Exam 2II75 (04-08-’08, 9:00-12:00)
Give concise answers. Unnecessary details may lower your score. Indicate assumptions if needed.
Relative weight of the exam questions 1: 2 pts, 2: 3 pts, 3: 2 pts, 4: 3 pts.
1. Comment on the problem sketched below and suggest improvements, extensions and additions to the
two questions, with a brief indication of the reason why.
Jobshop J offers two versions (P1 and P2) of a certain product.. Production of P1 and P2 differs only slightly,
where P1 needs a specialized resource R1 and P2 needs resource R2. Demand for P1 is higher than for P2; for
this reason, usage of R1 is rather high,.whereas R2 has unused capacity. In order to obtain a better balance,
management wants to shift the demand of J’s customers towards P2. Marketing experts have proposed certain
incentives, that they estimated will increase the demand for P2 by 10% and diminish the demand for P1 by 5%.
Management wants to conduct a simulation study to assess whether the demand shift can be realized and how
this demand shift will affect the production process.The following questions need to be answered.
i) Can the estimated demand shift be realized?
ii) How will the demand shift affect the utilization of R1 and R2 and the average lead times of P1 and P2?
2. Production of a certain item in a job shop requires two consecutive processing steps. The first step
S1 needs resource R1 and has a duration distribution D1. The second step S2 needs another resource
R2 and has a duration distribution D2. The resources R1,R2 are not needed for any other step. Either
resource can fail while the step is being executed. The chance of failure during step S1/S2 equals f1/f2.
You should assume that failure occurs at the end, after having fully spent the production time.
If a failure occurs, the production of that specific item has to restart and the resource must be repaired,
which requires a repair duration with distribution Dr for either resource. Create conceptual and Arena
models for the sketched situation. Give details for any modeled “choice” or “assign” blocks.
3. In the depicted Arena model, the blocks arrive1 and arrive2 have negative exponential distributions
with average interarrival times 20, resp. 30 minutes. All steps are seize-delay-release steps seizing
resource A. The durations of step1, step2 and stepA are normally distributed, with averages of
respectively 4, 5 and 6 minutes and standard deviation of 1 minute each. Decide1 and Decide2 are
both random decisions with a success rate of 50%.
a) What occupation rate for resource A do you derive from the model?
b) Derive an equivalent simpler model. Give the parameters for the new block(s).
4. Study the following excerpt from a simulation report, that refers back to exam question (1). You are
asked to criticize the hypotheses formulated, their assessment and the conclusions drawn. Even if you
do not agree with a hypothesis, you still are asked to check the assessment techniques used. Suggest
extensions, alternatives and/or improvements for errors you have discovered. The z-table for the
normal distribution is included for reference purposes.
In order to answer the questions, a simulation model M has been constructed for the current situation, which
has been validated and found correct by experts; its simulation results have also been found to match recorded
measurements. By modifying the arrival intensities, a model N has been obtained for the shifted demand
situation. We conducted simulation experiments to test the following hypotheses, derived from simulation
question i), where, A1(X) / A2(X) denote the “true” average number of requests for P1 / P2 per day in model X
(which would have been obtained as a limit of simulation runs of unbounded increasing duration):
H1: A1(N) is less than 0.95 * A1(M),
H2: A2(M) is less than 0.9 * A2(N).
Simulation runs, each consisting of thirty subruns of M and N have been conducted; for each subrun the
average number of requests for P1 and P2 have been recorded. The resulting run averages A1(M), A2(M),
A1(N), A2(N) and correspondig run standard deviations s1(M), s2(M), s1(N), s2(N) are listed below.
A1(M) s1(M) A2(M) s2(M)
A1(N) s1(N) A2(N) s2(N)
358.6
1.95
247.3
1.28
341.1
1.87
271.6
1.31
From these data, assuming normally distributed run averages and using the confidence assessment
s

s

z( )    x 
z ( ) , we derive 95% confidence intervals for A1(M), A2(M),
formula x 
2
n 2
A1(N), A2(N) of 358 .6  0.70 , 247 .3  0.49 , 351 .7  0.69 , 271 .6  0.47 respectively. The expression
n
A1(N)-0.95*A1(M) thus is approximated by A1(N) - 0.95*A1(M) = 0.43 with standard deviation
(1.28 2  1.95 2 ) / 1.96  1.19 . The confidence in hypothesis H1 therefore becomes z(0.43/1.19)= 64%. H1 is
therefore rejected. Since we cannot confirm H1, we test a weaker version namely H1’: A1(N) is less than
A1(M). The expression A1(N)-A1(M) is approximated by A1(N)-A1(M) = 17.5 with standard
(1.95 2  1.28 2 ) / 1.96  1.19 , so confidence in H1’ exceeds 99.99%. We conclude that we cannot
give an affirmative answer to question i); however we can confirm that some less extreme demand shift is
possible.
...
(Discussions of H2 and question ii) skipped)
deviation