Deck 20: Transportation Models

Question 1

A transportation problem with a total supply of 500 and a total demand of 400 will have an optimal solution that leaves 100 units of supply unused.

Answer 1

True

Question 2

Degeneracy occurs when the number of occupied squares is less than the number of rows plus the number of columns minus one.

Answer 2

True

Question 3

If demand exceeds supply in a transportation problem, the problem must be balanced by adding a dummy source with additional supply.

Answer 3

True

Question 4

When using the stepping stone method for a minimization problem, the number of units that reallocates corresponds to the smallest number found in the cells containing minus signs.

Question 5

The transportation model is a special class of linear programming models.

Question 6

A transportation problem with 8 sources and 6 destinations will have an optimal solution that uses at most 13 of the 48 possible routes.

Question 7

Neither the northwest corner rule nor the intuitive method considers shipping cost in making initial allocations.

Question 8

When using the stepping-stone method, the improvement index for an unused cell equals the shipping cost associated with that cell.

Question 9

Which of the following is not needed in order to use the transportation model?

A) the source points and their capacity
B) the fixed costs of source points
C) the destination points and their demand
D) the cost of shipping one unit from each source to each destination
E) All of these are needed.

Question 10

Degeneracy in a transportation problem is when no closed path exists for evaluating an unused cell.

Question 11

The transportation model seeks satisfactory, but not necessarily optimal, solutions for shipping goods from several origins to several destinations.

Question 12

A transportation problem requires exactly as many origins as destinations.

Question 13

The transportation model is an excellent tool for minimizing shipping costs among existing facilities, but it is not useful when firms consider new facility locations.

Question 14

A feasible solution in transportation models is one in which all of the supply and demand constraints are satisfied.

Question 15

When using the stepping stone method, the closed path sometimes has the shape of a triangle as diagonal moves are permitted.

Question 16

In a transportation minimization problem, the negative improvement index associated with a cell indicates that reallocating units to that cell would lower costs.

Question 17

The stepping-stone method frequently achieves an optimal solution as soon as it calculates an initial feasible solution.

Question 18

The intuitive method of generating an initial solution has different results for each problem solver because "intuition" varies from person to person.

Question 19

The information needs of a transportation problem include

A) the cost of shipping one unit from each origin to each destination.
B) the set of destinations and the demand of each.
C) the set of origins.
D) the supply at each origin.
E) all of the above.

Question 20

To handle degeneracy, a very small quantity is placed in one of the unused squares.

Question 21

For the problem below, what is the quantity assigned to the cell Source 1-Destination 2 using the intuitive method for an initial feasible solution? $$\begin{array} { | l | r | r | r | r | } \hline \text { COSTS } & \text { Dest. 1 } & \text { Dest. 2 } & \text { Dest. 3 } & \text { Supply } \\\hline \text { Source 1 } & 2 & 1 & 3 & 50 \\\hline \text { Source 2 } & 4 & 7 & 5 & 40 \\\hline \text { Source 3 } & 3 & 12 & 6 & 30 \\\hline \text { Demand } & 50 & 45 & 25 & 120 \backslash 120 \\\hline\end{array}$$

A) 1
B) 5
C) 30
D) 45
E) 50

Question 22

The purpose of the transportation approach for location analysis is to minimize

A) total costs.
B) total variable costs.
C) total fixed costs.
D) total shipping costs.
E) the number of shipments.

Question 23

For the problem data set below, what is the northwest corner allocation to the cell Source 3-Destination 3? $$\begin{array} { | l | r | r | r | r | } \hline \text { COSTS } & \text { Dest. 1 } & { \text { Dest. 2 } } & { \text { Dest. 3 } } &{ \text { Supply } } \\\hline \text { Source 1 } & 2 & 1 & 3 & 30 \\\hline \text { Source 2 } & 4 & 2 & 1 & 40 \\\hline \text { Source 3 } & 3 & 8 & 6 & 20 \\\hline \text { Demand } & 15 & 50 & 25 & 90 \backslash 90 \\\hline\end{array}$$

A) 0
B) 15
C) 20
D) 35
E) 45

Question 24

Upon completion of the northwest-corner rule, which source-destination cell is guaranteed to be occupied?

A) top-left
B) top-right
C) bottom-left
D) bottom-right
E) the cell with the lowest shipping cost (or at least one of them if there's a tie)

Question 25

A transportation problem has a feasible solution when

A) all of the improvement indices are greater than or equal to zero.
B) all demand and supply constraints are satisfied.
C) the number of filled cells is one less than the number of rows plus the number of columns.
D) all the squares are used.
E) the solution yields the lowest possible cost.

Question 26

The Northwest-Corner rule's biggest flaw is that

A) it ignores costs.
B) it cannot generate feasible solutions.
C) it never generates ideal solutions.
D) it ignores shipping time.
E) There are no major flaws.

Question 27

The initial solution to a transportation problem can be generated several ways, so long as

A) it minimizes cost.
B) it ignores cost.
C) all supply and demand conditions are satisfied.
D) degeneracy does not exist.
E) all cells are filled.

Question 28

For the problem data set below, what is the northwest corner allocation to the cell Source 1 -Destination 2? $$\begin{array} { | l | r | r | r | r | } \hline \text { COSTS } & \text { Dest. 1 } & { \text { Dest. 2 } } & { \text { Dest. 3 } } &{ \text { Supply } } \\\hline \text { Source 1 } & 2 & 1 & 3 & 30 \\\hline \text { Source 2 } & 4 & 2 & 1 & 40 \\\hline \text { Source 3 } & 3 & 8 & 6 & 20 \\\hline \text { Demand } & 15 & 50 & 25 & 90 \backslash 90 \\\hline\end{array}$$

A) 0
B) 15
C) 25
D) 45
E) 35

Question 29

Consider the transportation problem and its optimal solution in the tables below. The cell Source 3 - Destination 3 is currently empty. What would be the change in the objective function if the largest possible amount were shipped using that route, leaving all the supply and demand conditions unchanged? $\begin{array}{|lr|r|r|r|r|}\hline \text { COSTS } & \text { Dest. 1 } & \text { Dest. 2 } & \text { Dest. 3 } & \text { Dest. 4 } & \text { Supply } \\\hline \text { Source 1 } & 12 & 18 & 9 & 11 & 105 \\\hline \text { Source 2 } & 19 & 7 & 30 & 15 & 145 \\\hline \text { Source 3 } & 8 & 10 & 14 & 16 & 50 \\\hline \text { Demand } & 80 & 60 & 70 & 90 & 300/300 \\\hline\end{array}$

$\begin{array}{|lrrrrr|}\hline \text { Shipments } & \text { Dest. 1 } & \text { Dest. 2 } & \text { Dest. 3 } & \text { Dest. 4 } & \text { Row Total } \\\text { Source 1 } & 30 & 0 & 70 & 5 & 105 \\\text { Source 2 } & 0 & 60 & 0 & 85 & 145 \\\text { Source 3 } & 50 & 0 & 0 & 0 & 50 \\\text { Col. Total } & 80 & 60 & 70 & 90 & 300 \backslash 300 \\\hline\end{array}$

A) 0
B) fifty units
C) a decrease of $9
D) an increase of $450
E) an increase of $630

Question 30

The transportation method is a special case of the family of problems known as

A) regression problems.
B) decision tree problems.
C) linear programming problems.
D) simulation problems.
E) statistical problems.

Question 31

A transportation problem has an optimal solution when

A) all of the improvement indices are greater than or equal to zero.
B) all demand and supply constraints are satisfied.
C) the number of filled cells is one less than the number of rows plus the number of columns.
D) all the squares are used.
E) all origin-destination combinations have been made equally low in cost.

Question 32

For the problem below, what is the quantity assigned to the cell Source 3-Destination 1 using the intuitive method for an initial feasible solution? $$\begin{array} { | l | r | r | r | r | } \hline \text { COSTS } & \text { Dest. 1 } & \text { Dest. 2 } & \text { Dest. 3 } & \text { Supply } \\\hline \text { Source 1 } & 2 & 1 & 3 & 50 \\\hline \text { Source 2 } & 4 & 7 & 5 & 40 \\\hline \text { Source 3 } & 3 & 12 & 6 & 30 \\\hline \text { Demand } & 50 & 45 & 25 & 120 \backslash 120 \\\hline\end{array}$$

A) 3
B) 13.333
C) 30
D) 45
E) 50

Question 33

The main difference between the intuitive lowest-cost method and the Northwest-corner rule is that

A) the intuitive lowest-cost method always generates a better solution.
B) the Northwest-corner rule is faster.
C) the Northwest-corner rule ignores cost.
D) only the intuitive lowest-cost method generates an optimal solution.
E) intuitive lowest-cost method is more accurate.

Question 34

The total cost of the optimal solution to a transportation problem

A) is calculated by multiplying the total supply (including any dummy values) by the average cost of the cells.
B) cannot be calculated from the information given.
C) can be calculated based only on the entries in the filled cells of the solution.
D) can be calculated from the original northwest corner solution.
E) is found by multiplying the amounts in each cell by the cost for that cell for each row and then subtracting the products of the amounts in each cell times the cost of each cell for the columns.

Question 35

In transportation model analysis the stepping-stone method is used to

A) obtain an initial optimum solution.
B) obtain an initial feasible solution.
C) evaluate empty cells for possible degeneracy.
D) balance supply and demand.
E) evaluate empty cells for potential solution improvements.

Question 36

Consider the transportation problem and its optimal solution in the tables below. What is the value of the objective function? $\begin{array}{|lr|r|r|r|r|}\hline \text { COSTS } & \text { Dest. 1 } & \text { Dest. 2 } & \text { Dest. 3 } & \text { Dest. 4 } & \text { Supply } \\\hline \text { Source 1 } & 12 & 18 & 9 & 11 & 105 \\\hline \text { Source 2 } & 19 & 7 & 30 & 15 & 145 \\\hline \text { Source 3 } & 8 & 10 & 14 & 16 & 50 \\\hline \text { Demand } & 80 & 60 & 70 & 90 & 300/300 \\\hline\end{array}$

$\begin{array}{|lrrrrr|}\hline \text { Shipments } & \text { Dest. 1 } & \text { Dest. 2 } & \text { Dest. 3 } & \text { Dest. 4 } & \text { Row Total } \\\text { Source 1 } & 30 & 0 & 70 & 5 & 105 \\\text { Source 2 } & 0 & 60 & 0 & 85 & 145 \\\text { Source 3 } & 50 & 0 & 0 & 0 & 50 \\\text { Col. Total } & 80 & 60 & 70 & 90 & 300 \backslash 300 \\\hline\end{array}$

A) 0
B) $169
C) 300 units
D) $2,100
E) $3,140

Question 37

In a minimization problem, a negative improvement index in a cell indicates that the

A) solution is optimal.
B) total cost will increase if units are reallocated to that cell.
C) current iteration is worse than the previous one.
D) total cost will decrease if units are reallocated to that cell.
E) problem has no feasible solution.

Question 38

For the problem data set below, what is the northwest corner allocation to the cell Source 1-Destination 1? $$\begin{array} { | l | r | r | r | r | } \hline \text { COSTS } & \text { Dest. 1 } &{ \text { Dest. 2 } } & { \text { Dest. 3 } } & { \text { Supply } } \\\hline \text { Source 1 } & 2 & 1 & 3 & 30 \\\hline \text { Source 2 } & 4 & 2 & 1 & 40 \\\hline \text { Source 3 } & 3 & 8 & 6 & 20 \\\hline \text { Demand } & 15 & 50 & 25 & 90 \backslash 90 \\\hline\end{array}$$

A) 0
B) 2
C) 15
D) 90
E) 30

Question 39

The northwest-corner rule is best used

A) to minimize the total shipping cost from several origins to several destinations.
B) to calculate whether a feasible solution is also an optimal solution.
C) to calculate how much to transfer from one shipping route to another.
D) to generate an initial feasible solution to a transportation problem.
E) to resolve cases of degeneracy in transportation problems.

Question 40

Which of the following is not information needed for a transportation problem?

A) the cost of shipping one unit from each origin to each destination
B) the set of destinations and the demand at each
C) the set of origins and the demand at each origin
D) the list of sources and the capacity at each
E) None of the above are needed.

Question 41

An improvement index indicates

A) whether a method other than the stepping-stone should be used.
B) whether a method other than the northwest-corner rule should be used.
C) how much total cost would increase or decrease if a single unit was reallocated to that cell.
D) whether the transportation cost in the upper left-hand corner of a cell is optimal.
E) how much total cost would increase or decrease if the largest possible quantity were reallocated to that cell.

Question 42

The elements of a transportation problem that supply goods are referred to as ________.

Question 43

A large transportation problem has 220 origins and 1360 destinations. The optimal solution of this problem will fill no more than about ________ of cells with quantities to be shipped.

A) 0.5%
B) 5%
C) 10%
D) 25%
E) All cells will be occupied.

Question 44

A transportation model fills one-half of its cells under the optimal solution. Which of the following most closely describes the number of sources compared to the number of destinations?

A) 1 to 1
B) 2 to 2
C) 3 to 3
D) 4 to 3
E) 4 to 4

Question 45

In a transportation problem, degeneracy means that

A) the problem was improperly constructed, and must be reformulated.
B) the assumptions of the transportation model have not been met.
C) the number of filled cells is too small to allow the calculation of improvement indices.
D) the total supply and the total demand are unbalanced.
E) the number of origins is not equal to the number of destinations.

Question 46

Which of the following combinations of sources and destinations would fill no more than 25% of cells with quantities to be shipped for the optimal solution?

A) 1 to 1
B) 1 to 4
C) 4 to 1
D) 8 to 2
E) 5 to 16

Question 47

A transportation problem has 8 origins and 6 destinations. The optimal solution of this problem will fill no more than ________ cells with quantities to be shipped.

A) 2
B) 13
C) 14
D) 48
E) cannot be calculated without knowing the supply and demand totals

Question 48

Consider the transportation problem and its initial solution in the table below. What is the improvement index for the empty cell Source 1 - Destination 3? $\begin{array}{|l|r|r|r|r|r|}\hline \text { COSTS } & \text { Dest. 1 } & \text { Dest. 2 } & \text { Dest. 3 } & \text { Dest . 4 } & \text { Supply } \\\hline \text { Source 1 } & 12 & 18 & 9 & 11 & 45 \\\hline \text { Source 2 } & 19 & 7 & 30 & 15 & 145 \\\hline \text { Source 3 } & 8 & 10 & 14 & 16 & 50 \\\hline \text { Demand } & 80 & 30 & 70 & 60 & 240 /240 \\\hline\end{array}$

$\begin{array}{|lrrrrr|}\hline \text { Shipments } & \text { Dest. 1 } & \text { Dest. 2 } & \text { Dest. 3 } & \text { Dest 4 } & \text { Row Total } \\\text { Source 1 } & 45 & 0 & 0 & 0 & 45 \\\text { Source 2 } & 35 & 30 & 70 & 10 & 145 \\\text { Source 3 } & 0 & 0 & 0 & 50 & 50 \\\text { Col. Total } & 80 & 30 & 70 & 60 & 240/240 \\\hline\end{array}$

A) 0
B) -14
C) -23
D) -70
E) 115

Question 49

When the number of shipments in a feasible solution is less than the number of rows plus the number of columns minus one

A) the solution is optimal.
B) a dummy source must be created.
C) a dummy destination must be created.
D) there is degeneracy, and an artificial allocation must be created.
E) the closed path has a triangular shape.

Question 50

A transportation problem has two origins: A can supply 20 units and B can supply 30 units. This problem has two destinations: C requires 25 units and D requires 35 units. Which of the following is true?

A) The problem will require a dummy demand with a capacity of 10 units.
B) The problem is unbalanced and cannot be solved by the transportation method.
C) The problem will require a dummy supply with a capacity of 10 units.
D) Destinations C and D must each receive 5 units less than they require.
E) The problem will require a dummy supply with a capacity of 50 units.

Question 51

The ________ finds the least-cost means of shipping supplies from several origins to several destinations.

Question 52

Total demand for a transportation model is 15 while total supply is 20. Which of the following should be included?

A) a dummy source of 20 units
B) a dummy destination of 5 units
C) a dummy source of 5 units
D) a dummy destination of 20 units
E) a dummy destination of 35 units

Question 53

One difference between the stepping-stone method compared to the Northwest-corner and intuitive method is that

A) the stepping-stone method never finds a feasible solution.
B) the stepping-stone method ignores costs.
C) the stepping-stone method ALWAYS finds a lower-cost solution.
D) the stepping-stone method guarantees an optimal solution.
E) the Northwest-corner and intuitive method guarantees sub-optimal solutions.

Question 54

A transportation problem has improvement indices of 5, 4, 0, and -3. Which of the following is always true?

A) The transportation model is at its ideal solution.
B) The transportation model is at a feasible solution.
C) The total cost saved by changing one unit on the route with an improvement index of 5 would be $5.
D) The transportation matrix has four empty squares.
E) The transportation model is at a feasible solution and the transportation matrix has four empty squares.

Question 55

Suppose the solution for a transportation model fills 5 cells with quantities to be shipped. Which of the following combinations of sources and destinations would be degenerate?

A) 2 sources, 4 destinations
B) 3 sources, 3 destinations
C) 4 sources, 2 destinations
D) 5 sources, 1 destination
E) None of the above are degenerate.

Question 56

A transportation problem has 4 origins and 2 destinations. The optimal solution of this problem will fill no more than ________ cells with quantities to be shipped.

A) 5
B) 6
C) 8
D) 20
E) All cells will be occupied.

Question 57

Which of the following is true for degenerate problems?

A) They do not contain a feasible solution.
B) They cannot contain an optimal solution.
C) Supply is greater than demand.
D) Demand is greater than supply.
E) None of the above

Question 58

The three information needs of a transportation problem are the origin points and the capacity at each, the destination points and the demand at each, and ________.

Question 59

The stepping-stone method

A) is an alternative to using the northwest-corner rule.
B) often involves tracing closed paths with a triangular shape.
C) is used to identify the relevant costs in a transportation problem.
D) is used to evaluate the cost effectiveness of shipping goods via transportation routes not currently in the solution.
E) helps determine whether a solution is feasible or not.

Question 60

In a minimization problem, a positive improvement index in a cell indicates that

A) the solution is optimal.
B) the total cost will increase if units are reallocated to that cell.
C) there is degeneracy.
D) the total cost will decrease if units are reallocated to that cell.
E) the problem has no feasible solution.

Question 61

The number of routes filled by a solution to a transportation problem is no larger than ________.

Question 62

When is it necessary to add dummy sources or destinations to a transportation problem?

Question 63

A transportation problem has 6 origins and 12 destinations. How many possible routes are there for this problem? How many routes will be used in the optimal solution?

Question 64

State, in order, the three steps in making an initial allocation with the northwest-corner rule.

Question 65

How might the transportation method be used to help a firm add a facility to an existing distribution network? You may wish to describe a simple example.

Question 66

Consider the transportation problem in the data set and optimal solution below. Verify by hand or by calculator (show your work) the value of the objective function.

Question 67

For the data below, construct an initial feasible solution using the northwest-corner rule.

Question 68

A transportation problem has 10 origins and 32 destinations. How many possible routes are there for this problem? How many routes will be used in the optimal solution?

Question 69

In formulating a transportation problem, you discover that one of the route combinations is forbidden by contract, or prohibited by law, or ruled out by company policy. How would you indicate the cost of that cell in preparing the problem to be solved?

Question 70

A transportation problem that has more units supplied than demanded will require a(n) ________ to balance the problem.

Question 71

For the transportation problem below, construct an initial feasible solution using the intuitive method.

Question 72

What does the stepping-stone method do?

Question 73

The larger a transportation problem (that is, as the problem has more rows and more columns), the smaller the fraction of all possible routes that will be filled in a solved problem. Explain.

Question 74

What purpose does the northwest-corner rule serve?

Question 75

What is the difference between a feasible solution and an optimal solution?

Question 76

________ is an occurrence in transportation problems when too few shipping routes are being used to allow calculation of improvement indices.

Question 77

The ________ develops an initial feasible solution for a transportation model by starting at the upper left-hand cell of a table and systematically allocating units to shipping points.

Question 78

What is transportation modeling?

Question 79

The ________ is an iterative technique for moving from an initial feasible solution to an optimal solution in the transportation method.

Question 80

When does degeneracy occur in a transportation model?