The document describes the results of several linear programming optimization problems involving the allocation of resources with constraints. The problems include selecting radio station licenses, installing traffic monitoring devices, and determining locations for distribution centers and transporting natural gas. For each problem, the optimal solution and objective value are provided.

1. 11-7) select the highest quality combination of applications for low power radio station licences.max 45x1+30x2+84x3+73x4+80x5+70x6+61x7+91x8stx1+x2≤1x2+x5≤1x3≤1x4+x7≤1x5+x7+x6≤1x6+x7≤1x7+x4≤1x8≤1endinte x1inte x2inte x3inte x4inte x5inte x6inte x7inte x8LP OPTIMUM FOUND AT STEP 12 OBJECTIVE VALUE = 373.000000 NEW INTEGER SOLUTION OF 373.000000 AT BRANCH 0 PIVOT 12 RE-INSTALLING BEST SOLUTION... OBJECTIVE FUNCTION VALUE 1) 373.0000 VARIABLE VALUE REDUCED COST X1 1.000000 -45.000000 X2 0.000000 -30.000000 X3 1.000000 -84.000000 X4 1.000000 -73.000000 X5 1.000000 -80.000000 X6 0.000000 -70.000000 X7 0.000000 -61.000000 X8 1.000000 -91.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 0.000000 3) 0.000000 0.000000 4) 0.000000 0.000000 5) 0.000000 0.000000 6) 0.000000 0.000000 7) 1.000000 0.000000 8) 0.000000 0.000000 9) 0.000000 0.000000 NO. ITERATIONS= 12 BRANCHES= 0 DETERM.= 1.000E 011-9) Installing automatic traffic monitoring devices to providing full coveragemin 40x1+65x2+43x3+48x4+72x5+36x6stx1+x2≥1x1+x4≥1x2+x3≥1x2+x5≥1x3+x5≥1x3+x6≥1x4+x5≥1x5+x6≥1endinte x1inte x2inte x3inte x4inte x5inte x6LP OPTIMUM FOUND AT STEP 20 OBJECTIVE VALUE = 152.000000 NEW INTEGER SOLUTION OF 155.000000 AT BRANCH 0 PIVOT 20 RE-INSTALLING BEST SOLUTION... OBJECTIVE FUNCTION VALUE 1) 155.0000 VARIABLE VALUE REDUCED COST X1 1.000000 40.000000 X2 0.000000 65.000000 X3 1.000000 43.000000 X4 0.000000 48.000000 X5 1.000000 72.000000 X6 0.000000 36.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 0.000000 3) 0.000000 0.000000 4) 0.000000 0.000000 5) 0.000000 0.000000 6) 1.000000 0.000000 7) 0.000000 0.000000 8) 0.000000 0.000000 9) 0.000000 0.000000 NO. ITERATIONS= 20 BRANCHES= 0 DETERM.= 1.000E 011-9) Installing automatic traffic monitoring devices to providing full coverage(minimizing the number of uncovered road links while using at most 2 stations)min y1+y2+y3+y4+y5+y6+y7+y8stx1+x2+y1>1x1+x4+y2>1x2+x3+y3>1x2+x5+y4>1x3+x5+y5>1x3+x6+y6>1x4+x5+y7>1x5+x6+y8>1x1+x2+x3+x4+x5+x611-9) Installing automatic traffic monitoring devices to providing full coverage(maximizing the number of covered road links while using at most 2 stations)max y1+y2+y3+y4+y5+y6+y7+y8stx1+x2+y1>0x1+x4+y2>0x2+x3+y3>0x2+x5+y4>0x3+x5+y5>0x3+x6+y6>0x4+x5+y7>0x5+x6+y8>0y1-x111-25) locations for new distribution centers to serve it customersmin 66x11+90x12+135x13+75x14+44x21+56x22+75x23+150x24+55x31+144x32+30x33+100x34+200y1+400y2+225y3stx11+x21+x31=1x12+x22+x32=1x13+x23+x33=1x14+x24+x34=111x11-69y10x12>0x13>0x14>0x21>0x22>0x23>0x24>0x31>0x32>0x33>0x34>0endinte y1inte y2inte y3LP OPTIMUM FOUND AT STEP 69 OBJECTIVE VALUE = 358.086945 NEW INTEGER SOLUTION OF 554.000000 AT BRANCH 0 PIVOT 102 BOUND ON OPTIMUM: 554.0000 ENUMERATION COMPLETE. BRANCHES= 0 PIVOTS= 102 LAST INTEGER SOLUTION IS THE BEST FOUND RE-INSTALLING BEST SOLUTION... OBJECTIVE FUNCTION VALUE 1) 554.0000 VARIABLE VALUE REDUCED COST Y1 0.000000 -76.000000 Y2 0.000000 -6.333333 Y3 1.000000 225.000000 X11 0.000000 11.000000 X12 0.000000 0.000000 X13 0.000000 0.000000 X14 0.000000 0.000000 X21 0.000000 0.000000 X22 0.000000 0.000000 X23 0.000000 45.000000 X24 0.000000 0.000000 X31 1.000000 0.000000 X32 1.000000 0.000000 X33 1.000000 0.000000 X34 1.000000 0.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 -55.000000 3) 0.000000 -144.000000 4) 0.000000 -30.000000 5) 0.000000 -100.000000 6) 0.000000 0.000000 7) 0.000000 1.000000 8) 58.000000 0.000000 9) 0.000000 3.000000 10) 0.000000 4.888889 11) 51.000000 0.000000 12) 0.000000 0.000000 13) 0.000000 0.000000 14) 54.000000 0.000000 15) 0.000000 1.000000 16) 0.000000 0.000000 17) 44.000000 0.000000 18) 0.000000 0.000000 19) 0.000000 0.000000 20) 0.000000 -105.000000 21) 0.000000 0.000000 22) 0.000000 0.000000 23) 0.000000 0.000000 24) 0.000000 0.000000 25) 0.000000 -50.000000 26) 1.000000 0.000000 27) 1.000000 0.000000 28) 1.000000 0.000000 29) 1.000000 0.000000 NO. ITERATIONS= 109 BRANCHES= 0 DETERM.= 1.000E 011-27) moving natural gas from fields to storage areasMin 2000x13+2000x21+2000x23+2000x24+2000x34+8y13+6y21+10y23+14y24+2y34stx13-x21=800x21+x23+x24=600x35+x34-x23-x13=0x45-x34-x24=0x35+x45=1400x13-1000y130x21>0x23>0x24>0x34>0x35>0x45>0endinte y13inte y21inte y23inte y24inte y3454231LP OPTIMUM FOUND AT STEP 59 OBJECTIVE VALUE = 2800014.00 FIX ALL VARS.( 2) WITH RC > 0.000000E+00 SET Y23 TO wastwater problemMin 21x12+30x13+22x23+58x24+43x34+x39+49x43+63x48+44x56+51x57+56x67+94x68+82x74+x79+2x89+240y12+350y13+200y23+750y24+610y34+3800y39+1840y43+780y48+620y56+800y57+500y67+630y68+1120y74+3800y79+2500y89stx12+x13=27-x12+x23+x24=3-x13-x23-x43+x34+x39=14-x24-x34-x74+x43+x48=36x56+x57=21-x56+x67+x68=8-x57-x67+x74+x79=13x48+x68-x89=0x39+x79+x89=122x12-27y120x13>0x23>0x24>0x34>0x39>0x43>0x48>0x56>0x57>0x67>0x68>0x74>0x79>0x89>0endinte y12inte y13inte y23inte y24inte y34inte y39inte y43inte y48inte y56inte y57inte y67inte y68inte y74inte y79inte y89LP OPTIMUM FOUND AT STEP 143 OBJECTIVE VALUE = 12055.2783 SET Y39 TO >= 1 AT 1, BND= -0.1359E+05 TWIN=-0.1718E+05 170 SET Y43 TO = 1 AT 4, BND= -0.1608E+05 TWIN=-0.1000E+31 178 DELETE Y89 AT LEVEL 4 DELETE Y74 AT LEVEL 3 FLIP Y43 TO >= 1 AT 2 WITH BND= -14816.918 SET Y74 TO = 1 AT 6, BND= -0.1539E+05 TWIN=-0.1000E+31 185 SET Y12 TO = 1 AT 8, BND= -0.1539E+05 TWIN=-0.1000E+31 185 SET Y23 TO >= 1 AT 9, BND= -0.1557E+05 TWIN=-0.1570E+05 188 NEW INTEGER SOLUTION OF 15571.0000 AT BRANCH 20 PIVOT 188 BOUND ON OPTIMUM: 15571.00 DELETE Y23 AT LEVEL 9 DELETE Y13 AT LEVEL 8 DELETE Y12 AT LEVEL 7 DELETE Y67 AT LEVEL 6 DELETE Y56 AT LEVEL 5 DELETE Y89 AT LEVEL 4 DELETE Y74 AT LEVEL 3 DELETE Y43 AT LEVEL 2 DELETE Y39 AT LEVEL 1 ENUMERATION COMPLETE. BRANCHES= 20 PIVOTS= 188 LAST INTEGER SOLUTION IS THE BEST FOUND RE-INSTALLING BEST SOLUTION... OBJECTIVE FUNCTION VALUE 1) 15571.00 VARIABLE VALUE REDUCED COST Y12 0.000000 240.000000 Y13 1.000000 -1.000000 Y23 1.000000 200.000000 Y24 0.000000 750.000000 Y34 0.000000 610.000000 Y39 1.000000 3800.000000 Y43 1.000000 1840.000000 Y48 0.000000 -2148.000000 Y56 0.000000 620.000000 Y57 1.000000 800.000000 Y67 1.000000 500.000000 Y68 0.000000 630.000000 Y74 0.000000 1120.000000 Y79 1.000000 3800.000000 Y89 0.000000 2500.000000 X12 0.000000 0.000000 X13 27.000000 0.000000 X23 3.000000 0.000000 X24 0.000000 85.000000 X34 0.000000 92.000000 X39 80.000000 0.000000 X43 36.000000 0.000000 X48 0.000000 0.000000 X56 0.000000 49.000000 X57 21.000000 0.000000 X67 8.000000 0.000000 X68 0.000000 0.000000 X74 0.000000 131.000000 X79 42.000000 0.000000 X89 0.000000 39.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 6.000000 3) 0.000000 27.000000 4) 0.000000 49.000000 5) 0.000000 0.000000 6) 0.000000 -2.000000 7) 0.000000 -7.000000 8) 0.000000 49.000000 9) 0.000000 -87.000000 10) 0.000000 -50.000000 11) 0.000000 0.000000 12) 0.000000 13.000000 13) 27.000000 0.000000 14) 0.000000 0.000000 15) 0.000000 0.000000 16) 42.000000 0.000000 17) 72.000000 0.000000 18) 0.000000 24.000000 19) 0.000000 0.000000 20) 0.000000 0.000000 21) 21.000000 0.000000 22) 0.000000 0.000000 23) 0.000000 0.000000 24) 0.000000 0.000000 25) 0.000000 0.000000 26) 0.000000 0.000000 27) 27.000000 0.000000 28) 3.000000 0.000000 29) 0.000000 0.000000 30) 0.000000 0.000000 31) 80.000000 0.000000 32) 36.000000 0.000000 33) 0.000000 0.000000 34) 0.000000 0.000000 35) 21.000000 0.000000 36) 8.000000 0.000000 37) 0.000000 0.000000 38) 0.000000 0.000000 39) 42.000000 0.000000 40) 0.000000 0.000000 NO. ITERATIONS= 199 BRANCHES= 20 DETERM.= 1.000E 0NCB circuit boardS= proper subset of the points to be routedDecision variables(i&j=1,2,…,10)xij=1 if the route includes a leg between i &and &j0 otherwise Objective functionMin 3.6x12+5.1x13+10x14+15.3x15+20x16+16x17+14.2x18+23x19+26.4x110+3.6x23+6.4x24+21.1x25+18.1x26+13.2x27+10.6x28+19.7x29+23x210+7.1x34+10.6x35+15x36+15.8x37+10.8x38+18.4x39+21.9x310+7x45+15.7x46+10x47+4.2x48+13.9x49+17x410+9.9x56+15.3x57+5x58+7.3x59+11.3x510+25x67+14.9x68+12x69+15x610+10.3x78+19.2x79+21x710+10.2x89+13x810+3.6x910stx12+x13+x14+x15+x16+x17+x18+x19+x110=2x12+x23+x24+x25+x26+x27+x28+x29+x210=2x13+x23+x34+x35+x36+x37+x38+x39+x310=2x14+x24+x34+x45+x46+x47+x48+x49+x410=2x15+x25+x35+x45+x56+x57+x58+x59+x510=2x16+x26+x36+x46+x56+x67+x68+x69+x610=2x17+x27+x37+x47+x57+x67+x78+x79+x710=2x18+x28+x38+x48+x58+x68+x78+x89+x810=2x19+x29+x39+x49+x59+x69+x79+x89+x910=2(number of legs between points in S & points not in S)x110+x210+x310+x410+x510+x610+x710+x810+x910=2i∈Sj∉SXij+i∉Sj∈SXij≥2Xij=0 or 1 for all i&j=1,2,…,10NCB circuit boardDecision variables(i&k&j=1,2,…,10)yki=1 if the route includes a leg between i &and &j0 otherwise Objective functionMin 3.6(y11y22+y21y32+y31y42+y41y52+…+y91y102)+5.1(y11y23+y21y33+y31y43+…+y91y103)+…+3.6(y110y29+y210y39+y310y49+…+y910y109)st(for k=1,2,…,10)iYki=1(for i=1,2,…,10)kYki=1Xij=0 or 1 for all i&j=1,2,…,10