Question

2 Bill has just completed his exams for the academie year and wants to celebrate by seeing every movie showing in theaters in his town and in six other neighboring cities If he travels to another town, he will stay there until he has seen all the movies he wants The ollowing table provides the information about the movie offerings and the round-trip distance to the neighboring town 7.9 550 1,3 1,6,8 City City B City C 7.00 195 525 2.4.7 135,10 City E City P The cost of driving is 75 cents per mile. Bill wishes to determine the towns he needs lo visit to see all the movies while minimizing his total cost

Answer 1

Let X0 = 1, if visits in-town to see all the movies, otherwise X0 = 0

X1 = 1, if visits City A to see all the movies, otherwise X1 = 0

X2 = 1, if visits City B to see all the movies, otherwise X2 = 0

X3 = 1, if visits City C to see all the movies, otherwise X3 = 0

X4 = 1, if visits City D to see all the movies, otherwise X4 = 0

X5 = 1, if visits City E to see all the movies, otherwise X5 = 0

X6 = 1, if visits City F to see all the movies, otherwise X6 = 0

Xij = 1, if he watches movie j in town/city i , otherwise Xij = 0

Minimize .75*25X1+.75*30X2+.75*28X3+.75*40X4+.75*35X5+.75*32X6+7.95X01+7.95X03+5.5X11+5.5X16+5.5X18+5X22+5X25+5X27+7X31+7X38+7X39+4.95X42+4.95X44+4.95X47+5.25X51+5.25X53+5.25X55+5.25X510+6.75X64+6.75X65+6.75X66+6.75X69

s.t.

2X0-X01-X03>=0

3X1-X11-X16-X18>=0

3X2-X22-X25-X27>=0

3X3-X31-X38-X39>=0

3X4-X42-X44-X47>=0

4X5-X51-X53-X55-X510>=0

4X6-X64-X65-X66-X69>=0

X01+X11+X31+X51>=1

X22+X42>=1

X03+X53>=1

X44+X64>=1

X25+X55+X65>=1

X16+X66>=1

X27+X47>=1

X18+X38>=1

X39+X69>=1

X510>=1

Xi, Xij binary

Solution using LINGO is following:

Lingo 17.0-Lindo Model-Lingo! File Edit Solver Window Help Lindo Model - Lingo1 Solution Report Lingol Global optimal solution found Objective value: Objective bound: Infeasibi1ities: Minimize 18.75X1+22.5X2+21X3+30X4+26.25x5 +24X6+7.95X01+7.95XO3+5.5X11+5.5X16+5. 5X18+5X22 +5X25+5X27+7X31+7X38+7X39+4.95X42+4.95X44+4.95X47 +5.25XS1+5.25X53+5.25X55+5.25XS10+6.75X64+6.75X65 +6.75X66+6.75X69 146.7500 146.7500 0.000000 Total solver iterations: Elapsed runtime seconds: PILP Model Class: 2X0-X01-X03>-0 3X1-X11-X16-X18 >=0 3X2-X22-X25-X27-0 3X3-X31-X38-X39-0 3X4-X42-X44-X47-0 4X5-X51-X53-X55-X510>-0 4X6-X64-X65-X66-X69-0 Total variables: Nonlinear variables: Integer variables: Total constraintS: Nonlinear constraints: Total nonzeros: Nonlinear nonzeros: X22+X42>=1 X03+X53-1 X44+X64-1 X25+X55+X65-1 X16+X66-1 X27+X47-1 Value 1.000000 1.000000 0.000000 0.000000 1.000000 1.000000 0.000000 0.000000 0.000000 1.000000 1.000000 1.000000 1.000000 1.000000 0.000000 0.000000 Variable X1 X2 Reduced Cost 18.75000 22.50000 21.00000 30.00000 26.25000 24.00000 7.950000 7.950000 5.500000 5.500000 5.500000 5.000000 5.000000 5.000000 7.000000 7.000000 X510-1 END INT XO INT X INT X2 INT X3 INT X INT XS INT X INT X01 INT X03 INT X11 INT X16 X18 X22 X25 X38 Ln 14, Col 19 4:34 pm For Help, press F1 CAP NUM MOD

Variable X1 Reduced Cost 18.75000 22.50000 21.00000 30.00000 26.25000 24.00000 7.950000 7.950000 5.500000 5.500000 5.500000 5.000000 5.000000 5.000000 7.000000 7.000000 7.000000 4.950000 4.950000 4.950000 5.250000 5.250000 5.250000 5.250000 6.750000 6.750000 6.750000 6.750000 0.000000 2 value 1.000000 1.000000 0.000000 0.000000 1.000000 1.000000 0.000000 0.000000 0.000000 1.000000 1.000000 1.000000 1.000000 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 1.000000 0.000000 1.000000 1.000000 0.000000 0.000000 1.000000 1.000000 X3 X4 X5 X6 X01 X03 X11 X16 X18 X22 X25 X27 X31 X38 X39 X42 X44 X47 X51 X53 X55 X510 X64 X65 X66 X69 ХО

Dual Price -1 000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 Slack or Surplus 146.7500 2.000000 1.000000 0.000000 0.000000 0.000000 1.000000 2.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 Row 2 7 10 12 13 14 15 16 17 18

Optimal solution:

X1, X2, X5, X6 = 1

Therefore, Bill should visit towns A, B, E, F

He should watch movies as per following schedule:

Theater location	Movies to be watched
In-town
City A	6,8
City B	2,5,7
City C
City D
City E	1,3,10
City F	4,9

Total cost of travel and movies = 146.75