Operations Research · Unit 3
Transportation Problems
Exam-focused notes for Transportation Problems (Operations Research, ORS255): what the TU syllabus asks and how it has actually been tested, with 11 solved past questions from this unit.
What this unit covers
- Formulation of transportation problems
- Initial solution methods: Northwest Corner Rule
- Vogel's Approximation Method (VAM)
- Modified Distribution (MODI) method
- Testing optimality of transportation solutions
- Minimization and maximization of transportation costs
- Demand and supply constraints
Minimization and maximization of transportation costs
The table below represent the profit of a company earned from different plants to different market. Develop a transportation schedule that maximizes the profit of the company.
$$\begin{array}{|c|ccc|c|}\hline \text{Plants/Market} & M1 & M2 & M3 & \text{Supply (units)} \ \hline P1 & 22 & 25 & 24 & 170 \ \hline P2 & 15 & 20 & 18 & 130 \ \hline P3 & 30 & 21 & 20 & 100 \ \hline \text{Demand (units)} & 200 & 130 & 120 & 400/450 \ \hline \end{array}$$
[10]
Profit matrix (units per unit shipped): Plant/Market M1 M2 M3 Supply --------------- P1 22 25 24 170 P2 15 20 18 130 P3 30 21 20 100 Demand 200 130 120 - Total Supply = $170 + 130 + 100 = 400$ - Total Demand = $200 + 130 + 120 = 450$ Since Supply (400) < De...
Full solved answer →Transportation Problem - Minimum Cost Solution
Obtain the minimum transportation cost for the following transportation problem.
$$\begin{array}{|c|ccc|c|}\hline \text{Destination/Source} & S1 & S2 & S3 & \text{Units Demanded} \ \hline D1 & 12 & 16 & 10 & 200 \ \hline D2 & 15 & 12 & 10 & 200 \ \hline D3 & 11 & 12 & 12 & 50 \ \hline \text{Units Available} & 200 & 150 & 100 & 450 \ \hline \end{array}$$
[10]
I must read the matrix carefully. The LaTeX has two conflicting layouts. The trailing (cleaner) array is: $$ \begin{array}{ccccc}\hline \text{Destination/Source} & D1 & D2 & D3 & \text{Units Available} \\ \hline S1 & 12 & 15 & 11 & 200 \\ S2 & 16 & 12 & 12 ...
Full solved answer →Transportation Problem
Find transport schedule to minimize the transportation cost for the following transportation problem. The transportation cost per unit and units demanded and available are given in the table.
$$\begin{array}{|c|c|c|c|c|}\hline & \text{A} & \text{B} & \text{C} & \text{Units demanded} \ \hline X & 9 & 10 & 10 & 5 \ Y & 10 & 14 & 8 & 20 \ Z & 13 & 10 & 8 & 20 \ \hline \text{Units available} & 20 & 15 & 10 & 45 \ \hline \end{array}$$
[10]
The table structure needs careful reading. The headers list A, B, C, and "Units demanded" across the top, and the bottom row is "Units available." The row labels X, Y, Z are the supply sources. Cost matrix (rows = X, Y, Z; columns = A, B, C): A B C Units de...
Full solved answer →Determine the minimum transportation cost from the following matrix.
| Warehouse/Store Cost per unit | P₁ | P₂ | P₃ | P₄ | Supply |
|---|---|---|---|---|---|
| W₁ | 45 | 60 | 45 | 30 | 70 |
| W₂ | 35 | 15 | 35 | 35 | 60 |
| W₃ | 30 | 25 | 45 | 55 | 90 |
| Demand | 60 | 40 | 60 | 20 | 220/180 |
[10]
Cost matrix (per unit): W/S P₁ P₂ P₃ P₄ Supply ----------------------------- W₁ 45 60 45 30 70 W₂ 35 15 35 35 60 W₃ 30 25 45 55 90 Demand 60 40 60 20 Note on data: The supply column shows "$\frac{220}{180}$" which represents total supply = 220 and total dem...
Full solved answer →Modified Distribution
Describe modified distribution (MODI) method of obtaining the optimal solution of transportation problem. [5]
The MODI method (also called the Multiplier method or u-v method) is an iterative technique used to find the optimal solution to a transportation problem after an initial basic feasible solution has been obtained. The MODI method works by calculating opport...
Full solved answer →Explain the algorithm of the Modified Distribution (MODI) method for testing the optimality of the transportation problem. [5]
The MODI method (also called the u-v method) tests whether a basic feasible solution to a transportation problem is optimal. Here is the algorithm: For each basic variable (occupied cell) in the current solution, assign dual variables ui and vj such that: $...
Full solved answer →Describe modified distribution (MODI) method used for testing the optimality of initial solution of transport problem. [5]
The MODI method (also called the u-v method) is a technique used to test whether an initial basic feasible solution of a transportation problem is optimal, and if not, to improve it systematically. The MODI method is based on the duality theory of linear pr...
Full solved answer →Find the optimum transportation schedule from the following data in order to minimize transportation costs by using modified distribution method.
| Plant | X | Y | Z | Supply (units) |
|---|---|---|---|---|
| A | 5 | 2 | 8 | 150 |
| B | 4 | 3 | 5 | 150 |
| C | 2 | 4 | - | 200 |
| D | 6 | 3 | 4 | 250 |
| Demand (units) | 250 | 200 | 175 | 625/750 |
[10]
Cost matrix (Plant → Destination): Plant X Y Z Supply ------------------------ A 5 2 8 150 B 4 3 5 150 C 2 4 - 200 D 6 3 4 250 Demand 250 200 175 - Total Supply = 150+150+200+250 = 750 - Total Demand = 250+200+175 = 625 - Cell C-Z is prohibited (marked "-")...
Full solved answer →Vogel's Approximation Method
Write short notes on: (a) Vogel's Approximation Method (VAM) (b) Objectives of operations research [0+2.5+2.5]
Definition: Vogel's Approximation Method is an improved initial solution technique for the Transportation Problem that generally produces a better starting solution than the North-West Corner Method or Least Cost Method. Principle: VAM is based on the conce...
Full solved answer →Write short notes on: (a) Vogel's Approximation Method (VAM) (b) Dominance rule of game theory [0+2.5+2.5]
Definition: Vogel's Approximation Method is an iterative procedure for finding an initial basic feasible solution to the Transportation Problem. It is more efficient than the North-West Corner Rule as it produces solutions closer to the optimal solution. Pr...
Full solved answer →Find the initial solution by using Vogel's Approximation Method (VAM).
$$\begin{array}{|c|c|c|c|c|c|}\hline \text{From} & P & Q & R & S & \text{Supply} \ \hline A & 19 & 30 & 50 & 10 & 7 \ B & 70 & 30 & 40 & 60 & 9 \ C & 40 & 8 & 70 & 20 & 18 \ \hline \text{Demand} & 5 & 8 & 7 & 14 & 34 \ \hline \end{array}$$
[5]
Cost matrix: From P Q R S Supply ------------------ A 19 30 50 10 7 B 70 30 40 60 9 C 40 8 70 20 18 Demand 5 8 7 14 Total supply = 7 + 9 + 18 = 34; total demand = 5 + 8 + 7 + 14 = 34. Balanced. Penalty = difference between two smallest costs in each row/col...
Full solved answer →Make Unit 3 stick
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