Question

Please state what is the answer and how you get it, thanks.

Problem 7-21 The Bayside Art Gallery is considering installing a video camera security system to reduce its insurance premiums. A diagram of the eight display rooms that Bayside uses for exhibitions is shown in the figure below, the openings between the rooms are numbered 1-13. A security firm proposed that two-way cameras be installed at some room openings. Each camera has the ability to monitor the two rooms between which the camera is located. For example, if a rooms 1 covered; if a camera were located at opening 11, rooms 7 and 8 would be covered; and camera were located at opening number 4, and 4 would be so on. Management decided not to locate a camera system at the entrance to the display rooms. The objective is to provide security coverage for a eight rooms using the minimum number of two-way cameras.

Answer 1

This is a set covering problem. There is a simple logic to solve. here, 'xi' are binary integers and are equal to 1 when opening '1' is selected and '0' if not selected.

The objective function is simply the number of cameras so sum of all xi

Constraints are such that every room is occupied by at least one camera.

With these, the formulation will be:

Solution

X11 X2+1 X31 X41 X5 X61 X101 X111 X121 X13 Room 1 Room 2 Room 3 Room 4 Room 5 Room 6 Room 7 Room 8 X12 X51 X91 X10 1 X10 1 x21 X111 X12 1 X13 1 X2+ X51 X11

x1 x2 x3 x4 x5 x6 X7 x8 x9 x10 x11 x12 x13 No. of cameras Set Objective 1 |0 |0 |0 |0 |0 |1 |0 |0 |0 1 1 10 i:SUM(82-2 RHSTo:Max O Value of: 3 Subject to, 4 Room 1 1 SUMPRODUCT(SB$2:SN$2,B4:N4)1 -SUMPRODUCT(SB$2:SN$2,B5:N5)1 -SUMPRODUCT(SB$2:SN$2,B6:N6>1 -SUMPRODUCT(SB$2:$N$2,B7:NT) SUMPRODUCT$B$2:$N$2,BS:N8)1 By Changing Variable Cells: SBS2.SNS Room 2 6 Room 7 Room 4 8 Room 5 9 Room 6 10 Room 7 11 Room 8 12 13 1 1 1 Subject to the Constraints: SBS2 SNS2 binary SOS4 SOS11 -$Q54:SQS11 Add 1 1 SUMPRODUCT(SB$2:SN$2,B9:N9)1 1 >_ 1 SUMPRODUCT(SB$2:SN$2,B10:N10) :SUMPRODUCT(SBS2:SN$2,B11:N11) 丨 Delete Reset All 15 Load/Save 16 Make Unconstrained Variables Non-Negative Select a Solving Method: Simplex LP 19 20 21 Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear, Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth. 23 Sheet1Sheet2 Sheet3 Sheet4

So, the number of cameras required = 4 -----(b)

And they will be installed at opening 1, 7, 11, and 12.------------(b)

(c)

(d)

0 P a x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 No. of cameras Set Objective: To: O Max ⓔMin。Yolue Of: 3 Subject to, 4 Room 1 5 Room2 6 Room 3 1 1 1 7 Room 4 8 Room 5 9 Room 6 10 Room 7 11 Room 8 12 13 14 15 16 17 RHS :SUMPRODUCT($B$2:$N$2,84:N4) >= 1 :SUMPRODUCT($B$2:$N$2,B5:N5) >= 1 SUMPRODUCT(SBS2:SN$2,86:N6) >= 1 -SUMPRODUCT(SBS2:SNS2,B7:N7) >= 1 SUMPRODUCT(SB$2:SN$2,B8:N81 -SUMPRODUCT(SBS2:SN$2,89:N9) >= 1 SUMPRODUCT(SB$2:SN$2,B10:N10)2 =SUMPRODUCT($B$2:SN$2,B11:N11) >=1 By Changing Variable Cells: SB$2:SNS2 Subject to the Constraints: SBS2 SNS2 - binary SOS4:SOS11 >= SQ54 SQS11 Add 1 1 1 Delete Reset All Load/Save Make Unconstrained Variables Non-Negative Select a Soing Simplex LFP Method: Options 20 21 Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth. 23

So, the number of cameras required = 5 -----(b)

And they will be installed at opening 1, 5, 6, 10, and 11.------------(b)