Exam 3: Introduction to Optimization Modeling

The proportionality property of LP models implies that the sum of the contributions from the various activities to a particular constraint equals the total contribution to that constraint.

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

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 an optimization model,there can only be one:

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 dessert 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. Table Dessert Labor (Hours) 0.20 0.60 Process Time (Hours) 0.30 0.10 ​ -[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

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. -[Part 2] Refer to Exhibit 3-2.Obtain a sensitivity report for the solution reported in Part 1.Which constraints are binding

When formulating a linear programming spreadsheet model,there is a set of designated cells that play the role of the decision variables.These are called the changing variable cells.

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. -[Part 1] Refer to Exhibit 3-2.Find an optimal solution to Western's problem.What is the production plan,and what is the total revenue

The condition of nonnegativity requires that:

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 dessert 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. Table Dessert Labor (Hours) 0.20 0.60 Process Time (Hours) 0.30 0.10 ​ -[Part 2] Refer to Exhibit 3-1.Using the graphical solution method,find an optimal solution to the model in Part 1 and determine the maximum profit.

If the objective function has the equation {4x₁+ 2x₂},then the slope of the objective function line is −2.

The equation of the line representing the constraint 4x₁+ 2x₂ ≤ 100 passes through the points:

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.

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 dessert 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. Table Dessert Labor (Hours) 0.20 0.60 Process Time (Hours) 0.30 0.10 ​ -[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.

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 dessert 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. Table Dessert Labor (Hours) 0.20 0.60 Process Time (Hours) 0.30 0.10 ​ -[Part 3] Refer to Exhibit 3-1.Implement the model in Part 1 in Excel Solver and obtain an answer report.Which constraint(s)are binding on the optimal solution

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

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.An advertising agency has devised a marketing plan for the Western Chassis Company that will increase the market for Deluxe chassis.The plan will increase demand by 75 Deluxe chassis per month at a cost of $100 per month.Should Western adopt the plan Briefly explain why.

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

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

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