Exam 6: Integer Linear Programming

Exhibit 6.1 The following questions pertain to the problem,formulation,and spreadsheet implementation below. A research director must pick a subset of research projects to fund over the next five years.He has five candidate projects,not all of which cover the entire five-year period.Although the director has limited funds in each of the next five years,he can carry over unspent research funds into the next year.Additionally,up to $30K can be carried out of the five-year planning period.The following table summarizes the projects and budget available to the research director. Project Funds Required in $000s)Benefit Project 1 2 3 4 5 in \ 000s) 1 \ 70 \ 40 \ 30 \ 15 \ 15 \ 160 2 \ 82 \ 35 \ 20 \ 20 \ 10 \ 190 3 \ 55 \ 10 \ 10 \ 5 \ 125 4 \ 69 \ 17 \ 15 \ 12 \ 8 \ 139 5 \ 75 \ 20 \ 25 \ 30 \ 45 \ 174 Budget $225K $60K $60K $50K $50K The following is the ILP formulation and a spreadsheet model for the problem. -Refer to Exhibit 6.1.What values would you enter in the Analytic Solver Platform task pane for the above Excel spreadsheet? Objective Cell: Variables Cells: Constraints Cells:

A wedding caterer has several wine shops from which it can order champagne.The caterer needs 100 bottles of champagne on a particular weekend for 2 weddings.The first supplier can supply either 40 bottles or 90 bottles. The relevant decision variable is defined as X₁ = the number of bottles supplied by supplier 1 Which set of constraints reflects the fact that supplier 1 can supply only 40 or 90 bottles?

A company has four projects,numbered 1 through 4.If any project is selected for implementation,each lower- numbered project must also be selected for implementation.Formulate the constraints to enforce these conditions.

For minimization problems,the optimal objective function value to the LP relaxation provides what for the optimal objective function value of the ILP problem?

One way to find an optimal solution to the IP problem is to:

A company wants to select no more than 2 projects from a set of 4 possible projects.Which of the following constraints ensures that no more than 2 will be selected?

Pure IP formulation requires that:

A company needs to hire workers to cover a 7 day work week.Employees work 5 consecutive days with 2 days off.The demand for workers by day of the week and the wages per shift are: Days af Week Workers Required Shift Days off Wage Sunday 54 1 Sun a Man 900 Munday 50 2 Man a Tue 1000 Tuesday 36 3 Tue d Wed 1000 Wednestay 38 4 Wed a Thur 1000 Thursday 42 5 Thur a Fri 1000 Friday 40 6 Fri a Sat 900 Saturday 48 7 Sat a Sun 850 Formulate the ILP for this problem.

What are binary integer variables?

How are general integrality requirements indicated in the Analytic Solver Platform?

Binary variables are useful for modeling

The following ILP is being solved by the branch and bound method.You have been given the initial relaxed IP solution.Complete the entries for the 3 nodes and label the arcs when you branch on X₁. MAX: ^{35 X}₁ ^{+ 45 X}₂ Subject to: ^{35 X}₁ ^{+ 55 X}₂ ^{≤ 250} 65 X₁ + 25 X₂ ≤ 340 X₁,X₂ ≥ 0 and integer Initial solution X₁ = 4.6X₂ = 1.6 Obj = 233.9

The setup cost incurred in preparing a machine to produce a batch of product is an example of a

The concept of an upper bound in IP is associated with:

One approach to solving integer programming problems is to ignore the integrality conditions and solve the problem with continuous decision variables.This is referred to as

A company produces three products which must be painted,assembled,and inspected.The machinery must be cleaned and adjusted before each batch is produced.They want to maximize their profits for the amount of operating time they have.The unit profit and setup cost per batch are: Product Profit per unit Setup cost per batch 1 9 500 2 10 600 3 12 650 The operation time per unit and total operating hours available are: Operating Time per Unit Operation Product 1 Product 2 Product 3 Operating Hours Available Paint 1 2 2 400 Assemble 2 3 2 600 Inspection 2 4 3 540 Formulate the ILP for this problem.

The concept of a lower bound in IP is associated with:

Exhibit 6.2 The following questions pertain to the problem,formulation,and spreadsheet implementation below. A certain military deployment requires supplies delivered to four locations.These deliveries come from one of three ports.Logistics planners wish to deliver the supplies in an efficient manner,in this case by minimizing total ton-miles.The port-destination data,along with destination demand is provided in the following table. Destination Port D1 D2 D3 D4 A 75 88 103 56 105 76 101 85 43 80 95 82 Demand 500 600 450 700 The ports are supplied by one of two supply ships.These ships travel to a particular port where their supplies are off-loaded and shipped to the requesting destinations.Ship S1 carries 1200 tones of supplies while Ship S2 carries 1120 tons of supplies.These ships can only go to a single port and each port can only accommodate one ship. Assume the costs for a ship to travel to a port are not part of the objective function.The following is the ILP formulation and a spreadsheet model for the problem. Let ^Y_ij ^{be 1 if ship i travels to port j,for i = S1,S2 and j = A,B,C} X_jk be the tons shipped from port j = A,B,C to Destination k = D1,D2,D3,D4 Minimize ^75X₁₁ ^{+ 88X}₁₂ ^{+ 103X}₁₃ ^{+ 56X}₁₄ ^{+ 105X}₂₁ ^{+ 76X}₂₂ ^{+ 101X}₂₃ ^{+ 85X}₂₄ + 43X₃₁ + 80X₃₂ + 95X₃₃ + 62X₃₄ Subject to: ^Y₁₁ ^{+ Y}₂₁ ^{? 1} Y₁₂ + Y₂₂ ? 1 Y₁₃ + Y₂₃ ? 1 Y₁₁ + Y₁₂ + Y₁₃ = 1 Y₂₁ + Y₂₂ + Y₂₃ = 1 X₁₄ + X₁₅ + X₁₆ + X₁₇ ? 1200Y₁₁ + 1120Y₂₁ X₂₄ + X₂₅ + X₂₆ + X₂₇ ? 1200Y₁₂ + 1120Y₂₂ X₃₄ + X₃₅ + X₃₆ + X₃₇ ? 1200Y₁₃ + 1120Y₂₃ X₁₄ + X₂₄ + X₃₄ ? 500 X₁₅ + X₂₅ + X₃₅ ? 600 X₁₆ + X₂₆ + X₃₆ ? 450 X₁₇ + X₂₇ + X₃₇ ? 700 Y_ij.X_jk ? 0 -Refer to Exhibit 6.2.What formula would go into cell E14?

Binary variables are:

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 Formulate the ILP for this problem.