Basic variable Coefficient of X5 Right side 0 10+20 X4 1000 0 0010 5-50 0 10+20 So, when<0 <1, the optimal solution is x1=10-+2 0, x2=10+2 0, maxZ=30+60 When <0< 5. the final tableau is Coefficient of Basic variable Eq X4 X5 Right side 1000 X0100 10- 0010 10+26 (2) 56-5 X2 15-30 When1<0<5. the final tableau is Basic variable E Coefficient of XI X4 Right side 000 2 0 50-20 1)0 0 0 (2)0 20+10 When6> 25, the problem has no optimal solution 8. At a small but growing airport, the local airline company is purchasing a new tractor for a tractor-trailer train to bring luggage to and from the airplanes. A new mechanized luggage system will be installed in 3 years, so the tractor will not be needed after that. However, because it will receive heavy use, so that the running and maintenance costs will increase rapidly as the tractor ages, it may still be more economical to replace the tractor after 1 or 2 years. The following table gives the total et discounted cost associated with purchasing a tractor (purchase price minus trade-in allowance, plus running and maintenance costso at the end of year I and trading it in at the end of year j ( where year 0 is now) 0 31 12 The problem is to determine at what times(if any)the tractor should be replaced to minimize the total cost for the tractors over 3 years. (10 points) Solution: Applying the shortest algorithm, we get the following graph, and the shortest path0→1→3, the total um cost is 296 Coefficient of : Basic variable Eq. Z X1 X2 X3 X4 X5 Right side Z (0) 1 0 0 2 0 1 30+6θ X1 (1) 0 1 0 1 0 0 10+2θ X4 (2) 0 0 0 -1 1 -1 5-5θ X2 (3) 0 0 1 0 0 1 10+2θ So, when0<θ≤1,the optimal solution is x1=10+2θ,x2=10+2θ,maxZ=30+6θ When1 < θ ≤ 5 , the final tableau is Coefficient of : Basic variable Eq. Z X1 X2 X3 X4 X5 Right side Z (0) 1 0 0 1 1 0 35+θ X1 (1) 0 1 0 1 0 0 10+2θ X5 (2) 0 0 0 1 -1 1 5θ-5 X2 (3) 0 0 1 -1 1 0 15-3θ When1 < θ ≤ 5 , the final tableau is Coefficient of : Basic variable Eq. Z X1 X2 X3 X4 X5 Right side Z (0) 1 0 1 0 2 0 50-2θ X1 (1) 0 1 1 0 1 0 25-θ X5 (2) 0 0 1 0 0 1 2θ+10 X3 (3) 0 0 -1 1 -1 0 3θ-15 Whenθ ≥ 25 , the problem has no optimal solution. 8. At a small but growing airport, the local airline company is purchasing a new tractor for a tractor-trailer train to bring luggage to and from the airplanes. A new mechanized luggage system will be installed in 3 years, so the tractor will not be needed after that. However, because it will receive heavy use, so that the running and maintenance costs will increase rapidly as the tractor ages, it may still be more economical to replace the tractor after 1 or 2 years. The following table gives the total net discounted cost associated with purchasing a tractor (purchase price minus trade-in allowance, plus running and maintenance costs0 at the end of year I and trading it in at the end of year j (where year 0 is now). j 1 2 3 0 8 18 31 i 1 10 21 2 12 The problem is to determine at what times (if any) the tractor should be replaced to minimize the total cost for the tractors over 3 years. (10 points) Solution: Applying the shortest algorithm, we get the following graph, and the shortest path 0→1→3, the total minimum cost is 29