...Select and appropriately use technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches related to the selected modules for this unit from the ‘Applications’ area of study. Students need to demonstrate: Key knowledge This knowledge includes • key features of straight line, line segment and step graphs and the form of related tables of values; • the concept of break-even analysis and its relation to graphic and tabular representation of relations; • non-linear relations, constant of proportionality and key features; • linear inequalities, systems of linear inequalities and their properties; • the role of variables, constraints and objective functions...
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...stadium selling Slices of cheese pizza, hot dogs, and barbecue sandwiches. This covers the items X1, X2, and X3 respectively. The costs of these items are $4.50, $0.50 and $1.00 respectively. As one knows the objective is to maximize Julia Robertson’s profit. The method of doing this is by evaluating subtracted costs from the selling price. Things to consider are constraints and the cost of everything involved. Being that cost always seems the most important this paper will start there. Profit on pizza slice = $1.50 - $0.75 = $ 0.75 Profit on hot dog = $1.50 – 0.45 = $1.05 Profit on sandwich = $2.25 - $0.90 = $1.35 The overall profit would be the following: Z=0.75 X1+ 1.05 X2 + 1.35 X3 Constraints: Budget Constraints would go as follows: 0.75X1 + 0.45X2 + 0.90X3 <=$1500 Space Constraints: Space available = 3 x 4 x 16 = 192 sq. feet = 192 x 12 x 12 =27648 sq. inches The oven will be refilled during half time. Thus total space available = 27648 x 2 = 55296 Space required for pizza = 14 x 14 = 196 sq. inches Space required for pizza slice = 196/ 8 = 24.5 sq. inches Total Oven Space required: 24.5 X1 + 16 X2 + 25 X3 Constraint: 24.5 X1 + 16 X2 + 25 X3 ≤ 55296 sq. Inches Another area to consider is the demand or potential demand for each item and what the availability is. Julia can sell at least as many slices of pizza as she is able to sell hot dogs and sandwiches represented by: X2 ≥2 X3 X2 - 2 X3 ≥ 0 LP programming model : Maximize Z = 0.75 X1 + 1.05 X2 + 1.35 X3 Subject to:...
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...Chapter 2 Introduction to Optimization & Linear Programming 1. If an LP model has more than one optimal solution it has an infinite number of alternate optimal solutions. In Figure 2.8, the two extreme points at (122, 78) and (174, 0) are alternate optimal solutions, but there are an infinite number of alternate optimal solutions along the edge connecting these extreme points. This is true of all LP models with alternate optimal solutions. 2. There is no guarantee that the optimal solution to an LP problem will occur at an integer-valued extreme point of the feasible region. (An exception to this general rule is discussed in Chapter 5 on networks). 3. We can graph an inequality as if they were an equality because the condition imposed by the equality corresponds to the boundary line (or most extreme case) of the inequality. 4. The objectives are equivalent. For any values of X1 and X2, the absolute value of the objectives are the same. Thus, maximizing the value of the first objective is equivalent to minimizing the value of the second objective. 5. a. linear b. nonlinear c. linear, can be re-written as: 4 X1 - .3333 X2 = 75 d. linear, can be re-written as: 2.1 X1 + 1.1 X2 - 3.9 X3 0 e. nonlinear 6. [pic] 7. [pic] 8. [pic] 9. [pic] 10. [pic] 11. [pic] 12. [pic] 13. X1 = # of TV spots, X2 = # of magazine ads MAX 15 X1 + 25 X2 (profit) ...
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...Fruit and Chocolate CompanyThis assignment is due for online students by midnight on Sunday of Week 3 and for on-ground students before Week 4’s class meeting. Submit your assignment to the course shell.Read the Harriet’s Fruit and Chocolate Company case study found in Chapter 2 of the textbook. Then, answer the questions below. Each response should be one (1) paragraph in length submitted in MS Word and the submission is not to exceed two (2) pages.Write a one to two (1-2) page paper that addresses: * What investigation will you do with regard to the physical infrastructure of the orchards, orchard shacks, and the cold storage building? * Make a list of business goals for Harriet’s Fruit and Chocolate Company. What are some constraints that will affect these goals? * Make a list of technical goals for Harriet’s Fruit and Chocolate Company. What tradeoffs might you need to make to meet these goals? * Will a wireless solution support the low delay that will be needed to meet the needs of the applications? Defend your answer. * What security concerns should you bring up as you design the network upgrade?The format of the paper is to be as follows: * Typed, double-spaced, Times New Roman font (size 12), one-inch margins on all sides, APA format. * Type the question followed by your answer to the question. * In addition to the one to two (1-2) pages required, a title page is to be included. The title page is to contain the title of the assignment...
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...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 that show the limits of what you can do, and (3) non-negativity restrictions, because you cannot turn outputs back into inputs. LINEAR PROGRAMMING II Graph method of solution Graph the constraints as equalities, like before. The constraints are now $ rather than #, so the feasible area is everything to the right and above all of the constraint lines. You want to find the lowest...
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...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 4,600 2. Flour consumed should not exceed 9,400 3. Herbs consumed should not exceed 2,200 In mathematical terms, let’s say X1 to be the number of Lime bags and X2 to be the number of Vinegar bags. LP is: Maximize: 0.48 X1 + 0.59 X2 Subject to: 1.5X1 + 4 X2 <=4,600 5X1 + 6X2 <= 9,400 2 X1 +...
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... Calculations & Statistical Analysis 1. Choose the unknowns. |X1 = Ice Cream | |X2 = Specialties | |2. Write the objective function. | |f( x, y) = 900X1 + 1500X2 | |3. Write the constraints as a system of inequalities | | |Product | | | | |X1 = Ice Cream |X2 = Specialties |X3 = Labor Time | |Machine |2 |1 |40 | |Packaging Line |1 |1 |40 | |Labor (hours) |3 |6 |150 | As the numbers of products are natural numbers, there are two more constraints. |X1 ≥ 0 | |X2 ≥ 0 | |4. Find the set of feasible solutions that graphically represent the constraints. | | | | | |...
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...it to the Week 2 iLab Dropbox by the due date. Have fun while learning. Your Name Christopher White NETW410, Professor’s Name Current Date Lab 2: Application of the Top-Down Network Design Methodology Lab Report 1. What are the business goals? (10 points) Redundancy is not required but it is highly recommended to reduce any data loss so the network automatically uses a secondary path if one goes down. The redundancy would be located in layer 3. 2. What are the business constraints? (10 points) The school is utilizing its revenue from property taxes and bonds that must deliver the required improvements but with a very strict budget, suggesting that affordability is more important than the best technical solution. 3. What are the technical goals? (10 points) Improve network performance by increasing the amount of users that can access the network and speed, making the network fast as possible is key to a network administrator. 4. What are the technical constraints? (10 points) The existing equipment such as the CAT5 cable should be replace with CAT6 at every drop and Fiber Optics should replace each uplink at the core. The campus should replace or upgrade its existing ISP or service so that its bandwidth is consist with campus needs for more speed and easy access. 5. Diagram the existing network. (10 points) There was a workstation in the library as well. 6. Describe the existing network traffic. (10 points) The network setup makes for a...
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...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: 1. Organize the data for the model on the spreadsheet. – Target cell - the cell in the spreadsheet that represents the objective function – Changing cells - the cells in the spreadsheet representing the decision variables – Constraint cells - the cells in the spreadsheet representing the LHS formulas on the constraints PANKAJ DUTTA IMTCDL Chapter – 11 Linear Programming...
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...In the summer of 1988, Merton Truck Company (“Merton”) was struggling with financial performance. A manufacturer of two specialized trucks (model 101 and model 102), Merton was conflicted with how to manage production and improve financial results. The executive team offered many different options to improve performance, but many were conflicted. For example, the sales manager suggested cancelling all production of model 101 trucks, whereas the controller suggested increasing model 101 production and curtailing model 102 production. To determine the best course of action for the company, an in-depth analysis of their current practices and optimal position is required. Currently, Merton produces 1,000 of Model 101 trucks and 1,500 of Model 102 trucks. The company is constrained by monthly machine hours available in their production facilities for each activity (SEE EXHIBIT 1). The contribution margins for each model are $3,000 for Model 101, $5,000 for Model 102 (SEE EXHIBIT 2). Each model provides contribution margin, so increasing production in any capacity should increase overall profits. However, the decision on which product to increase production of has caused some internal conflict because of the allocation of shared resources. At the moment, the production facilities are maximizing the engine assembly capacity. This is a shared resource by both Model 101 production and Model 102 production. On one hand, Model 101 production has lower contribution margin ($3,000), but...
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...Quiz 4[pic] Question 1 2 out of 2 points | | | |[pic] |The standard form for the computer solution of a linear programming problem requires all variables to the right and all | | | | | |numerical values to the left of the inequality or equality sign | | | | | | | | | | | |Answer | | | | | |Selected Answer: | | | | | |[pic] False | | | | | | | | | | | |Correct Answer: | | | | | |[pic] False | |...
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...Merton Trucks Case Note Abstract We discuss Merton Trucks [Dhe90a] as a case to introduce linear programming in the MBA program. This case adapted from Sherman Motor Company case, was used to introduce Linear Programming formulations as well as duality. Refer to the teaching note [Dhe90b]. Our approach differs from the approach suggested by Dhebar [Dhe90b]. First, our audience consists pre-dominantly of engineers with not too much work experience. As a result, handling math and algebra is relatively easy. Explaining the algebraic formulation, graphical approach and using the Excel solver do not consume that much time. Second, because this case is used during the first week of the MBA program, students are still unfamiliar with the case methodology and we spend significant time in understanding case facts. The circular logic used in allocating fixed costs based on the product mix that in turn is used in deciding the product mix takes some time to understand. Third, because of the participant background, they have difficulty in translating the model to the specific business situation and interpreting the trade-offs involved in various what-if analyses that are prompted by the case questions. We return to the case when we teach duality. After explaining duality, we analyze the case to show how some of the questions and what-if analyses can be simplified using duality. This note is based on our experiences with teaching three large batches of students in our MBA programs. 1 1 Without...
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...representing a firm's decisions given an objective and resource constraints. Answer: TRUE Diff: 2 Page Ref: 31 Section Heading: Model Formulation Keywords: model formulation AACSB: Analytic skills 2) The objective function always consists of either maximizing or minimizing some value. Answer: TRUE Diff: 2 Page Ref: 31 Section Heading: Model Formulation Keywords: objective function AACSB: Analytic skills 3) The objective function is a linear relationship reflecting the objective of an operation. Answer: TRUE Diff: 1 Page Ref: 31 Section Heading: Model Formulation Keywords: model formulation AACSB: Analytic skills 4) A constraint is a linear relationship representing a restriction on decision making. Answer: TRUE Diff: 1 Page Ref: 31 Section Heading: Model Formulation Keywords: model formulation AACSB: Analytic skills 5) A linear programming model consists of only decision variables and constraints. Answer: FALSE Diff: 1 Page Ref: 56 Section Heading: Characteristics of Linear Programming Problems Keywords: model formulation AACSB: Analytic skills 6) A parameter is a numerical value in the objective function and constraints. Answer: TRUE Diff: 1 Page Ref: 31 Section Heading: Model Formulation Keywords: parameter AACSB: Analytic skills 7) A feasible solution violates at least one of the constraints. Answer: FALSE Diff: 2 Page Ref: 34 Section Heading: Model...
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...Question 1 a) Min: 260EM+220EO+290ET+230CM+240CO+310CT CONSTRAINT: EM+ET=10, EO+CO=15,ET+CT=10, EM+EO+ET<=20,CM+CO+CT<=20 b) c) the optimal solution is when em,eo,et,cm,co,ct= (0,10,10,10,5,0 ) and the least cost is 8600. d) No. as there is no “zero” for the allowable increase and decrease in any constraint e) Yes ,it is unique ,as there is no zero for the allowable I/D in any constrain. f) There is no impact on the solution as there are only 15 of Clerment are used g) As the allowable decrease of E is 5, the 5 tons decrease of c is within the range. And the △opt=shadowPriceE*△E=100 h) The cost of E should be reduced 210=260-250(the reduced cost) to then it could be counted in the solution. Question 2 a) 0 for wiring and 10 for testing As the allowable Increase of wiring capacity is 20 so the △ opt=shadowPriceW*△wiring=1300 250 increase on the profit of alternator. As the allowable decrease of g is 166.7 and the allowable decrease of a is 250 so the optimal solution is unchanged There is no impact on the solution as the alternator are not included in the solution as the profit of a is too low As the △c is 0.5 ，according to 100% rule the optimal solution will not change. Question 3 a ) Memory Resistor and H are binding constraint as their shadow price are not zero b) HyperLink could be eliminated , the reduced cost of it is the lowest one, which means the profit has to be much higher than now, then it could be used more....
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...Analyze the techniques discussed in the Guldemond et al. article (2008) and propose how you would apply them to your own skill set as a Project Manager. The article discusses a new methodology for scheduling problems with strict deadlines. Time Constrained Project Scheduling Problem (TCPSP) is typically dealt in different ways. The article discussed different methods and techniques. One of the techniques is to schedule the different tasks for the fraction of the time needed to complete them, then later to add as needed. This method helps in getting the different tasks down and later dealing with time constraints instead of first dealing with the time constraint then try to schedule them. This is also called partial schedule based on the article. Another method is to try to keep the work in normal working hours instead of scheduling in overtime at first. Based on that and the calculations within the article, it appears that project timing would eventually be condensed down by at least 10%. The article discusses some formulas on calculating and getting the best out of a project with constrained time. I was able to understand some of the calculations, but because I was never good with Math, most of them I could not understand. Being a project manager, these skills can be applied differently. I would first schedule everything as normal and within regular business hours of operation. In doing this step, it helps in breaking down the project into all the different tasks...
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