Consider the problem in the below figure. Your goal is to find the places of the digits from 1 to 9 in grey cells i...

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Consider the problem in the below figure. Your goal is to find the places of the digits from 1 to 9 in grey cells in such a w

math Advanced-Math

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Answer 1

Answer #1

IF YOU HAVE ANY DOUBTS COMMENT BELOW I WILL BE TTHERE TO HELP YOU..ALL THE BEST..

AS FOR GIVEN DATA.

a)

Let us use following notations to identify each of the nine positions in the grid

A1 13 8 12 B2 14 B3 15 7 C3 C2 C1

Let XJK = 1, if value assigned to grid XJ is K, otherwise XJK = 0 , where X = {A,B,C}, J = {1,2,3} and K = {1,2,3,...9}

For example, A11 = 1, if value of 1 is assigned to cell A1 in the above grid, otherwise A11 = 0

Mixed-interger linear programming model is following:

Max 0

s.t.

A11+A21+A31+B11+B21+B31+C11+C21+C31 = 1

A12+A22+A32+B12+B22+B32+C12+C22+C32 = 1

A13+A23+A33+B13+B23+B33+C13+C23+C33 = 1

A14+A24+A34+B14+B24+B34+C14+C24+C34 = 1

A15+A25+A35+B15+B25+B35+C15+C25+C35 = 1

A16+A26+A36+B16+B26+B36+C16+C26+C36 = 1

A17+A27+A37+B17+B27+B37+C17+C27+C37 = 1

A18+A28+A38+B18+B28+B38+C18+C28+C38 = 1

A19+A29+A39+B19+B29+B39+C19+C29+C39 = 1

A11+A12+A13+A14+A15+A16+A17+A18+A19 = 1

A21+A22+A23+A24+A25+A26+A27+A28+A29 = 1

A31+A32+A33+A34+A35+A36+A37+A38+A39 = 1

B11+B12+B13+B14+B15+B16+B17+B18+B19 = 1

B21+B22+B23+B24+B25+B26+B27+B28+B29 = 1

B31+B32+B33+B34+B35+B36+B37+B38+B39 = 1

C11+C12+C13+C14+C15+C16+C17+C18+C19 = 1

C21+C22+C23+C24+C25+C26+C27+C28+C29 = 1

C31+C32+C33+C34+C35+C36+C37+C38+C39 = 1

(1A11+2A12+3A13+4A14+5A15+6A16+7A17+8A18+9A19)+(1B11+2B12+3B13+4B14+5B15+6B16+7B17+8B18+9B19) = 12

(1B11+2B12+3B13+4B14+5B15+6B16+7B17+8B18+9B19)+(1C11+2C12+3C13+4C14+5C15+6C16+7C17+8C18+9C19) = 7

(1A21+2A22+3A23+4A24+5A25+6A26+7A27+8A28+9A29)+(1B21+2B22+3B23+4B24+5B25+6B26+7B27+8B28+9B29) = 8

(1A21+2A22+3A23+4A24+5A25+6A26+7A27+8A28+9A29)+(1A31+2A32+3A33+4A34+5A35+6A36+7A37+8A38+9A39) = 7

(1B21+2B22+3B23+4B24+5B25+6B26+7B27+8B28+9B29)+(1B31+2B32+3B33+4B34+5B35+6B36+7B37+8B38+9B39) = 14

(1A31+2A32+3A33+4A34+5A35+6A36+7A37+8A38+9A39)+(1B31+2B32+3B33+4B34+5B35+6B36+7B37+8B38+9B39) = 13

(1B31+2B32+3B33+4B34+5B35+6B36+7B37+8B38+9B39)+(1C31+2C32+3C33+4C34+5C35+6C36+7C37+8C38+9C39) = 15

All variables are binary

Solution using Excel Solver is following:

Value l A1 | A2 | A3 | B1 | B2 | B3 | C1 | C2 | C3 | TOTAL SSolver Parameters SKS11 Set Objective: O Max O Min Value Of: 9 By

EXCEL FORMULAS:

Value	A1	A2	A3	B1	B2	B3	C1	C2	C3	TOTAL
1	0	0	0	0	0	0	0	1	0	=SUM(B2:J2)
2	0	1	0	0	0	0	0	0	0	=SUM(B3:J3)
3	0	0	0	1	0	0	0	0	0	=SUM(B4:J4)
4	0	0	0	0	0	0	1	0	0	=SUM(B5:J5)
5	0	0	1	0	0	0	0	0	0	=SUM(B6:J6)
6	0	0	0	0	1	0	0	0	0	=SUM(B7:J7)
7	0	0	0	0	0	0	0	0	1	=SUM(B8:J8)
8	0	0	0	0	0	1	0	0	0	=SUM(B9:J9)
9	1	0	0	0	0	0	0	0	0	=SUM(B10:J10)
TOTAL	=SUM(B2:B10)	=SUM(C2:C10)	=SUM(D2:D10)	=SUM(E2:E10)	=SUM(F2:F10)	=SUM(G2:G10)	=SUM(H2:H10)	=SUM(I2:I10)	=SUM(J2:J10)	=SUM(B11:J11)


	1	2	3
A	=SUMPRODUCT(B2:B10,A2:A10)	=SUMPRODUCT(C2:C10,A2:A10)	=SUMPRODUCT(D2:D10,A2:A10)	=SUM(B14+B16)	=	12
				=SUM(B16+B18)	=	7
B	=SUMPRODUCT(E2:E10,A2:A10)	=SUMPRODUCT(F2:F10,A2:A10)	=SUMPRODUCT(G2:G10,A2:A10)	=SUM(D14+D16)	=	8
				=SUM(D14+F14)	=	7
C	=SUMPRODUCT(H2:H10,A2:A10)	=SUMPRODUCT(I2:I10,A2:A10)	=SUMPRODUCT(J2:J10,A2:A10)	=SUM(D16+F16)	=	14
				=SUM(F14+F16)	=	13
				=SUM(F16+F18)	=	15

I HOPE YOU UNDERSTAND..

PLS RATE THUMBS UP..ITS HELPS ME ALOT..

THANK YOU...!!

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