1. (a) State the Fundamental Theorem of Linear Programming. What are itsmain implications for solution of Linear Programmes?(b) (i) Briefly outline the two main ideas of duality, and give an economicinterpretation of duality.(ii) LetA1 = (² 8-1¹)40b-2= (-3³²)and C = 14Write down (in matrix notation) the dual of the Linear Programme:Maximise cx subject to Ax ≤ b and x ≥ 0.(c) Define the shadow price of a resource. Suppose that, in a Linear Pro-gramme with two decision variables x₁ and x2, you are given a budgetconstraint 7x₁ + 3x2 ≤ €80 and told that for this constraint, the shadowprice is €0.45. Explain what this means.

Linear Programming (LP) is a mathematical technique used to optimize a linear objective function…

Answered: (a) State the Fundamental Theorem of…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 23EQ: 23. Consider a simple economy with just two industries: farming and manufacturing. Farming consumes...

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23. Consider a simple economy with just two industries: farming and manufacturing. Farming consumes 1/2 of the food and 1/3 of the manufactured goods. Manufacturing consumes 1/2 of the food and 2/3 of the manufactured goods. Assuming the economy is closed and in equilibrium, find the relative outputs of the farming and manufacturing industries.
An economy is based on three sectors: Agriculture,Manufacturing, and Energy. Production of a cedi’s worthof agriculture requires inputs of GH¢0.2 from agriculture,GH¢0.15 from manufacturing, and GH¢0.2 from energy.Production of a Ghana cedi’s worth of manufacturing 3requires inputs of GH¢0.40 from agriculture, GH¢0.10from manufacturing, and GH¢0.10 from energy.Production of a Ghana cedi’s worth of energy requiresinputs of GH¢0.3 from agriculture, GH¢0.10 frommanufacturing, and GH¢0.10 from energy.There is a final demand of GH¢10 billion for agriculture,GH¢15 billion for manufacturing, and GH¢20 billion forenergy. (i) Formulate the system of linear equations whichmust be solved to give the combination of outputfor the three sectors that will satisfy the finaldemand.(ii) Write the equations in matrix form(iii) Determine the output for each sector using theinverse matrix method.
Q. 3 (a) Consider complete pivoting strategy to discuss the following system of 1 equation. 4x, - 4x, - 3x, +7x, 1.3 8x, - 3x, - 8x, +17x, 6.6 12x, - 12x, - 16.x, + 29x, -2.1 - 8x, + 33 x, - 25x, +36x, = 10.4 %3D (b) Discuss the real roots of polynomial equation x*-5x' + 5x + 5x-7 = 0, correct to 4 dp, given that the equation has roots near 3 and -1.
P.nil
(1 b) Formulate a linear programming problem (product mix) for four products and three raw materials. Write a numerical example of such a task (without decision).
Solve the problem. Suppose an economy consists of three sectors: Energy (E), Manufacturing (M), and Agriculture (A). Sector E sells 70% of its output to M and 30% to A. Sector M sells 30% of its output to E, 50% to A, and retains the rest. Sector A sells 15% of its output to E, 30% to M, and retains the rest. Denote the prices (dollar values) of the total annual outputs of the Energy, Manufacturing, and Agriculture sectors by Pe. Pm, and Pa respectively. If possible, find equilibrium prices that make each sector's income match its expenditures. Find the general solution as a vector, with pa free. 0.356 Pal Pm-0.686 Pa Pe Pa Pe Pm Pa Pe Pm - Pa Pe Pm Pa Pa [0.465 Pa = 0.593 Pa Pa [0.308 Pa 0.716 Pa Pa 0.607 Pa 0.481 Pa Pa
A Sporting Goods Company makes basketballs and footballs. Each product is produced from two resources-rubber and leather. The resource requirements for each product and the total resources available are as follows: Resource requirements per Unit Product Rubber (Ib) Leather ft? Basketball 3 4 Football 2 Total resources 500lb 800ft? available Each basketball produced results in a profit of $12 and each football earns $16 in profit. a) Formulate a linear programming model to determine the number of basketballs and footballs to produce in order to maximize profit. b) Transform this model into standard form. c) Solve the model formulated by using graphical analysis d) Are there any binding constraint(s)? If Yes/No Explain
1. Leather Limited manufactures two types of belts: the deluxe model and the regular model. Each type requires 1 square meter of leather. A regular belt requires 1 hour of skilled labor, and a deluxe belt requires 2 hours. Each week, 40 square meter of leather and 60 hours of skilled labor are available. Each regular belt contributes USD 3 to profit and each deluxe belt, USD 4. (a) State the problem in a formal Linear Programming equations! (b) Solve the problem using Simplex method! (c) Write down your conclusion from the optimal solution that you have. (d) To improve the total profit of the company, is it better to increase the inventory of leather per week, or to increase the availability of skilled labor per week? Explain!
A company makes ceramic and porcelain tiles. They produce three different grades of tile, each of which requires different amounts of materials and production time and generates different contributions to profit. The accompanying table shows the percentage of materials needed for each grade and the profit per square foot. Each week, the company receives raw material shipments, which must be used efficiently to maximize profitability. Currently, inventory consists of 6,000 pounds of clay, 3,500 pounds of silica, 5,000 pounds of sand, and 8,500 pounds of feldspar. At most 8,000 square feet of grade III tiles should be produced and at least 2,000 square feet of grade I tiles are required. Each square foot of tile weighs approximately 2 pounds. Develop and solve a linear optimization model to determine how many of each grade of tile the company should make next week to maximize profit contribution. Click here to view the tile data. Complete the table below to indicate the number of square…
An economy has only two industries: the electric company and the gas company Production of a dollar's worth of electricity requires an input of 0.25$ of electricity and 0.25$ of gas. For the gas company each dollar's production requires 0.20$ of gas and 0.40$ electricity. What should the production of electricity and gas be (in dollars) if there is a 16$ million demand for electricity and a 7$ million demand for gas?
Formulate a linear programming problem that can be used to solve the following question. A firm has plants in Boston and Baltimore that manufacture three models of hot tubs: regular, fancy, and super. In one day the Boston plant can manufacture 44 of the regular model, 32 of the fancy model, and 20 of the super model, and costs $4500 per day to operate, whereas the Baltimore plant can manufacture 12 of the regular model, 16 of the fancy model, and 58 of the super model, and costs $2000 per day to operate. At least 300 of the regular model, 320 of the fancy model, and 680 of the super model are needed. How many days must each plant operate in order to minimize the cost? ---1pəjəs-- = X y = ---Select--- ---Select--- v F = (objective function) Subject to (regular models) (fancy models) (super models) 0A---1pajas--
A chemical manufacturing plant can produce z units of chemical Z given p units of chemical P and r units of chemical R, where: z = 100p-75,0.25 Chemical P costs $100 a unit and chemical R costs $700 a unit. The company wants to produce as many units of chemical Z as possible with a total budget of $112,000. A) How many units each chemical (P and R) should be "purchased" to maximize production of chemical Z subject to the budgetary constraint? Units of chemical P, p = Units of chemical R, r = B) What is the maximum number of units of chemical Z under the given budgetary conditions? (Round your answer to the nearest whole unit.) Max production, z= units

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