Lecture 21 Optimization Problems PDF

Year: 2023-2024 Spring Semester Introduction to Operations Research and Decision Support Dr. Amany Magdy Dr. Hussien Shafei Dr. Asmaa Awad  Final Exam MCQ  10 questions T/F  15 questions Essay  3 questions...

Year: 2023-2024 Spring Semester Introduction to Operations Research and Decision Support Dr. Amany Magdy Dr. Hussien Shafei Dr. Asmaa Awad  Final Exam MCQ  10 questions T/F  15 questions Essay  3 questions Branch and Bound Method Shortest Route Problem Transportation Model (2 methods) 2  Revision 3 Q1. Given the following integer programming model, the branch and bound method was applied and two solution subsets were created from one of its steps as shown in the following figure. Write the mathematical model that will be used to find the solution at node 5 and node 6. Maximize 𝑍 = 5𝑥1 + 4𝑥2 Subject to 𝑥1 + 𝑥2 ≤ 5 10𝑥1 + 6𝑥2 ≤ 45 𝑥1 , 𝑥2 ≥ 0 and integer Q1. Given the following integer programming model, the branch and bound method was applied and two solution subsets were created from one of its steps as shown in the following figure. Write the mathematical model that will be used to find the solution at node 5 and node 6. Maximize 𝑍 = 5𝑥1 + 4𝑥2 Subject to 𝑥1 + 𝑥2 ≤ 5 10𝑥1 + 6𝑥2 ≤ 45 𝑥1 , 𝑥2 ≥ 0 𝑥1 =3.75 , 𝑥2 =1.25 upper bound z= 23.75 lower bound z=19  Maximize 𝑍 = 5𝑥1 + 4𝑥2 Maximize 𝑍 = 5𝑥1 + 4𝑥2 Subject to Subject to 𝑥1 + 𝑥2 ≤ 5 𝑥1 + 𝑥2 ≤ 5 10𝑥1 + 6𝑥2 ≤ 45 10𝑥1 + 6𝑥2 ≤ 45 𝑥1 ≥4 𝑥1 ≤ 3 𝑥1 , 𝑥2 ≥ 0 𝑥1 , 𝑥2 ≥ 0  Maximize 𝑍 = 5𝑥1 + 4𝑥2 Maximize 𝑍 = 5𝑥1 + 4𝑥2 Subject to Subject to 𝑥1 + 𝑥2 ≤ 5 𝑥1 + 𝑥2 ≤ 5 10𝑥1 + 6𝑥2 ≤ 45 10𝑥1 + 6𝑥2 ≤ 45 𝑥1 ≥4 𝑥1 ≥4 𝑥1 ≥ 1 𝑥2 ≤ 0 𝑥1 , 𝑥2 ≥ 0 𝑥1 , 𝑥2 ≥ 0 Maximize 𝑍 = 5𝑥1 + 4𝑥2 Subject to Maximize 𝑍 = 5𝑥1 + 4𝑥2 𝑥1 + 𝑥2 ≤ 5 Subject to 10𝑥1 + 6𝑥2 ≤ 45 𝑥1 + 𝑥2 ≤ 5 𝑥1 ≥4 10𝑥1 + 6𝑥2 ≤ 45 𝑥2 ≤ 0 𝑥1 ≥4 𝑥1 ≤ 4 𝑥2 ≤ 0 𝑥1 , 𝑥2 ≥ 0 𝑥1 ≥ 5 𝑥1 , 𝑥2 ≥ 0 Q2. Consider the following transportation problem: The total cost of the initial basic feasible solution obtained from the Northwest corner method is _________. Q2. Consider the following transportation problem: / 2 5 | | / 0 The total cost of the initial basic feasible solution obtained from the Northwest corner method is _________. Q2. Consider the following transportation problem: / 2/ 0 5 2 ___ ___ | | / 0 / 6 The total cost of the initial basic feasible solution obtained from the Northwest corner method is _________. Q2. Consider the following transportation problem: / 2/ 0 5 2 ___ ___ | 6 / 3 | | / 0 / 6/ 0 The total cost of the initial basic feasible solution obtained from the Northwest corner method is _________. Q2. Consider the following transportation problem: / 2/ 0 5 2 ___ ___ | 6 3 ___ / 3/ 0 | | / 0 / 6/ 0 / 4 The total cost of the initial basic feasible solution obtained from the Northwest corner method is _________. Q2. Consider the following transportation problem: / 2/ 0 5 2 ___ ___ | 6 3 ___ / 3/ 0 | | 4 14 / 0 / 0 / 6/ 0 / 4 / 0 / 0 The total cost of the initial basic feasible solution obtained from the Northwest corner method is =5*19+2*30+6*30+3*40+4*70+14*20=1015. Q2. Consider the following transportation problem: | | / 10 8 / 0 The total cost of the initial basic feasible solution obtained from the Minimum cell cost method is _________. Q2. Consider the following transportation problem: / 0 ___ | ___ 7 | / 10 8 / 0 / 7 The total cost of the initial basic feasible solution obtained from the Minimum cell cost method is _________. Q2. Consider the following transportation problem: / 0 ___ | ___ 7 | | / 10 / 3 8 7 / 0 / 7/ 0 The total cost of the initial basic feasible solution obtained from the Minimum cell cost method is _________. Q2. Consider the following transportation problem: / 0 ___ | ___ 7 / 2 | 7 | / 10 / 3 8 | 7 / 0 / 0 / 7/ 0 The total cost of the initial basic feasible solution obtained from the Minimum cell cost method is _________. Q2. Consider the following transportation problem: / 0 ___ | ___ 7 / 2 | 7 | / 10 / 3/ 0 3 8 | 7 / 2 / 0 / 0 / 7/ 0 The total cost of the initial basic feasible solution obtained from the Minimum cell cost method is _________. Q2. Consider the following transportation problem: / 0 ___ | ___ 7 / /2 0 2 | 7 | / 10 / 3/ 0 3 8 | 7 / 2/ 0 / 0 / 0 / 7/ 0 The total cost of the initial basic feasible solution obtained from the Minimum cell cost method is =7*10+2*70+7*40+3*40+8*8+7*20=814 Q2. Consider the following transportation problem: 9 | 10 | / 10 12 8 / 0 21 22 10 10 The total cost of the initial basic feasible solution obtained from the Vogel’s approximation model is _________. Q2. Consider the following transportation problem: 9 | / 20 10 | / 10 / 20 12 8 / 0 21 / 22 10 10 The total cost of the initial basic feasible solution obtained from the Vogel’s approximation model is _________. Q2. Consider the following transportation problem: / 2 9 5 | / 20 10 | | / 10 / 20 12 | 8 / 0 / 0 21 / 22 10 10 The total cost of the initial basic feasible solution obtained from the Vogel’s approximation model is _________. Q2. Consider the following transportation problem: / 2 /9 40 5 | / 20 10 | | / 0 12 / 10 / 50 / 20 | 8 ___ 10 / 0 / 0 / 4 / 21 / 22 10 10 The total cost of the initial basic feasible solution obtained from the Vogel’s approximation model is _________. Q2. Consider the following transportation problem: / 2/ 0 /9 40 5 | ___ 2 / 20 10 | | / 50 / 20 / 0 12 / 10 / | 8 ___ 10 / 0 / 0 / 4/ 2 / 21 / 22 10 10/ 50 The total cost of the initial basic feasible solution obtained from the Vogel’s approximation model is _________. Q2. Consider the following transportation problem: / 2/ 0 / /9 40 5 | ___ 2 / 0 / 20 10 | | 7 2 / 50 / 20 / 0 12 / 10 / | 8 ___ 10 / 0 / 0 / 0 / 4/ 2/ 0 / 21 / 22 10 10/ 50 The total cost of the initial basic feasible solution obtained from the Vogel’s approximation model is =5*19+2*10+7*40+2*60+8*8+10*20=779 Q5. Given the following network with the indicated distances between nodes (in miles), determine the shortest route from node 1 to node 6. The Shortest Route Problem - Solution Approach (1 of 8) Given the following network with the indicated distances between nodes (in miles), determine the shortest route from node 1 to node 6. The permanent set indicates the nodes for which the shortest route to has been found. Network with node 1 in the permanent set The Shortest Route Problem - Solution Approach (2 of 8) Given the following network with the indicated distances between nodes (in miles), determine the shortest route from node 1 to node 6. Network with nodes 1 and 3 in the permanent set The Shortest Route Problem - Solution Approach (3 of 8) Given the following network with the indicated distances between nodes (in miles), determine the shortest route from node 1 to node 6. Network with nodes 1, 2, and 3 in the permanent set The Shortest Route Problem - Solution Approach (4 of 8) Given the following network with the indicated distances between nodes (in miles), determine the shortest route from node 1 to node 6. Network with nodes 1, 2, 3, and 4 in the permanent set Path  1-3-6 Distance =31