BIT Exam 2

If the number of TVs required in New York goes up by 10, what will be the total cost of the optimal shipping plan?

$7460

Transportation Model characteristics

-A product is transported from a number of sources to a number of destinations at the *minimum possible cost.* -Each source is able to supply a fixed number of units of the product, and each destination has a fixed demand for the product. -The linear programming model has constraints for supply at each source and demand at each destination. -All constraints are equalities in a *balanced* transportation model where *supply equals demand.* -Constraints contain inequalities in *unbalanced* models where *supply does not equal demand.*

Transshipment Model characteristics

-extension of the transportation model -intermediate transshipment points are added between the sources and destinations -items may be transported from: --sources thru transshipment points to destinations --one source to another --one transshipment point to another --one destination to another --directly from sources to destinations

Assignment Problem characteristics

-special form of linear programming model similar to the transportation model -*supply* at each source *and demand* at each destination *limited to one unit* -in a balanced model supply equals demand -in an unbalanced model supply does not equal demand

Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8000 gallons of component 1 available, and the demand gasoline types 1 and 2 are 11,000 and 14,000 gallons, respectively. Write the constraint stating that the component 1 cannot account for more than 35% of the gasoline type 1.

.65x11 -.35x21 <= 0

The optimal solution has XB3 = 1 and XA4 = 1. What are the optimal values of XA1, XB1, and XC1:

0,0,0

In an assignment model, all supply and demand values are equal to:

1

For the city you identified in the previous question, the most additional amount of this demand we could require before the marginal cost changes is

100 tv sets

If the overall amount of chicken available went down by 100 lbs., how much will the total profit be?

1062.5 - 100*2.8

The formula in cell C11 for cost has several terms in it. One of them is:

16*C5

For the nutrient you identified in the previous question, the most additional amount of this nutrient we could require before the marginal cost changes is

2.5 units

Suppose the landlord really wants the back door to be installed. How should the constraint for the back door be written if he uses the following scheme for decision variables? x1 = contractor works on wood floors x2 = landlord works on wood floors x3 = contractor works on kitchen tile x4 = landlord works on kitchen tile x5 = contractor works on back door x6 = landlord works on back door x7 = contractor works on garage door x8 = landlord works on garage door

5+ x6 = 1

If the calories required in the diet go up by 50, what will be the total cost of the optimal diet?

56 cents

The formula in cell G13 is

=.8*C4 - .2*C5 - .2*C6 - .2*C7

What is the formula for the transshipment constraint for the plant in Indiana in cell C20? [Hint: Flow in must equal flow out]

=B9 - F14

The formula in cell C8 is

=SUM(C5:C7)

A correct formula for cell C9 for profit is:

=SUMPRODUCT(C4:E7,C14:E17)

The diet problem in chapter 4 of your text involved minimizing the cost of a combination of up to 10 foods. The decision variables xi represented the number of servings of the ith food included in the diet. Seven nutrition constraints were included, as indicated in the formulation and excel worksheet below. The problem was solved using solver, and the solver dialog box and sensitivity report follow. The formula for C19 for cost is:

=SUMPRODUCT(C5:C14,E5:E14)

The formula in cell F15 is

=SUMPRODUCT(C5:C14,F5:F14)

The formula for the objective function in cell B13 is:

=sum(B11:G11)

What is the formula for computers shipped from Pennsylvania in cell E7?

=sum(B7:C7)

the formula in cell H4 is:

=sumproduct(B4:G4,B11:G11)

What is the formula for total cost in cell C24?

=sumproduct(B6:C8, I6:J8) + sumproduct(C14:E15, J14:L15)

The formula that should be entered into the "By Changing Variable Cells" cell is:

B11:G11

The next 4 problems deal with problem 6-32. Recall that this transshipment problem concerns a fruit company who contracts with farms, plants, and distribution centers. They want to determine the optimal shipments from farms to plants to distribution centers to minimize total transportation costs. The screenshot of the Excel worksheet for this problem is: The variables to be entered into the "By Changing Variable Cells" box in the Excel solver are:

B6:C8, C14:E15

The formula for the model decision variables that should be entered into the "By Changing Variable Cells:" window in the Solver window above is

C4:E7

The formula for the model decision variables that should be entered into the "By Changing Variable Cells:" window in the Solver window above is

C5:C14

The formula for the model decision variables that should be entered into the "By Changing Variable Cells:" window in the Solver window above is

C5:E7

One of the formulas for the constraints that should be entered in the "Subject to the Constraints:" window in the Solver window is

C8 = C10

The nutrient requirements are going to be modified so that one more unit of Iron or calcium or protein will be required. Which one of these nutrient changes would have the greatest impact on the total cost of the diet?

Calcium

One of the formulas for the constraints that should be entered in the "Subject to the Constraints:" window in the Solver window is

F15=>F17

One of the formulas for the constraints that should be entered in the "Subject to the Constraints:" window in the Solver window is

F4:F7>=G4:G7

The Excel formulas for the Fat and Cholesterol constraints can be reflected in the "Subject to the constraints:" window as G15:H15 >= F17:G17.

False

The formula that should be entered into the "Subject to the Constraints" cell is:

H4:H9 => 1 (sum column)

How much can the amount of Pork available go down before the marginal profit changes?

It can go down BY 25

Mixed Integer Model

Some of the decision variables (but not all) required to have integer values

Cost/benfit information is available. Wood floors could increase the monthly rent about $100 and an upgrade to the kitchen tile would fetch $80 per month. Garage door replacement wouldn't net more than $20 per month. A safety door would cost $250 and add only $15 to the monthly rent. The garage door would cost $350, the kitchen tile update would cost $1000, and the wood floor refinish job would cost $400 to rent the buffer and buy the chemicals. It wouldn't be easy doing these upgrades; the garage door would take a half week, the back door one week, the floors two weeks and the tile three weeks. There was another way around these jobs though; instead of doing them himself, the landlord could always hire a professional in each field that could finish the job in half the time but would charge a pretty penny for that speed. Refinishing floors would cost $2700, upgrading the kitchen tile would cost $2500, replacing the back window with a door would cost $600, and installing a garage door opener would cost $350. The landlord ran the model in Excel and received the answer report contained in the table. Which of the following statements is correct? Variable Cells Cell Name Original Value Final Value $G$4 wood floors contractor 1 0 $H$4 wood floors self 1 1 $G$5 kitchen tile contractor 1 1 $H$5 kitchen tile self 1 0 $G$6 back door contractor 1 0 $H$6 back door self 1 0 $G$7 garage door opener contractor 1 0 $H$7 garage door opener self 1 0

The rent will be $180 higher and the project will cost $2900.

The constraint for the North American supply region is:

X11 + X12 + X13 + X14 - 5Y11 - 10Y12 ≤ 0.

The constraint for the South Asia demand region is:

X13 + X23 + X33 + X43 = 7

The constraint for Philadelphia is:

X13 + X23+ X43- X34 - X35 - X36 - X37 = 0

Madlantic Devices designs and manufactures high-end medical devices. The facilities in Madison and Atlanta serve as design and component manufacturing facilities. Components are then shipped to warehouses in Philadelphia or Knoxville, where they are held until final assembly is completed at either Dayton, Bloomington, or Albany. Manufacturing capacity in Madison and Atlanta is 1000 units. Demand at Dayton, Bloomington, and Albany is 450, 500, and 610, respectively. The network representing the shipping routes is shown below. The constraint for the quantity shipped from Madison is:

X13+ X 14 ≤ 1000

Use the following information for the next 3 questions. A professor needs help from three student helpers to complete four tasks. The first task is grading, the second is scanning, the third is copying, and the fourth is organizing student portfolios. The estimated time for each student to do each task is given in the matrix below. Each student is to be assigned exactly one task, but not all tasks will be assigned. None of the tasks require more than one student. Total student time used is to be minimized. Which of the following constraints represents the assignment for task 2, scanning?

XA2 + XB2 + XC2 ≤ 1

Student B is not experienced enough to grade. The constraint to reflect this is:

XB1 = 0

Which of these constraints will ensure that a low capacity facility is not built in South America?

Y21 = 0

Which of these constraints will ensure that either a low capacity or a high facility capacity facility is built in the European supply region?

Y41 + Y42 = 1

Lewis Gale is deciding where to put medical centers in Square City. They want to make sure that the residents of Square City all have access to health care. A resident has access to health care if he or she lives in a zone with a medical center, or a neighboring zone has medical center. The map of Square City is: Suppose Lewis Gale puts a medical center in Zone 5. Citizens in which zones will have access to health care?

Zone 4, Zone 5, and Zone 6

The assignment problem constraint x31 + x32 + x33 + x34 ≤ 2 means:

agent 3 can be assigned to no more than 2 tasks.

Total Integer Model

all decision variables required to have integer solution values

0-1 Integer Model

all decision variables required to have integer values of 0 or 1 -selection constraint: either swimming pool or tennis center (not both)

If additional ingredients were available, which would be most profitable for the manufacturer?

chicken

The demand requirements are going to be modified so that one more TV set will be required at either New York or Dallas or Detroit. Which one of these demand changes would have the greatest impact on the total cost?

dallas

Assume that we are using 0-1 integer programming model to solve a capital budgeting problem and xj = 1 if project j is selected and xj = 0, otherwise. The constraint (x1 + x2 + x3 + x4 = 2) means that ________ out of the ________ projects must be selected.

exactly 2, 4

In a balanced transportation problem, a demand constraint (the amount of product demanded at a given destination) is a less-than-or equal-to constraint (≤).

false

In a mixed integer model, the solution values of the decision variables must be 0 or 1.

false

The Excel formulas for the ingredient constraints can be reflected in the "Subject to the constraints:" window as F4:F7=G4:G7

false

a feasible solution is ensured by rounding down non-integer solution values but may result in a less than optimal (sub optimal) solution. (for < constraints with positive coefficients)

hey

In an assignment problem:

one task can be done by only one agent.

Let: rj = regular production quantity for period j, oj =overtime production quantity in period j, ij = inventory quantity at the end of period j, and dj = demand quantity in period j. Correct formulation of the demand constraint for a multiperiod scheduling problem is:

r2 + o2+ i1 - i2 = d2

Branch and Bound Method

traditional approach to solving integer programming problems -feasible solutions can be partitioned into smaller subsets -smaller subsets evaluated until best solution is found -method is a tedious and complex math process

The Excel formulas for the demand constraints can be reflected in the "Subject to the constraints:" window as C8:E8 = C10:E10.

true

In an integer programming problem, if we were choosing between two locations (location 1 and location 2) to build a facility, this would be written as (assume the facility must be built in one of the two locations):

x1 + x2 = 1

Let xi indicate whether Lewis Gale puts a medical center in zone i. That is, xi = 0 if there is no medical center in zone i and xi = 1 if there is a medical center in zone i. 10. Which is the constraint that ensures the residents of zone 6 have access to health care.

x3 + x4 + x5 + x6≥ 1

A landlord is considering upgrades and repairs, including wood floors, kitchen tile, a new back door, and a repaired garage door. The landlord could do these things himself or hire a contractor. The contractor could finish the job in half the time but would charge more. The landlord wants to leave the choice of whether to actually upgrade the kitchen tile up to the optimization algorithm. How should this constraint be written if he uses the following scheme for decision variables? x1 = contractor works on wood floors x2 = landlord works on wood floors x3 = contractor works on kitchen tile x4 = landlord works on kitchen tile x5 = contractor works on back door x6 = landlord works on back door x7 = contractor works on garage door x8 = landlord works on garage door

x3 + x4 <= 1

Rounding non-integer solution values up to the nearest integer value can result in an infeasible solution to an integer programming problem.

yeah