Find the optimal level of production for the following minimization problem using graphical linear programming method. use the following information Minimization function is Z=8x1+12x2 Subject to: 5x1+2x2≥20 4x1+3x2≥24 X2≥2
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- An automobile manufacturer is considering whether to introduce a new model called the Racer. The profitability of the Racer depends on the following factors: The fixed cost of developing the Racer is triangularly distributed with parameters 3, 4, and 5, all in billions. Year 1 sales are normally distributed with mean 200,000 and standard deviation 50,000. Year 2 sales are normally distributed with mean equal to actual year 1 sales and standard deviation 50,000. Year 3 sales are normally distributed with mean equal to actual year 2 sales and standard deviation 50,000. The selling price in year 1 is 25,000. The year 2 selling price will be 1.05[year 1 price + 50 (% diff1)] where % diff1 is the number of percentage points by which actual year 1 sales differ from expected year 1 sales. The 1.05 factor accounts for inflation. For example, if the year 1 sales figure is 180,000, which is 10 percentage points below the expected year 1 sales, then the year 2 price will be 1.05[25,000 + 50( 10)] = 25,725. Similarly, the year 3 price will be 1.05[year 2 price + 50(% diff2)] where % diff2 is the percentage by which actual year 2 sales differ from expected year 2 sales. The variable cost in year 1 is triangularly distributed with parameters 10,000, 12,000, and 15,000, and it is assumed to increase by 5% each year. Your goal is to estimate the NPV of the new car during its first three years. Assume that the company is able to produce exactly as many cars as it can sell. Also, assume that cash flows are discounted at 10%. Simulate 1000 trials to estimate the mean and standard deviation of the NPV for the first three years of sales. Also, determine an interval such that you are 95% certain that the NPV of the Racer during its first three years of operation will be within this interval.Long-Life Insurance has developed a linear model that it uses to determine the amount of term life Insurance a family of four should have, based on the current age of the head of the household. The equation is: y=150 -0.10x where y= Insurance needed ($000) x = Current age of head of household b. Use the equation to determine the amount of term life Insurance to recommend for a family of four of the head of the household is 40 years old. (Round your answer to 2 decimal places.) Amount of term life insurance thousandsA manufacturer has the capability to produce both chairs and tables. Both products use the same materials (wood, nails and paint) and both have a setup cost ($100 for chairs, $200 for tables). The firm earns a profit of $20 per chair and $65 per table and can sell as many of each as it can produce. The daily supply of wood, nails and paint is limited. To manage the decision-making process, an analyst has formulated the following linear programming model:Max 20x1 + 65x2 – 100y1 – 200y2s.t. 5x1 + 10x2 ≤ 100 {Constraint 1}20x1 + 50x2 ≤ 250 {Constraint 2}1x1 + 1.5x2 ≤ 10 {Constraint 3}My1 ≥ x1 {Constraint 4}My2 ≥ x2 {Constraint 5}yi={1, if product j is produced0, otherwiseyi=1, if product j is produced0, otherwiseWhich of the constraints limit the amount of raw materials that can be consumed? A. Constraint 1 B. Constraint 4 C. Constraint 5 D. Constraint 1 and 4 E. Constraint 1, 2 and 3
- Solve the following problem using Simplex Method and find the P value. Maximize: P = 3x + 4y subject to: x+ y 0Graph the feasible region for the system of inequalities. 5x+y< -3 x-y > 3Dickie Hustler has $2 and is going to toss an unfair coin(probability .4 of heads) three times. Before each toss, hecan bet any amount of money (up to what he now has). Ifheads comes up, Dickie wins the number of dollars he bets;if tails comes up, he loses the number of dollars he bets.Use dynamic programming to determine a strategy thatmaximizes Dickie’s probability of having at least $5 afterthe third coin toss.
- TRUE OR FALSE In the simplex maximization method, optimality is signaled by the presence of all negative values in the Cj-ZjFor your next 10k race of the athletics season, you consider competing in either Sheffield, Cardiff, or Birmingham. After considering results from past years and the three different routes, you estimate your prize money, which will also depend on your form on the day. These estimated prizes are shown in the following table: Good form Average form Bad form 0 Sheffield 1000 Cardiff 700 Birmingham 800 400 500 600 200 100 Which condition must a satisfy such that Birmingham is the optimal city to go compete, according to the optimism-pessimism rule?A chemical company must produce exactly 1,000 kilograms of a special mixture of phosphate and potassium for a customer. The phosphate costs $5 per kilogram and potassium costs $6 per kilogram. No more than 300 kilograms of phosphate can be used, and at least 150 kilograms of potassium must be used. The problem is to determine the least-cost blend of the two ingredients. Restate the problem mathematically by: a) Showing the decision variablesb) Showing the objective functionc) Showing the constraintsd) Solve the linear programming problem by graphing
- Graph the following systems of linear Inequalities, shade the solution/feasible region and indicate if solution/feasible region is bounded or unbounded Sy-3x <3 (3y 2 x+3 10 1. 8 2 2. 8 10 -10 -8 -6 -4 -2 -2 46 -4 -6 -8 -10Do the following problems using either TreePlan A student is deciding which scholarships (out of two) to accept. The first scholarship is worth $10,000 but carries the condition that recipients cannot accept another other forms of income (such as other scholarships). The second scholarship is awarded in a competition, where this student has a 50% chance of earning $7,000, a 40% chance of earning $10,000, and a 10% chance of earning $15,000. The student must inform the administrator of the first scholarship whether she will be accepting their offer today. A. Develop a decision tree to determine which scholarship this student should accept (using our normal decision criteria). B. Under what circumstance might the student accept the other scholarship?Find the dual of the problem.