1. (TCO 8) Name three of the measures of a queue’s performance. (Points : 15)
Ans) Queuing models help managers make decisions that balance service costs with waiting-line costs.
Queuing analysis can obtain many measures of a waiting-line system’s performance, including the
following:
1. Average time that each customer or object spends in the queue
2. Average queue length
3. Average time that each customer spends in the system (waiting time plus service time)
4. Average number of customers in the system
5. Probability that the service facility will be idle
6. Utilization factor for the system
7. Probability of a specific number of customers in the system
2. (TCOs 11 and 12) Describe the steps of the simplex method that follow setting up the initial
tableau. (Points : 15)
Ans) The simplex method changes constraints (inequalities) to equations in linear programming
problems, and then solves the problem by matrix manipulation. The solution set for the altered
problem is of higher dimension than the solution set of the original problem, but it is easier to study
with matrices.
STEP 1: Rewrite each constraint (inequality) as an equation.
STEP 2: Write the revised problem as a tableau, with the objective row (= bottom row) consisting of
negatives of the coefficients of the objective function z ; z will be maximized. The lower right corner is
the value of z when x, y,... are zero ; thus, z usually starts out as zero. Simplex method will move the
, ISM, one column at a time; after each such move, we arrive at (or "hop" to) a new corner point (basic
solution) with bigger objective value. Since the solution set has only finitely many corners, this process
ultimately yields the biggest value of the objective function.
STEP 3: The pivot column is that column containing the most negative indicator. If no indicator is
negative, the tableau is a FINAL TABLEAU.
STEP 4: Form RATIOS (quotients) for each row: divide the right-most number by the number in the pivot
column of that row.
STEP 5: The PIVOT ROW is the row with the smallest NON-NEGATIVE ratio (quotient).
Note that 0(+1) and 0(-1) are both numerically zero, but in calculating RATIOS, consider 0(+1) as
positive (OK), and 0(-1) as negative (not OK)
STEP 6: Apply a pivot operation (Rolf 8th ed., pg. 98) to the tableau, including the bottom (objective)
row. The pivot column will become a column of a new ISM in the new tableau. Note which column is
replaced, and where the new ISM is located; its columns may not be in the usual order: not to worry.
Check out the PIVOT ENGINE to speed up practice.
STEP 7: (optional) Note the new basic solution (corner point) for the new tableau.
STEP 8: If all indicators (in the bottom row) are non-negative, your tableau is a FINAL TABLEAU. The
basic solution of step 7 is the maximal solution you have been seeking! Note that if you correctly
reached this step from a Non-Standard Problem, then all right-side numbers above the objective row
will also be non-negative.
STEP 9: Otherwise, if some indicator remains negative, repeat steps 3 through 9 WITH YOUR NEW
TABLEAU. After a finite number of such repetitions (usually 2-3), simplex method must terminate at
step 8.
3. (TCOs 15 and 16) List the two approaches to setting up a PERT or CPM network, and describe the
difference between the two. (Points : 15)
Ans) Queuing models help managers make decisions that balance service costs with waiting-line costs.
Queuing analysis can obtain many measures of a waiting-line system’s performance, including the
following:
1. Average time that each customer or object spends in the queue
2. Average queue length
3. Average time that each customer spends in the system (waiting time plus service time)
4. Average number of customers in the system
5. Probability that the service facility will be idle
6. Utilization factor for the system
7. Probability of a specific number of customers in the system
2. (TCOs 11 and 12) Describe the steps of the simplex method that follow setting up the initial
tableau. (Points : 15)
Ans) The simplex method changes constraints (inequalities) to equations in linear programming
problems, and then solves the problem by matrix manipulation. The solution set for the altered
problem is of higher dimension than the solution set of the original problem, but it is easier to study
with matrices.
STEP 1: Rewrite each constraint (inequality) as an equation.
STEP 2: Write the revised problem as a tableau, with the objective row (= bottom row) consisting of
negatives of the coefficients of the objective function z ; z will be maximized. The lower right corner is
the value of z when x, y,... are zero ; thus, z usually starts out as zero. Simplex method will move the
, ISM, one column at a time; after each such move, we arrive at (or "hop" to) a new corner point (basic
solution) with bigger objective value. Since the solution set has only finitely many corners, this process
ultimately yields the biggest value of the objective function.
STEP 3: The pivot column is that column containing the most negative indicator. If no indicator is
negative, the tableau is a FINAL TABLEAU.
STEP 4: Form RATIOS (quotients) for each row: divide the right-most number by the number in the pivot
column of that row.
STEP 5: The PIVOT ROW is the row with the smallest NON-NEGATIVE ratio (quotient).
Note that 0(+1) and 0(-1) are both numerically zero, but in calculating RATIOS, consider 0(+1) as
positive (OK), and 0(-1) as negative (not OK)
STEP 6: Apply a pivot operation (Rolf 8th ed., pg. 98) to the tableau, including the bottom (objective)
row. The pivot column will become a column of a new ISM in the new tableau. Note which column is
replaced, and where the new ISM is located; its columns may not be in the usual order: not to worry.
Check out the PIVOT ENGINE to speed up practice.
STEP 7: (optional) Note the new basic solution (corner point) for the new tableau.
STEP 8: If all indicators (in the bottom row) are non-negative, your tableau is a FINAL TABLEAU. The
basic solution of step 7 is the maximal solution you have been seeking! Note that if you correctly
reached this step from a Non-Standard Problem, then all right-side numbers above the objective row
will also be non-negative.
STEP 9: Otherwise, if some indicator remains negative, repeat steps 3 through 9 WITH YOUR NEW
TABLEAU. After a finite number of such repetitions (usually 2-3), simplex method must terminate at
step 8.
3. (TCOs 15 and 16) List the two approaches to setting up a PERT or CPM network, and describe the
difference between the two. (Points : 15)
