Exam 6: Integer Linear Programming

An investor has $500,000 to invest and wants to maximize the money they will receive at the end of one year.They can invest in condos,apartments and houses.The profit after one year,the cost and the number of units available are shown below. Profit Cost Number Variable Investment \ 1,000) \ 1,000) Available Condos 6 50 10 Apartments 12 90 5 Houses 9 100 7 Based on this ILP formulation of the problem and the indicated optimal integer solution values what values should go in cells B5:F12 of the following Excel spreadsheet? MAX: ^{6 X}₁ ^{+ 12 X}₂ ^{+ 9 X}₃ Subject to: ^{50 X}₁ ^{+ 90 X}₂ ^{+ 100 X}₃ ^{? 500} X₁ ? 10 X₂ ? 5 X₃ ? 7 X_i ? 0 and integer

The feasible region for the pure ILP problem

A popular solution technique to IP problems is called:

An investor has $500,000 to invest and wants to maximize the money they will receive at the end of one year.They can invest in condos,apartments and houses.The profit after one year,the cost and the number of units available are shown below. Formulate the ILP for this problem. Profit Cost Number Variable Investment \ 1,000) \ 1,000) Available Condos 6 50 10 Apartments 12 90 5 Houses 9 100 7

A company is planning next month's production.It has to pay a setup cost to produce a batch of X₄'s so if it does produce a batch it wants to produce at least 100 units.Which of the following pairs of constraints show the relationships)between the setup variable Y₄ and the production quantity variable X₄?

A manufacturing company has costs associated with production preparation and with per unit production.The per unit production costs are referred to as

The ILP problems are computationally

A company will be able to obtain a quantity discount on component parts for its three products,X₁,X₂ and X₃ if it produces beyond certain limits.To get the X₁ discount it must produce more than 50 X₁'s.It must produce more than 60 X₂'s for the X₂ discount and 70 X₃'s for the X₃ discount.How many decision variables are required in the formulation of this problem?

A company will be able to obtain a quantity discount on component parts for its three products,X₁,X₂ and X₃ if it produces beyond certain limits.To get the X₁ discount it must produce more than 50 X₁'s.It must produce more than 60 X₂'s for the X₂ discount and 70 X₃'s for the X₃ discount.Which of the following pair of constraints enforces the quantity discount relationship on X₃?

A cellular phone company wants to locate two new communications towers to cover 4 regions.The company wants to minimize the cost of installing the two towers.The regions that can be covered by each tower site are indicated by a 1 in the following table: Tower Sites Region 1 2 3 4 A 1 1 B 1 1 1 C 1 1 1 D 1 1 COST \ 000 s) 200 150 190 250 MIN: ^{200 X}₁ ^{+ 150 X}₂ ^{+ 190 X}₃ ^{+ 250 X}₄ Subject to: ^X₂ ^{+ X}₄ ^{? 1} X₁ + X₃ + X₄ ? 1 X₁ + X₂ + X₃ ? 1 X₁ + X₄ ? 1 X₁ + X₂ + X₃ + X₄ = 2 X_i = 0,1 Based on this ILP formulation of the problem and the solution X₁,X₂,X₃,X₄)= 1,1,0,0)what values should go in cells B6:G14 of the following Excel spreadsheet?

A company is developing its weekly production plan.The company produces two products,A and B,which are processed in two departments.Setting up each batch of A requires $60 of labor while setting up a batch of B costs $80)Each unit of A generates a profit of $17 while a unit of B earns a profit of $21.The company can sell all the units it produces.The data for the problem are summarized below. Hours required by Operation A B Hours Cutting 3 4 48 Welding 2 1 36 The decision variables are defined as X_i = the amount of product i produced Y_i = 1 if X_i > 0 and 0 if X_i = 0 What is the objective function for this problem?

The B & B algorithm solves ILP problems

An integrality condition indicates that some or all)of the

A company wants to build a new factory in either Atlanta or Columbia.It is also considering building a warehouse in whichever city is selected for the new factory.The following table shows the net present value NPV)and cost of each facility.The company wants to maximize the net present value of its facilities,but it only has $16 million to invest. Variable Decision NPV $million) Cost $million) ^X₁ Factory in Columbia 3 10 ^X₂ Factory in Atlanta 4 8 ^X₃ Warehouse in Columbia 2 6 ^X₄ Warehouse in Atlanta 1 5 Based on this ILP formulation of the problem and the indicated optimal solution what formulas should go in cells F6:F14 of the following Excel spreadsheet? MAX: ^{3 X}₁ ^{+ 4 X}₂ ^{+ 2 X}₃ ^{+ X}₄ Subject to: ^{10 X}₁ ^{+ 8 X}₂ ^{+ 6 X}₃ ^{+ 5 X}₄ ^{≤ 15} X₁ + X₂ = 1 X₃ + X₄ ≤ 1 X₃ − X₁ ≤ 0 X₄ − X₂ ≤ 0 X_i = 0,1

Suppose you want to minimize an objective function z=2x₁+3x₂.Both decision variables must be integer.The optimal solution to the LP relaxation will:

Which of the following is not a benefit of using binary variables?

A company is developing its weekly production plan.The company produces two products,A and B,which are processed in two departments.Setting up each batch of A requires $60 of labor while setting up a batch of B costs $80.Each unit of A generates a profit of $17 while a unit of B earns a profit of $21.The company can sell all the units it produces.The data for the problem are summarized below. Hours required by Operation A B Hours Cutting 3 4 48 Welding 2 1 36 The decision variables are defined as X_i = the amount of product i produced Y_i = 1 if X_i > 0 and 0 if X_i = 0 Which of the following constraints creates the link between setting up to produce A's and making some A's for this problem?

A company is developing its weekly production plan.The company produces two products,A and B,which are processed in two departments.Setting up each batch of A requires $60 of labor while setting up a batch of B costs $80.Each unit of A generates a profit of $17 while a unit of B earns a profit of $21.The company can sell all the units it produces.The data for the problem are summarized below. Hours required by Operation A B Hours Cutting 3 4 48 Welding 2 1 36 The decision variables are defined as X_i = the amount of product i produced Y_i = 1 if X_i > 0 and 0 if X_i = 0 Using the approach discussed in the text,what is the appropriate value for M₁ in the linking constraint for product A?

If a company produces Product 1,then it must produce at least 150 units of Product 1.Which of the following constraints enforces this condition?

A company will be able to obtain a quantity discount on component parts for its three products,X₁,X₂ and X₃ if it produces beyond certain limits.To get the X₁ discount it must produce more than 50 X₁'s.It must produce more than 60 X₂'s for the X₂ discount and 70 X₃'s for the X₃ discount.How many binary variables are required in the formulation of this problem?