Exam 8: Nonlinear Programming and Evolutionary Optimization

An investor is developing a portfolio of stocks.She has identified 3 stocks in which to invest.She wants to earn at least 11% return but with minimum risk. Let: P_i = proportion of total funds invested in i,i = A,B,C The NLP for this problem is: What formulas should go in cells G4:J14 of the spreadsheet for this problem? NOTE: Formulas are not required in all of these cells.

A company makes products A and B from 2 resources,labor and material.The company wants to determine the selling price which will maximize profits.A unit of A costs 30 to make and demand is estimated to be 50 ? .09 * Price of A.A unit of B costs 20 to make and demand is estimated to be 30 ? .14 * Price of B.The utilization of labor and materials and the available quantity of resources is shown in the table.A reasonable price for the products is between 90 and 140. Product A B Available resources Labor hr/unit) 2 4 150 Material ounces/unit) 2 8 220 Manufacturing cost \ / unit) 30 20 Demand units) 50-0.0 30-0.1 Let X₁ = demand for As and X₂ = demand for Bs.Let P₁ = price for As and P₂ = price for Bs. Formulate the NLP for this company

Exhibit 8.2 The following questions pertain to the problem and spreadsheet below. A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building.They need at least 60,000 ft² of office floor space.Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height.Further,parking space must equal at least 30% of the total floor space available.The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost in millions of dollars). The following is the NLP formulation for the problem. The spreadsheet implementation of this formulation applies to the following questions. -Refer to Exhibit 8.2.What formula would you place in cell D13 to calculate total cost?

Exhibit 8.1 The following questions pertain to the problem and spreadsheet below. A company makes products A and B from 2 resources,labor and material.The company wants to determine the selling price which will maximize profits.A unit of A costs 25 to make and demand is estimated to be 20 ? .10 * Price of A.A unit of B costs 18 to make and demand is estimated to be 30 ? .07 * Price of B.The utilization of labor and materials and the available quantity of resources is shown in the table.A reasonable price for the products is between 100 and 200. Product A B Avalable resources Labor hr/unit) 3 2 150 Material ounces/unit) 1 2 200 Manufacturing cost \/ unit) 25 18 Demand units) 20-.1 30-.0 Let X₁ = demand for As and X₂ = demand for Bs.Let P₁ = price for As and P₂ = price for Bs -Refer to Exhibit 8.1.What formula is used in cell B9 of the spreadsheet for this problem?

A company makes products A and B from 2 resources,labor and material.The company wants to determine the selling price which will maximize profits.A unit of A costs 30 to make and demand is estimated to be 50 ? 0.09 * Price of A.A unit of B costs 20 to make and demand is estimated to be 30 ? 0.14 * Price of B.The utilization of labor and materials and the available quantity of resources is shown in the table.A reasonable price for the products is between 90 and 140. Product A B Avalable resources Labor hr/unit) 2 4 150 Material ounces/unit) 2 8 220 Manufacturing cost \/ unit) 30 20 Demand units) 50-0.0 30-1 Let X₁ = demand for As and X₂ = demand for Bs.Let P₁ = price for As and P₂ = price for Bs. The NLP for the problem is: MAX: 52.70-0.09+32.80-0.14-2100 Subject to: -50+0.09=0 -30+0.14=0 2+4\leq150 2+8\leq220 90\leq,\leq140 ,\geq0 and the solution , =140.0,117.14 What values should go in cells B3:E18 of the spreadsheet for this problem?

Find the maximum solution on this graph of a function starting from X = 12.Mark its location on the graph.

Calculate the annual inventory costs for the following data. Order quantity = 400 units Annual demand = 12,500 units Unit purchase cost = 50 Fixed cost of placing an order = 75 Percentage cost of holding one unit in inventory for a year = 20%

In solving the NLP problem,Solver found a degenerate model.This means that:

Exhibit 8.2 The following questions pertain to the problem and spreadsheet below. A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building.They need at least 60,000 ft² of office floor space.Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height.Further,parking space must equal at least 30% of the total floor space available.The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost in millions of dollars). The following is the NLP formulation for the problem. The spreadsheet implementation of this formulation applies to the following questions. -Refer to Exhibit 8.2.What formula would you place in cell B6 to calculate Total Lot Area?

Which point or points are global optima in this diagram? The dashed line represents the objective function and the objective is to maximize the value of the objective function.

The Analytic Solver Platform solution strategy for NLP problems is the GRG method.What does GRG stand for?

In solving the NLP problem,Solver produced a message "Solver found a solution.All constraints and optimality conditions are satisfied." This means that Solver found:

The Sweet Water beverage company is designing a new soft drink can.The designers wish to minimize the manufacturing cost of the can,a cost that is directly related to the amount of aluminum used in the can.The can must hold at least 350 ml or cm³)of beverage,have a diameter between 3 and 7 cm,and have a height between 7 and 19 cm. Formulate the NLP for Sweet Water.

Exhibit 8.2 The following questions pertain to the problem and spreadsheet below. A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building.They need at least 60,000 ft² of office floor space.Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height.Further,parking space must equal at least 30% of the total floor space available.The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost in millions of dollars). The following is the NLP formulation for the problem. The spreadsheet implementation of this formulation applies to the following questions. -Refer to Exhibit 8.2.The company wishes to have a relatively square building.Thus,they wish neither the building length nor the building width exceed the other by more than 25%.Add constraints)to enforce this design constraint.

The optimal solution to a NLP problem can occur at an) I.corner point. II.interior point.

What is the search path for the following feasible solution space? The dashed line represents the objective function and the objective is to maximize the value of the objective function.

Exhibit 8.1 The following questions pertain to the problem and spreadsheet below. A company makes products A and B from 2 resources,labor and material.The company wants to determine the selling price which will maximize profits.A unit of A costs 25 to make and demand is estimated to be 20 ? .10 * Price of A.A unit of B costs 18 to make and demand is estimated to be 30 ? .07 * Price of B.The utilization of labor and materials and the available quantity of resources is shown in the table.A reasonable price for the products is between 100 and 200. Product A B Avalable resources Labor hr/unit) 3 2 150 Material ounces/unit) 1 2 200 Manufacturing cost \/ unit) 25 18 Demand units) 20-.1 30-.0 Let X₁ = demand for As and X₂ = demand for Bs.Let P₁ = price for As and P₂ = price for Bs -How much must the objective function coefficient of the variable Pumpkin change before any Pumpkins are Produced based on the following sensitivity report? Changing Cells Final Reduced Cell Name Value Gradient $B$4 Corn 9.52 0 $C$4 Pumpkin 0 ?499.99 $D$4 Beans 10.79 0 Constraints Final Lagrange Cell Name Value Multiplier $E$8 Corn 200000.00 0.016 $E$9 Pumpkin 99.00 12.000 $E$10 Beans 37777.78 0 $E$11 Water 29.84 0 $E$12 Fertilizer 8000.00 3.490

The main difference between shadow prices and Lagrange multipliers is

The optimal solution to a LP problem is always at I.a corner point. II.an interior point III.the origin.

The Reduced Gradient is similar to which of these terms from linear programming?