Economic/Statistic Problems

          1. Consider the following LP problem: MIN W = 4Y1 +Y2 S.i. 3Y1 +Y2 = 3 4Y1 + 3Y2 6 Yi + 2Y2 5 4 and 111,Y2 0 i) Solve this LP problem using the simplex algorithm. (Use the Big-M Method, if necessary, to obtain an initial tableau.) NOTE: Be sure to give a complete written statement of the optimal solution.ii) Check your answer to part i) using LINDO. (Be sure to attach the relevant LINDO output and to highlight the optimal solution.) 2. Ethelred’s new Mediterranean diet requires that all his food come from one of the four “basic food groups”, namely, Greek salad, figs, red wine, and chocolate. Each serving of Greek salad costs $6, each bag of figs costs $2, each bottle of red wine costs $3, and each bar of chocolate costs $7. Each day, Ethelred must ingest at least 800 calories, at least 8 units of nutrient 1, at least 12 units of nutrient 2, and at least 10 units of nutrient 3. The nutritional content per unit of each food is given below: Calories Nutrient 1 Nutrient 2 Nutrient 3Greek Salad (GS) 400 3 2 2Figs (FG) 200 2 2 4Red Wine (RW) 150 0 4 1Chocolate (CH) 500 0 4 5 i) Formulate an LP problem to help Ethelred to minimize his daily food costs. (Be sure to define the decision variables, state the objective function, state and label the constraints, and include the sign restrictions.)  ii) Solve your LP problem using LINDO. (Be sure to attach the relevant LINDO output and to highlight the optimal solution.)iii) Using your LINDO output from part ii), answer the following questions:a) What is Ethelred’s optimal daily diet and how much does it cost?b) Which constraints are binding? How do you know? Explain.c) By how much would the price of chocolate have to fall before chocolate became part of an optimal daily diet for Ethelred? Explain. d) If a bag of figs were to cost $3, rather than the current $2, what would be the optimal daily diet and how much would it cost? Explain.e) If a serving of Greek salad were to cost $5, rather than the current $6, what would be the optimal daily diet and how much would it cost? Explain.f) If Ethelred’s doctor insisted that his daily diet must contain at least 900 calories, rather than the current 800 calories, what would be the optimal daily diet and how much would it cost? Explain.g) If Ethelred’s doctor insisted that his daily diet must contain at least 14 units of nutrient 2, rather than the current 12 units, what would be the optimal daily diet and how much would it cost? Explain.h) Graph the relationship between the minimum cost of Ethelred’s optimal daily diet and the price of a bag of figs.i) Graph the relationship between the minimum cost of Ethelred’s optimal daily diet and the number of units of nutrient 1 in his diet. HINT: For parts h) and i), see Section 5-4. 3. Consider the following normal max primal LP problem: i) Write down the corresponding dual LP problem. ii) Solve the dual LP problem using the simplex algorithm. (Use the Big-M Method, if necessary, to obtain an initial tableau.) NOTE: Be sure to give a complete written statement of the optimal solution.iii) Use your answer from part ii), together with the results from the Dual Theorem, to write down the optimal solution to the primal LP problem. (If you should need to invert a matrix, then use the MINVERSE function in Excel.) NOTE: Be sure to give a complete written statement of the optimal solution.iv) Check your answer to part iii) using LINDO. (Be sure to attach the relevant LINDO output and to highlight the optimal solution.)