Problem 1 Consider the following LP: min 5x₁ + 4x2 s.t. x₁ + x₂ ≥ 1 2x1 - x₂ ≥ 1 3x2 ≤ 2 x1, x₂ > 0. 1. Write down the dual of this LP. 2. We know that by strong duality, we can either solve this LP or its dual form to get the same optimal objective value. If you apply the simplex algorithm, which one would you rather solve? Make your choice and apply the simplex algorithm to solve it.
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- Problem: Acme Manufacturing produces widgets on an assembly line. The assembly process has five tasks, each with its own processing time: Task Processing Time (Seconds) A 10 B 8 C 12 D 9 E 7 drive_spreadsheetExport to Sheets Acme currently has two workers assigned to the assembly line, and they work simultaneously. The company wants to improve line efficiency and reduce waste. Objective: Balance the assembly line by assigning tasks to each worker in a way that minimizes idle time and maximizes output.Suppose Jack like to solve the following formulation in Excel. And the problem is setup in Excel like this: See attached picture. A B C D E 1 X1 X2 2 decision variable objective 3 coefficient of objective function 2 3 4 Constraints LHS RHS 5 Constraints 1 2 1 3 6 Constraints 2 4 5 20 7 Constraints 3 2 8 16 8 Constraints 4 5 6 60 We want to put our objective function in cell D3. What should we type in D3? In cell D5 to D8, we will put the left hand side of our constraint. What should we type in D8? In OpenSolver, what should we assign to Variable Cell? What should you do to input the non-negativity constraint in OpenSolver?Consider the following LP problem: Min 6X+ 27Y Subject to : 2 X + 9Y => 25, and X + Y <= 75. Pick a suitable statement for this problem: a. X=37.5, Y=37.5 is the only optimal solution. b. Optimal Obj. function value is 75 c. X = 0, Y = 0 is the only optimal solution. d. Optimal Obj. function value is 0
- Consider the following LP problem: Min 6X+ 18Y; Subject to : 3 X + 9Y <= 47, and X + Y <= 141. Which one of the following is true?: a. Slack for each constraint is zero. b. Optimal Obj. function value is 94 c. X=70.5, Y=70.5 is the only optimal solution. d. Optimal Obj. function value is 0Task 4In this task, you are asked to formulate the linear programming problem below andthen solve it.A carpenter produces Tables (T) and Chairs (C).Each Table unit requires 6kg of wood and 1kg of plastic.Each Chair unit requires 4kg of wood and 4kg of plastic.The carpenter has only 100kg of wood and only 50kg of plastic in stock.On each sale, the carpenter makes a profit of £15 per Table unit sold and a profit of£30 per Chair unit sold.Required:Formulate the Linear Programming problem above by defining the variables,stating the object function and the constraints.You are required to produce the graph for all inequalities (copy image to Word)Find the quantity to make of Tables and Chairs to maximise profit.Problem: A farmer is planning to raise wheat and barley. Each acre of wheat yields a profit of $50, and each acre of barley yields a profit of $70. To sow the crop, two machines, a tractor and a tiller are rented. The tractor is available for 150 hours, and the tiller is available for 200 hours. Sowing an acre of wheat requires 4 hours of tractor time and 1 hour of tilling. Sowing an acre of barley requires 3 hours of tractor time and 2 hours of tilling. How many acres of each crop should be planted to maximize the farmer’s profit? (Let W be the number of acres of wheat to be planted, B the number of acres of barley to be planted and P the profit) (a) What is the objective function for the problem? (b) Excluding the non-negative constraint, how many constraints does the problem have? (c) What is the linear programming model of the problem?
- Instruction: Formulate the dual problems for X2 the following linear programming model No need to compute/doing a tableau 1. Maximize Z = 2X1 + X2 Subject to: X2 < 10 2X1 + 5X2 < 60 X1 + X2 <18 3X1 + X2 < 44 2. Maximize Z = 2X1 + X2 +3X3 Subject X1 + X2 + 2X3 < 400 2X1 + X2 + X3 < 500 3. Maximize Z = 80X1 + 60X2 Subject X1 + X2 = 200 X1 < 50 X2 > 80-For this problem clearly derive the Linear program-Graph this problem and clearly indicate the feasible options- Clearly determine the solution to the problem using the method of points (be smart about which points you have to evaluate based on the graph of the objective functionDon't use chatgpt, I will 5 upvotes Alan wants to bake blueberry muffins and bran muffins for the school bake sale. For a tray of blueberry muffins, Alan uses 1/3 cup of oil and 2 eggs. For a tray of bran muffins, Alan uses 1/2 cup of oil and 1 egg. Alan has 4 cups of oil and 12 eggs on hand. He sells trays of blueberry muffins for $12 each and trays of bran muffins for $9 each. Alan wants to maximize the money raised at the bake sale. Let x represent the number of blueberry muffins and y represent the number of bran muffins Alan bakes.
- Find solution using BigM (penalty) method.Maximize Z = x1 + 2x2 + 3x3 - x4subject to the constraintsx1 + 2x2 + 3x3 = 152x1 + x2 + 5x3 = 20x1 + 2x2 + x3 + x4 = 10and x1, x2, x3, x4 ≥ 0A goldsmith makes two types of jewelry. A model A ring is made with 1 g of gold and 1.5 g of silver and sells for 25 UM.Another model B ring sells for 30 UM and is made of 1.5 g of gold and 1 g of silver. If you only have 750 gof each metal, how many rings should be made of each type to obtain maximum profit?Requested:- Make Initial Table of the problem.- Obtain the Case Variables- Obtain the Objective Function- Get Restrictions- Create the Simplex Table- Obtain the Optimal Solution and the Slack Variables.Solve this operational research exercise.The minimum-cost flow problem could be applied_________ Option- 1. by a telephone company attempting to lay out wires in a new housing development. 2. by a forest ranger seeking to minimize the risk of forest fires. 3. None of the choices. 4. by an airline laying out flight routes. 5. All of the choices are correct. Which of the following is not an assumption of a shortest path problem? 1. Associated with each link or arc is a nonnegative number called its length, 2. The objective is to find a shortest path from the origin to the destination. 3. The lines connecting certain pairs of nodes always allow travel in either direction. 4. None of the choices. 5. A path through the network must be chosen going from the origin to the destination.