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r;lwluvly v The study of how changes in the coefficients of a linear programming problem affect the optimal solution.
Range of optimality The range of values over which an objective function coefficient may vary without causing any change in the values of the decision variables in the optimal solution. Dual price The improvement in the value of the objective function per unit increase in the righthand side of a constraint. Reduced cost The amount by which an objective function coefficient would have to improve (increase for a maximization problem, decrease for a minimization problem) before it would be possible for the corresponding variable to assume a positive value in the optimal solution. Range of feasibility The range of values over which the dual price is applicable.
100 percent rule A rule indicating when simultaneous changes in two or more objective function coefficients will not cause a change in the optimal solution. It can also be applied to indicate when two or more righthandside changes will not cause a change in any of the dual prices.
Sunk cost A cost that is not affected by the decision made. It will be incurred no matter what values the decision variables assume. Relevant cost A cost that depends upon the decision made. The amount of a relevant cost will vary depending on the values of the decision variables.
PROBLEMS
1 Consider the following linear program: .
SELFm
Max 3A
+ 2B
Chapter 3 Linear Programming: SensitivityAnalysis and Interpretation of Solution a. Use the graphical solution procedure to find the optimal solution. b. Assume that the objective function coefficient for A changes from 3 to 5 . Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution. c. Assume that the objective function coefficientforA remains 3, but the objective function coefficient for B changes from 2 to 4. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution. d. The Management Scientist computer solution for the linear program in part (a) provides the following objective coefficient range information:
Variable A
Lower L i t 2
Current Value 3
Upper L i t 6
Use this objective coefficient range information to answer parts (b) and (c). 2. Consider the linear program in Problem 1. The value of the optimal solution is 27. S up pose that the righthand side for constraint 1 is increased from 10 to 11. a. Use the graphical solution procedure to find the new optimal solution. b. U the solution to part (a) to determine the dual price for constraint 1. se c. The Management Scientist computer solution for the linear program in Problem 1 provides the following righthandside range information:
2 3
Lower Limit 8 18 13
Current Value 10 24 16
Upper Limit 11.2 30 No Upper Limit
What does the righthandside range information for constraint 1 tell you about the dual price for constraint l ? d. The dual price for constraint 2 is 0 .5. Using this dual price and the righthandside range information in part (c), what conclusion can be drawn about the effect of changes to the righthand side of constraint 2? Consider the following linear program: Min s.t. 8X
+ 12Y
lX+ 3Y1 9 2 X + . 2 Y 1 10 6X 2Y 1 18 X,Y1O
+
Use the graphical solution procedure to find the optimal solution. Assume that the objective function coefficient for X changes from 8 to 6. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution.
Problems
131
Assume that the objective function coefficient for X remains 8, but the objective function coefficient for Y changes from 12 to 6. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution. d. The Management Scientist computer solution for the linear program in part (a) provides the following objective coefficient range information:
c.
Variable X
Lower Limit 4 8
Current Value 8
Upper Limit 12
How would this objective coefficient range (c) prior to resolving the problem?
4. Consider the linear program in Problem 3 . The value of the optimal solution is 48. Suppose that the righthand side for constraint 1 is increased from 9 to 10. a . Use the graphical solution procedure to find the new optimal solution. b. Use the solution to part (a) to determine the dual price for constraint 1. c. The Management Scientistcomputer solution for the linear program in Problem 3 provides the following righthandside range information:
Lower L i t
5
9 No Lower Limit
7
Current Value 9 10 18
Upper Limit 11 18
,.
.
.
..
''I
+L,
.>;
+

1test
What does the righthandside range information for constraint 1 tell you about the dual price for constraint l ? d. The dual price for constraint 2 is  3. Using this dual price and the righthandside range information in part (c), what conclusion can be drawn about the effect of changes to the righthand side of constraint 2? 5. Refer to the Kelson Sporting Equipment problem (Chapter 2, Problem 24). Letting
R = number of regular gloves C = number of catcher's mitts
leads to the following formulation: Max S.t.
5R
+
8C
R + 8/1C 5 9 0 0 Cutting and sewing % R + l/gC 5 3 00 Finishing l/gR + % C S 1 00 Packaging and shipping R,CrO
The computer solution obtained using The Management Scientistis shown in Figure 3 .13. a. What is the optimal solution, and what is the value of the total profit contribution? b. Which constraints are binding?
Chapter 3
Linear Programming: Sensitivity Analysis and Interpretation
of Solution
FIGURE 3.13 THE MANAGEMENT SCIENTIST SOLUTION FOR THE KELSON SPORTING EQUIPMENT PROBLEM
Objective Function Value = Variable  R C Constraint
3700.00146
I
 500.00153 149.99924
Value
0.00000 0.00000
Reduced Costs
Slack/Surplus

Dual Prices
3
,I
0.00000
j
OBJECTIVE COEFFICIENT RANGES Variable Lower Limit
  5.00000 8.00000' 12.00012 10.00000
Current Value
Upper Limit
RIGHT HAND SIDE RANGES Constraint Lower Limit
 900.00000 300.00000 100.00000
Current Value
Upper Limit
No Upper Limit 400.00000 134.99982
SELF
t est
What are the dual prices for the resources? Interpret each. If overtime can be scheduled in one of the departments, where would you recommend doing so? 6. Refer to the computer solution of the Kelson Sporting Equipment problem in Figure 3.13 ' (see Problem 5). a. Determine the objective coefficient ranges. b. Interpret the ranges in part (a). c. Interpret the righthandside ranges. d. How much will the value of the optimal solution improve if 20 extra hours of pac aging and shipping time are made available?
'
7. Investment Advisors, Inc., is a brokerage firm that manages stock portfolios for a number of clients. A particular portfolio consists of U shares of U.S. Oil and H shares of Huber Steel. The annual return for U.S. Oil is $3 per share and the annual return for Huber Steel is $5 per share. U.S. Oil sells for $25 per share and Huber Steel sells for $50 per share. The portfolio has $80,000 to be invested. The portfolio risk index (0.50 per share of U.S. Oil and 0.25 per share for Huber Steel) has a maximum of 700. In addition, the portfolio is
Problems
&.*urld
limited to a maximum of 1000 shares of U Oil. The linear programming formation that .S. will maximize the total annual return of the portfolio is as follows: Max s.t.
3U+
5H
Maximize total annual return
25U + 50H 5 80,000 Funds available 0.50U + 0.25H 5 700 Risk maximum 1U 5 1000 U .S.Oil maximum U,HZ0
The computer solution of this problem is shown in Figure 3.14. a. What is the optimal solution, and what is the value of the total annual return? b. Which constraints are binding? What is your interpretation of these constraints in I terms of the problem? ' c. What are the dual prices for the constraints? Interpret each. d. Would it be beneficial to increase the maximum amount invested in U Oil? W hy .S. or why not?
FIGURE 3.14 THE MANAGEMENT SCIENTIST SOLUTION FOR THE INVFSTMENT ADVISORS PROBLEM
Objective F'unction Value = Variable
.
8. 400 . OOO . . .. . .
,
..
 
800.000 1,200 :ooo .
. ..
Value
.
.
..

Reduced Costs
.,
.. .
U
H
'
.
.
"
.
.
 .I
2
Constraint
 .
Slack/Surplus
EJECTIVE COEFFICIENT RANGES
   ',U
H
Variable

3
.
'. 0 .000 0 .000. 200.000.
.
'
.
.
.
.. .
~ o k e r imit L
2 ,500 1 .500
. ..
Current Value
,
,
Upper,Limit
'3.000, 5 .000
1 0.000 6.000
. ..
IGHT HAND SIDE RANGES Constraint 1.
2 .3
.
.

'
 65000.000 4 00.000 8 00. O QO
.
'Lower Limit
.
.
'
W r e n t Value
.
.
1 40000.000 775.000
.
, .
.UpperLimit
'
 8 0000.000
:
700.000 1000; 0 00
,
No  Upper Limit
Chapter 3
Linear Programming: Sensitivity Analysis and Interpretation of Sdution
8. Refer to Figure 3.14, which shows the computer solution of Problem 7. a. How much would the return for U.S.Oil have to increase before it would be benefi. cial to increase the investment in this stock? b. How much would the return for Huber Steel have to decrease before it would be ben. eficial to reduce the investment in this stock? c. How much would the total annual return be reduced if the U.S. il maximum were re O duced to 900 shares? 9. Recall the Tom's, Inc., problem (Chapter 2, Problem 28). Letting
W = jars of Western Foods Salsa M = jars of Mexico City Salsa leads to the formulation: Max s.t. l W + 1.25M 5W 7M 5 4480 Whole tomatoes 1 S 2080 Tomato sauce M 3W + 2W + 2M 5 1600 Tomato paste W,M?O The Management Scientist solution is shown in Figure 3.15. a. What is the optimal solution, and what are the optimal production quantities? b. Specify the objective function ranges. c. What are the dual prices for each constraint? Interpret each. d. Identify each of the righthandside ranges.
+
10. Recall the Innis Investments problem (Chapter 2, Problem 39). Letting
SELFm
S = units purchased in the stock fund M = units purchased in the money market fund leads to the following formulation: Min S.t. 8S 5 0s 5S
+
3M Funds available Annual income 3,000 Units in money market
+ lOOM 5 1,200,000 + 4M 2 60,000
Mr
S,MrO The computer solution is shown in Figure 3.16. a. What is the optimal solution, and what is the minimum total risk? b. Specify the objective coefficient ranges. . How much annual income will be earned by the portfolio? c d. What is the rate of return for the portfolio? e. What is the dual price for the funds available constraint? f. 'What is the marginal rate of return on extra funds added to the portfolio? 1 . Refer to Problem 10 and the computer solution shown in Figure 3.16, 1 a. Suppose the risk index for the stock fund (the value of C,) increases from its curren value of 8 to 12. How does the optimal solution change, if at all?
Problems
135
FIGURE 3.15 THE MANAGEMENT SCIENTIST SOLUTION FOR THE TOM'S, INC., PROBLEM
II
I
OPTIMAL SOLUTION v ~jectiveFunction Value =
.         8 60.000
L
 Reduced Costs
Variable
 
Value

Constraint

Slack/Surplus
0 .125 0. Q OO 0.187
Dual Prices
OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit
RIGHT HAND SIDE RANGES Constraint Lower Limit
4 320.000 1920.000 1 280.000
Current Value
4480.000 2080.000 1600.000
U m e r Limit
5600.000
No Upper Limit
1640.000
.
SELF
Suppose the risk index for the money market fund (the value of C,) increases from its current value of 3 to 3.5. How does the optimal solution change, if at all? Suppose C , increases to 12 and C , increases to 3.5. How does the optimal solution change, if at all?
test 
Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $63, $95, and $135, respectively. The production requirements per unit are as follows:
Number of
Fans
Economy Standard Deluxe
1
1 1
Number of Cooling C o b 1 2 4
Manufacturing Time (hours) 8 12 14
Chapter 3
Linear Programming: Sensitivity Analysis and interpretationof Solution
FIGURE 3.16 THE MANAGEMENT SCIENTIST SOLUTION FOR THE INNIS
INVESTMENTS PROBLEM Objective Function Value = Variable
2
62000.000
4000.000 10000.000
Value
 0.000 0.000
Reduced Costs
S M
1 2 3
Constraint
0.000 0.000 7000.000
Slack/Surplus

Dual Prices
I
.
0.057 2.167 0.000
i
OBJECTIVE COEFFICIENT RANGES
1
,
~
!
 S M
Variable
3.750 No Lower Limit
Lower Limit
8.000 3.000
Current Value
No Upper Limit 6.400
Upper Limit
RIGHT HAND SIDE RANGES
1 2 3
Constraint
 780000.000 48000,000 No Lower Limit
Lower Limit
1200000.000 60000.000 3000.000
Current Value
1500000.000 102000.00Q 10000.000
Upper Limit
For the coming production period, the company has 200 fan motors, 320 cooling coils, and 2400 hours of manufacturing time available. How many economy models ( E), standard models ( S), and deluxe models ( D) should the company produce in order to maximize profit? The linear programming model for the problem is as follows. Max 6 3E s.t.
+ 9 5 s + 1 350
1D 5 2 00 4 D 5 3 20 1 4 0 1 2 400 Fan motors Cooling coils Manufacturing time
+ + 8 E + 1 2s +
1E + IS 1E + 2S
lr i
E,S , D r o
f The computer solution using The Management Scientist is shown in Figure 3.17. a. What is the optimal solution, and what is the value of the objective function? b. Which constraints are c. binding? Which constraint shows extra capacity? How much? d. If the profit for the deluxe model were increased to $ 150 per unit, would the optimal ' solution change? Use the information in Figure 3.17 to answer this question.
Problems
137
FIGURE 3.17 T HE MANAGEMENT SCIENTIST SOLUTION FOR THE QUALITY AIR
CONDITIONING PROBLEM
Objective Function Value = Variable Value
1 6440.000
Reduced Costs
Constraint
Dual Prices
OBJECTIVE COEFFICIENT RANGES
 E
Variable
 4 7.500 8 7.000
Lower Limit
63.000 95.000 1 35.000
Current Value
75.000 126.000 159.000
Upper Limit
S D
No Lower Limit
RIGHT HAND SIDE RANGES
 1 2 3
Constraint
 1 60.000 2 00.000 2080.000
Lower Limit
 200.000 320.000 2400.000
Current Value
280.000 400.000
Upper Limit
No Upper Limit
1
t est
13. Refer to the computer solution of Problem 12 in Figure 3.17. Identify the range of optimality for each objective function coefficient. b Suppose the profit for the economy model is increased by $6 per unit, the profit for . the standard model is decreased by $2 per unit, and the profit for the deluxe model is increased by $4 per unit. What will the new optimal solution be? e. Identify the range of feasibility for the righthandside values. 4. If the number of fan motors available for production, is, U L . _  by 100, will the dual increased price for that constraint change? Explain.
trl
$4. Digital Controls, Inc. (DCI), manufactures two models of a radar gun used by police to
monitor the speed of automobiles. Model A has an accuracy of plus or minus 1 mile per hour, whereas the smaller model B has an accuracy of plus or minus 3 miles per hour. For the next week, the company has orders for 100 units of model A and 150 units of model B. Although DCI purchases all the electronic components used in both models, the plastic cases for both models are manufactured at a DCI plant in Newark, New Jersey. Each model A case requires 4 minutes of injectionmolding time and 6 minutes of assembly time.
Chapter 3
Linear Programming: Sensitivity Analysis and Interpretation of Solution
Each model B case requires 3 minutes of injectionmolding time and 8 minutes of assembly time. For next week, the Newark plant has 600 minutes of injectionmoldingtime available and 1080 minutes of assembly time available. The manufacturing cost is $ 10 per case for model A and $ 6 per case for model B. Depending upon demand and the time available at the Newark plant, DCI occasionally purchases cases for one or both models from an outside supplier in order to fill customer orders that could not be filled otherwise. The purchase cost is $ 14 for each model A case and $ 9 for each model B case. Management wants to develop a minimum cost plan that will determine how many cases of each model should be produced at the Newark plant and how many cases of each model should be purchased. The following decision variables were used to formulate a linear programming model for this problem: AM BM AP BP
= number of cases of model A manufactured = number of cases of model B manufactured = number of cases of model A purchased = number of cases of model B purchased
The linear programming model that can be used to solve this problem is as follows: Min s.t. lOAM 1AM 4 AM 6 AM
3
I
+ 6BM + 14AP + 9 BP
+ + +
IBM 3BM 8BM
I
+ +
1AP
+
= 1 00 I BP = 1 50 5 6 00 5 1 080
Demand for model A Demand for model B Injection molding time Assembly time
A M, BM, AP, BP r o The computer solution developed using The Management Scientistis shown in Figure 3 .18. a. What is the optimal solution and what is the optimal value of the objective function? b. Which constraints are binding? c. What are the dual prices? Interpret each. d. If you could change the righthand side of one constraint by one unit, which one would you choose? Why? 15. Refer to the computer solution to Problem 14 in Figure 3 .18. a. Interpret the ranges of optimality for the objective function coefficients. b. Suppose that the manufacturing cost increases to $ 11.20 per case for model A. What is the new optimal solution? c. Suppose that the manufacturing cost increases to $1 1.20 per case for model A and the manufacturing cost for model B decreases to $ 5 per unit. Would the optimal solution change? Use the 100 percent rule and discuss. 16. Tucker Inc. produces highquality suits and sport coats for men. Each suit requires 1.2 hours of cutting time and 0 .7 hours of sewingtime, uses 6 yards of material, and provides a profit contribution of $ 190. Each sport coat requires 0 .8 hours of cutting time and 0 .6 hours of sewing time, uses 4 yards of material, and provides a profit contribution of $ 150. For the coming week,2 00 hours of cutting time, 180 hours of sewing time, and 1200 yards of fabric are available. Additional cutting and sewing time can be obtained by scheduling overtime for these operations. Each hour of overtime for the cutting operation increases the hourly cost by $ 15, and each hour of overtime for the sewing operation increases the hourly cost
Problems
139
FIGURE 3.18 THE MANAGEMENT SCIENTIST SOLUTION FOR THE DIGITAL
CONTROLS, INC., PROBLEM
Objective Function Value =
2170.000
 AM BM AP BP
Variable
Value
 *_
 0.000 0.000 1.750
Reduced Costs
100.000 60.000 0.000 90.000
Constraint
Dual Prices
3 4
20.000 0.000
OBJECTIVE COEFFICIENT RANGES
AM
]&M
Variable
 No Lower Limit 3.667 12.250 6.000
Lower Limit
 10.000 6.000 14.000 9.000
Current Value
 11.750 9.000 No Upper Limit 11.333
Upper Limit
A P BP
RIGHT HAND SIDE RANGES
 1 2 3 4
Constraint
 0.000 60.000 580.000 600.000
Lower Limit
 100.000 150.000 600.000 1080.000
Current Value
 111.429 No Upper Limit No Upper Limit 1133.333
Upper Limit
by $10. maximum of 1 0 hours of overtime can be scheduled. Marketing requirements A 0 0 specify a minimum production of 1 0 suits and 75 sport coats. Let
S = number of suits produced SC = numlier of sport coats produced Dl = hours of overtime for the cutting operation 0 2 = hours of overtime for the sewing operation
The computer solution developed using The Management Scientist is shown in Figure 3 .19. a. What is the optimal solution, and what is the total profit? What is the plan for the use of overtime?
140
Chapter 3
Linear Programming:Sensitivity Anatysis and Interpretationof Solution
FIGURE 3.19 THE MANAOEMENT SCIENTIST SOLUTION FOP THE TUCKER INC. PROBLEM
ion Value =
40900

100.000 150.000 40.000 0.000
Value
Reduced Costs  0.000 0.000 0.000 10.000
15.000 0.000 34.500 0.000 35.000 0.000
Dual Prices
1
OWECTIVE COEFFICIENT RANGES
Variable
I
No Lower Limit 126.667 187.500 No Lower Limit
Lower Limit
190.000 150.000 15.000 10.000
Current Value

Upper Limit
H Rl
I C )1
225,000 No upper Limit 0 000 0,000
I1
I
I
RIGHT HAND SIDE RANGES :onstraint Lower Limit
140.000 160.000 1000.000 40.000 0.000 No Lower Limit
Current Value
200.000 180.000 1200.000 100.000 100.000 75.000
Upper Limit
240.000 No Upper Limit 1333 333 No Upper Limit 150.000 150.000
2 3 4 5 6 .
b. A price increase for suits is being considered that would result in a profit contribution of $210 per suit. If this price increase is undertaken, how will the optimal solution change? Discuss the need for additional material during the coming week. If a rush order for material can be placed at the usual price plus an extra $8 per yard for handling, would you recommend the company consider placing a rush order for material? What is the maximum price Tucker would be willing to pay for an additional yard of material? How many additional yards of material should Tucker consider ordering? d. Suppose the minimum production requirement for suits is lowered to 75. Would this change help or hurt profit? Explain. 17. The Porsche Club of America sponsors driver education events that provide highperformance driving instruction on actual race tracks. Because safety is a primary considt.
Problems
eration at such events, many owners elect to install roll bars in their cars. Deegan Industries manufactures two types of roll bars for Porsches. Model DRB is bolted to the car using existing holes in the car's frame. Model DRW is a heavier roll bar that must be welded to the car's frame. Model DRB requires 20 pounds of a special high alloy steel, 40 minutes of manufacturing time, and 60 minutes of assembly time. Model DRW requires 25 pounds of the special high alloy steel, 100 minutes of manufacturing time, and 40 minutes of assembly time. Deegan's steel supplier indicated that at most 40,000 pounds of the high alloy steel will be available next quarter. In addition, Deegan estimates that 2000 hours of manufacturing time and 1600 hours of assembly time will be available next quarter. The profit contributions are $200 per unit for model DRB and $280 per unit for model DRW. The linear programming model for this problem is as follows: Max 200DRB + 280DRW s.t. 20DRB + '25DRW I 40,000 Steel available 40DRB + 1 OODRW 5 120,000 Manufacturing minutes 60DRB + 40DRW 5 96,000 Assembly minutes DRB, DRW 2 0 The Management Scientist solution is shown in Figure 3.20. a. What are the optimal solution and the total profit contribution? b. Another supplier offered to provide Deegan Industries with an additional 500 pounds of the steel alloy at $2 per pound. Should Deegan purchase the additional pounds of the steel alloy? Explain. c. Deegan is considering using overtime to increase the available assembly time. What would you advise Deegan to do regarding this option? Explain. d. Because of increased competition, Deegan is considering reducing the price of model DRB such that the new contribution to profit is $175 per unit. How would this change in price affect the optimal solution? Explain. e. If the available manufacturing time is increased by 500 hours, will the dual price for the manufacturing time constraint change? Explain.
18. Davison Electronics manufactures two LCD television monitors, identified as model A and model B. Each model has its lowest possible production cost when produced on Davison's new production line. However, the new production line does not have the capacity to handle the total production of both models. As a result, at least some of the production must be routed to a highercost, old production line. The following table shows the minimum production requirements for next month, the production line capacities in units per month, A d the produEtion cost per unit for each production line.
Model A B Production Line Capacity
Production Cost per Unit New Line Old Line $30 $50 $25 $40 80,000 60,000
. $
pQ
AN = Units of model A produced on the new production line A 0 = Units of model A produced on the old production line Lm _ BN = Units of model B produced on the new production line ', BO = Units of model B produced on the old production line .
3
%
Chapter 3
Linear Programming: Sensitivity Analysis and Interpretation of Solution
FIGURE 3.20 THE MANAGEMENT SCIENTIST SOLUTION FOR THE DEEGAN
'
.
.O..OOQ
.
.
11428.5.714
120000.000 96000.000
160000.000 . Noupper Limit
.
Davison's objective is to determine the minimum cost production plan. The computer solution obtained using The Management Scientist is shown in Figure 3.21. a. Formulate the linear programming model for this problem using the following four constraints: Constraint 1: Minimum production for model A Constraint 2: Minimum production for model B Constraint 3: Capacity of the new production line Constraint 4: Capacity of the old production line Using The Management Scientist solution in Figure 3.21, what is the optimal solution, and what is the total production cost associated with this solution? c. Which constraints are binding? Explain. d. The production manager noted that the only constraint with a positive dual price is the constraint on the capacity of the new production line. The manager's interpretation of the dual price was that a oneunit increase in the righthand side of this constraint would actually increase the total production cost by $15 per unit. Do you agree with this
b.
Problems
143
FIGURE 3.21 THE MANAGEMENT SCIENTIST SOLUTION TO THE DAVISON ELECTRONICS PROBLEM
OPTIMAL SOLUTION Objective Function Value =
3 850000.000
 AN
Variable
 50000.000 0.000 30000.000 40000 . OOO
Value
 0.000 5.000 0.000 0 .000
Reduced Costs
A0 BN
BO
 1 2 3 4
Constraint
0 .000 0.000 0.000 20000.000
Slack/Surplus
45.000 40.000 15.000 0.000
Dual Prices
OBJECTIVE COEFFICIENT RANGES
 AN
Variable
Lower Limit
 30.000 50.000 25.000 40.000 35.000
Current Value
Upper Limit
A0 BN BO
 15.000 4 5.000 2 0.000 2 5.000
No Upper Limit
40.000 45.000
RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit
4
40000.000
60000.000
No Upper Limit
interpretation? Would an increase in capacity for the new production line be desirable? Explain. e. Would you recommend increasing the capacity of the old production line? Explain. f. The production cost for model A on the old production line is $50 per unit. How much would this cost have to change to make it worthwhile to produce model A on the old production line? Explain. g. Suppose that the minimum production requirement for model B is reduced from 70,000 units to 60,000 units. What effect would this change have on the total production cost? Explain.
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Case Problem R. C. COLEMANR. C. Coleman distributes a variety of food products that are sold through grocery store and supermarketoutlets. The company receives orders directly from the individual outlets, with a typical order requesting the delivery of s
Langara  MATH  2310
A SIMPLE MAXIMIZATION PROBLEM (text pp.3334) Par, Inc., is a small manufacturer of golf equipment and supplies whose managementhas decided to move into the market for medium and highpriced golf bags. Par's distributor is enthusiastic about the new pro
Langara  MATH  2310
532INTRODUGION TO MANAGEMENT SCIENCELeadtime demand Cycle timeThe number of units demanded during the leadtime period.The length of time between the placing of two consecutive orders.Constant supply rate A situation in which the inventory is built
Langara  MATH  2310
Case Problem 1 WAGNER FABRICATING COMPANYManagers at Wagner Fabricating Company are reviewing the economic feasibility of manufacturing a part that it currently purchases from a supplier. Forecasted annual demand for the part is 3200 units. Wagner operat
Langara  MATH  2310
Chapter 14Decision Analysis687PROBLEMS1. The following payoff table shows profit for a decision analysis problem with two decision alternatives and three states of nature.State of Nature Decision Alternativeddz8 250 1008283100 10025 75a. b.C
Langara  MATH  2310
PLEASE NOTE THE FOLLOWING: NOTHING ON YOUR TABLE EXCEPT A HELPSHEET AND CALCULATOR. ONLY PAPER DICTIONARY IS ALLOWED. SWITCH OFF AND REMOVE YOUR CELL PHONE SANCTION: YOU WILL BE REMOVED FROM THE EXAM ROOM AND FAIL THE COURSE WRITE YOUR NAME ON YOUR
Langara  MATH  2310
Langara  MATH  2310
Langara  MATH  2310
Langara  MATH  2310
Langara  MATH  2310
Langara  MATH  2310
Langara  MATH  2310
Langara  MATH  2310
Langara  MATH  2310
Langara  MATH  2310
Langara  MATH  2310
Langara  MATH  2310
Langara  MATH  2310
Langara  MATH  2310
Langara  MATH  2310
UIllinois  STAT  430
Stat 430/Math 468 Notes #1Aspects of Multivariate AnalysisIntroductionChapter 1We will analyze the data which include simultaneous measurement on many variables, and the methodology is called multivariate analysis. We will try to provide explanations
UIllinois  STAT  430
Stat 430/Math468 Notes #2Matrix Algebra and Random VectorsChapter 2SAS code for histogram, normality test and normal QQ plot (dataset is in table 1.5 on page 39.).options formdlim="*"; data table1_5; infile 'F:\teachingS2010\T15.dat'; input wind sol
UIllinois  STAT  430
Stat 430/Math468 Notes #3Matrix Algebra and Random Vectors (continued)Chapter 2Orthogonal Matrix The square matrix Q is called an orthogonal matrix if Q ' Q = QQ ' = I , where I is the identity matrix of the same dimension. Remark: 1. From the definiti
UIllinois  STAT  430
Stat 430/Math468 Notes #4Random Vectors and MatricesChapter 2A random vector (or matrix) is a vector (or matrix) whose elements are random variables. Suppose X = ( X 1 ,., X p ) ' is a random vector with joint PDF/PMF f ( x1 ,., x p ) and each element
UIllinois  STAT  430
Stat 430/Math468 Notes #5Sample Geometry and Random SamplingChapter 3A single multivariate observation is the collection of measurements of p different variables taken on the same item or trial. If n observations have been obtained, the entire data set
UIllinois  STAT  430
Stat 430/Math468 Notes #6The Multivariate Normal DistributionChapter 4Univariate Normal Distribution Recall the univariate normal distribution N ( , 2 ) has probability density function (PDF) 1 f ( x) = e 2 ( x )2 2 2=1 2 ( 2 ) 21e1 ( x )( 2 )1 (
UIllinois  STAT  430
Stat 430/Math468 Notes #7The Multivariate Normal Distribution (Continued)Chapter 4If a random vector X has multivariate normal distribution, then it has the following properties: 1. Linear combinations of the components of X are normally distributed. 2
UIllinois  STAT  430
Stat 430/Math468 Notes #8The Multivariate Normal Distribution (Continued)Chapter 4Result: (Conditional distribution) Suppose X ~ N p ( , ) . Make the following 12 partitioning X = X1 , = 1 , and = 11 . Assume that  21 22 2 X2 conditional distribution
UIllinois  STAT  430
Stat 430/Math468 Notes #10Inference about a Mean VectorChapter 5Suppose X1 ,., Xn are random sample from a normal population N p (, ) . The sample mean and sample variancecovariance matrix are, respectively,X = 1 ( X1 + X 2 + . + X n ) and S = n1 n
UIllinois  STAT  430
Stat 430/Math 468 Notes #11Chapter 5: Inference about a Mean Vector (Continued)Simultaneous Confidence Interval for a ' (i.e. for ai i )i =1 pChapters 5, 8Result: Let X1 ,., Xn be random sample from a normal population N p (, ) . The sample mean and
CUNY Hunter  ECONOMICS  100
INTRODUCTION TO ECONOMICS/ECO 100 Professor Timothy J. Goodspeed Hunter College Spring 2010 Office: Hunter West 1527, phone 7725434 Graduate Center, Room 5306 Office Hours: Hunter: Mon. and Thurs: 11:30 12:30 Graduate Center: Tuesday: 10:3011:30 Email:
Los Angeles City College  BIO  1291
BIOL 1202 Biology for Science Majors II Spring 2010 Section 001 102 Williams Hall M W F 1:40 2:30pm Purpose and content of course This course covers the basic information, concepts and methods of modern biological science. It is intended for students in s
Los Angeles City College  BIO  1291
Chapter Learning Objectives Chapter 22: Descent with Modification: a Darwinian View of Life Define evolution and adaptation.Concept 22.1: The Darwinian revolution challenged traditional views of a young Earth inhabited by unchanging species Describe the
Los Angeles City College  BIO  1291
What is evolution and adaptation? Two main ideas 1. Change over time of the genetic compositions of a populations 2. Decent of modern organisms with modification from preexisting organisms Evolutionary adaptation Accumulation of inherited characteristics
Los Angeles City College  BIO  1291
ENGL 2000: Writing from the MoviesSection 160 T Th 1:403:00 Instructor: Al Dixon Office: 26 Allen Hall Spring 2010 226 Tureaud adixon@lsu.edu Office hours: T Th 9:3010:30 and by appointmentEnglish 2000 builds upon the skills emphasized in English 1001.
Kean  MBA  5785
Chengyan Liu Professor Afriyie Kofi GMBA5785 March 1, 2009 Case1: Cultural Intelligence A training program for American ManagersCultural intelligence (CQ) is more and more important in today's global business world and has become the key factor to the su
Clarkson  STAT  383
Clarkson  STAT  383
Clarkson  STAT  383
Name: Assignment: 1Reading Read sections 1.1 through 1.5 in your book. Suggested Problems 1. Section 1.2: 13,6, 8, 1016 2. Section 1.3: 1921, 2324, 26 3. Section 1.4: 3032, 34, 40STAT 383Section: 1 Date: 15 Jan 2010Instructions:1 This set of prob
Clarkson  STAT  383
Name: Assignment: 2Reading Read sections 2.1 through 2.2 in your book. Suggested Problems 1. Section 2.1: 13, 6, 8, 10 2. Section 2.2: 1517, 20, 24, 26STAT 383Section: 12 Date: 22 Jan 2010Instructions:1 This set of problems is not due. 2 I encoura
Clarkson  STAT  383
Clarkson  STAT  383
Name: Assignment: 3Reading Read sections 2.3, 2.4, and 3,1 in your book. Suggested Problems 1. Section 2.3: 32, 33, 34, 36 2. Section 2.4: 44, 45, 46 3. Section 3.1: 1, 2, 4, 8STAT 383Section: 12 Date: 29 Jan 2010Instructions:1 This set of problems
Clarkson  STAT  383
Clarkson  STAT  383
Name: Assignment: 4Reading Read chapter 4 in your book. Suggested Problems 1. Section 3.2: 9, 10, 12 2. Section 3.3: 18, 20 3. Section 5.1: 14STAT 383Section: 12 Date: 5 Feb 2010Instructions:1 This set of problems is not due. 2 I encourage you to w
Clarkson  STAT  383
Clarkson  STAT  383
Name: Assignment: 5Reading Read chapter 5 in your book. Suggested Problems 1. Section 5.2: 8, 10 2. Section 5.3: 12, 14, 16,21 3. Section 5.4: 25, 28, 30, 32STAT 383Section: 12 Date: 12 Feb 2010Instructions:1 This set of problems is not due. 2 I enc
Clarkson  STAT  383