Homework
1. Complete the following two problems.
Problem 1
Activity Mean Duration (Days) Std. Dev. (Days)
A 11 0.9
B 13 1.1
C 7 0.2
D 9 0.8
E 6 1
F 7 1.2
G 10 0.7
H 9 0.6
I 8 0.8
a) Calculate the project completion time.
b) Indicate the critical path activities.
c) What is the probability of completing this project between 38 and 40 days? d) What are the slack values for activities C and F? Interpret the meaning of their slack values?
Problem 2
A registered nurse is trying to develop a diet plan for patients. The required nutritional elements are the total daily requirements of each nutritional element are as indicated in table 2:
Table 2
(Required Nutritional Element Total and Daily Requirements)
Calories Not more than 2,700 calories
Carbohydrates Not less than 300 grams
Protein Not less than 250 grams
Vitamins Not less than 60 units
The nurse has four basic types to use when planning the menus. The units of nutritional elements per unit of food type are shown in the table below. Note than the cost associated with a unit of ingredient also appears at the bottom of table 3. Table 3

(Required Nutritional Element and Units of Nutritional Elements Per Unit of Food Type) Element Milk Chicken Bread Vegetables
Calories 160 210 120 150
Carbohydrates 110 130 110 120
Protein 90 190 90 130
Vitamins 50 50 75 70
Cost per unit $0.42 $0.68 $0.32 $0.17
Moreover, due to dietary restrictions, the following aspects should also be considered when the developing the diet plan:
i) The chicken food type should contribute at most 25% of the total calories intake that will result from the diet plan.
ii) The vegetable food type should provide at least 30% of the minimum daily requirements for vitamins.
Provide a linear programming formulation for the above case. (No need to solve the problem.)

...RESEARCH PAPER ON
LINEARPROGRAMMING
Vikas Vasam
ID: 100-11-5919
Faculty: Prof. Dr Goran Trajkovski
CMP 561: Algorithm Analysis
VIRGINIA INTERNATIONAL UNIVERSITY
Introduction:
One of the section of mathematical programming is linearprogramming.
Methods and linearprogramming models are widely used in the optimization of processes in all sectors of the economy: the development of the production program of the company, its distribution on the performers, when placing orders between the performers and the time intervals, to determine the best range of products, in problems of perspective, current and operational planning and management, traffic planning, defining a plan of trade and distribution, in the problems of development and distribution of productive forces, bases and depots of material handling systems, resources, etc. especially widely used methods and linearprogramming model for solving problems are savings (choice of resource-saving technologies, preparation of mixes, nesting materials), production, transportation and other tasks.
Beginning of linearprogramming was initiated in 1939 by the Soviet mathematician and economist Kantorovich in his paper "Mathematical methods of organizing and planning production." The appearance of...

...LINEARPROGRAMMING
INTRODUCTION:
The term ‛programming′ means planning and it refers to a particular plan of action amongst several alternatives for maximizing profit or minimizing cost etc. Programming problems deal with determining optimal allocation of limited resources to meet the given objectives, such as cost, maximum profit, highest 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 linearprogramming deals with that class of problems for which all relations among the variables involved are linear.
Formally, linearprogramming 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 form of a linearprogramming problem is
Optimize (Maximize or Minimize) Z = c1x1 + c2x2 + ……..+ cnxn
Subject to
a11 x1 + a12x2 + ….. + a1n xn (≤ , = , ≥) b1
a21 x1+ a22x2+ ….. + a2nxn (≤ , = , ≥ ) b2
. . . .
am1 x1+ am2 x2 +...

... LINEARPROGRAMMING
DATE;
5 JUNE, 14
UNIVERSITY OF CENTRAL PUNJAB
INTRODUCTION TO LINEARPROGRAMMINGLinearprogramming (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. Linearprogramming is a special case of mathematical programming.
It is a mathematical technique used in computer modeling to find the best possible solution in allocating limited resources (energy, materials, machines, money etc) to achieve maximum profit or minimum cost.
LinearProgramming is a method of expressing and optimizing a business problem with a mathematical model. It is one of the most powerful and widespread business optimization tools.
Linearprogramming can be used in very large variety of business problems. They include:
transportation distribution problems
production scheduling in oil & gas, manufacturing, chemical, etc industries
financial and tax planning
human resource planning
facility planning
fleet scheduling.
LINEARPROGRAMMING; an optimization technique capable of solving an amazingly...

...LinearProgramming Concept Paper
There are two types of linearprogramming:
1. LinearProgramming- involves no more than 2 variables, linearprogramming 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. Linearprogramming 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. The general nature of the business problems in which linearprogramming can be effectively used are multifaceted. They include purchasing, transportation, job assignments, production scheduling and mixing. Linearprogramming provides a method of maximizing or minimizing a first degree function subject to certain environmental restrictions or constraints which are usually in the form of equations and inequalities.
2. Simplex method- is an algorithm for solving linearprogramming with any number of variables. Most real-world linearprogramming problems have more than two variables and thus are too complex for graphical solution. A...

...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
Linearprogramming 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
The formulation step deals with displaying the problem in a mathematical form. Once that is developed the solution stage solves the problem and finds the variable values. During the interpretation stage the sensitivity analysis gives managers the opportunity to answer hypothetical questions regarding the solutions that are generated.
There are four basic assumptions of linearprogramming and they are as follows:
Certainty
Proportionality
Additivity
Divisibility
Linearprogramming is the development of modeling and solution procedures which employ mathematical techniques to optimize the goals and objectives of the decision-maker. Programming problems determine the optimal allocation of scarce resources to meet certain objectives. LinearProgramming Problems are mathematical programming...

...Karen Boyd,
Matt Beaumont.
Executive Summary:
Filatoi Riuniti is expanding to meet growing demand, and we have used outsourcing to keep up. Currently, we outsource only coarse and medium-sized yarn, but we believe that it would be more efficient to look at outsourcing all types. There are so many potential suppliers and constraints to consider that we constructed a linearprogramming model to identify our best option and check our solution's sensitivity to changes in our situation.
We've analyzed our potential suppliers for each gauge, taking into account their capacity, cost of production for each plant, and transportation costs (The model and our objective function can be found in the appendix.) Our goal was to allocate spinning production (at Filatoi Riuniti and six local mills) in a manner that would minimize overall costs, while meeting the demand and operating within the capacity constraint of each plant. Given the output of our optimization model, we should be outsourcing the spinning of our yarn in this way:
Sensitivity Analysis for Linear Optimization model:
Keep in mind that this model is sensitive to changes in each constraint, and there are ways that we can reduce our costs in the long run. We took into account several specific changes that management identified as probable and sought to see how they would change our optimal production strategy.
First, we wanted to...

...INVESTMENT STRATEGY REPORT
Submitted to J. D. Williams, Inc.
By
Mizar Gonzalez
Industrial Engineering Department
Southern Polytechnic State university
404-519-2792
February 20, 2008
EXECUTIVE SUMMARY
This report is our recommendation for an optimal investment strategy that would allow J. D. Williams, Inc. to maximize the annual yield of an investment of $800,000 in a diversified portfolio of funds.
To find the investment that would result in the greatest annual yield we have formulated a linear program that takes into account the requirements for the client of J. D. Williams, Inc. The requirements for the investment portfolio can be found on the section titled “Problem Description”
The greatest annual yield that can be expected while subject to the requirements of the different funds and the prospective client is $94,133.33. The money has to be invested in the following manner to achieve this result: The amount to be invested in the growth fund must be $ 248,889. The income fund must have an investment of $ 160,000 and the money market fund must have an investment of $ 391, 112.
PROBLEM DESCRIPTION
J. D. Williams, Inc. has a client who wishes to invest $800,000 with the firm in order to maximize his yield after a period of one year. The firm wants to allocate the funds while accommodating some requirements related to portfolio composition and the risk index of the funds as well as the client.
The portfolio must have...

...LinearProgramming Using Excel
Subject: LinearProgramming using Excel Application: Microsoft Excel 2007 Task: Solving a Linear Program Using Excel Tutorial Date: 25th February, 2010 by Nathan Smith
Install the Solver Add-In
1. In the Microsoft Office button, go to excel options to click Add-ins 2. In the Add-Ins box, select Solver Add-In and click Go...
LinearProgramming Using Excel - 1
Install the Solver Add-In (continue)
3. In the Add-Ins available box, check the Analysis ToolPak and then OK
LinearProgramming Using Excel - 2
Setting Up the Problem on the Spreadsheet
Example Min Z = 6X + 7Y s.t 2X + 6Y â‰¥ 10 5X + 3Y â‰¥ 10 X,Y â‰¥ 0
LinearProgramming Using Excel - 3
(continued)
1. Enter the coefficients of the objective function Z i.e., (6, 7) in cells E5 and F5. 2. Enter the coefficients of the Constraint-1 i.e., (2,6) and RHS value 10 in cells E9, F9 and H9 respectively 3. Enter the coefficients of the Constraint-2 i.e., (5,3) and RHS value 10 in cells E10, F10 and H10 respectively
LinearProgramming Using Excel - 4
(continued)
1. For the Objective function value, enter the formula for computing Z = SUMPRODUCT(E5:F5,E6:F6). This formula uses the coefficient values and also the solution values for variables X and Y, which are supposed to be...