1. (20 points) Linear Programming can be used to solve very large problems with thousands of variables and constraints. Smaller problems can be easily solved with Excel, which is available on virtually all desktop computers with Microsoft Office or with online application tools. Good linear programming formulations represent as much of an art as a science.
A) Create a real world scenario and develop a linear program problem (max or min) explaining in detain what you are trying to accomplish. Your model must include at least three constraints (excluding the nonlinearity constraints) and two variables. (3 points)
B) Explain the meaning of the numbers on the right hand side of your constraints. (3 points)
C) Explain the significance of the coefficients in your objective function. (3 points)
D) Solve your problem graphically and show the critical region along with the corner points. Indicate the value of the objective function at each corner point. Identify which corner point is optimal. (5 points)
E) Explain the meaning of your solution. (3 points)
F) Increase the value of your first variable in your objective function by 50%. Does this change your optimal solution? Explain why the increase did or did not change your optimal solution. (3 points)

2. (20 points)St. John’s hospital is a small hospital in Lakeview, FL. St. John’s began a new process to make sure patients receive the hottest meals possible.St. John’s food service manager plans to deliver meals in bulk to three recently installed serving stations in the hospital. From the serving stations, the food will be reheated, and meals will be placed on individual trays, placed on a cart, and served to the patients on the different floors and wings of the hospital.
The three serving stations have been strategically placed throughout the hospital for optimal efficiency. The table below presents the location name and the meal capacity at each location.
Station Location Meal Capacity
SL1 225
SL2 225
SL3 250
Saint John’sHospital has six wings with the number of patients each as follows:
Wing 1 Wing 2 Wing 3 Wing 4 Wing 5 Wing 6
Patients   100      100       150       200         70          80
Saint John’sHospital wants to increase the temperature of the meals served to patients. Thus, the amount of time needed to deliver a tray from a serving station to a patient determines the proper distribution of food to each wing. The time associated with each possible distribution from station to wing is listed in the table below.
Delivery Time in Minutes
To:
Wing 1 Wing 2 Wing 3 Wing 4 Wing 5 Wing 6
From:
5A                       12           11          8           9             9            6
3G                        6              12         7          7             5            8
1S                         8                9           6         6             7             9
What is your recommendation for handling the distribution of trays from the three serving stations (Show all work for maximum credit)? (20 points)

3. (50 points) Lyndon’s custom design shop is adding a new product line. The advertisements and announcements will go out in two weeks. At that time, he will expect his customers to request the new product. However, his work area and shop must be redesigned to meet his needs. A number of activities must be accomplished to complete the redesign of the work area and shop.
The table below lists the activities about his project:
ACTIVITY IMMEDIATE PREDECESSOR TIME (hours)
a m b

ACTIVITY IMMEDIATE PREDECESSOR TIME (hours)
a m b

A                  –              9                   11                       13
B                   –              5                    7                          9
C                    –             4                    6                          8
D                   A           10                 15                        20
E                   C               6                   7                         8
F                   B                8                  10                      12
G                   D,F            5                   7                          9
H                   F ,E            13              17                      21
I                       D              9                 11                         13
J                    G,H             5                  7                           9
K                       I                4                 6                           8
L                      J,K            1                   3                           5
A) Construct a network for this problem. (5 points)Hint: consider constructing in MS Power Point or some other tool and paste into your submission document.

B) Determine the expected time and variance for each activity.(5 points)(Complete and copy table into your submission.)
ACTIVITY Expected (Time in Hours) Variance
A
B
C
D
E
F
G
H
I
J
K
L
C) Determine the ES, EF, LS, LF, and slack for each activity. (5 points)(Complete and copy table into your submission.)
ACTIVITY ACTIVITY TIME
ES EF LS LF Slack
A
B
C
D
E
F
G
H
I
J
K
L
D) Determine the critical path and project completion time. (10 points)

E) Determine the probability that the project will be done in 44 hours or less.(2 points)
F) Determine the probability that the project will be done in 52 hours or less.(2 points)

G) Determine the probability that the project will be done in 54 hours or less. (2 points)

H) Determine the probability that the project will be done in 41 hours or less.(2 points)

I) Based on the information provided and the calculations done, make a final report for Lyndon that explains to him the situation at hand. (17 points)

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1. The Scrod Manufacturing Co. produces two key items – special-purpose Widgets (W) and more generally useful Frami (F).
Management wishes to determine that mix of W & F which will maximize total Profits (P).

Data W F

Unit profit contributions $ 30 $ 20

Demand estimates (unit/week) 250 500

Average processing rates – each
product requires processing on
both machines (units/hour)

Machine #1 2 4

Machine #2 3 3
The two products compete for processing time using the same limited plant capacity. Only 160 hours are available on each of two machines (1 and 2) during each week, barring unexpected equipment breakdowns. Management has established a desired minimum production level of 200 units per week (total output: W + F) in order to maintain distribution outlets.

As a newly hired management analyst for Scrod, you have been asked to analyze the available options and recommend an appropriate product mix. Your boss has suggested that you structure a model of the underlying constrained optimization problem and test to be sure that a feasible solution exists before proceeding to analyze the alternatives. (You do not have to solve this problem; just set it up and make sure that a feasible solution exists. You should try this both with and without the demand estimates included as constraints).
2. The Ace Manufacturing Company produces two lines of its product, the super and the regular. Resource requirements for production are given in the table. There are 1,600 hours of assembly worker hours available per week, 700 hours of paint time, and 1200 hours of inspection time. Regular customers will demand at least 150 units of the regular line and at least 90 of the super line.
Profit Assembly Paint Inspection
Product Line Contribution time (hr.) time (hr.) time (hr.)

Regular 50 1.2 .8 1.5

Super 75 1.6 .5 .7

a) Formulate an LP model which the Ace Company could use to determine the optimal product mix on a weekly basis. Use two decision variables (units of regular and units of super). Suggest any feasible solution and explain what “feasible solution” means.

b) Find the optimal solution by using the graphical solution technique. What is the value of the objective function? What are the values of all variables?

c) By how many units can the demand for the super product increase before the optimal intersection point changes? Explain. For the regular product?

d) How much would it be worth to the Ace Company if it could obtain an additional hour of paint time? Of assembly time? Of inspection time? Explain fully. Show all calculations.

e) Find the upper and lower bounds for assembly time by identifying the corner points on either end of the line and substituting these points into the assembly equation. What do these bounds mean? Explain.

f) Solve this problem with LINDO or POM and verify that your answers are correct.
3. Matchpoint Company produces 3 types of tennis balls: Heavy Duty, Regular, and
Extra Duty, with a profit contribution of $24, $12, and $36 per gross (12 dozen),
respectively.
The linear programming formulation is:

Max. 24×1 + 12×2 + 36×3

Subject to: .75×1 + .75×2 + 1.5×3 < 300 (manufacturing)

.8×1 + .4×2 + .4×3 < 200 (testing)

x1 + x2 + x3 < 500 (canning)

x1, x2, x3 > 0

where x1, x2, x3 refer to Heavy Duty, Regular, and Extra Duty balls (in gross). The LINDO solution is on the following page.

a) How many balls of each type will Matchpoint product?
b) Which constraints are limiting and which are not? Explain.
c) How much would you be willing to pay for an extra man-hour of testing capacity? For how many additional man-hours of testing capacity is this marginal value valid? Why?
d) By how much would the profit contribution of Regular balls have to increase to make it profitable for Matchpoint to start producing Regular balls?
e) By how much would the profit contribution of Heavy Duty balls have to decrease before Matchpoint would find it profitable to change its production plan?
f) Matchpoint is considering producing a low-pressure ball, suited for high altitudes, called the Special Duty. Each gross of Special Duty balls would require 1 ½ and ¾ man-hours of manufacturing and testing, respectively, and would give a profit contribution of $33 per gross. Special Duty balls would be packed in the same type of cans as the other balls.

Should Matchpoint produce any of the Special duty balls? Explain; provide support for
your answer.

Max 24×1 + 12×2 + 36×3
Subject to
.75×1 + .75×2 + 1.5×3 <300
.8×1 + .4×2 + .4×3 <200
x1 + x2 + x3 < 500
end

LP OPTIMUM FOUND AT STEP 2

OBJECTIVE FUNCTION VALUE

1) 8400.000

VARIABLE VALUE REDUCED COST
X1 200.000000 0.000000
X2 0.000000 8.000000
X3 100.000000 0.000000
ROW SLACK OR SURPLUS DUAL PRICES
2) 0.000000 21.333334
3) 0.000000 10.000000
4) 200.000000 0.000000

NO. ITERATIONS= 2
RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X1 24.000000 48.000000 6.000000
X2 12.000000 8.000001 INFINITY
X3 36.000000 12.000000 24.000000

RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 300.000000 450.000000 112.500000
3 200.000000 120.000000 120.000000
4 500.000000 INFINITY 200.000000

1. The Scrod Manufacturing Co. produces two key items – special-purpose Widgets (W) and more generally useful Frami (F).
Management wishes to determine that mix of W & F which will maximize total Profits (P).

Data W F

Unit profit contributions $ 30 $ 20

Demand estimates (unit/week) 250 500

Average processing rates – each
product requires processing on
both machines (units/hour)

Machine #1 2 4

Machine #2 3 3
The two products compete for processing time using the same limited plant capacity. Only 160 hours are available on each of two machines (1 and 2) during each week, barring unexpected equipment breakdowns. Management has established a desired minimum production level of 200 units per week (total output: W + F) in order to maintain distribution outlets.

As a newly hired management analyst for Scrod, you have been asked to analyze the available options and recommend an appropriate product mix. Your boss has suggested that you structure a model of the underlying constrained optimization problem and test to be sure that a feasible solution exists before proceeding to analyze the alternatives. (You do not have to solve this problem; just set it up and make sure that a feasible solution exists. You should try this both with and without the demand estimates included as constraints).
2. The Ace Manufacturing Company produces two lines of its product, the super and the regular. Resource requirements for production are given in the table. There are 1,600 hours of assembly worker hours available per week, 700 hours of paint time, and 1200 hours of inspection time. Regular customers will demand at least 150 units of the regular line and at least 90 of the super line.
Profit Assembly Paint Inspection
Product Line Contribution time (hr.) time (hr.) time (hr.)

Regular 50 1.2 .8 1.5

Super 75 1.6 .5 .7

a) Formulate an LP model which the Ace Company could use to determine the optimal product mix on a weekly basis. Use two decision variables (units of regular and units of super). Suggest any feasible solution and explain what “feasible solution” means.

b) Find the optimal solution by using the graphical solution technique. What is the value of the objective function? What are the values of all variables?

c) By how many units can the demand for the super product increase before the optimal intersection point changes? Explain. For the regular product?

d) How much would it be worth to the Ace Company if it could obtain an additional hour of paint time? Of assembly time? Of inspection time? Explain fully. Show all calculations.

e) Find the upper and lower bounds for assembly time by identifying the corner points on either end of the line and substituting these points into the assembly equation. What do these bounds mean? Explain.

f) Solve this problem with LINDO or POM and verify that your answers are correct.
3. Matchpoint Company produces 3 types of tennis balls: Heavy Duty, Regular, and
Extra Duty, with a profit contribution of $24, $12, and $36 per gross (12 dozen),
respectively.
The linear programming formulation is:

Max. 24×1 + 12×2 + 36×3

Subject to: .75×1 + .75×2 + 1.5×3 < 300 (manufacturing)

.8×1 + .4×2 + .4×3 < 200 (testing)

x1 + x2 + x3 < 500 (canning)

x1, x2, x3 > 0

where x1, x2, x3 refer to Heavy Duty, Regular, and Extra Duty balls (in gross). The LINDO solution is on the following page.

a) How many balls of each type will Matchpoint product?
b) Which constraints are limiting and which are not? Explain.
c) How much would you be willing to pay for an extra man-hour of testing capacity? For how many additional man-hours of testing capacity is this marginal value valid? Why?
d) By how much would the profit contribution of Regular balls have to increase to make it profitable for Matchpoint to start producing Regular balls?
e) By how much would the profit contribution of Heavy Duty balls have to decrease before Matchpoint would find it profitable to change its production plan?
f) Matchpoint is considering producing a low-pressure ball, suited for high altitudes, called the Special Duty. Each gross of Special Duty balls would require 1 ½ and ¾ man-hours of manufacturing and testing, respectively, and would give a profit contribution of $33 per gross. Special Duty balls would be packed in the same type of cans as the other balls.

Should Matchpoint produce any of the Special duty balls? Explain; provide support for
your answer.

Max 24×1 + 12×2 + 36×3
Subject to
.75×1 + .75×2 + 1.5×3 <300
.8×1 + .4×2 + .4×3 <200
x1 + x2 + x3 < 500
end

LP OPTIMUM FOUND AT STEP 2

OBJECTIVE FUNCTION VALUE

1) 8400.000

VARIABLE VALUE REDUCED COST
X1 200.000000 0.000000
X2 0.000000 8.000000
X3 100.000000 0.000000
ROW SLACK OR SURPLUS DUAL PRICES
2) 0.000000 21.333334
3) 0.000000 10.000000
4) 200.000000 0.000000

NO. ITERATIONS= 2
RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X1 24.000000 48.000000 6.000000
X2 12.000000 8.000001 INFINITY
X3 36.000000 12.000000 24.000000

RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 300.000000 450.000000 112.500000
3 200.000000 120.000000 120.000000
4 500.000000 INFINITY 200.000000