and how to solve it?

• Introduction
• Transportation Problem
• Balanced Problem
• Unbalanced Problem

Contributed by: Patrick

Operations Research (OR) is a state of art approach used for problem-solving and decision making. OR helps any organization to achieve their best performance under the given constraints or circumstances. The prominent OR techniques are,

• Linear programming
• Goal programming
• Integer programming
• Dynamic programming
• Network programming

One of the problems the organizations face is the transportation problem. It originally means the problem of transporting/shipping the commodities from the industry to the destinations with the least possible cost while satisfying the supply and demand limits.  It is a special class of linear programming technique that was designed for models with linear objective and constraint functions. Their application can be extended to other areas of operation, including

• Scheduling and Time management
• Network optimization
• Inventory management
• Enterprise resource planning
• Process planning
• Routing optimization  

The notations of the representation are:

m sources and n destinations

(i , j) joining source (i) and destination (j)

c ij 🡪  transportation cost per unit

x ij 🡪  amount shipped

a i   🡪 the amount of supply at source (i)

b j   🡪 the amount of demand at destination (j)

Transportation problem works in a way of minimizing the cost function. Here, the cost function is the amount of money spent to the logistics provider for transporting the commodities from production or supplier place to the demand place. Many factors decide the cost of transport. It includes the distance between the two locations, the path followed, mode of transport, the number of units that are transported, the speed of transport, etc. So, the focus here is to transport the commodities with minimum transportation cost without any compromise in supply and demand. The transportation problem is an extension of linear programming technique because the transportation costs are formulated as a linear function to the supply capacity and demand. Check out the course on transportation analytics .

Transportation problem exists in two forms. 

• Balanced 

It is the case where the total supply equals the total demand.

It is the case where either the demand is greater than the supply, or vice versa.

In most cases, the problems take a balanced form. It is because usually, the production units work, taking the inventory and the demand into consideration. Overproduction increases the inventory cost whereas under production is challenged by the demand. Hence the trade-off should be carefully examined. Whereas, the unbalanced form exists in a situation where there is an unprecedented increase or decrease in demand.

Let us understand this in a much simpler way with the help of a basic example. 

Let us assume that there is a leading global automotive supplier company named JIM. JIM has it’s production plants in many countries and supplies products to all the top automotive makers in the world. For instance, let’s consider that there are three plants in India at places M, N, and O. The capacity of the plants is 700, 300, 550 per day. The plant supplies four customers A, B, C, and D, whose demand is 650, 200, 450, 250 per day. The cost of transport per unit per km in INR and the distance between each source and destination in Kms are given in the tables below.

Here, the objective is to determine the unknown while satisfying all the supply and demand restrictions. The cost of shipping from a source to a destination is directly proportional to the number of units shipped.

Many sophisticated programming languages have evolved to solve OR problems in a much simpler and easier way. But the significance of Microsoft Excel cannot be compromised and devalued at any time. It also provides us with a greater understanding of the problem than others. Hence we will use Excel to solve the problem.

It is always better to formulate the working procedure in steps that it helps in better understanding and prevents from committing any error.

Steps to be followed to solve the problem:

• Create a transportation matrix (define decision variables)
• Define the objective function
• Formulate the constraints
• Solve using LP method 

Creating a transportation matrix:

A transportation matrix is a way of understanding the maximum possibilities the shipment can be done. It is also known as decision variables because these are the variables of interest that we will change to achieve the objective, that is, minimizing the cost function.

Define the objective function: 

An objective function is our target variable. It is the cost function, that is, the total cost incurred for transporting. It is known as an objective function because our interest here is to minimize the cost of transporting while satisfying all the supply and demand restrictions.

The objective function is the total cost. It is obtained by the sum product of the cost per unit per km and the decision variables (highlighted in red), as the total cost is directly proportional to the sum product of the number of units shipped and cost of transport per unit per Km.

The column “Total shipped” is the sum of the columns A, B, C, and D for respective rows and the row “Total Demand” is the sum of rows M, N, and O for the respective columns. These two columns are introduced to satisfy the constraints of the amount of supply and demand while solving the cost function. 

Formulate the constraints:

The constraints are formulated concerning the demand and supply for respective rows and columns. The importance of these constraints is to ensure they satisfy all the supply and demand restrictions.

For example, the fourth constraint, x ma + x na + x oa = 650 is used to ensure that the number of units coming from plants M, N, and O to customer A should not go below or above the demand that A has. Similarly the first constraint x ma + x mb + x mc + x md  = 700 will ensure that the capacity of the plant M will not go below or above the given capacity hence, the plant can be utilized to its fullest potential without compromising the inventory. 

Solve using LP method:

The simplest and most effective method to solve is using solver. The input parameters are fed as stated below and proceed to solve. 

This is the best-optimized cost function, and there is no possibility to achieve lesser cost than this having the same constraints.

From the solved solution, it is seen that plant M ships 100 units to customer A, 350 units to C and 250 units to D. But why nothing to customer B? And a similar trend can be seen for other plants as well. 

What could be the reason for this? Yes, you guessed it right! It is because some other plants ship at a profitable rate to a customer than others and as a result, you can find few plants supplying zero units to certain customers. 

So, when will these zero unit suppliers get profitable and can supply to those customers? Wait! Don’t panic. Excel has got away for it too. After proceeding to solve, there appears a dialogue box in which select the sensitivity report and click OK. You will get a wonderful sensitivity report which gives details of the opportunity cost or worthiness of the resource.

Basic explanation for the report variables,

Cell: The cell ID in the excel

Name: The supplier customer pairing

Final value: Number of units shipped (after solving)

Reduced cost: How much should the transportation cost per unit per km should be reduced to make the zero supplying plant profitable and start supplying

Objective coefficient: Current transportation cost per unit per Km for each supplier customer pair

Allowable Increase: It tells us the maximum cost of the current transportation cost per unit per Km can be increased which doesn’t make any changes to the solution

Allowable Decrease: It tells how much maximum the current transportation cost per unit per Km can be lowered which doesn’t make any changes to the solution

Here, look into the first row of the sensitivity report. Plant M supplies to customer A. Here, the transportation cost per unit per Km is ₹14 and 100 units are shipped to customer A. In this case, the transportation cost can increase a maximum of ₹6, and can lower to a maximum of ₹1. For any value within this range, there will not be any change in the final solution. 

Now, something interesting. Look at the second row. Between MB, there is not a single unit supplied to customer B from plant M. The current shipping cost is ₹22 and to make this pair profitable and start a business, the cost should come down by ₹6 per unit per Km. Whereas, there is no possibility of increasing the cost by even a rupee. If the shipping cost for this pair comes down to ₹16, we can expect a business to begin between them, and the final solution changes accordingly.

The above example is a balanced type problem where the supply equals the demand. In case of an unbalanced type, a dummy variable is added with either a supplier or a customer based on how the imbalance occurs.

Thus, the transportation problem in Excel not only solves the problem but also helps us to understand how the model works and what can be changed, and to what extent to modify the solution which in turn helps to determine the cost and an optimal supplier. 

If you found this helpful, and wish to learn more such concepts, head over to Great Learning Academy and enroll in the free online courses today.

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Assignment Problem

• Jun 15, 2023

The assignment problem is a special case of transportation problem.

Suppose there are $n$ jobs to be performed and $n$ persons are available for these jobs. Assume that each person can do each job at a time, though with varying degree of efficiency.

Let $c_{ij}$ be the cost if the $i^{th}$ person is assigned the $j^{th}$ job. Then the problem is to find an assignment so that the total cost of performing all jobs is minimum. (i.e., which job should be assigned to which person with minimum cost). Such problem is called an Assignment Problem (AP).

The tabular form of the assignment problem is as follows

The above table is called the $n\times n$ cost-matrix, where $c_{ij}$ are real numbers.

Thus the objective in the Assignment Problem is to assign a number of jobs to the equal number of persons at a minimum cost or maximum profit. e.g. assigning men to offices, classes to rooms, drivers to trucks, problems to research teams etc.

Mathematical Formulation of AP

Mathematically, the AP can be stated as $$ \begin{equation}\label{eq2.4} \min z = \sum_{i=1}^n\sum_{j=1}^n x_{ij} c_{ij}. \end{equation} $$ subject to $$ \begin{equation}\label{eq2.5} \sum_{j=1}^n x_{ij} =1,; \text{ for } i=1,2,\ldots, n \end{equation} $$

$$ \begin{equation}\label{eq2.6} \sum_{i=1}^n x_{ij} =1,; \text{ for } j=1,2,\ldots,n \end{equation} $$ where $$ \begin{equation*} x_{ij}=\left{ \begin{array}{ll} 1, & \hbox{if $i^{th}$ person is assigned $j^{th}$ job;} \ 0, & \hbox{otherwise.} \end{array} \right. \end{equation*} $$ Constraint \eqref{eq2.5} indicate that one job is done by $i^{th}$ person $i=1,2,\ldots, n$ and constraint \eqref{eq2.6} indicate that one person should be assigned $j^{th}$ job $j=1,2,\ldots, n$.

It may be observed from the above formulation that AP is a special type of Linear programming problem.

Unbalanced Assignment Problem

If the cost matrix of an assignment problem is not a square matrix, the assignment problem is called an Unbalanced Assignment Problem . In such a case, add dummy row or dummy column with zero cost in the cost matrix so as to form a square matrix. Then apply the usual assignment method to find the optimal solution.

Maximal Assignment Problem

Sometimes, the assignment problem deals with the maximization of an objective function instead of minimization.. For example, it may be required to assign persons to jobs in such a way that the expected profit is maximum. In such a case, first convert the problem of maximization to minimization and apply the usual procedure of assignment.

The assignment problem of maximization type can be converted to minimization type by subtracting all the elements of the given profit matrix from the largest element.

Restrictions on Assignment

Sometimes due to technical or other difficulties do not permit the assignment of a particular facility to a particular job. In such a situation, the difficulty can be overcome by assigning a very high cost (say, infinity i.e. $\infty$) to the corresponding cell.

Because of assigning an infinite penalty to such a cell the activity will be automatically excluded from the optimal solution. Then apply the usual procedure to find the optimal assignment.

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Solving fuzzy linear programming problem, fuzzy transportation and fuzzy assignment problem using MATLAB: A new approach

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P. Duraisamy , M. Kaliyappan; Solving fuzzy linear programming problem, fuzzy transportation and fuzzy assignment problem using MATLAB: A new approach. AIP Conference Proceedings 31 August 2023; 2852 (1): 090006. https://doi.org/10.1063/5.0164530

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This article used MATLAB to resolve the fuzzy assignment, fuzzy transportation, and fuzzy linear programming problems. Cell arrays in MATLAB are used to hold fuzzy numbers that are later converted to crisp numbers in the Fuzzy Assignment Problem, Fuzzy Transportation Problem, and Fuzzy Linear Programming Problem. The obtained Crisp Linear Programming Problem, Transportation Problem and Assignment Problem are solved using the built-in function linprog and other MATLAB codes. The techniques are well elucidated in detail.

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Transportation problems and assignment problems are two types of linear programming problems that arise in different applications.

The main difference between transportation and assignment problems is in the nature of the decision variables and the constraints.

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Decision Variables:

In a transportation problem, the decision variables represent the flow of goods from sources to destinations. Each variable represents the quantity of goods transported from a source to a destination.

In contrast, in an assignment problem, the decision variables represent the assignment of agents to tasks. Each variable represents whether an agent is assigned to a particular task or not.

Constraints:

In a transportation problem, the constraints ensure that the supply from each source matches the demand at each destination and that the total flow of goods does not exceed the capacity of each source and destination.

In contrast, in an assignment problem, the constraints ensure that each task is assigned to exactly one agent and that each agent is assigned to at most one task.

Objective function:

The objective function in a transportation problem typically involves minimizing the total cost of transportation or maximizing the total profit of transportation.

In an assignment problem, the objective function typically involves minimizing the total cost or maximizing the total benefit of assigning agents to tasks.

In summary,

The transportation problem is concerned with finding the optimal way to transport goods from sources to destinations,

while the assignment problem is concerned with finding the optimal way to assign agents to tasks.

Both problems are important in operations research and have numerous practical applications.

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Norfolk Southern is recovering from a technology problem that shut down its railroad

FILE - A Norfolk Southern freight train rolls through downtown Pittsburgh, on March 26, 2018. Regulators say Norfolk Southern has made improvements since a fiery Ohio derailment but still falls well short of being the “gold standard for safety” it is striving to be. The Federal Railroad Administration released a report on the railroad’s safety culture Wednesday, Aug. 9, 2023. (AP Photo/Gene J. Puskar, File)

FILE - Cleanup of portions of a Norfolk Southern freight train that derailed Friday night in East Palestine, Ohio, continues on Feb. 9, 2023. (AP Photo/Gene J. Puskar, File)

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OMAHA, Neb. (AP) — Norfolk Southern railroad is recovering from a “hardware-related technology outage” that disrupted train traffic across its network in the Eastern United States Monday, but there may be lingering effects for at least a couple of weeks.

The railroad said there is no indication that the outage was related to any cybersecurity incident. But spokesman Connor Spielmaker said Tuesday morning that “we’re continuing our investigation into the root cause of these issues.”

All system functionality was restored by 7 p.m. Eastern on Monday, the company said, and it is bringing the rail network back online.

Norfolk Southern has been in contact with its customers and is working Tuesday to get shipments where they need to be. The railroad is one of the biggest in North America with nearly 20,000 miles of track across 22 eastern states. Businesses of all kinds rely on Norfolk Southern to deliver the raw materials and goods they need while helping get their finished products to their destinations.

Norfolk Southern Corp., based in Atlanta, is in the crosshairs of federal regulators after a derailment in Ohio earlier this year led to a fiery, toxic chemical spill .

In a 143-page report this month, the Federal Railroad Administration said that Norfolk Southern has made some improvements , but is nowhere near the “gold standard for safety” its CEO is striving for. The railroad is too often only willing to meet minimum safety requirements, regulators said.

The agency has promised to conduct similar safety culture reviews at all the major freight railroads, including CSX, Union Pacific, BNSF, Canadian National and Canadian Pacific Kansas City, but it hasn’t set a timeline for those reviews.

Congress and regulators have called for all the major freight railroads to make a number of changes to improve safety but proposed legislation has stalled in the Senate and failed to get started in the Republican-controlled House.

Paying parents to drive? JCPS seeks 'sustainable' solution to bus transportation issues

It is becoming more clear that leaders of Kentucky's largest school system will be making massive changes to its policies regarding which students are transported and to where.

As Jefferson County Public Schools Superintendent Marty Pollio puts it, their hands are tied - the task of transporting 68,000 students to and from schools is simply not possible anymore, given the national bus driver shortage that has forced education leaders to alter their transportation systems.

"We have some tough decisions ahead of us, there's no doubt about it," Pollio told board members during their meeting this week. "We will have to make some decisions around who gets transportation and possibly who doesn't, or what type of transportation we do provide."

Changes to the district's transportation system this year were meant to remedy two issues: Students getting to and from school late and the lack of sleep middle and high school students were enduring with a 7:40 a.m. start time. But then on the first day of classes, students were stuck after school for hours and some didn't get home till nearly 10 p.m.

Over the course of several days that school was canceled, district leaders made changes that have significantly improved efficiency - with the last students now regularly getting home about 7 p.m. - which is less than three hours after the last set of schools dismiss. Still, Pollio and board members are looking toward a long-term solution that will significantly reduce the number of students transported by school buses each day.

Pollio told board members whatever change is made, he wants it to be impactful.

"Any change we make is going to have to reduce the number of routes significantly," he said. "I don't want to make a major change and only reduce by 75 routes. We need to say, 'If we make this change, it is a sustainable change and it will be a sustainable change for the foreseeable future.'"

How could JCPS curb bus ridership?

JCPS has several options, including reducing its ridership solely to the students who are federally required to receive transportation to incentivizing families to carry more of the burden.

Pollio said he wants to "produce a menu of options to our board and community" at the board's next meeting in late September.

Paying families to transport their own children to schools is one remedy that large urban districts have turned to amid the driver shortage.

Philadelphia's school district began paying eligible parents $300 a month to take their kids to school this year following a pilot program launched in 2020. More than half of the he district's approximately 200,000 students are eligible for transportation, per its policies, and about 33,000 are transported. But, about 13,000 students' families have taken advantage of the monthly stipends, according to a report by an ABC affiliate .

The details of how those payments work if a student needs to switch back to busing is unclear - an issue Pollio brought up when asked if JCPS would consider a similar program.

"The logistics of it are a major concern for me – who exactly do you give it to and does it mean they can't get on a bus after," he said.

Chicago Public Schools has implemented a similar program, but the monthly stipend is only offered to the parents of children who are required to receive transportation.

Federal law requires districts to provide transportation for homeless students and those with a disability plan, known as an Individualized Education Plan, that includes transportation services.

Chicago's district is now only transporting those groups, which equates to about 7,100 students, according to a Chicago NBC affiliate.

The district offers those families $500 a month to transport their children. The district offers city transportation vouchers for other children.

In Louisville, community members have suggested such parent stipends as a solution to JCPS' busing woes. Some have also suggested that JCPS stop transporting students who don't qualify for free or reduced lunches or who choose to attend a school outside their neighborhood.

Which students does JCPS have to bus?

JCPS students are eligible for busing if they live more than one mile from their schools, but there are exceptions the district can make to pick up students within that range.

Meanwhile, district spokesman Mark Hebert said JCPS had about 3,300 homeless students and about 1,500 students with Individualized Education Plans that required transportation last year. If the district chose to limit riders to only these two groups − the only students it is required to transport − ridership would decrease by 91%.

However, such a move could eliminate students' opportunity to be taken to a magnet school that they otherwise could not attend. That would create another issue as the district seeks to diversify its magnet programs.

School board member James Craig echoed Pollio, stressing that today's system is simply unsustainable.

"Everyone in the district knows we are patching this system together on a daily basis depending on who is available to pitch in and I think the system needs more predictability than we are able to deliver right now," Craig told The Courier Journal. "It's wonderful we have offered transportation to any child who requested it, up until this time."

When asked who to exclude, though, Craig didn't have an answer.

He mentioned Chicago's move to solely follow federal law, the discontinuation of magnet student busing, limiting busing to economically disadvantaged students, and expanding the radius of how close students can live to a school and still get a bus.

Each of these, though, still create concerns. For example, he said economically disadvantaged students comprise nearly 70% of the district, so its unclear if this choice would significantly decrease ridership.

"Whatever the solution is," he said, "we just need to hear from the operations specialist as to what could deliver a functioning, sustainable and predictable system."

More: Some Kentucky lawmakers want to consider splitting up JCPS. Could that really happen?

Contact reporter Krista Johnson at  [email protected] .