operation research

pls answer all the questions

Problem \#1: Consider the following nonlinear production problem for Acme Industries and answer the questions following it. Acme Industries produces two goods: widgets (w) and gadgets (g). Widgets are made from steel. and the production process being described by the following production function: xv=xi2+1, where xw denotes the quantity of widgets produced and xi denotes the quantity (in pounds) of steel used during production. Gadgets are produced using large quantities of plastic, the production process being represented by the following additional production function: xr=xp5, where xz denotes the quantity of gadgets produced and xp denotes the quantity (in pounds) of plastic used during production. Acme Industries is able to purchase steel for $1.00 per pound and plastic for $0.25 per pound. The marketing department at Acme Industries has researched the market for widgets and gadgets. They have determined that the inverse demand curve for widgets can be represented by the following functional relationship: pw=e(x/40) where pv is the market price (in dollars) for widgets; and the inverse demand curve for gadgets can be represented by the following functional relationship: py=e(x/20), where pr is the market price (in dollars) for gadgets. Part a: State the (nonlinear) profit maximization problem for Acme Industries to determine the optimal levels of pg,pv,xg,x2,x8, and xw. In constructing the above optimization problem, st. the inverse demand curves and production functions as explicit constraints. (8 points) Part b: Restate the profit maximization problem for Acme Industries in part (a) so that the only two variables that need to be solved for are xr and x2. ( 6 points) Part c: Explicitly state the KKT conditions for the optimization problem stated in part (b). (8 points) Part d: Interpret the KKT conditions for the optimization problem derived in part (b). Focus on the first order conditions, using the complementarity conditions and nonnegativity constraints to support your explanation. (Hint: What is true at a local (global) maximum of the profit function? Using the answer to this question and the complementary conditions in part (c), provide an interpretation of the above constraints.) ( 4 points) Part :: Perform up to four iterations of the steepest descent algorithm covered in class. Let the step size be equal to 3 for each iteration (i.c. do not perform a line search procedure, just set t= 3). Use the initial starting point (xg,xw)=(2,2) and let =0.000001. Summarize your results in a table, indicating if an optimal solution has been found. In the table, include the gradient, norm of the gradient, search direction xi+1, and objective function value. ( 10 points) Part f : Solve the problem using EXCEL or GAMS. What is the optimal solution? ( 8 points) Part g. Verify that the KKT conditions in derived in part (c) are approximately met for the optimal solution found in part (f). The use of the word "approximately" means that the KKT conditions could be violated, but by only a very minute amount (eg by an amount less than 0.001). (2 points) Part h: Graph the solution path of the points found after each iteration using the steepest descent algorithm outlined in part (d) onto the objective function contoars of the objective function in part (b) in the (xf,xw) space provided below. Using this graph and the solunion provided in part (f), what would be the likely effect on the solution path for the steepest descent algorithm if a quadratic interpolation or other line search procedure was used to determine the step size n ( 6 points) Countour Plot of Objective Function for Optimization Problem in Part (b)