Operations Management: Transportation Problems

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Questions and Answers

What is the total supply available from source S1?

50 (correct)

Which row has the highest penalty for unmet demand?

D3 (correct)

How much supply from S2 is allocated to demand D3?

7 (correct)

What is the total demand to be satisfied across all demands?

34 Signup and view all the answers

What indicates whether the allocation is a basic feasible solution?

There are no unmet demands within the supply limits. Signup and view all the answers

What is the total cost calculated for the given allocation?

Rs 1,015 Signup and view all the answers

Which statement accurately describes the supply status based on the given constraints?

Demand exceeds supply in at least one category. Signup and view all the answers

What does the 'Non-negativity condition' imply in this context?

All allocations must be positive or zero. Signup and view all the answers

What is indicated by the equation m(3) + n(4) – 1 = 6 in the context of the allocation problem?

The number of basic variables in the solution. Signup and view all the answers

In this scenario, which term refers to the starting point for the solution process?

Basic feasible solution Signup and view all the answers

What are the transportation costs per unit from S1 to D1?

19 Signup and view all the answers

Which facility has the highest production capacity?

S3 Signup and view all the answers

How many units does D2 require?

8 Signup and view all the answers

What is the total demand across all warehouses?

34 Signup and view all the answers

What is the cost incurred for shipping from S2 to D4?

60 Signup and view all the answers

What does the penalty in transportation cost indicate?

Additional cost incurred if the best route is not used Signup and view all the answers

Which production facility provides the least transportation cost overall?

S3 Signup and view all the answers

What is the initial basic solution total cost calculated?

Rs 814 Signup and view all the answers

What should be done if one or more unoccupied cells have a negative value of $d_{ij}$?

Select the unoccupied cell with the largest negative value for entering the solution mix. Signup and view all the answers

In the MODI method, what is the next step after selecting an unoccupied cell with the largest negative opportunity cost?

Construct a closed-path (or loop) starting from the selected unoccupied cell. Signup and view all the answers

How should signs be marked when tracing a path in the closed-path construction of the MODI method?

Start with a plus sign and alternate with minus signs. Signup and view all the answers

What does a negative value of $d_{ij}$ indicate in the context of the transportation problem?

An unoccupied cell can be improved upon by entering it into the basis. Signup and view all the answers

What is the purpose of the closed-path loop in step 5 of the MODI method?

To facilitate the adjustment necessary to improve the transportation schedule. Signup and view all the answers

What is the purpose of calculating opportunity costs in the MODI method?

To determine if an alternative solution exists Signup and view all the answers

In the context of the MODI method, what does a positive $d_{ij}$ value indicate?

The current basic feasible solution is optimal Signup and view all the answers

What equation is used to evaluate the opportunity costs of unused routes in the MODI method?

$d_{rs} = c_{rs} - (u_r + v_s)$ Signup and view all the answers

Why might a basic feasible solution remain unaffected despite a $d_{ij}$ value of zero?

It reveals that an alternative solution exists Signup and view all the answers

What does the variable $u_i$ represent in the context of the transportation problem?

The potential value at supply nodes Signup and view all the answers

Which of the following statements about the transportation tableau is true?

The sum of the supply must equal the sum of the demand Signup and view all the answers

What is represented by the $v_j$ values in the MODI method?

The potential value at destination nodes Signup and view all the answers

What scenario describes an unbalanced problem in transportation?

Quantity demanded is larger than supply Signup and view all the answers

How many supply sources are represented in the transportation problem outlined?

4 Signup and view all the answers

In the transportation problem, how is degeneracy resolved?

By allocating a small quantity to unoccupied cells Signup and view all the answers

When quantity demanded supposedly equals quantity supplied, what term represents the excess supply?

S excess Signup and view all the answers

If the supply is 27 and the demand is 23, what conclusion can be drawn?

There is a surplus of 4 units of supply Signup and view all the answers

What is the effect of allocating a very small quantity denoted by epsilon in the initial solution?

It helps to achieve the required number of occupied cells Signup and view all the answers

In a transportation problem, what does it mean when excessive supply exists?

Supply must be reduced to meet demand Signup and view all the answers

If the demand at D1 is 5 units and S1 can supply 3 units, what does this indicate?

S1 cannot meet D1's full demand Signup and view all the answers

What can be inferred about the rows and columns in an unbalanced transportation problem?

The total supply will always exceed the total demand Signup and view all the answers

Study Notes

Operations Management MBA 2024-2025

Course offered at ICFAI Business School, Hyderabad
Taught by Dr. Hasanuzzaman, Assistant Professor, Operations & Information Technology

Transportation

Transportation is a core topic in operations management.
Different scenarios for transportation are used for analysis.
Transportation problems can be balanced or unbalanced
Minimizing cost and/or time is a primary objective.

Transportation Problem (LP Model)

Objective: Determine the optimal number of units to ship from origins to destinations at minimum cost or time, maximizing profit.
Assumptions:
- Total quantity available at sources equals total requirement at destinations.
- Convenient transportation from all sources to all destinations.
- Unit transportation costs are accurately known.
- Transportation cost is directly proportional to the number of units shipped.
- Objective focuses on the organization as a whole, not individual supply/distribution centers.
Transportation Table: Presents supply (availability) at sources, demand (requirement) at destinations, and transportation costs between sources and destinations.
Variables: x_ij represents the number of units shipped from source i to destination j.
Objective Function: Minimize total transportation cost: Z = Σ_i=1^mΣ_j=1ⁿ c_ijx_ij
Constraints:
- Supply Constraint: Σ_j=1ⁿ x_ij = a_i for each source i
- Demand Constraint: Σ_i=1^m x_ij = b_j for each destination j
- Non-negativity Constraint: x_ij ≥ 0 for all i and j
Feasible Solution: Any set of non-negative allocations satisfying both supply and demand constraints.
Basic Feasible Solution: A feasible solution with exactly (m + n - 1) non-zero allocations.
Non-degenerate Solution: A basic feasible solution with exactly (m + n - 1) allocated routes..
Degenerate Solution: A basic feasible solution with fewer than (m + n - 1) allocations.

Transportation Problem - Algorithms

Methods for initial basic feasible solution:
- North-West Corner Method (NWCM)
- Least Cost Method (LCM)
- Vogel's Approximation Method (VAM)
Optimality Test:
- Stepping Stone Method
- Modified Distribution Method (MODI)

Transportation - Example

Illustrative problem of a company with three production facilities and four warehouses, with given transportation costs and capacities.
Students are expected to formulate an LP model to minimize total transportation costs.

Initial Basic Solution

Different methods to find the initial basic feasible solution (NWCM, LCM, VAM)

Transportation Problem - Important Definitions

Feasible solution: A set of non-negative allocations that satisfies all supply and demand constraints.
Basic feasible solution: A feasible solution with exactly (m+ n -1) non zero allocations, where m is number of rows and n is number of columns..
Non-degenerate basic feasible solution: Has exactly (m + n - 1) allocations
Degenerate basic feasible solution: Has less than (m + n - 1) allocations.
Optimal solution: A feasible solution that minimizes the total transportation cost

Transportation Problems – Special Cases

Unbalanced Transportation Problems: When total supply is not equal to total demand
Degeneracy in Initial Solutions: When the number of occupied cells in the initial basic feasible solution is fewer than (m + n – 1)
Degeneracy in Subsequent Iterations: When the number of occupied cells in an intermediate or final solution is fewer than (m + n – 1)
Alternative Optimal Solution: When two or more feasible solutions have the same transportation cost.

Test for Optimality (MODI Method)

Method for determining if a transportation solution is optimal.
Requires calculation of opportunity costs for unoccupied cells.
An unoccupied cell with the lowest negative opportunity cost is chosen.
Iterative process to find the best allocation of resources that minimizes transportation costs.

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Description

This quiz explores the core concepts of transportation in Operations Management, focusing on the transportation problem as a linear programming model. It covers scenarios for balancing shipping needs and aims to minimize costs and maximize profits while adhering to specific assumptions regarding sources and destinations.