Consider the following optimization problem: Minimize Z-6x1-5x2 Subject to 3x₁ + x₂ ≤ 11 -X1+2x₂55 X1, X220, integer Solve the above integer model using the branch and bound method: • Show the formulation of each subproblem • Give the solution of the linear relaxation of each sub-problem (you can use excel solver or the graphical method to solve the linear relaxations) When branching occurs, briefly explain why • When a subproblem is fathomed, briefly explain why When the best integer solution is updated, briefly explain why State the optimum solution
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- Seas Beginning sells clothing by mail order. An important question is when to strike a customer from the companys mailing list. At present, the company strikes a customer from its mailing list if a customer fails to order from six consecutive catalogs. The company wants to know whether striking a customer from its list after a customer fails to order from four consecutive catalogs results in a higher profit per customer. The following data are available: If a customer placed an order the last time she received a catalog, then there is a 20% chance she will order from the next catalog. If a customer last placed an order one catalog ago, there is a 16% chance she will order from the next catalog she receives. If a customer last placed an order two catalogs ago, there is a 12% chance she will order from the next catalog she receives. If a customer last placed an order three catalogs ago, there is an 8% chance she will order from the next catalog she receives. If a customer last placed an order four catalogs ago, there is a 4% chance she will order from the next catalog she receives. If a customer last placed an order five catalogs ago, there is a 2% chance she will order from the next catalog she receives. It costs 2 to send a catalog, and the average profit per order is 30. Assume a customer has just placed an order. To maximize expected profit per customer, would Seas Beginning make more money canceling such a customer after six nonorders or four nonorders?Consider the following linear programming model: maximize Z = 3x1 + 2x2 subject to : x1 +x2 ≤ 1 x1 + x2 ≥ 2 x1,x2 ≥ 0 a) Write this model in a standard (augmented) form. (i.e. Introduce slack/surplus, artificial etc.)b) Constract the initial simplex tableau and carry on your calculations to solve this model using the simplex method. Interpret your result.Simplex Method Write in normal form and solve by the simplex method, assuming x, to be nonnegative. 1. The owner of a shop producing automobile trailers wishes to determine the best mix for his three products: flat-bed trailers, economy trailers, and luxury trailers. His shop is limited to working 24 days/month on metal- working and 60 days/month on woodworking for these products. The following table indicates production data for the trailers. Usage per unit of trailer Flat-bed Economy Luxury Available resources 1 2 1 24 Metal work days Wood work days Contribution (Rx100) 60 4 6 2 14 13
- Solve the following Linear Programming model using the graphical method (USING EXCEL){Write the steps of construction} Q1)MaximizeH = x + 3y Objective functionsubject tox + y ≤ 502x + y ≤ 60 x ≥ 0, y ≥ 0Find a solution using the Simplex method (BigM method) MIN Z= 5x1 + 3x2 subject to 3x1 +x2 = 3 4x1 + 3x2>= 6 x1 +2x2 = 0 A. Construct the initial Tableau for the above simplex problem using the Big M B. Identify; the Leaving, Entering Variable, and the Pivot NumberUse simplex method for linear programming. Show solution for each iteration table. Problem: A small firm that assembles computers is about to start production of two new types of personal computers. Each type will require assembly time, inspection time, and storage space. The amount of each of these resources that can be devoted to the production of the computers is limited. The manager of the firm would like to determine the quantity of each computer to produce in order to maximize the profit generated by sales of these computers. In order to develop a suitable model of the problem, the manager has met with design and manufacturing personnel. As a result of those meetings, the manager has obtained the following information: Type 1 Type 2 Profit per unit $60 $50 Assembly time per unit 4 hours 10 hours Inspection time per…
- Consider the following problem: Maximize subject to = w = 5x12x2 + x3 w 1+ 4x2 + x3 ≤ 6 2x1 + x2 + 3x3 ≥ 2 1, 220, 03 ≤0 a) Put it in standard and canonical form. b) Solve the problem using the simplex method.Solve the following problem using Simplex Method and find the P value. Maximize: P = 3x + 4y subject to: x+ y 0Facility Location. A paper products manufacturer has enough capital to build and manage some additional manufacturing plants in the United States in order to meet increased demand in three cities: New York City, NY; Los Angeles, CA; and Topeka, KS. The company is considering building in Denver, CO; Seattle, WA; and St. Louis, MO. Max Operating Capacity 400 tons/day 700 tons/day Denver Seattle $10/ton $17/tor $5/ton $11/ton.... $18/ton.... $28/ton Los Angeles Topeka New York City Figure 1: Graphical representation of the given data = • The cost fi of building plants in these cities is fi $10,000,000 in Seattle. Unmet Demand 300 tons/day 100 tons/day 500 tons/day • Due to geographic constraints, plants in Denver and Seattle would have a maximum operating capacity kį of 400 tons/day and 700 tons/day respectively. $5,000,000 in Denver and f2 = • The cost cij per ton of transporting paper from city i to city j is outlined in Figure 1. • The unmet demand d, for Los Angeles, Topeka, and New…
- In problems involving maximization and minimization, what is the objective function? States intended outcome in equation form Incorporates constraint of maximum budget Lists set of potential restrictions on the solution Cites constraints to ensure objectivityA manufacturing company has an aggregate planning of selling its beds in the following sizes (the proportion of production for each is given within the parenthesis): twin (30%), queen (40%), and king (30%). The company produces each one of these bed sizes in the following colors (the proportion of production for each is given within the parenthesis): dark wood (40%), white(35%), and beige (25%). The company will produce a total of 10,000 beds. How many white twin beds will they produce? 3000 1200 750 1050Analyze algebraically what special case in simplex application is present in each of the LP model below. Give an explanation to support your answer. a) Maximize z = 4x1 + 2x2 Subject to: 2x1 - x2 ≤ 2 3x1 - 4x2 ≤ 8 x1, x2 ≥ 0b) Maximize z = 3x1 + 2x2 Subject to: 4x1 - x2 ≤ 8 4x1 + 3x2 ≤ 12 4x1 + x2 ≤ 8 x1, x2 ≥ 0