Exam test BPM AOME, 5. 6. 2020
Name: Score
1 2 3 4 5 6 7 8
Problem 1:[10b] Consider following LP problem:
2x1 + 3x2 + 7x3 + 9x4 → max
subject to
x1 + x2 + x3 + x4 ≤ 9,
x1 + 2x2 + 4x3 + 8x4 ≤ 24,
xi ≥ 0, i = 1, . . . 4.
After several steps of simplex method we got the table:
x1 x2 x3 x4 x5 x6 bi
3/4 1/2 0 -1 1 -1/4 3
1/4 1/2 1 2 0 1/4 6
-1/4 5/4 0 5 0 7/4 42
• Check the optimality of the table and if it is not yet optimal, perform the
next steps of the Simplex method.
• Find the basic variables in the optimal table.
• Find the optimal value of objective function.
• Determine which of the constraints are binding and find the shadow prices.
Problem 2:[10b] The office needs to outsource four jobs, selecting from the offer
of five companies. Each company is able to complete only one job. Estimated
costs are in the table. Use Hungarian method to assign jobs to companies so
that the total cost is minimal.
F1 F2 F3 F4 F5
Z1 15000 14000 20000 17000 16000
Z2 18000 17000 19000 16000 15000
Z3 17000 15000 18000 15000 14000
Z4 17000 19000 20000 16000 18000
Problem 3: [10b] The project consists of nine activities. The duration of activities
in days and their predecessors are listed in the following table:
activity A B C D E F G H I
duration 7 9 6 5 5 4 9 8 6
predecessors - - A A B C,D C C,D E,F
Draw a project network and use CPM to find minimal realization time of the
project, critical activities and total time reserves for non-critical activities.
Problem 4:[10b] The DEA model for 6 units with two inputs and one output
is defined by the table.
U1 U2 U3 U4 U5 U6
I1 12 2 2 10 16 14
I2 9 16 12 10 4 2
O 3 4 2 5 4 2
• Solve the problem graphically constant returns to scale and draw efficient
frontier.
• Compute efficiency of U1. Find virtual unit for U1 (its projection on efficient
frontier) and its peer units.
Problem 5:[15b] Suppose that you live in the city depicted in the plan.
a) Find the shortest path from home to other destinations.
b) Design which paths are to be repaired so there was a repaired connection
between each of the two places and the total number of repaired km was
minimal.
Problem 6:[15b] Optimize the delivery plan from three plants D1-D3 to four
customers O1-O4. Plant capacities [t], customer requirements [t] and distance
between suppliers and customers [km] are shown in the table below.
O1 O2 O3 O4 capacities
D1 6 10 4 2 100
D2 8 14 10 6 140
D3 4 18 12 8 180
requirements 60 80 80 200
• Introduce a dummy row or column to balance the the transportation problem
of minimizing total tonne-km.
• Find any feasible solution.
• Check the optimality of the solution using dual table. If the condition is
not satisfied, perform steps of the MODI method necessary to obtain the
optimal solution.
Problem 7:[15b] 5 cars are evaluated according to three criteria - price (CZK
thousand), consumption (l per 100 km) and safety (in points). The first two
criteria are considered as minimizing, the last one is maximizing.
Alternatives price consumption safety
X1 400 7.5 7
X2 500 7.2 9
X3 340 6.8 6
X4 300 6.5 4
X5 380 7 8
• Find all dominated alternatives
• Find basal and ideal alternative.
• Normalize criteria matrix and rank the alternatives using WSA method for
the weights vector v = (0.4, 0.2, 0.4).
Problem 8:[15b] The investor decides to divide his investment between shares,
mutual funds and government bonds. The total amount of the investment is
equal to CZK 10 million. The investor assumes the return of individual assets
of 6, 3, 2 percent p. a. The risk of individual assets was rated by 10, 6 and 2
points.
• Formulate a mathematical model to maximize expected return and minimize
the risk.
• Find both partial optimal solutions.
• Write down the aggregate objective function and find the optimal solution
using Solver when we consider the risk to be a three times more important
criterion than the return.
• Convert the criterion return to the constraint by allowing a 10-percent
deviation from the optimal value. Identify the minimum risk of the resulting
single-criterion problem.