Problem 2. Given the following linear programming problem: max 2x1 + x₂ + 4x3 s.t. ₁4x2x3 ≥ 2 - 2x12x2 + 8x3 ≤ 1 T1, T2, T3 20 . (a) Write down the Dual LP for the provided problem. (b) Find the optimal value of the objective of (a).

Consider the following linear programming problem: Min   A + 2B s.t.             A + 4B ≤ 21     2A + B ≥ 7     3A + 1.5B ≤ 21     -2A + 6B ≥ 0         A, B ≥ 0 Determine the amount of slack or surplus for each constraint. If required, round your answers to one decimal place.         (1) A + 4B ≤ 21  Slack or surplus (2) 2A + B ≥ 7  Slack or surplus (3) 3A + 1.5B ≤ 21  Slack or surplus (4) -2A + 6B ≥ 0  Slack or surplus

1. Consider an instance of the Knapsack Problem without repetitions with 4 items, having weights and values as follows. The weights (in pounds) are w1=2, w2 =7, w3 =10, w4 =12. The dollar values of these items are respectively v1 = 12, v2 = 28, v3 = 30, v4 = 5. The capacity of the knapsack is 12. (a) Find the optimal solution for Fractional Knapsack. (b) Find the optimal solution for 0-1 Knapsack.

5. Given a transportation problem with the following costs, supply, and demand, find the initial solu- tion using the minimum cell cost method and Vogel's approximation model. Is the VAM solution optimal? To From 1 2 3 Supply A $ 7 $ 4 100 В 3 180 C 5 7 200 Demand 135 175 170 658

In this problem we have n jobs j1, j2, ..., jn, each has an associated deadline d1, d2, ..., dn and profit p1, p2, ..., pn. Profit will only be awarded or earned if the job is completed before the deadline. We assume that each job takes 1 unit of time to complete. The objective is to earn maximum profit when only one job can be scheduled or processed at any given time.Provide the pseudocode of an algorithm to find the sequence of jobs to do with the maximum total profit. Also describe the main idea of your algorithm using plain language.[Hint: You can select the jobs in a greedy way. You can use the following example to help your analysis.] Job J1 J2 J3 J4 J5 Deadline 2 1 3 2 1 Profit 60 100 20 40 20 The best job sequence would be J2 →J1 →J3.

Problem: Solve the following equality-constrained optimiza- tion problem using Newton descent algorithm with the initial point (1, 4, 0): min f(x, y, z) = e" + 2y? + 3z2 x,y,z subject to x – 5z = 1 | y + z = 4 Compute the optimal dual variables as well. You have to answered above problem By using Python Programming Language.

uses the python program 1- Sketching the optimal level set of the objective function and find optimal value 1) using ploting objective function contours and also 2) checking vertex points. maximize subject to (Hint: optimal value = 42) 2x + 3y -3x + y ==0 1:06 ص //

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