Find the optimal solution for the assignment problem with following cost matrix. | Salesmen \ Area | 1 | 2 | 3 | 4 | | --- | --- | --- | --- | --- | | P | 11 | 17 | 8 | 16 | | Q | 9 | 7 | 12 | 6 | | R | 13 | 16 | 15 | 12 | | S | 14 | 10 | 12 | 11 |

> > [!TLDR] > Hungarian method: after row and column reduction the zeros allow the assignment $P\!-\!3,\ Q\!-\!4,\ R\!-\!1,\ S\!-\!2$; minimum cost $=37$. > > **Step 1 — Row reduction** (subtract each row's minimum: $8,6,12,10$): > > | | 1 | 2 | 3 | 4 | > | --- | --- | --- | --- | --- | > | P | 3 | 9 | 0 | 8 | > | Q | 3 | 1 | 6 | 0 | > | R | 1 | 4 | 3 | 0 | > | S | 4 | 0 | 2 | 1 | > > **Step 2 — Column reduction** (column 1 minimum is $1$; other columns already have a zero): > > | | 1 | 2 | 3 | 4 | > | --- | --- | --- | --- | --- | > | P | 2 | 9 | **0** | 8 | > | Q | 2 | 1 | 6 | **0** | > | R | **0** | 4 | 3 | 0 | > | S | 3 | **0** | 2 | 1 | > > **Step 3 — Assign zeros so that each row and column has exactly one.** > - P has its only zero in column 3 → **P → Area 3**. > - S has its only zero in column 2 → **S → Area 2**. …

Find the optimal solution for the assignment probl – Class 12 Business Mathematics Statistics

Question 38 of 38

Find the optimal solution for the assignment problem with following cost matrix.

Salesmen \ Area	1	2	3	4
P	11	17	8	16
Q	9	7	12	6
R	13	16	15	12
S	14	10	12	11

Tamil Nadu DgeTamil Nadu HSC (DGE) Commerce Board 2026Subjective· 3mImportance★★★★★

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Hungarian method: after row and column reduction the zeros allow the assignment $P\!-\!3,\ Q\!-\!4,\ R\!-\!1,\ S\!-\!2$ ; minimum cost $=37$ .

Step 1 — Row reduction (subtract each row's minimum: $8,6,12,10$ ):

	1	2	3	4
P	3	9	0	8
Q	3	1	6	0
R	1	4	3	0
S	4	0	2	1

Step 2 — Column reduction (column 1 minimum is $1$ ; other columns already have a zero):

	1	2	3	4
P	2	9	0	8
Q	2	1	6	0
R	0	4	3	0
S	3	0	2	1

Step 3 — Assign zeros so that each row and column has exactly one.

P has its only zero in column 3 → P → Area 3.
S has its only zero in column 2 → S → Area 2. …

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