The Simplex Method: Learning Team A Mike Smith, Todd Jones Math212/Introduction to Finite Mathematics February 1, 2011 The Simplex Method: Learning Team A Sam’s Hairbows and Accessories is a small company preparing for the next scheduled craft fair. The owners, Sam and Todd, both have full-time jobs in addition to owning the company so they are only able to spend a combined total of 80 hours labor to prepare for the fair in four weeks. Sam’s offers five main product lines: basic bows, elaborate...
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Phase Simplex Method Consider the following LP problem. max z = 2x1 + 3x2 + x3 s.t. x1 + x2 + x3 · 40 2x1 + x2 ¡ x3 ¸ 10 ¡x2 + x3 ¸ 10 x1; x2; x3 ¸ 0 It can be transformed into the standard form by introducing 3 slack variables x4, x5 and x6. max z = 2x1 + 3x2 + x3 s.t. x1 + x2 + x3 + x4 = 40 2x1 + x2 ¡ x3 ¡ x5 = 10 ¡x2 + x3 ¡ x6 = 10 x1; x2; x3; x4; x5; x6 ¸ 0 There is no obvious initial basic feasible solution, and it is not even known whether there exists one. We can use Phase I method...
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Lesson 9 : The Big M Method Learning outcomes • The Big M Method to solve a linear programming problem. In the previous discussions of the Simplex algorithm I have seen that the method must start with a basic feasible solution. In my examples so far, I have looked at problems that, when put into standard LP form, conveniently have an all slack starting solution. An all slack solution is only a possibility when all of the constraints in the problem have or = constraints, a starting basic feasible...
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questions 1. a. Explain how and why Operation Research methods have been valuable in aiding executive decisions. [5 Marks] b. Discuss the usefulness of Operation Research in decision making process and the role of computers in this field. [5 Marks] 2. Explain how the linear programming technique can be helpful in decision-making in the areas of Marketing and Finance. [10 Marks] 3. a. How do you recognise optimality in the simplex method? b. Write the role of pivot element in simplex table? ...
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respectively. Formulate the problem as a linear programming problem. Solution: Let x and y denote the number of production runs of the two processes, respectively. Then the appropriate mathematical formulation of the problem is: Maximize Z=200x+300y Subject to the constraints: 4x+5y≤225 3x+6y≤200 4x+5y≥150 7x+5y≥120 x,y≥0 The standard weight of a special purpose brick is equal to 5kg and it contains two basic ingredients B1 and B2. B1 costs Nu. 5 per kg and B2 costs...
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research which is very easy to use. Further, TORA is menu-driven and Windows-based which makes it very user friendly. The software can be executed in automated or tutorial mode. The automated mode reports the final solution of the problem, usually in the standard format followed in commercial packages, while the tutorial mode keeps on giving step-wise information about the methodology and solution. TORA tutorial software deals with the following algorithms: •Solution...
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OFFICE EXCEL2 3.1.1 EXCEL DATA INPUT AND SOLVE THE PROBLEM2 3.1.2 ANSWER ANALYSIS3 3.1.3 SENSITIVITY ANALYSIS4 3.2 XPRESS-IVE ANALYSIS6 4. CONCLUSION6 APPENDIX Operational Research Methods Report 1. Introduction Linear programming (LP) model is a significant and popular used model of operational research technique. It helps to optimize the objective value with constraints. LP model have three essential assumptions when use this model to solve problem. Firstly, proportionality and...
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6S-2 Solving LP Models Using Excel, 6S-17 SUPPLEMENT TO CHAPTER LEARNING OBJECTIVES Linear Programming Model, 6S-2 Formulating Some Other Types of Constraints, 6S-3 Sensitivity Analysis, 6S-19 Objective Function Coefficient Changes, 6S-20 Changes in the Right-Hand Side (RHS) Value of a Constraint, 6S-21 After completing this supplement, you should be able to: 1 Describe the type of problem Graphical Solution Method, 6S-4 Outline of Graphical Solution Method, 6S-4 Plotting...
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Recognize special cases such as infeasibility, unboundedness and degeneracy. 5. Use the simplex tables to conduct sensitivity analysis. 6. Construct the dual problem from the primal problem. Linear Programming: The Simplex Method LEARNING OBJECTIVES After completing this chapter, students will be able to: 1. Convert LP constraints to equalities with slack, surplus, and artificial variables. 2. Set up and solve LP problems with simplex tableaus. 3. Interpret the meaning of every number in a simplex...
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Simplex Method Paper Simplex Method Paper Many people may be wondering exactly what the simplex method is. The simplex method definition is a method for solving linear programming problems. According to Barnett, Byleen, and Karl (2011) the simplex method is used routinely on applied problems involving thousands of variables and problem constraints. George B. Dantzig developed the simplex method in 1947. In this paper the topic of discussion includes how to solve a simplex method problem that...
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Formulate and solve this LP production mix situation to find the best combination of air conditioners and fans that yields the highest profit. Use the corner point graphical approach. Let X1 = the number of air conditioners scheduled to be produced X2 = the number of fans scheduled to be produced Maximize | 25X1 | + | 15X2 | | | (maximize profit) | Subject to: | 3X1 | + | 2X2 | ≤ | 240 | (wiring capacity constraint) | | 2X1 | + | X2 | ≤ | 140 | (drilling capacity constraint) | |...
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Solutions to LP Practice Problems[1] 1. Furnco manufactures desks and chairs. Each desk uses 4 units of wood, and each chair uses 3 units of wood. A desk contributes $40 to profit, and a chair contributes $25. Marketing restrictions require that the number of chairs produced be at least twice the number of desks produced. There are 20 units of wood available. Using the graph below, determine a production plan that maximizes Furnco’s profit. a) Draw isoprofit lines where the total profit equals...
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Consider the following linear programming problem: Min z = 3x12x23x3 Subject to x1+2x2 +x3 14 x1+2x2+4x3 12 x1 x2 +x3 = 2 x3 3 x1; x2 unrestricted (a) [6 points] Reformulate the problem so it is in standard format. (b) [6 points] Reformulate the problem so it is in canonical format. (c) [3 points] Convert the problem into a maximization problem. 2. [20 points] A lathe is used to reduce the diameter of a steel shaft whose length is 36 inches (in.) from 14 in. to 12 in. The speed x1 (in revolutions...
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Course MBA – 2nd Semester Subject Operations Research Assignment MB0048 – Set 1 Q.1. (a) what is linear programming problem? (b) A toy company manufactures two types of dolls, a basic version doll- A and a deluxe version doll-B. Each doll of type B takes twice as long to produce as one of type A, and the company would have time to make maximum of 1000 per day. The supply of plastic is sufficient to produce 1000 dolls per day (both A & B combined). The deluxe version requires...
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as 3 separate constraints in an integer program. Answer Selected Answer: False Correct Answer: False . Question 2 2 out of 2 points Rounding non-integer solution values up to the nearest integer value will result in an infeasible solution to an integer linear programming problem. Answer Selected Answer: False Correct Answer: False . Question 3 2 out of 2 points If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a conditional...
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a.|objectives, resources, goals.| b.|decisions, constraints, an objective.| c.|decision variables, profit levels, costs.| d.|decisions, resource requirements, a profit function.| ___A_ 2. What is the goal in optimization? a.|Find the best decision variable values that satisfy all constraints.| b.|Find the values of the decision variables that use all available resources.| c.|Find the values of the decision variables that satisfy all constraints.| d.|None of the above.| ____B 3. Limited resources...
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In the following expression, which is (are) the dependent variable(s)? PROFIT = REVENUE – EXPENSES. Profit The categories of modeling techniques presented in this book include all of the following except: preventive models. Which of the following categories of modeling techniques includes optimization techniques? Prescriptive models What is the goal in optimization? Find the decision variable values that result in the best objective function and satisfy all constraints. The following linear...
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Linear Programming: Using the Excel Solver Outline: We will use Microsoft Excel Solver to solve the four LP examples discussed in last class. 1. The Product Mix Example The Outdoor Furniture Corporation manufactures two products: benches and picnic tables for use in yards and parks. The firm has two main resources: its carpenters (labor) and a supply of redwood for use in the furniture. During the next production period, 1200 hours of manpower are available under a union agreement. The firm...
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The simplex method is used regularly on applied problem involving thousands of variables and problem constrains (Barnett, Ziegler, & Byleen, 2011). This was the method chosen when Wintel Technologies needed to figure out the best way to utilize time and schedule for their field engineers. With a busy schedule and being needed on several different areas in the United States, the simplex method was brought in to decide where and when employees should be spending their time and the company’s money....
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Quantitative Methods: MAT540 Quiz 5 • Question 1 If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a conditional constraint. Answer Selected Answer: True Correct Answer: True • Question 2 If we are solving a 0-1 integer programming problem with three decision variables, the constraint x1 + x2 + x3 ≤ 3 is a mutually exclusive constraint. Answer Selected Answer: False Correct Answer: False • Question 3 If we...
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Assignment #3: Julia’s Food Booth Quantitative Methods 540 Buddy L. Bruner, Ph.D. Shirley Foster 11/25/2012 Assignment 3: Case problem “Julia’s Food Booth” Page 1 A. Julia Robertson is making an allowance for renting a food booth at her school. She is seeking ways to finance her last year and believed that a food booth outside her school’s stadium would be ideal. Her goal is to earn the most money possible thus increasing her earnings. In this case...
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Calculate Basic Feasible Solution using Simplex Method Abstract: The problem of maximization/minimization deals with choosing the ideal set of values of variables in order to find the extrema of an equation subject to constraints. The simplex method is one of the fundamental methods of calculating the Basic Feasible Solution (BFS) of a maximization/minimization. This algorithm implements the simplex method to allow for quick calculation of the BFS to maximize profit or minimize loss, depending on the...
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margin or least time, when resources have alternative uses. The term ‛linear’ means that all inequations or equations used and the function to be maximized or minimized are linear. That is why linear programming deals with that class of problems for which all relations among the variables involved are linear. Formally, linear programming deals with the optimization (maximization or minimization) of a linear function of a number of variables subject to a ¹equations in variables involved. The general...
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Computer Methods 2.1 Chapter Questions 1) Consider the following linear programming model: Max X12 + X2 + 3X3 Subject to: X1 + X2 ≤ 3 X1 + X2 ≤ 1 X1, X2 ≥ 0 This problem violates which of the following assumptions? A) certainty B) proportionality C) divisibility D) linearity E) integrality Answer: D Page Ref: 22 Topic: Developing a Linear Programming Model Difficulty: Easy 2) Consider the following linear programming model: Min 2X1 + 3X2 Subject to: X1 + 2X2 ≤ 1 X2 ≤ 1 X1 ≥ 0, X2 ≤...
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QUIZ LP Formulation 1. ACME Co. is a manufacturer of X’s which they sell to OR/MS teachers to be used as variables. For this week they intend to manufacture two types of X’s, namely, X1’s & X2’s. The profit contribution for each unit of X1 & X2 are $4 and $5, respectively. Mr. Loony, the manager of ACME, found out that his available supply of raw material 1 (RM1) and raw material 2 (RM2) are 18 units and 24, respectively. Each unit of X1 will require 3 units of RM1 and 3 units of RM2 while...
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of interior and exterior paints that maximizes the daily profit. TAHA Example 2.1-1 (Page 47) : The Reddy Mikks Company LP models have three basic components (like all OR models) – 1. 2. 3. Decision variables that we seek to determine. An objective that we need to optimize (minimize or maximize). This involves constructing an objective function. Constraints that the solution must satisfy. The variables of the model for solving this problem are : x1 = Tons of exterior paint to be produced...
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on times when demand is low due to the fixed costs of the company. Probability of a favorable market is 60%. The company has a few couple of choices in the plans for expanding the company, and after some deliberation, they have come up with the following decisions: 1) Expand with a Large Factory, having an estimate increase in income of P11,000 on a good demand and a loss of P1000 on a bad demand. 2) Expand with a Medium Factory, having an estimate increase in income of P8,000 on a good demand and...
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Graphical Solution Method Linear Programming Model Simplex method Solution Solving Linear Programming Problems with Excel Dr A Lung Student exercises Kingston University London 1 Linear Programming (LP) • A model consisting of linear relationships representing a firm’s objective and resource constraints • LP is a mathematical modeling technique used to determine a level of operational activity in order to achieve an objective, subject to restrictions called constraints • Pioneered by George...
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Online Tutorial The Simplex Method of Linear Programming Tutorial Outline CONVERTING THE CONSTRAINTS TO EQUATIONS SOLVING MINIMIZATION PROBLEMS SETTING UP THE FIRST SIMPLEX TABLEAU KEY TERMS SIMPLEX SOLUTION PROCEDURES SOLVED PROBLEM SUMMARY OF SIMPLEX STEPS FOR MAXIMIZATION PROBLEMS DISCUSSION QUESTIONS ARTIFICIAL AND SURPLUS VARIABLES SUMMARY PROBLEMS 3 T 3-2 O N L I N E T U T O R I A L 3 THE SIMPLEX METHOD OF L I N E A R P RO G R A M M I N...
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Julia’s Food Booth Case Problem Assignment 3 Max Z =Profit1x1+ Profit2x2+ Profit3x3 A - Formulation of the LP model x1 - number of pizza slice x2 - number of hot dogs x3 - number of barbecue sandwiches Constraints Cost Maximum fund available for food = $1500 Cost per pizza $6 ÷08 (slices) = $0.75 Cost for a hot dog = $0.45 Cost for a barbecue sandwich = $0.90 Constraint: 0.75x1+0.45x2+0.90x3 ≤1500 Oven space Space available 16.3.4.2 = 384ft^2 384...
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fund and a money market fund. Each unit of the stock fund costs $50 and provides an annual rate of return of 10%; each unit of the money market fund costs $100 and provides an annual rate of return of 4%. The client wants to minimise risk subject to the requirement that the annual income from the investment be at least $60,000. According to Innis’s risk measurement system, each unit invested in the stock fund has a risk index of 8, and each unit invested in the money market fund has a risk...
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Problem 4. (25 Points) Solve the following problem graphically (Please be neat). Draw the polytope on the x-y coordinate system (can be done either by hand or computer). Show all intersection of the polytope and identify the point (x,y coordinate) where the objective function is maximized and provide that value. Maximize Z = 3x1 + 2x2 Subject to: 1x1 + 1x2 ≤ 10 8x1 + 1x2 ≤ 24 and x1, x2 ≥ 0 Solution : Point (a) is the origin (0,0) where Z(a) = 3*0 + 2*0 = 0 Point (b) is the...
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Quantitative Methods BITS Pilani Pilani Campus Course handout BITS Pilani Pilani Campus Session-1 Instructor Details Dr. Remica Aggarwal 1214 C ; FD-1 Department of Management Email: remica_or@rediffmail.com Mobile: 09772054839 BITS Pilani, Pilani Campus Course Details • • • • • • • Management Science Use of QM/QA Modelling Techniques Data Analysis Techniques MS Excel QM for Windows Test BITS Pilani, Pilani Campus Quantitative Methods • • • • • • • Operations...
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MATH 4321 Spring 2013 Assignment Solution 0-Sum Games 2 1. Reduce by dominance to 2x2 games and solve. 5 4 4 3 (a) 0 1 1 2 1 0 2 1 4 3 1 2 10 0 7 1 (b) 2 6 4 7 6 3 3 5 Solution: (a). Column 2 dominates column 1; then row 3 dominates row 4; then column 4 dominates column 3; then row 1 dominates row 2. The resulting submatrix consists of row 1 and 3 vs. columns 2 and 4. Solving this 2 by 2 game and moving back to the original game we find that value is...
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its third chapter. Key words: linear programming; simplex method. I. Fundamentals and scope Based on a prototype example, Linear Programming is presented, as well as the simplex method of resolution. This method was first presented by G. B. Dantzig in 1947 [MacTutor, 2007]. The text is based on the book by Hillier and Lieberman [2005], and begins with segments of the third chapter of the book. II. Explanation of the simplex method 3 Introduction to Linear Programming (H&L 25) The development...
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Statistics and Quantitative Methods ASSIGNMENT: 2 PREPARED FOR: CHOWDHURY PREPARED BY: VALLEY STATE UNIVERSITY DATE: 13. The Electrotech Corporation manufactures two industrial-sized electrical devices: generators and alternators. Both of these products require wiring and testing during the assembly process. Each generator requires 2 hours of wiring and 1 hour of testing and can be sold for a $250 profit. Each alternator requires 3 hours of wiring and 2 hours of testing and can...
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and solve a linear programming model for Julia that will help you advise her if she should lease the booth. Let, X1 =No of pizza slices, X2 =No of hot dogs, X3 = barbeque sandwiches Formulation: 1. Calculating Objective function co-efficients: The objective is to Maximize total profit. Profit is calculated for each variable by subtracting cost from the selling price. • For Pizza slice, Cost/slice=$6/8=$0.75 | |X1 ...
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1. Discuss why and how you would use a liner programming model for a project of your choice, either from your own work or as a hypothetical situation. Be sure that you stae your situation first, before you develpp the LP model Linear programming is a modeling technique that is used to help managers make logical and informed decisions. All date and input factors are known with certainty. Linear program models are developed in three different steps: Formulation Solution Interpretation ...
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2,5 9,10,15 Standard Form 1 14 Slack/Surplus Variables 1 16 Equal-to Constraints 3,5 14 Redundant Constraints 5,7 12,13 Extreme Points 2,5 11,16 Alternative Optimal Solutions 7...
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JIT versus the Theory of Constraints | AMB303 International Logistics | Theory of Constraints | Name : Hui LuStudent Number: N8035636Date: 02/09/2012Word Count:1007 | Contents 1.0 Definition……………………………..…………………….3 2.0 Discussion…………………………………………...……..3 2.1Core concept…………………………………..….3 2.2Five Steps of TOC………………………………..4 2.3 Evaluation………………………………………..4 2.3.1 Advantages…………………………...4 2.3.2 Disadvantages……………………...…4 2.4. Example……………………..…………………..5 3.0 Conclusion 6 ...
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Mat 540 Quiz 3 Question 1 .2 out of 2 points Correct The following inequality represents a resource constraint for a maximization problem: X + Y ≥ 20 Answer Selected Answer: False Correct Answer: False Question 2 .2 out of 2 points Correct Graphical solutions to linear programming problems have an infinite number of possible objective function lines. Answer Selected Answer: True Correct Answer: True Question 3 .2 out of 2 points Correct Surplus variables are only associated...
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SENSITIVITY ANALYSIS The solution obtained by simplex or graphical method of LP is based on deterministic assumptions i.e. we assume complete certainty in the data and the relationships of a problem namely prices are fixed, resources known, time needed to produce a unit exactly etc. However in the real world, conditions are seldom static i.e. they are dynamic. How can such discrepancy be handled? For example if a firm realizes that profit per unit is not Rs 5 as estimated but instead closer...
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mathematical method for determining a way to achieve the best outcome such as maximum profit or lowest cost in a given mathematical model for some list of requirements represented as linear relationships. Linear programming is a specific case of mathematical programming The Primary Purpose of the present investigation is to develop an interactive spreadsheet tool to aid in determining a maximum return function in 401K plan. In this paper, we discuss how the Excel spreadsheet can be used to solve a linear...
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is a mathematical procedure for determining optimal allocation of scarce resources. Requirements of Linear Programming • all problems seek to maximize or minimize some quantity • The presence of restrictions or constraints • There must be alternative courses of action • The objective and constraints in linear programming must be expressed in terms of linear equations or inequalities Objective Function it maps and translates the...
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Operations Research Unit 4 Unit 4 Simplex Method Structure: 4.1 Introduction Objectives 4.2 Standard Form of LPP Fundamental theorem of LPP 4.3 Solution of LPP – Simplex Method Initial basic feasible solution of an LPP To solve an LPP in canonical form by simplex method 4.4 The Simplex Algorithm Steps 4.5 Penalty Cost Method or Big M-method 4.6 Two Phase Method 4.7 Solved Problems on Minimisation 4.8 Summary 4.9 Glossary 4.10 Terminal Questions 4.11 Answers 4.12 Case Study 4.1 Introduction ...
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TOPIC; LINEAR PROGRAMMING DATE; 5 JUNE, 14 UNIVERSITY OF CENTRAL PUNJAB INTRODUCTION TO LINEAR PROGRAMMING Linear programming (LP; also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming. ...
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through the simplex method step by step to demonstrate that the following problem is unbounded. (5 marks) max 5x1 + x2 + 3x3 + 4x4 s.t. x1 – 2x2 + 4x3 + 3x4 ≤ 20 –4x1 + 6x2 + 5x3 – 4x4 ≤ 40 2x1 – 3x2 + 3x3 + 8x4 ≤ 50 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 Problem 2. Work through the simplex method step by step to find all optimal basic feasible solutions for the following problem. (5 marks) max x1 + x2 + x3 + x4 s.t. x1 + x2 ≤ 3 x3 + x4 ≤ 2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0,...
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programming: 1. Linear Programming- involves no more than 2 variables, linear programming problems can be structured to minimize costs as well as maximize profits. Due to the increasing complexity of business organizations, the role of the management executive as a decision maker is becoming more and more difficult. Linear programming is a useful technique to solve such problems. The necessary condition is that the data must be expressed in quantitative terms in the form of linear equations and inequalities...
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easily connect to existing libraries. That is why optimization is an important and practical topic in Scilab, which provides tools to solve linear and nonlinear optimization problems by a large collection of tools. Overview of the industrial-grade solvers available in Scilab and the type of optimization problems which can be solved by Scilab. Objective Linear Bounds y Equality l Inequalities l Problem size m m l l y Nonlinear s Gradient needed y n Solver linpro quapro qld qpsolve optim neldermead...
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A: Formulation of the LP Model X1(Pizza), X2(hotdogs), X3(barbecue sandwiches) Constraints: Cost: Maximum fund available for the purchase = $1500 Cost per pizza slice = $6 (get 8 slices) =6/8 = $0.75 Cost for a hotdog = $.45 Cost for a barbecue sandwich = $.90 Constraint: 0.75X1 + 0.45X2+ 0.90(X3) ≤ 1500 Oven space: Space available = 3 x 4 x 16 = 192 sq. feet = 192 x 12 x 12 =27648 sq. inches The oven will be refilled before half time- 27648 x 2 = 55296 ...
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Show your work. Required Problems: 1) Find the slope of the following functions at X = 3. a. Y = 4 + 3X2 b. Y = 5X + 6X3 c. Y = 6X d. Y = (6X + 3)2 / 4X 2) If a firm’s Total Cost equation is TC = 200 + 3Q + 7Q2: a. What is the equation for the firm’s marginal cost? b. What is the firm’s marginal cost when Q =1? Q= 5? 3) At the Peoria Company, the relationship between profit and output is as follows: ...
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Page 1 7 MODULE Linear Programming: The Simplex Method LEARNING OBJECTIVES After completing this chapter, students will be able to: 1. Convert LP constraints to equalities with slack, surplus, and artificial variables. 2. Set up and solve LP problems with simplex tableaus. 3. Interpret the meaning of every number in a simplex tableau. 4. Recognize special cases such as infeasibility, unboundedness and degeneracy. 5. Use the simplex tables to conduct sensitivity analysis. 6...
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the Theory of Constraints? The Theory of Constraints is an organizational change method that is focused on profit improvement. The essential concept of TOC is that every organization must have at least one constraint. A constraint is any factor that limits the organization from getting more of whatever it strives for, which is usually profit. The Goal focuses on constraints as bottleneck processes in a job-shop manufacturing organization. However, many non-manufacturing constraints exist, such as...
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APPLIED OPERATIONAL RESEARCH FOR MANAGEMENT NOTES 1 ANNA UNIVERSITY CHENNAI UNIT I INTRODUCTION TO LINEAR PROGRAMMING (LP) INTRODUCTION Operations Research (OR) (a term coined by McClosky and Trefthen in 1940) was a technique that evolved during World War II to effectively use the limited military resources and yet achieve the best possible results in military operations. In essence you can state that OR is a technique that helps achieve best (optimum) results under the given set of limited...
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the stepping-stone method is that if, at a particular iteration, we accidentally choose a route that is not the best, the only penalty is to perform additional iterations. 4) The transportation algorithm can be used to solve both minimization problems and maximization problems. 5) In the assignment problem, the costs for a dummy row will be equal to the lowest cost of the column for each respective cell in that row. Multiple Choice Questions: 6) Which of the following techniques can be...
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Big-M method The Big-M method of handling instances with artificial variables is the “common sense approach”. Essentially, the notion is to make the artificial variables, through their coefficients in the objective function, so costly or unprofitable that any feasible solution to the real problem would be preferred....unless the original instance possessed no feasible solutions at all. But this means that we need to assign, in the objective function, coefficients to the artificial variables that...
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. . . . . . . . . . . . . Shortest path problem . . . . . . . . . . . . . . . . . . . . . . . . . Pricing a Tech Gadget . . . . . . . . . . . . . . . . . . . . . . . . . Finding a closest point feasible in an LP . . . . . . . . . . . . . . . . Finding a “central” feasible solution of an LP . . . . . . . . . . . . . 7 8 9 10 12 13 21 22 24 35 35 37 40 40 42 48 48 50 50 52 53 55 55 56 Linear programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integer programs . . . . . . ...
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Vandenberghe January 4, 2006 Chapter 2 Convex sets Exercises Exercises Deﬁnition of convexity 2.1 Let C ⊆ Rn be a convex set, with x1 , . . . , xk ∈ C, and let θ1 , . . . , θk ∈ R satisfy θi ≥ 0, θ1 + · · · + θk = 1. Show that θ1 x1 + · · · + θk xk ∈ C. (The deﬁnition of convexity is that this holds for k = 2; you must show it for arbitrary k.) Hint. Use induction on k. Solution. This is readily shown by induction from the deﬁnition of convex set. We illustrate the idea for k = 3, leaving the...
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standard tool that has saved many thousands or millions of dollars for most companies or businesses of even moderate size in the various industrialized countries of the world; and its use in other sectors of society has been spreading rapidly. A major proportion of all scientific computation on computers is devoted to the use of linear programming. Dozens of textbooks have been written about linear programming, and published articles describing important applications now number in the hundreds. What...
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problem: n Maximize j=1 cjxj, subject to: n j=1 ai j x j = bi xj ≥ 0 x j integer (i = 1, 2, . . . , m), ( j = 1, 2, . . . , n), (for some or all j = 1, 2, . . . , n). This problem is called the (linear) integer-programming problem. It is said to be a mixed integer program when some, but not all, variables are restricted to be integer, and is called a pure integer program when all decision variables must be integers. As we saw in the preceding chapter, if the constraints are of a network...
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