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1.         A Head of Department has five jobs, A, B, C, D, E and five people V, W, X, Y, Z who
are capable of carrying out the jobs. He assesses the number of hours each person
would take to perform each job:

V             W              X             Y            Z

A              3             5             10            15            8
JOB         B              4             7             15            18            8
C              8            12             20            20           12
D              5             5              8            10            6
E             10            10             15            25           10

How should the jobs be allocated to the men to minimise the total time taken?

2.         Glenaff Ltd have appointed five new clerks A, B, C, D, E to fill the five positions of
cashier, invoice clerk, payroll clerk, cost clerk and general ledger clerk.

In order to find the optimal assignment of people to posts all the clerks were given a
test that covered the five areas of work involved. Each section of the test was marked
out of ten and the scores were as follows:

Cashier         Invoicing        Payroll         Costing          General
Ledger

A                         3               3                1               3               5
B                         5               8                9               4               4
C                         1               9               10               2               7
D                         5               5                9               6               7
E                         5               5                4               5               6

What is the optimum assignment of people to posts?

3.         A market research firm has three clients who have each requested that the firm conduct
a sample survey. Four available statisticians can be assigned to these three projects;
however, all four statisticians are busy, and therefore each can handle only one of the
clients. The following data show the number of hours required for each statistician to
complete each job; the differences in time are based on experience and ability of the
statisticians.

Tutorial 4.doc
Client
Statistician              A                 B                    C
1                   150               210                  270
2                   170               230                  220
3                   180               230                  225
4                   160               240                  230

a.        Formulate a linear programming model for this assignment problem.
b.        Solve the assignment problem using the Hungarian method.
b.        Suppose that the time it takes statistician 4 to complete the job for client A is
increased from 160 to 165 hours. What effect will this have on the solution?
c.         Suppose that the time it takes statistician 4 to complete the job for client A is
decreased to 140 hours. What effect will this have on the solution?
d.         Suppose that the time it takes statistician 3 to complete the job for client B
increases to 250 hours. What effect will this have on the solution?

(ASW : Ch 7, Qn 21 – min - sensitivity)

4.         The Department head of a management science department at a major university will
be scheduling faculty staff to teach courses during the coming Autumn term. Four
core courses need to be covered. The four courses are at the UG, MBA, MS and PhD
levels. Four professors will be assigned to the courses, with each professor receiving
one of the courses. Student evaluations of professors are available from previous
terms. Based on a rating scale of 4 (excellent), 3 (very good), 2 (average), 1 (fair) and
0 (poor), the average student evaluations for each professor are shown. Professor D
does not have a PhD, and cannot be assigned to teach the PhD level course. If the
department head makes teaching assignments based on maximising the student
evaluation ratings over all four course, what staffing assignments should be made?

Course
Professor       UG              MBA            MS             PhD

A             2.8            2.2            3.3              3.0
B             3.2            3.0            3.6              3.6
C             3.3            3.2            3.5              3.5
D             3.2            2.8            2.5               -

(ASW : Ch 7, Qn 20 – max - prohibited)

5.         Steel mills in three cities produce the following amounts of steel:

Location                    Weekly Production (tons)
A. Bethlehem                150
B. Birmingham               210
C. Gary                     320

Tutorial 4.doc
680

These mills supply steel to four cities where manufacturing plants have the following
demand:

Location                Weekly Demand (tons)
1. Detroit              130
2. St Louis              70
3. Chicago              180
4. Norfolk              240
620

Shipping costs (£) per ton of steel are as follows:

To
From      1           2         3           4
A      14           9        16          18
B      11           8         7          16
C      16          12        10          22

Because of a lorry drivers’ strike, shipments are presently prohibited from Birmingham
to Chicago.

a.        Set up a transportation tableau for this problem and determine the initial
solution using minimum cost method.
b.       Solve this problem using the MODI method.
c.       Are there any alternate optimum solutions? Explain. If so, identify them.
d.       Formulate this problem as a general linear programming model.

(Taylor : Ch 6, Qn 9 – min – unbalanced - alternative)

Tutorial 4.doc
6.         Coal is mined and processed at the following four mines in Kentucky, West Virginia
and Virginia:

Location                 Capacity (tons)
A. Cabin Creek            90
B. Surry                  50
C. Old Fort               80
D. McCoy                  60
280

These mines supply the following amount of coal to utility power plants in three cities:

Plant                    Demand (tons)
1. Richmond              120
2. Winston-Salem         100
3. Durham                110
330

The railroad shipping costs (£1000s) per ton of coal are shown in the following table.
Because of railroad construction, shipments are presently prohibited from Cabin Creek
to Richmond:

To
From         1           2          3
A          7          10          5
B         12           9          4
C          7           3         11
D          9           5          7

a.        Set up the transportation tableau for this problem, determine the initial solution
using the least cost method, and compute total cost.
b.       Solve using the MODI method.
c.       Are there multiple optimal solutions? Explain. If there are alternative optimum
solutions, identify them.

(Taylor : Ch 6, Qn 11 – min – prohibited - unbalanced)

7.         A manufacturing firm produces diesel engines in four cities - Phoenix, Seattle, St
Louis and Detroit. The company is able to produce the following numbers of engines
per month:

Plant                   Production
1. Phoenix              5
2. Seattle              25
3. St Louis             20
4. Detroit              25

Tutorial 4.doc
Three trucking firms purchase the following numbers of engines for their plants in
three cities:

Firm                   Demand
A. Greensboro          10
B. Charlotte           20
C. Louisville          15

The transportation costs per engine (£’00) from sources to destinations are shown in
the following table. However, the Charlotte firm will not accept engines made in
Seattle, and the Louisville firm will not accept engines from Detroit; therefore, these
routes are prohibited.

To
From        A          B           C
1          7          8           5
2          6         10           6
3         10          4           5
4          3          9          11

a.        Set up the transportation tableau for this problem. Find the initial solution
using VAM.
b.        Solve for the optimal solution using the stepping-stone method. Compute the
total minimum cost.
c.       Formulate this problem as a linear programming model.

(Taylor: Ch 6, Qn 13 – min – unbalanced - prohibited)

Tutorial 4.doc

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