Posted: September 13th, 2017

Linear Programming

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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