14.6 County forest trail system. The network is illustrated in 14-6.ppt, a Microsoft Powerpoint file. Here's how we formulated the problem with LINDO: Min 0.16 X1 + 0.14 X2 + 0.07 X3 + 0.19 X4 + 0.07 X5 + 0.17 X6 + 0.17 X7 + 0.11 X8 + 0.17 X9 + 0.19 X10 + 0.11 X11 + 0.39 X12 + 0.11 X13 + 0.39 X14 + 0.39 X15 + 0.17 X16 + 0.11 X17 + 0.17 X18 + 0.17 X19 + 0.39 X20 + 4670.45 Y1 + 2863.64 Y2 + 2267.99 Y3 + 5160.51 Y4 + 5186.36 Y5 + 3200.76 Y6 + 5013.83 Y7 + 9161.74 Y8 subject to 2) X1 + X2 = 20000 3) -X1 + X3 + X4 = 0 4) -X2 + X5 + X6 = 0 5) -X3 + X7 = 0 6) -X4 + X8 + X9 = 0 7) -X5 + X10 = 0 8) -X6 + X11 + X12 = 0 9) -X7 + X13 + X14 = 0 10) -X8 + X15 = 0 11) -X10 + X16 + X17 = 0 12) -X11 + X18 = 0 13) -X13 + X19 = 0 14) -X17 + X20 = 0 15) X19 + X14 + X15 + X9 + X16 + X20 + X18 + X12 = 20000 16) X1 - 99999 Y1 <= 0 17) X2 - 99999 Y2 <= 0 18) X3 - 99999 Y3 <= 0 19) X4 - 99999 Y4 <= 0 20) X5 - 99999 Y3 <= 0 21) X6 - 99999 Y5 <= 0 22) X7 - 99999 Y5 <= 0 23) X8 - 99999 Y6 <= 0 24) X9 - 99999 Y7 <= 0 25) X10 - 99999 Y4 <= 0 26) X11 - 99999 Y6 <= 0 27) X12 - 99999 Y8 <= 0 28) X13 - 99999 Y6 <= 0 29) X14 - 99999 Y8 <= 0 30) X15 - 99999 Y8 <= 0 31) X16 - 99999 Y7 <= 0 32) X17 - 99999 Y6 <= 0 33) X18 - 99999 Y7 <= 0 34) X19 - 99999 Y7 <= 0 35) X20 - 99999 Y8 <= 0 end inte Y1 inte Y2 inte Y3 inte Y4 inte Y5 inte Y6 inte Y7 inte Y8 We sought to minimize the cost of contruction (the "Y" variables) and maintenance (the "X" variables). The coefficient for Y1 was calculated as $15,000 x (1644/5280) = $4,670.45. The coefficient for X1 was calculated as $0.50 x (1644/5280) = $0.16. Other coefficients are calculated similarly. Rows 2-15 account for the flow of people along the paths from PL to PA. Rows 16-35 make sure that the problem knows that if some people are expected to have hiked down a trail, the trail is built. Variables Y1-Y8 are all integers, since we do not expect to build only part of a trail in this situation. The LP output consisted of the following: NEW INTEGER SOLUTION OF 25244.7891 AT BRANCH 14 PIVOT 987 OBJECTIVE FUNCTION VALUE 1) 25244.79 VARIABLE VALUE REDUCED COST Y1 1.000000 0.000000 Y2 0.000000 0.000000 Y3 0.000000 0.000000 Y4 1.000000 -nan Y5 0.000000 0.000000 Y6 0.000000 0.000000 Y7 1.000000 0.000000 Y8 0.000000 0.000000 X1 20000.000000 0.000000 X2 0.000000 0.000000 X3 0.000000 0.000000 X4 20000.000000 0.000000 X5 0.000000 0.000000 X6 0.000000 0.000000 X7 0.000000 0.000000 X8 0.000000 0.000000 X9 20000.000000 0.000000 X10 0.000000 0.000000 X11 0.000000 0.000000 X12 0.000000 0.000000 X13 0.000000 0.000000 X14 0.000000 0.000000 X15 0.000000 0.000000 X16 0.000000 0.000000 X17 0.000000 0.000000 X18 0.000000 0.000000 X19 0.000000 0.000000 X20 0.000000 0.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 0.000000 3) 0.000000 0.393523 4) 0.000000 0.283182 5) 0.000000 0.463522 6) 0.000000 0.656582 7) 0.000000 0.466581 8) 0.000000 0.546582 9) 0.000000 0.666707 10) 0.000000 0.436582 11) 0.000000 0.656582 12) 0.000000 0.656582 13) 0.000000 0.776707 14) 0.000000 0.436582 15) 0.000000 -0.826581 16) 79999.000000 0.000000 17) 0.000000 0.000000 18) 0.000000 0.000000 19) 79999.000000 0.000000 20) 0.000000 0.000000 21) 0.000000 0.000000 22) 0.000000 0.033184 23) 0.000000 0.032008 24) 79999.000000 0.000000 25) 99999.000000 0.000000 26) 0.000000 0.000000 27) 0.000000 0.000000 28) 0.000000 0.000000 29) 0.000000 0.091618 30) 0.000000 0.000000 31) 99999.000000 0.000000 32) 0.000000 0.000000 33) 99999.000000 0.000000 34) 99999.000000 0.000000 35) 0.000000 0.000000 36) 0.000000 -0.233523 37) 0.000000 -0.143182 38) 0.000000 -0.073059 39) 0.000000 -0.113399 40) 0.000000 0.000000 41) 20000.000000 0.000000 42) 20000.000000 0.000000 43) 0.000000 1867.989990 44) 0.000000 0.000000 45) 1.000000 0.000000 46) 0.000000 0.000000 47) 0.000000 0.000000 48) 0.000000 0.000000 49) 1.000000 0.000000 NO. ITERATIONS= 987 BRANCHES= 14 DETERM.= 1.000E 0 BOUND ON OPTIMUM: 25244.79 DELETE Y4 AT LEVEL 2 DELETE Y7 AT LEVEL 1 ENUMERATION COMPLETE. BRANCHES= 14 PIVOTS= 987 LAST INTEGER SOLUTION IS THE BEST FOUND RE-INSTALLING BEST SOLUTION... So the answers to Parts (a) through (d) are: (a). Construct trails 1, 4, and 7. (b). $25,244.79 (c). (20,000 x 0.16) + (20,000 x 0.19) + (20,000 x 0.17) = $10,400 (d). If trail 8 construction costs are reduced to $17,000 per mile, no, our answer wouldn't change. You need to reformulate your problem, specifically the coefficient for Y8, to check this result.