...LINEAR PROGRAMMING II 1 Linear Programming II: Minimization © 2006 Samuel L. Baker Assignment 11 is on page 16. Introduction A minimization problem minimizes the value of the objective function rather than maximizing it. Minimization problems generally involve finding the least-cost way to meet a set of requirements. Classic example -- feeding farm animals. Animals need: 14 units of nutrient A, 12 units of nutrient B, and 18 units of nutrient C. Learning Objective 1: Recognize problems that linear programming can handle. Linear programming lets you optimize an objective function subject to some constraints. The objective function and constraints are all linear. Two feed grains are available, X and Y. A bag of X has 2 units of A, 1 unit of B, and 1 unit of C. A bag of Y has 1 unit of A, 1 unit of B, and 3 units of C. A bag of X costs $2. A bag of Y costs $4. Minimize the cost of meeting the nutrient requirements. To solve, express the problem in equation form: Cost = 2X + 4Y objective function to be minimized Constraints: 2X + 1Y $ 14 nutrient A requirement 1X + 1Y $ 12 nutrient B requirement 1X + 3Y $ 18 nutrient C requirement 8 8 Read vertically to see how much of each nutrient is in each grain. X $ 0, Y $ 0 non-negativity Learning objective 2: Know the elements of a linear programming problem -- what you need to calculate a solution. The elements are (1) an objective function that shows the cost or profit depending on what choices you make, (2) constraint inequalities...
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...Linear Programming (LP) Linear programming, simply put, is the most widely used mathematical programming technique. It has a long history dating back to the 1930s. The Russian mathematical economist Leonid Kantorovich published an important article about linear programming in 1939. George Stigler published his famous diet problem in 1945 (“The Cost of Subsistence”). Of course, no one could actually solve these problems until George Dantzig developed the simplex method, which was published in 1951. Within a few years, a variety of American businesses recognized that they could save millions of dollars a year using linear programming models. And in the 1950s, that was a lot of money. In his book Methods of Mathematical Economics (Springer-Verlag, 1980), Joel Franklin talks about some of the uses of linear programming (LP). In fact, about half of his book is devoted to LP and its extensions. Today, we will analyze one of the examples provided in that book. The example comes from a 1972 article published in the Monthly Review of the Federal Reserve Bank of Richmond. Alfred Broaddus, the author, was trying to explain to bankers how Bankers Trust Company used linear programming models in investment management. His example was simple and effective. The bank has up to 100 million dollars to invest, a portion of which can go into loans (L), and a portion of which can go into securities (S). Loans earn 10%, securities 5%. The bank is required to keep 25% of its invested...
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...Linear programming solution examples Linear programming example 1997 UG exam A company makes two products (X and Y) using two machines (A and B). Each unit of X that is produced requires 50 minutes processing time on machine A and 30 minutes processing time on machine B. Each unit of Y that is produced requires 24 minutes processing time on machine A and 33 minutes processing time on machine B. At the start of the current week there are 30 units of X and 90 units of Y in stock. Available processing time on machine A is forecast to be 40 hours and on machine B is forecast to be 35 hours. The demand for X in the current week is forecast to be 75 units and for Y is forecast to be 95 units. Company policy is to maximise the combined sum of the units of X and the units of Y in stock at the end of the week. * Formulate the problem of deciding how much of each product to make in the current week as a linear program. * Solve this linear program graphically. Solution Let * x be the number of units of X produced in the current week * y be the number of units of Y produced in the current week then the constraints are: # 50x + 24y = 45 so production of X >= demand (75) - initial stock (30), which ensures we meet demand # y >= 95 - 90 # i.e. y >= 5 so production of Y >= demand (95) - initial stock (90), which ensures we meet demand The objective is: maximise (x+30-75) + (y+90-95) = (x+y-50) i.e. to maximise the number of units left in stock...
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...METHODS/ APPROACH This scheduling problem can be solved most expeditiously using linear programming. Let F denote the number of full-time employ- ees. Some number, F1, of them will work one hour of overtime between 5 PM and 6 PM each day and some number, F2, of the full- time employees will work overtime between 6 PM and 7 PM. There will be seven sets of part-time employees who begin their work day at hour j=j␣1,2,...,7,withP1beingthenumberofworkers beginning at 9 AM, P2 at 10 AM, . . . , P7 at 3 PM. Note that because part-time employees must work a minimum of four hours, none can start after 3 PM because the entire operation ends at 7 PM. Similarly, some number of part-time employees, Qj, leave at the end of hour j, j 4, 5, . . . , 9. The workforce requirements for the first two hours, 9 AM and 10 AM, are: F P1 14 F P1 P2 25 At 11 AM half of the full-time employees go to lunch; the remaining half go at noon. For those hours: 0.5F P1 P2 P3 26 0.5F P1 P2 P3 P4 38 Starting at 1 PM, some of the part-time employees begin to leave. For the remainder of the straight-time day: F P1 P2 P3 P4 P5 −Q4 55 F P1 P2 P3 P4 P5 P6 −Q4 −Q5 60 F P1 P2 P3 P4 P5 P6 P7 −Q4 −Q5 −Q6 51 F P1 P2 P3 P4 P5 P6 P7 −Q4 −Q5 −Q6 −Q7 29 For the two overtime...
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...in equation (1) =0, then 8x1 + 6(0) = 1920 X1=1920/8 X1=240 (240, 0) If x1 in equation (2) =0, then 3(0) + 6x2 = 1440 6x2 = 1440 X2 = 1440/6 X2 = 240 (0, 240) If x2 in equation (2) = 0, then 3x1 + 6(0) = 1440 3x1 = 1440 X1 = 1440/3 X1= 480 (480, 0) If x1 in equation (3) =0, then 3(0) + 2x2 = 720 2x2 =720 X2=720/2 X2=360 (0, 360) If x2 in equation (3) =0, then 3x1 + 2(0) =720 3x1 = 720 X1 =720 3 X1 = 240 (240, 0) If x1 in equation (4) = 0, then X2=288 (0, 288) If x2 in equation (4) = 0, then X1 = 288 (288, 0) Due to the multiple constraints, it is difficult to obtain the optimal solution from the graph. Therefore, the simultaneous equation would be used to the solve linear programming model. Using simultaneous equation, 8x1 + 6x2 = 1920 ounces…………………………. (1) 3x1 + 6x2 = 1440 ounces…………………………. (2) 3x1 + 2x2 = 720 ounces…………………………… (3) X1 + x2 =288 jars………………………………….. (4) Using the...
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...LINEAR PROGRAMING AND SIMPLEX METHOD Devharajan Rangarajan Department of Electronic Engineering National University of Ireland, Maynooth devharajan.rangarajan.2016@mumail.ie Abstract— An optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. This pays way to a new world of constrained optimization. This paper focuses on one such optimization technique known as Linear programming and one of its method known as Simplex method in detail with examples. cTx = c1x1 + · · · + cnxn The subject of linear programming can be defined quite concisely. It is concerned with the problem of maximizing or minimizing a linear function whose variables are required to satisfy a system of linear constraints, a constraint being a linear equation or inequality. The subject might more appropriately be called linear optimization. Problems of this sort come up in a natural and quite elementary way in many contexts but especially in problems of economic planning. (or Ax ≤ b) I. INTRODUCTION Linear programming is the process of taking various linear inequalities relating to some situation, and finding the "best" value obtainable under those conditions. A typical example would be taking the limitations of materials and labour, and then determining the "best" production levels for maximal profits under those conditions. In "real life", linear...
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...Linear Programming: Using Solver in Excel Linear Programming was conceptually developed before World War II by the outstanding Russian mathematician A.N.Kolmogorov and gained its popularity ever since the development of Simplex method by George B. Dantzig in 1947. Linear programming deals with problems of maximizing or minimizing a linear function in the presence of linear equality and/or inequality constraints. In these problems, we find the optimal, or most efficient way of using limited resources to achieve the objective of the situation. Linear Programming enables users to model large and complex problems and solve in a short amount of time by the use of effective algorithm, hence it is a powerful and widely used tool in various fields such as science, industrial engineering, financial planning and management decision making. Nowadays, with the development of technology, most of the real world Linear Programming problems are solved by computer programs. Excel Solver is a popular one. We work through different examples to demonstrate the applications of linear Programming model and the use of Excel Solver for various decision making in operation and supply chain management. Components of Linear Programming model To solve the linear programming problems, we first need to formulate the mathematical description called a mathematical model to represent the situation. Linear programming model usually consists of the following components * Decision variables: These represent...
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...The owner of Chips etc. produces 2 kinds of chips: Lime (L) and Vinegar (V). He has a limited amount of the 3 ingredients used to produce these chips available for his next production run: 4600 ounces of salt, 9400 ounces of flour, and 2200 ounces of herbs. A bag of Lime chips requires 1.5 ounces of salt, 5 ounces of flour, and 2 ounces of herbs to produce; while a bag of Vinegar chips requires 4 ounces of salt, 6 ounces of flour, and 2 ounces of herbs. Profits for a bag of Lime chips are $0.48, and for a bag of Vinegar chips $0.59. a) What is the formulation for this problem? b) For the production combination of 800 bags of Lime and 600 bags of Vinegar, which resource is not completely used up and how much is remaining? c) For the production combination of 800 bags of Lime and 600 bags of Vinegar, which resource is not completely used up and how much is remaining? d) Discuss: Slack (if any); shadow price, and sensitivity analysis results using the program of your choice. Above problem is a maximization problem as one is trying to maximize the profits by making different bags of chips. It takes salt, flour and herbs to make two different types of chips – Lime and Vinegar. There are constrained amounts of salt, flour and herb and the owner want to maximize his profits. The amount of profit per bag is given as well. The LP problem thus becomes: Maximize Profits from the sale of bags of both lime and vinegar chips Constraints: 1. Salt consumed should not exceed...
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...The development of linear programming has been ranked among the most important scientific advances of the mid 20th century. Its impact since the 1950’s has been extraordinary. Today it is a standard tool used by some companies (around 56%) of even moderate size. Linear programming uses a mathematical model to describe the problem of concern. Linear programming involves the planning of activities to obtain an optimal result, i.e., a result that reaches the specified goal best (according to the mathematical model) among all feasible alternatives. Linear Programming as seen by various reports by many companies has saved them thousands to even millions of dollars. Since this is true why isn’t everyone using Linear Programming? Maybe the reason is because there has never been an in-depth experiment focusing on certain companies that do or do not use linear programming. My main argument is that linear programming is one of the most optimal ways of resource allocation and making the most money for any company today. I used (in conjunction with another field supporter – My Dad) the survey method to ask 28 companies that were in Delaware, New Jersey, and Pennsylvania whether they were linear programming users. In addition, I wanted to examine the effect of the use of linear programming across three different but key decision support areas of the participating companies to include (1) Planning (2) Forecasting and (3) Resource Allocation. The companies were selected randomly from the Dunn...
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...LINEAR PROGRAMMING Definition. A mathematical technique for solving constrained maximization and minimization problems when there are many constraints and the objective function to be optimized, as well as the constraints faced, are linear (i.e., can be represented by straight lines) Assumptions. -LP is based on the assumption that the objective function that the organization seeks to optimize (maximize or minimize), as well as the constraints that it faces, are linear and can be represented GRAPHICALLY by straight lines. -Input and output prices are constant -Average and marginal costs are constant and equal (they are linear) -Profit per unit is constant; profit function is linear Applications of Linear Programming 1. Optimal process selection 2. Optimal product mix 3. Satisfying minimum product requirements 4. Long-run capacity planning 5. Other specific applications of linear programming Basic Linear Programming Concepts A. Production Process and Isoquants -where a production process or activity can be represented by a straight line ray from the origin in input space B. Optimal Mix of Production Process Procedure Used in Formulating and Solving Linear Programming Problems The steps followed in solving linear programming problem are: 1. Express the objective function of the problem as an equation and the constraints as inequalities. 2. Graph the inequality constraints and define the feasible region. 3. Graph the...
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...1. INTRODUCTION Linear programming, an operations research technique is widely used in finding solutions to complex managerial decision problems. The introduction of linear programming (LP) has produced remarkable benefits in a number of industries. The early experimental applications of LP techniques in the petroleum industry as a refinery management tool had such profound effects that LP is now standard in almost every aspect of that industry. The first application of LP in the textile industry was designed to produce optimal plant efficiency, that is, allocate plant resources to production problems so as to achieve the highest practical return. The purpose of this study is to demonstrate the application of LP model in the blending (mixing) of cotton to produce Acrylic yarn in case of Arbaminch Textile factory. Because the cotton blending process involves complex quality control, it is particularly responsive to LP techniques. In view of today’s technology, the process of cotton fiber selection should undergo an inevitable transition from the traditional pure art to a sound scientific technique. In order to achieve this transition, fiber selection should be integrated into a cotton fiber mixing program that attempts to optimize cotton fiber use with respect to cost and quality of end product. I attempted to examine the practical aspect of linear programming for optimization of cost of producing cotton blended yarn in Arbaminch textile factory without impinging the required...
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...Math 428 Operation Research Linear Programming Project This project is mainly to report on our analysis of the mathematical model established by the three-month production plan from the bicycle manufacturer. It includes what suggestions we’ve had for the client, details on how the model was set up and also how future change influences the results. Firstly, we’ll explain how and why the model was set up in this way: Our goal is to help the company minimize their cost on the production and the inventory. Let x4m be the amount of mountain bikes produced in April Let x5m be the amount of mountain bikes produced in May Let x6m be the amount of mountain bikes produced in June Let x4r be the amount of road bikes produced in April Let x5r be the amount of road bikes produced in May Let x6r be the amount of road bikes produced in June Since each mountain bike frame costs $200 and each road bike costs $250, then the total production cost would be 200x4m+x5m+x6m+250(x4r+x5r+x6r). The carried-over inventory costs $10 for each bike frame. We got 150 in store before the three-month production plan and it costs 150∙10=$1500. Then for each month, we have 50+x4m+100+x4r-1300 for the carryover from April to May; 50+x4m+100+x4r-1300+x5m+x5r-1700 for the carryover from May to June. Hence these carryovers are going to cost $10 for each as well. Put everything together and simplies it we have the objective function: Minimize Z=220x4m+210x5m+200x6m+270x4r+260x5r+250x6r-38500 For...
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...on the basis on which inventory management is done, the different type of inventory models in the context of the assignment given. The process in which each of them is used. This would help you decide the right inventory model. Pl go through this properly. As a growing corporation in the e-commerce retail apparel industry, XYZ has recently been under pressure to package and ship a higher volume of clothes. In order for the corporation to meet this demand, XYZ’s executive staff must decide either to increase its staffs’ hours in the logistics department and potentially hire another employee or bulk ship inventory to Amazon warehouses to be fulfilled through Amazon Prime. The executive team was able to recreate these two scenarios through linear programing and has determined that the company will seek more profit by utilizing Amazon’s fulfillment services. After analyzing the sales model of using Amazon Inc. fulfillment the recommendations for management are to focus on inventory limits and create patterns that will help forecast the inventory needed. By seeking out inventory models, which resolves dual problems of maintaining sufficient inventories to meet demand, we are able to successfully determine the appropriate inventory limits needed for this operation. Now we see that there could have been many solutions like for low values of demand, fulfillment should be done in-house and for high values direct shipping should be used. When demand is comparable to capacity, all three...
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...EMSE 6850 Introduction to Management Science Year 2013 – 2014 First deadline is 31st December 2013 Absolute deadline is 5th January 2014 Assessment You are required to prepare a business report on a Management Science related problem of your choice. The report should be a self-contained (3000 words max) document explaining the problem; the method of your choice with justification; application analysis and outcomes. The maximum number of words is 3000 words but you are allowed to add any appendices should you deem necessary. The contents should be as follows: Executive Summary One page description of the business problem tackled, the MS approached used, and outcomes. Document signposts Table of contents and tables of figures and tables (if needed). Use of citation and references as appropriate. Introduction A description of the business problem faced and the objectives as laid down by the management group. You may refer to Hillier and Hillier for help in describing the problem. Method used Present the MS method used and why you thought it was the most appropriate amongst other methods. Your justification of the choice is an important part of your assessment Implementation A description of how the raw problem is converted into a spreadsheet model. Please provide details of the raw data and the steps followed for populating it in Excel Analysis Provide alternative solutions and scenarios and their respective outcomes. This should be accompanied by a...
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...Data Analysis and Business Decision Making Session – IV Chapter 11: Linear Programming Chapter 12: Transportation, Transshipment, and Assignment Problems Chapter 13: Decision Analysis Faculty Pankaj Dutta Chapter – 11 Linear Programming LPP solution through Excel Solver: The Steps in Implementing an LP Model in a Spreadsheet: 1. 2. 3. 4. Organize the data for the model on the spreadsheet. Reserve separate cells in the spreadsheet for each decision variable in the model. Create a formula in a cell in the spreadsheet that corresponds to the objective function. For each constraint, create a formula in a separate cell in the spreadsheet that corresponds to the left-hand side (LHS) of the constraint. PANKAJ DUTTA IMTCDL Chapter – 11 Linear Programming LPP solution through Excel Solver: Max Z = 350X1 + 300X2 Subject To 1X1 + 1X2 ≤ 200 9X1 + 6X2 ≤ 1566 12X1+16X2 ≤ 2880 X1 , X 2 ≥ 0 1. Organize the data for the model on the spreadsheet. PANKAJ DUTTA IMTCDL Chapter – 11 Linear Programming LPP solution through Excel Solver: Max Z = 350X1 + 300X2 Subject To 1X1 + 1X2 ≤ 200 9X1 + 6X2 ≤ 1566 12X1+16X2 ≤ 2880 X1 , X 2 ≥ 0 1. Organize the data for the model on the spreadsheet. Changing cells Target cell PANKAJ DUTTA IMTCDL Constraint cells Chapter – 11 Linear Programming Max Z = 350X1 + 300X2 Subject To 1X1 + 1X2 ≤ 200 9X1 + 6X2 ≤ 1566 12X1+16X2 ≤ 2880 X1 , X 2 ≥ 0 LPP solution through Excel Solver: ...
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