Transportation Problem in Operations Research

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Questions and Answers

What is the initial total shipping cost before any adjustments?

$2020 (correct)

$1900

$-4

What procedure is formalized in this example as 'the transportation algorithm'?

Minimize z = ∑(c_ij * x_ij) subject to supply and demand constraints.

The cost is always reduced when units are sent along route (1,3).

True

When sending 20 units along route (1,3), the cost is reduced by ___.

$80 Signup and view all the answers

Match the following terms with their descriptions:

Supply = The total quantity available for shipment. Demand = The total quantity required at destinations. Entering Variable = A variable that enters the solution to improve cost. Leaving Variable = A variable that exits the solution to maintain feasibility. Signup and view all the answers

What does the equation $ ext{Minimize } z = rac{ ext{sum of costs}}{ ext{total shipment}}$ represent in the transportation problem?

Minimizing total shipping cost Signup and view all the answers

An optimal solution is reached when all entering variables are negative.

True Signup and view all the answers

What happens to the variable $x_{14}$ when $x_{13}=20$?

It becomes the leaving variable and is set to 0. Signup and view all the answers

What is the primary goal of formulating the transportation problem?

Minimize transportation cost Signup and view all the answers

What is the objective of the transportation problem formulated in the document?

Minimize transportation costs Signup and view all the answers

The transportation problem can only be solved using the simplex method.

False Signup and view all the answers

How many tons per week can the San Antonio plant supply?

100 tons Signup and view all the answers

The transportation problem is always feasible if the total supply equals the total demand.

True Signup and view all the answers

The San Antonio warehouse needs __________ tons per week.

120 Signup and view all the answers

What do the variables $x_{ij}$ represent in the transportation problem?

The amount shipped from the ith plant to the jth warehouse. Signup and view all the answers

Which shipping route has the lowest cost per ton from Salt Lake City?

Salt Lake City to San Antonio Signup and view all the answers

The total supply and total demand for the transportation model include _______ which must be equal for feasibility.

360 Signup and view all the answers

Match the shipping routes with their costs.

Salt Lake City to Los Angeles = $6 Denver to Los Angeles = $7 San Antonio to New York City = $80 Denver to San Antonio = $0 Signup and view all the answers

Which of the following constraints represents the supply from the first plant in the given model?

$x_{11} + x_{12} + x_{13} + x_{14} = 120$ Signup and view all the answers

The total supply of the plants matches the total demand of the warehouses in this transportation problem.

False Signup and view all the answers

Match the following terms with their correct descriptions:

Supply Vector = A representation of the availability of goods from sources Demand Vector = A representation of the required goods at destinations Feasibility = Condition where supply meets or exceeds demand Redundant Constraint = A constraint that is not necessary due to equality of supply and demand Signup and view all the answers

What type of system does the transportation problem apply to?

Logistical systems or distribution systems Signup and view all the answers

In the given model, all variables must be non-negative integers.

True Signup and view all the answers

Study Notes

Transportation Problem

The transportation problem involves finding the optimal way to transport goods from multiple origins (e.g., plants) to multiple destinations (e.g., warehouses).
The objective is to minimize the total transportation cost.
The problem is defined by constraints that ensure supply is equal to demand.

Example: Plastic Manufacturing Company

A plastic manufacturing company has three plants (Salt Lake City, Denver, San Antonio) and four warehouses (Los Angeles, Chicago, New York City, San Antonio).
The company wants to minimize the cost of shipping polyethylene from plants to warehouses.
The supply of each plant and the demand of each warehouse is known.

Model

Shipping costs are represented by a cost matrix (C).
Each cell in the matrix represents the cost of shipping one unit from the origin to the destination.
The model seeks to minimize total shipping cost by choosing the optimal shipping routes and amounts.
The model ensures that supply meets demand at each warehouse, and that no more than the plant's supply is shipped.

Transportation Algorithm

The transportation algorithm is used to solve the transportation problem.
The algorithm involves finding an initial basic feasible solution (using the minimum cost rule) and then iteratively improving it.
The improvement process involves identifying entering and departing variables, modifying the tableau, and re-evaluating the cost function.
The algorithm stops when no entering variable can be chosen, indicating an optimal solution has been found.

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Description

Explore the transportation problem which focuses on minimizing shipping costs while ensuring supply equals demand. This quiz will cover definitions, constraints, and application through a case study of a plastic manufacturing company. Test your understanding of optimal transportation methods and cost-effective solutions.