Q.6 (A)(ii) Three new machines M₁, M₂, M₃ are to be installed in a machine shop. There are four vacant places A, B, C, D. Due to limited space, machine M₂ can not be placed at B. The cost matrix (in hundred ₹) is as follows: Machine / PlacesABCD M₁ 13101211 M₂ 15–1320 M₃ 57106 Determine the optimum assignment schedule and find the minimum cost.

Chapter: Assignment Problem and Sequencing· Mathematics & Statistics (Commerce) Part 2

Three new machines \(M_1, M_2, M_3\) are to be installed in a machine shop. There are four vacant places A, B, C, D. Due to limited space, machine \(M_2\) can not be placed at B. The cost matrix (in hundred ₹) is as follows :Machine / PlacesABCD\(M_1…

Q.6 (A)(ii) Three new machines \(M_1, M_2, M_3\) are to be installed in a machine shop. There are four vacant places A, B, C, D. Due to limited space, machine \(M_2\) can not be placed at B. The cost matrix (in hundred ₹) is as follows :

Machine / Places	A	B	C	D
\(M_1\)	13	10	12	11
\(M_2\)	15	–	13	20
\(M_3\)	5	7	10	6

Determine the optimum assignment schedule and find the minimum cost.

Doubtly Asked 29 Oct 2026 Board exam · March 2024 hsc 12th Save WhatsApp Report

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Answer: \(M_1 \to B\), \(M_2 \to C\), \(M_3 \to A\) (place D left vacant); minimum cost = 28 hundred ₹ = ₹ 2,800.

The problem is unbalanced (3 machines, 4 places), so add a dummy machine \(M_4\) with zero costs. The prohibited cell \(M_2\)–B is given a very large cost M.

	A	B	C	D
\(M_1\)	13	10	12	11
\(M_2\)	15	M	13	20
\(M_3\)	5	7	10	6
\(M_4\) (dummy)	0	0	0	0

Row reduction (subtract 10, 13, 5, 0 from the rows):

	A	B	C	D
\(M_1\)	3	0	2	1
\(M_2\)	2	M	0	7
\(M_3\)	0	2	5	1
\(M_4\)	0	0	0	0

Column reduction changes nothing, since every column already has a zero (the dummy row). Minimum number of lines covering all zeros = 4 = order of the matrix, so an optimal assignment exists.

Row \(M_1\) has a single zero at B, row \(M_2\) a single zero at C and row \(M_3\) a single zero at A; the dummy \(M_4\) takes D.

Machine	Place	Cost (hundred ₹)
\(M_1\)	B	10
\(M_2\)	C	13
\(M_3\)	A	5
\(M_4\) (dummy)	D	0
Total		28

Minimum cost = 28 hundred rupees = ₹ 2,800. Place D remains vacant.

Doubtly AI · 29 Oct 2026 AI-verified · Nov 2026 Report

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Three new machines \(M_1, M_2, M_3\) are to be installed in a machine shop. There are four vacant places A, B, C, D. Due to limited space, machine \(M_2\) can not be placed at B. The cost matrix (in hundred ₹) is as follows :Machine / PlacesABCD\(M_1…

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