Transcript スライド 1 - New Mexico Institute of Mining and Technology

EMGT 501
Fall 2005
Midterm Exam
Due Day: Oct 17 (Noon)
Note:
(a) Do not send me after copying your computer results
of QSB. Answer what are your decision variables,
formulation and solution, only. See my HW answer on
my HP.
(b) Put your mailing address so that I will be able to
return your exam result via US postal service.
(c) Answer on a PPS series of slides.
(d) Do not discuss on the exam with other students.
(e) Return your answer attached to your e-mail.
1. Linear Programming (20%)
Benson Electronics manufactures three components to
produce cellular telephones and other communication
devices. In a given production period, demand for the three
components may exceed Benson’s manufacturing capacity.
In this case, the company meets demand by purchasing the
components from another manufacturer at an increased
cost per unit. Benson’s manufacturing cost per unit and
purchasing cost per unit for the three components are as
follows:
Source
Manufacture
Purchase
Component 1 Component 2 Component 3
$4.50
$5.00
$2.75
$6.50
$8.80
$7.00
Manufacturing times in minutes for Benson’s three
departments are as follows:
Department
Component 1 Component 2 Component 3
Production
2
3
4
Assembly
1
1.5
3
Testing and packaging
1.5
2
5
For instance, each unit of component 1 that Benson
manufactures requires 2 minutes of production time, 1
minute of assembly time, and 1.5 minutes of testing and
packaging time. For the next production period, Benson
has capacities of 360 hours in the production department,
250 hours in the assembly department, and 300 hours in
the testing and packaging department.
a. Formulate a linear programming model that can be used
to determine how many units of each component to
manufacture and how many units of each component to
purchase. Assume that component demands that must be
satisfied are 6000 units for component 1, 4000 units for
component 2, and 3500 units for component 3. The
objective is to minimize the total manufacturing and
purchasing costs.
b. What is the optimal solution? How many units of each
component should be manufactured and how many units
of each component should be purchased?
c. Show its dual formulation
d. Discuss complementary slackness condition between the
primal and dual models.
2. Linear Programming (15%)
Consider the following problem.
Maximize
subject to
Z  4x1  3x2  x3  2x4 ,
4 x1  2 x2  x3  x4  5
3x1  x2  2 x3  x4  4
x1  0, x2  0, x3  0, x4  0.
(a) Solve the problem.
(b) What is B-1? How about B-1b and CBB-1b?
(c) If the right hand side is changed from (5, 4) to (6, 5),
how is an optimal solution changed? How about an
optimal objective value?
3. Linear Programming (15%)
Suppose that in a product-mix problem x1, x2, x3,and x4
indicate the units of products 1,2,3, and 4, respectively,
and we have
Max
s.t.
4 x1  6x2  3x3  1x4 ,
1.5 x1  2 x2  4 x3  3x4  550 Machine A hours
4 x1  1x2  2 x3  1x4  700 Machine B hours
2 x1  3x2  1x3  2 x4  200 Machine C hours
x1 , x2 , x3 , x4  0
a. Formulate the dual model to this problem.
b. Solve the dual. Use the dual solution to show that the
profit-maximizing product mix is x1=0, x2=25, x3=125,
and x4=0.
c. Use the dual variables to identify the machine or
machines that are producing at maximum capacity. If
the manager can select one machine for additional
production capacity, which machine should have
priority? Why?
4. PERT/CPM (20%)
Building a backyard swimming pool consists of nine major
activities. The activities and their immediate predecessors
are shown. Develop the project network.
Activity
A
B
Immediate
Predecessor
-
-
C
D
A,B A,B
E
F
G
H
I
B
C
D
D,F E,G,H
Assume that the activity time estimates (in days) for the
swimming pool construction project are as follows:
Activity
A
B
C
D
E
F
G
H
I
a.
b.
c.
d.
Optimistic Most Probable
3
2
5
7
2
1
5
6
3
5
4
6
9
4
2
8
8
4
Pessimistic
6
6
7
10
6
3
10
10
5
What is an activity schedule?
What is a critical path?
What is the expected time to complete the project?
What is the probability that the project can be completed in
25 or fewer days?
5. Inventory (15%)
Wilson Publishing Company produces books for the retail
market. Demand for a current book is expected to occur at
a constant annual rate of 7200 copies. The cost of one copy
of the book is $14.50. The holding cost is based on an 18%
annual rate, and production setup costs are $150 per setup.
The equipment on which the book is produced has an
annual production volume of 25,000 copies. Wilson has
250 working days per year, and the lead time for a
production run is 15days. Use the production lot size
model to compute the following values:
a. Minimum cost production lot size
b. Number of production runs per year
c. Cycle time
d. Length of a production run
e. Maximum inventory
f. Total annual cost
g. Reorder point
6. Inventory (15%)
A well-known manufacturer of several brands of toothpaste
uses the production lot size model to determine production
quantities for its various products. The product known as
Extra White is currently being produced in production lot
sizes of 5000 units. The length of the production run for
this quantity is 10 days. Because of a recent shortage of a
particular raw material, the supplier of the material
announced that a cost increase will be passed along to the
manufacturer of Extra White. Current estimates are that the
new raw material cost will increase the manufacturing cost
of the toothpaste products by 23% per unit. What will be
the effect of this price increase on the production lot sizes
for Extra White?
EMGT 501
HW #4
Solutions
Chapter 11 - SELF TEST 3
Chapter 11 - SELF TEST 7
Chapter 11 - SELF TEST 17
Ch. 11 – 3
Q 
*
2 DCO

Ch
2(5000)(32)
 400
2
D
5000
d

 20 units per day
250 250
a. r = dm = 20(5) = 100
*
Since
both
inventory position and inventory
r , Q
on hand equal 100.
b. r = dm = 20(15) = 300
*
r , Q
Since
both
inventory position and inventory
on hand equal 300.
c. r = dm = 20(25) = 500
Inventory position reorder point = 500. One order
of Q* = 400 is outstanding. The on-hand inventory
reorder point is 500 - 400 = 100.
d. r = dm = 20(45) = 900
Inventory position reorder point = 900. Two orders
of Q* = 400 are outstanding. The on-hand
inventory reorder point is 900 - 2(400) = 100.
Ch. 11 – 7
2 DCo

Ch
Q 
*
2 DCo
IC
2 DCo
Q'
I 'C
Where Q ' is the revised order quantity for the new
carrying charge I ' . Thus
Q '/ Q 
*
Q ' 
Q' 
2 DCo / I ' C
2 DCo / IC
I *
Q
I'
0.22
(80)  72
0.27

I
I'
Ch. 11 – 17
EOQ Model
Q* 
2 DCo

Ch
2(800)(150)
 282.84
3
Total Cost
1
D
800
 282.84 
 QCh  C0  
3
(150)  $848.53

2
Q
282.84
 2 
Planned Shortage Model
2DCo  Ch  Cb 
2(800)(150)  3  20 
Q* 
 303.32




Ch  Cb 
3
 20 
 Ch
S*  Q * 
 Ch  Cb

 3 
 39.56
  303.32 

 3  20 

Total Cost
(Q  S ) 2
D
S2

Ch  Co 
Cb  344.02  395.63  51.60  $791.25
2Q
Q
2Q
Cost Reduction with Backorders allowed
$848.53 - 791.25 = $57.28 (6.75%)
Both constraints are satisfied:
1. S / Q = 39.56 / 303.32 = 0.13
Only 13% of units will be backordered.
2. Length of backorder period = S / d = 39.56 / (800/250)
= 12.4 days