Unit 03 β Transportation & Assignment
Recognising the two problem types, checking whether they are balanced, counting variables and constraints, and converting the cost matrix into a full LPP.
The course session plan lists the full Unit 3 as including North-West Corner Rule, Least Cost Method, VAM, the Modified Distribution (MODI) method, degeneracy, maximisation problems and the Hungarian Method Policy p. 7.
Your notes do not cover any of those solution algorithms. Every page of the Unit 3 material β the handwritten Assignment Problems set pp. 1β6 and the transportation section of Note 2 pp. 12β18 β stops at formulation: identify the type, check balance, count variables and constraints, write the LPP.
This page therefore covers formulation thoroughly and does not invent algorithm content you were not taught. That is also what the papers asked: both 2025-26 Q1B items and both Q5B/Q7B items are pure formulation and balance questions. If your class later covers NWCR/VAM/MODI/Hungarian, tell me and I will add them.
- Q1B (2.5 marks), both papers β a cost matrix, then three fixed sub-questions: Is the problem balanced? Number of constraints (excluding NNC)? Number of variables? Final p. 1 Re-Exam p. 1. This is the most formulaic 2.5 marks on the paper. Solved β
- Q5B (5 marks), both papers β flight scheduling: "formulate an assignment problem to minimise aircraft overnight idle time". Solved β
- Q7B (5 marks), Final β "formulate in transportation format and identify the number of constraints and variables", with a profit twist. Solved β
1. Telling the Two Problems Apart Core syllabus concept
1Understand the Concept
Both problems arrive as a cost matrix. One sentence separates them Class notes p. 3:
| Transportation problem | Assignment problem | |
|---|---|---|
| Rows / columns | Sources (plants, warehouses) Γ Destinations (markets, regions) | Workers / machines / flights Γ Tasks / jobs / return flights |
| Extra data | A supply column and a demand row | None β just the cost matrix |
| Balanced when | Total supply = total demand | Number of rows (m) = number of columns (n) |
| Xα΅’β±Ό means | Quantity shipped from i to j (any non-negative number) | 1 if row i is assigned to column j, else 0 |
| Row constraint | Ξ£β±Ό Xα΅’β±Ό β€ supplyα΅’ | Ξ£β±Ό Xα΅’β±Ό = 1 (each worker gets exactly one task) |
| Column constraint | Ξ£α΅’ Xα΅’β±Ό = demandβ±Ό | Ξ£α΅’ Xα΅’β±Ό = 1 (each task gets exactly one worker) |
2. Balance, Variables and Constraints Core syllabus concept
1The three counting rules
Q1B in both papers is exactly these three rules applied to an m Γ n matrix. Memorise them as a block:
2How to balance an unbalanced problem
Whichever side is less, add a corresponding DUMMY row or column, and set every cost in the dummy to zero Class notes p. 1.
- Assignment: if rows β columns, add a dummy row (or column) of zeros so the matrix becomes square.
- Transportation: if Ξ£ supply β Ξ£ demand, add a dummy source (if supply is short) or dummy destination (if demand is short), with zero costs and the missing quantity as its supply/demand p. 3.
3Worked β the exact Q1B pattern, both versions
| T1 | T2 | T3 | |
|---|---|---|---|
| W1 | 9 | 2 | 7 |
| W2 | 6 | 4 | 3 |
| W3 | 5 | 8 | 1 |
Answer: (1) Yes, balanced β number of rows = number of columns = 3. (2) Constraints m + n = 3 + 3 = 6. (3) Variables m Γ n = 3 Γ 3 = 9. Matches the synoptic key exactly Key p. 1.
| P1 | P2 | P3 | Supply | |
|---|---|---|---|---|
| W1 | 9 | 2 | 7 | 30 |
| W2 | 6 | 4 | 3 | 25 |
| W3 | 5 | 8 | 1 | 20 |
| Demand | 20 | 30 | 35 |
Answer: Ξ£ supply = 30 + 25 + 20 = 75; Ξ£ demand = 20 + 30 + 35 = 85. (1) No, not balanced β demand exceeds supply by 10, so add a dummy row (source) with supply 10 and zero costs. (2) Constraints m + n = 3 + 3 = 6. (3) Variables m Γ n = 3 Γ 3 = 9. Matches the key Key p. 1.
The synoptic key counts m + n and m Γ n on the original matrix, not on the balanced one. Answer the counting parts from the matrix as given, and mention the dummy separately when asked how to balance it. Also note the phrase "excluding NNC" in the question β the non-negativity restriction is not counted as a constraint.
Practice H1βH2 β one unbalanced assignment, one unbalanced transportation β
3. Converting the Matrix into an LPP Core syllabus concept
1The template β assignment problem
| Employees | Task 1 | Task 2 | Task 3 | Task 4 |
|---|---|---|---|---|
| A | 10 | 12 | 8 | 15 |
| B | 8 | 10 | 12 | 11 |
| C | 12 | 15 | 9 | 13 |
| D | 11 | 9 | 14 | 10 |
Step 1 β housekeeping (write these four lines first, they carry marks):
Step 2 β objective function. Read the matrix left to right, row by row:
Step 3 β constraints. Each employee gets exactly one task; each task goes to exactly one employee.
Eight constraints β exactly the m + n you counted in Step 1. That internal check is worth doing out loud in the answer.
2The template β transportation problem
| D1 | D2 | D3 | D4 | Supply | |
|---|---|---|---|---|---|
| Oβ | 6 | 4 | 1 | 5 | 14 |
| Oβ | 8 | 9 | 2 | 7 | 16 |
| Oβ | 4 | 3 | 6 | 2 | 5 |
| Demand | 6 | 10 | 15 | 4 | 35 / 35 |
Step 1 β housekeeping.
Step 2 β objective function.
Step 3 β constraints. Note the asymmetry: supply is a ceiling (β€), demand must be met exactly (=).
Seven constraints = m + n. When the problem is perfectly balanced you may write the supply constraints as = too; the class notes use β€, which is the safer general form.
- Writing Xα΅’β±Ό without saying what i and j range over. "i β row 1 to 3, j β column 1 to 4" is a marked line.
- Mixing up the row and column constraint directions. Supply β€, demand =. For assignment, both are = 1.
- Transposing the cost matrix in the objective. Xββ is row 1, column 3 β read row-wise and it is impossible to slip.
- Answering "how many constraints" with m Γ n. Variables are m Γ n; constraints are m + n. They are swapped surprisingly often under time pressure.
Practice H3βH5 β full LPP conversion, profit variant, forbidden pairing β