Unformatted text preview: 8/13/2010 Module 6: Linear Programming Applications
Diet Design
Investment Portfolio (2.25)
Make vs. Buy (3.14)
Work Scheduling (4.8) In Live Classroom Diet Design
A student wishes to design the lowest cost diet from the following basic food groups: D.V Food Cost X1
X2
X3
X4 Brownies
Chocolate ice cream
Soda
Cheesecake $.50/each
$.20/scoop
$.30/bottle
$.80/slice The diet requires at least the following per day:
500 cal
6 oz chocolate
10 oz sugar
8 oz fat Diet Design
Given the following amounts of nutrients and minimal daily requirements how many servings of each food will minimize the cost per day to eat?
Food Calories Chocolate
(oz) Sugar
(oz) Fat
(oz) Brownie
Chocolate Ice
Cream (per scoop) 400 3 2 2 200 2 2 4 Soda
(per bottle)
Cheesecake
(per slice)
Minimal
Requirement 150 0 4 1 500 0 4 5 500 6 10 8 1 8/13/2010 Example Problem, 2.25 Investment Analysis: George Johnson recently inherited a large sum of money; he wishes to use some of this money to set up a trust fund for his children. The trust fund has two investments: a bond fund and a stock fund. The estimated returns over the life of the investment are 6% for the bond fund and 10 % for the stock fund. Whatever portion of the inheritance he decides to commit to the trust fund, he wishes to invest at least 30% in the bond fund. In addition he wishes to have an investment mix that will provide a overall return of at least 7.5%
(a) Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of the possible investment alternatives.
(b) Solve the LP and determine the optimal value of the objective Investment Analysis
Max R(B,S) = 0.06 B+ 0.10 S s.t
.
C.1 B ≥ 0.3 Bond fund minimum
C.2 0.06 B+ 0.10 S ≥ 0.075 Minimum total return
C.3 B + S = 1 Percentage requirement SOLUTION TO INVESTMENT PROBLEM
MAX 0.06B+0.1S S.T.
1) 1B≥.30
2) 0.06B+.1S≥.075
3) 1B+1S=1
OPTIMAL SOLUTION
Objective Function Value = 0.088
Variable Value Reduced Costs B 0.300
0.000
S 0.700 0.000
Constraint Slack/Surplus Dual Prices 1 0.000 ‐0.040
2 0.013 0.000
3 0.000 0.100
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
B No Lower Limit 0.060 0.100
S 0.060 0.100 No Upper Limit
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
1 0.000 0.300 0.625
2 No Lower Limit 0.075 0.088
3 0.870 1.000 No Upper Limit 2 8/13/2010 Make vs. Buy Problem‐ Formulation and MS Software Output Interpretation
Digital Controls Inc. makes two models of a radar gun used by police to monitor the speed of automobiles. Model A has an accuracy of 1 mile per hour (mph) and model B has an accuracy of 3 mph. 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 it manufactures the plastic cases for both models at a DCI plant in Newark N.J. Each Model A case requires 4 minutes of injection molding time and 6 minutes of assembly time. Each Model B case requires 3 minutes of injection molding time and 8 minutes of assembly time. For the next week the Newark plant has 600 minutes of molding time available and 1080 minutes of assembly time. The manufacturing cost is $10 per case for model A and $6/case for model B. Depending on the availability of molding time and assemble time DCI occasionally purchases cases for one or both models from an outside supplier in order to meet the demand. The purchase cost for model A is $14 per case and $9 per case for model B. Management wishes to develop a plan to determine the number of plastic cases to make and the number to purchase from the outside supplier. The decision variables and the linear programming model are as follow: Work Scheduling The Clarke County sheriff’s department schedules police officers for 8 hours shifts. The beginning time for the shifts are 8:00am, 12:00noon, 4:pm, 8:00pm and 12:00 midnight. An officer beginning a shift works for the next 8 hours. During normal weekday operations the number of officers on duty varies with the time of day. The staffing guidelines require the following minimum officers on duty during the given time intervals:
8:00AM‐‐‐‐‐Noon
5 officers
Noon‐‐‐‐‐‐‐‐4:00PM 6
4:00PM‐‐‐‐‐8:00PM 10
8:00PM‐Midnight 7
Midnight—4:00AM 4
4:00AM‐‐‐8:00Am 6
Determine the number of officers to be scheduled for each 8‐hr shift to minimize the total offers on duty(which will minimize cost) Use X1‐‐‐ X6 for your decision variables. Work Scheduling Formulation 3 ...
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 Spring '10
 J.LOUTEN
 ice cream, brownie, Chocolate brownie, Ice cream soda

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