Exam 3: Introduction to Optimization Modeling

Exhibit 3-1 A winemaker in California's Napa Valley must decide how much of two types of wine she will produce from a particular variety of grapes.Each liter of table wine yields $8 profit,while each liter of desert wine produces $3 profit.The labor hours and bottling process time used for type of wine are given in the table below.Resources available include 200 labor hours and 80 hours of bottling process time.Assume the winemaker has more than enough grapes available to supply any feasible production plan. -[Part 5] Refer to Exhibit 3-1.Suppose the winemaker can obtain 100 addition labor hours.Can you use the sensitivity analysis obtained for Part 4 to determine her expected profit? Would her bottling plan change? Explain your answer.

The feasible region in a graphical solution of a linear programming problem will appear as some type of polygon,with lines forming all sides.

Suppose the allowable increase and decrease for an objective coefficient of a decision variable that has a current value of $50 are $25 (increase)and $10 (decrease).If the coefficient were to change from $50 to $65,the optimal value of the objective function would not change.

If a constraint has the equation 5x + 2y $\le$ 60,then the constraint line passes through the points (0,12)and (30,0).

All linear programming problems should have a unique solution,if they can be solved.

If a manufacturing process takes 4 hours per unit of x and 2 hours per unit of y and a maximum of 100 hours of manufacturing process time are available,then an algebraic formulation of this constraint is:

Common errors in LP models that exhibit unboundedness are a constraint that has been omitted or an input which is incorrect.

Exhibit 3-1 A winemaker in California's Napa Valley must decide how much of two types of wine she will produce from a particular variety of grapes.Each liter of table wine yields $8 profit,while each liter of desert wine produces $3 profit.The labor hours and bottling process time used for type of wine are given in the table below.Resources available include 200 labor hours and 80 hours of bottling process time.Assume the winemaker has more than enough grapes available to supply any feasible production plan. -[Part 2] Refer to Exhibit 3-1.Find an optimal solution to the model in Part 1 and determine the maximum profit.

Shadow prices are associated with nonbinding constraints,and show the change in the optimal objective function value when the right side of the constraint equation changes by one unit.

Consider the following linear programming problem: The above linear programming problem:

The condition of nonnegativity requires that:

Exhibit 3-2 Western Chassis produces high-quality polished steel and aluminum sheeting and two lines of industrial chassis for the rack mounting of Internet routers,modems,and other telecommunications equipment.The contribution margin (contribution toward profit)for steel sheeting is $0.40 per pound and for aluminum sheeting is $0.60 per pound.Western earns $12 contribution on the sale of a Standard chassis rack and $15 contribution on a Deluxe chassis rack.During the next production cycle,Western can buy and use up to 25,800 pounds of raw unfinished steel either in sheeting or in chassis.Similarly,20,400 pounds of aluminum are available.One standard chassis rack requires 16 pounds of steel and 8 pounds of aluminum.A Deluxe chassis rack requires 12 pounds of each metal.The output of metal sheeting is restricted only by the capacity of the polisher.For the next production cycle,the polisher can handle any mix of the two metals up to 4,000 pounds of metal sheeting.Chassis manufacture can be restricted by either metal stamping or assembly operations; no polishing is required.During the cycle no more than 2,500 total chassis can be stamped,and there will be 920 hours of assembly time available.The assembly time required is 24 minutes for the Standard chassis rack and 36 minutes for the Deluxe chassis rack.Finally,market conditions limit the number of Standard chassis racks sold to no more than 1,200 Standard and no more than 1,000 Deluxe.Any quantities of metal sheeting can be sold. -Refer to Exhibit 3-2.What is the value of additional capacity on the polisher? How much increase and decrease in this capacity is possible before a change occurs in the optimal production schedule?

In using Excel to solve linear programming problems,the changing cells represent the:

If the objective function has the equation {4x + 2y},then the slope of the objective function line is -2.

Exhibit 3-2 Western Chassis produces high-quality polished steel and aluminum sheeting and two lines of industrial chassis for the rack mounting of Internet routers,modems,and other telecommunications equipment.The contribution margin (contribution toward profit)for steel sheeting is $0.40 per pound and for aluminum sheeting is $0.60 per pound.Western earns $12 contribution on the sale of a Standard chassis rack and $15 contribution on a Deluxe chassis rack.During the next production cycle,Western can buy and use up to 25,800 pounds of raw unfinished steel either in sheeting or in chassis.Similarly,20,400 pounds of aluminum are available.One standard chassis rack requires 16 pounds of steel and 8 pounds of aluminum.A Deluxe chassis rack requires 12 pounds of each metal.The output of metal sheeting is restricted only by the capacity of the polisher.For the next production cycle,the polisher can handle any mix of the two metals up to 4,000 pounds of metal sheeting.Chassis manufacture can be restricted by either metal stamping or assembly operations; no polishing is required.During the cycle no more than 2,500 total chassis can be stamped,and there will be 920 hours of assembly time available.The assembly time required is 24 minutes for the Standard chassis rack and 36 minutes for the Deluxe chassis rack.Finally,market conditions limit the number of Standard chassis racks sold to no more than 1,200 Standard and no more than 1,000 Deluxe.Any quantities of metal sheeting can be sold. -Refer to Exhibit 3-2.What is the incremental contribution associated with adding an hour of assembly time? Over what range of increase is the marginal value valid?

When the profit increases with a unit increase in a resource,this change in profit will be shown in Solver's sensitivity report as the:

The feasible region in all linear programming problems is bounded by:

Linear programming models have three important properties.They are:

Exhibit 3-1 A winemaker in California's Napa Valley must decide how much of two types of wine she will produce from a particular variety of grapes.Each liter of table wine yields $8 profit,while each liter of desert wine produces $3 profit.The labor hours and bottling process time used for type of wine are given in the table below.Resources available include 200 labor hours and 80 hours of bottling process time.Assume the winemaker has more than enough grapes available to supply any feasible production plan. -[Part 4] Refer to Exhibit 3-1.Obtain a sensitivity report for the model in Part 1.How much should the winemaker be willing to pay for an additional labor hour?

Suppose the allowable increase and decrease for shadow price for a constraint are $25 (increase)and $10 (decrease).If the right hand side of that constraint were to increase by $10 the optimal solution would not change.