Transportation and Assignment Problems

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Questions and Answers

What must be true for the transportation simplex method to be applied?

Total supply and demand can be estimated

Total supply must equal total demand (correct)

Total supply must exceed total demand

Total demand must exceed total supply

What is typically used to find an initial feasible solution in transportation problems?

Graphical methods

Exact algorithms

Linear programming

Heuristics (correct)

What is the minimum cost method primarily designed to achieve?

Finding the optimal solution immediately

Quickly finding a feasible solution close to optimal (correct)

Minimizing the number of routes used

Balancing supply and demand accurately

What defines a balanced transportation problem?

Total supply equals total demand Signup and view all the answers

What happens if a transportation problem is unbalanced?

A dummy origin or destination is added Signup and view all the answers

What does the transportation tableau represent?

The cost associated with each route Signup and view all the answers

During Phase II of the transportation simplex method, what is the first step at each iteration?

Identify an incoming arc Signup and view all the answers

Why might some heuristics not find a good initial feasible solution quickly?

They prioritize speed over cost minimization Signup and view all the answers

What is the Hungarian method mainly used for?

Solving the assignment problem Signup and view all the answers

What should be done if it is not possible to cover all zeros with fewer lines in the matrix?

Return to check the previous step for better covering Signup and view all the answers

What is a common characteristic of the transportation simplex method that enhances its performance?

It relies solely on addition and subtraction operations Signup and view all the answers

In handling an assignment problem with an unequal number of agents and tasks, which approach is suggested?

Adjust the matrix to match the number of agents and tasks Signup and view all the answers

Which of the following represents a problem variation that can be addressed by the Hungarian method?

Maximization objective function Signup and view all the answers

Why is it problematic to draw too many lines while covering zeros in the reduced matrix?

It can prevent reaching an optimal solution Signup and view all the answers

What primary structure does each cell in the transportation tableau correspond to?

An arc or route in the network model Signup and view all the answers

What is the primary advantage of using a transportation tableau over a simplex tableau for a transportation problem?

It simplifies the calculations due to its smaller size. Signup and view all the answers

If a zero-value assignment cannot be found, what action should be taken?

Investigate if fewer lines can cover the zeros Signup and view all the answers

In the provided transportation tableau for the problem with four origins and four destinations, what is the total demand?

400 Signup and view all the answers

According to the MODI method, when does an alternative optimal solution exist?

When all penalties are equal. Signup and view all the answers

If the initial feasible solution to the given transportation problem is disrupted, what method is suggested to find the optimal solution?

Use the transportation simplex method. Signup and view all the answers

What is the function of the transportation tableau when solving transportation problems?

To represent supply, demand, and costs visually. Signup and view all the answers

Which route must be added to better optimize shipping if a constraint is provided in the transportation model?

Tucson to San Diego. Signup and view all the answers

What does it indicate if a transportation problem tableau has all its reduced costs equal to zero?

The problem has multiple optimal solutions. Signup and view all the answers

What is the significance of balancing supply and demand in a transportation tableau?

It ensures there are no excess shipments. Signup and view all the answers

What is the first phase of the transportation simplex method?

Finding an initial feasible solution Signup and view all the answers

Why might a general-purpose linear programming code be insufficient for large transportation problems?

It is less efficient due to the size of the problems Signup and view all the answers

What is the purpose of the incoming arc in the transportation simplex method?

To make a flow allocation leading to the largest per-unit reduction in total cost. Signup and view all the answers

What does the transportation tableau help to summarize in the transportation simplex method?

The supply at each origin and demand at each destination Signup and view all the answers

In the minimum cost method, what should be done if there is a tie in the lowest cost cell?

Choose the cell that can ship the most units. Signup and view all the answers

What is a significant feature of the transportation simplex method compared to general linear programming methods?

It is specifically designed for problems with a network structure Signup and view all the answers

What happens after allocating flow to a cell in the transportation tableau?

The row supply and column demand are reduced accordingly. Signup and view all the answers

How many variables could potentially be involved in a large transportation problem with 100 origins and 1000 destinations?

100,000 variables Signup and view all the answers

What is a key aspect involved in iterating to the optimal solution in the transportation simplex method?

Making improvements to the initial feasible solution Signup and view all the answers

What is indicated by eliminating a row from further consideration in the tableau?

The row supply has been exhausted completely. Signup and view all the answers

How can the transportation simplex method handle a situation where total supply does not equal total demand?

By allowing dummy supplies or demands. Signup and view all the answers

In the context of the transportation simplex method, what does the term 'problem variations' refer to?

Different types of transportation problems that can be solved Signup and view all the answers

What major characteristic distinguishes the assignment problem from the general transportation problem?

All supplies and demands are equal to 1. Signup and view all the answers

What best describes the entries in the right-hand margin of the transportation tableau?

The supply available at each origin Signup and view all the answers

During which phase of the transportation simplex method is the incoming arc utilized?

Phase II Signup and view all the answers

What should be done if not all row supplies and column demands are exhausted after an allocation?

Continue with the next allocation steps. Signup and view all the answers

What is the plant capacity for P₂?

500 Signup and view all the answers

Which origin has the lowest transportation cost to D₁?

O1 Signup and view all the answers

Using the Hungarian method, which project leader has the longest estimated completion time for their most efficient client?

Smith Signup and view all the answers

Which crew has the lowest cost for Job 3?

Blue Signup and view all the answers

What is the total warehouse demand across D₁, D₂, and D₃?

900 Signup and view all the answers

In a transportation problem, if the supply for O₁ is 250 and the total demand is 450, what can be inferred?

There is a surplus Signup and view all the answers

Which of the following best describes the purpose of the transportation simplex method?

Minimize transportation costs Signup and view all the answers

Which of the project leaders has the shortest estimated completion time for their least efficient client?

Jackson Signup and view all the answers

Study Notes

Transportation Problems

Transportation problems involve minimizing the cost of shipping goods from origins to destinations
Formulated using linear programming, involving variables for each route and constraints for each origin/destination node
Special-purpose solution procedures, like the transportation simplex method, streamline calculations for large-scale problems
Transportation tableau organizes data, entries show cost of moving unit from supply origin to demand destination
Minimum cost method is a common heuristic to find initial feasible solutions for transportation problems
It aims for a compromise between speed and optimality in solution

Assignment Problems

Assignment of agents to tasks, a special case of the transportation problem
All origins and destinations have supply/demand of 1 unit
Employ special procedures, such as Hungarian method, optimized for faster computations
Row and column reduction (Hungarian method) modifies matrix to find zero-cost solutions, ensuring an optimal assignment
Used when assigning resources (agents) to tasks (destinations) with the goal of minimizing cost or maximizing efficiency
Degeneracy can occur in assignment problems if the number of occupied cells in the tableau is less than the number of rows or columns. This case needs special handling to ensure appropriate solution.

Transportation Simplex Method

Two-phase iterative process (Phase I then Phase II) used for solving transportation problems
Phase I: Finds an initial feasible solution via heuristics (e.g., minimum cost method). Invariants: it satisfies supply/demand constraints of the problem.
Phase II: Iteratively improves the solution, one route at a time, to minimize transportation costs. It utilizes the MODI method and stepping-stone methods to find the incoming and outgoing routes at each step.
MODI and stepping-stone methods used to determine next optimal step; evaluate unoccupied routes to decide the route (cell) to allocate more flow, and decide which existing route (cell) to reduce

Important Terms

Incoming arc/cell: The unutilized route (cell) in transportation table whose usage will reduce the total cost most
Outgoing arc/cell: The utilized route (cell) to reduce to maintain feasibility in the transportation table. The used routes (cells) are adjusted to satisfy the incoming cell's new usage
Net evaluation index: Evaluate an unoccupied cell, the difference in cost of using it versus the cost of current cell
Degeneracy: A transportation solution is degenerate if the number of occupied cells (routes) is less than m + n - 1 where m is the number of origins and n is the number of destinations. This is handled by adding an artificial cell (route) with a zero cost in the tableau during calculation.
Tableau: A table used to organize data for linear optimization procedures (Transportation and Assignment Problems), visualizing routes and calculations steps. Visual representation of the problem and each iteration of the solution
Heuristic: A commonsense method to quickly find a solution, often used to find an initial feasible solution for large-scale transportation or assignment problems. This could be the minimum cost method to find initial starting feasible solution or a similar method

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Description

This quiz focuses on transportation and assignment problems, exploring methods like the transportation simplex and the Hungarian method for optimizing shipping and task assignment. It covers key concepts such as cost minimization and special procedures for efficient computations. Test your understanding of these essential topics in linear programming.