18) 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.
19) The ________ is an iterative technique for moving from an initial feasible solution to an optimal
solution in the transportation method.
20) What does the stepping-stone method do?
21) What is the difference between a feasible solution and an optimal solution?
22) 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 total shipping cost.
23) Find the minimum cost shipping solution for the transportation problem data set in the table below.
Provide a table of shipping quantities and the minimum value for the shipping cost.
24) Consider the transportation data set for a minimization problem below.
a. Calculate the initial solution using the northwest-corner rule.
b. Calculate improvement indices, iterate, and solve for the optimal shipping pattern.
25) Find the minimum cost solution for the transportation problem detailed in the table below. Explain
carefully the meaning of any quantity in a “dummy” row or column.
26) The Shamrock Transportation Company has four terminals: A, B, C, and D. At the start of a particular
day, there are 8, 8, 6, and 3 tractors available at those terminals, respectively. During the previous night,
trailers were loaded at plants R, S, T, and U. The number of trailers at each plant is 2, 12, 5, and 6,
respectively. The company dispatcher has determined the distances between each terminal and each
plant, as follows. How many tractors should be dispatched from each terminal to each plant in order to
minimize the total number of miles traveled?
27) Find the minimum cost solution for the transportation problem detailed in the table below.
Before your solution can be implemented, you discover that the combination Source 3 — Destination 1 is
unavailable, due to political turmoil in the country where Source 3 is located. Solve the revised problem.
How much is cost increased by this complication?
28) A firm has established a distribution network for the supply of a raw material critical to its
manufacturing. Currently there are two origins for this raw material, which must be shipped to three
manufacturing plants. The current network has the following characteristics:
The firm has identified two potential sites for a third raw material source; these are identified as
Candidate A and Candidate B. From A, the costs to ship would be $9 to Plant 1, $10 to Plant 2, and $12 to
Plant 3. From B, these costs would be $11, $14, and $8. The new source, wherever it is located, will have a
capacity of 500 units. Solve with the transportation method. Which site should be selected?
29) A manufacturer of semiconductor “wafers” has been attempting to convert its operations to practices
more in keeping with JIT principles. The firm is now paying much more attention to the transit time
between one processing stage and the next. The plant has a somewhat haphazard pattern of machine
locations, partly because the machines were purchased and installed at different times, partly from a
shortage of floor space, and partly from previous experiments with work cells. The bottom line is this:
there are four machines that perform a certain processing phase, and three machines that perform the
next phase. All units of a large class of wafers go through these two phases. The table below displays the
transit time, in minutes, from each machine of the first phase to each machine of the second. Machine 3 is
not really 100 minutes away from machine B; the company has prohibited that combination because of
quality problems associated with that specific pairing. Supply and demand quantities are in wafers
processed per week. Develop a transit time minimizing solution for this firm. What is the total transit
time of this solution? Which machines are fully utilized? Which machines have some capacity unused or
requirements unfilled? Was the prohibition on the 3-B combination honored?
Section 4 Special Issues in Modeling
1) Degeneracy in a transportation problem is when no closed path exists for evaluating an unused cell.
2) 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.
3) A transportation problem with 8 sources and 6 destinations will have an optimal solution that uses at
most 13 of the 48 possible routes.
4) To handle degeneracy, a very small quantity is placed in one of the unused squares.
5) Degeneracy occurs when the number of occupied squares is less than the number of rows plus the
number of columns minus one.
6) If demand exceeds supply in a transportation problem, the problem must be balanced by adding a
dummy source with additional supply.
7) 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) None of the above is true.
8) 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.
9) In a transportation problem, degeneracy means that:
A) the problem was improperly constructed, and it 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.
10) 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.
11) 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.
12) 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) one-half of one percent
B) five percent
C) ten percent
D) twenty-five
E) All cells will be occupied.
13) 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
14) 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.
15) A transportation model fills one-half of its cells under the non-degenerate 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
16) 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
17) A transportation problem that has more units supplied than demanded will require a(n) ________ to
balance the problem.
18) ________ is an occurrence in transportation problems when too few shipping routes are being used to
allow calculation of improvement indices.
19) The number of routes filled by a solution to a transportation problem is no larger than ________.
20) When does degeneracy occur in a transportation model?
21) When is it necessary to add dummy sources or destinations to a transportation problem?
22) 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.
23) A transportation problem has 6 origins and 12 destinations. How many possible routes are there for
this problem? What is the maximum number of routes that will be used in the optimal solution?
24) A transportation problem has 10 origins and 32 destinations. How many possible routes are there for
this problem? What is the maximum number of routes that will be used in the optimal solution?
25) A transportation model’s optimal solution uses no more than 8% of cells and has 100 sources. Find the
number of destinations.
26) Given the following feasible solution, determine if the problem is degenerate and then find the
optimal solution and its cost. Assume that capacity for source A is 10 and 30 for source B. Destination A
demands 10 units while destination B demands 30 units. Cost of shipping per unit is given as AA ($4),
AB ($1), BA ($3), and BB ($2).
Source A
Source B
Source A
Source B
27) Source A has capacity of 15, Source B has capacity of 30, Destination 1 has demand of 5 and
Destination 2 has demand of 20. Fill in the following table with the correct initial solution for a northwest–
corner method approach.
Destination 1
Destination 2
Source A
Source B
Destination 1
Destination 2
Source A
Source B
28) A transportation model uses no more than 5% of its cells. If the number of destinations is 1000,
determine the number of sources in the model.
29) A transportation model uses at least 10 sources and 100 destinations. If the ratio of sources to
destinations remains constant, does the maximum % of cells used by the optimum solution remain
constant? Why or why not?