Hi Good Morning, Greetings, Linear Programming is an optimization technique for
ID: 372923 • Letter: H
Question
Hi Good Morning,
Greetings,
Linear Programming is an optimization technique for a system of linear constriants and a linear objective function. An objective function defines the quantity to be optimized, and the goal of linear programming is to find the values of the variables that maximize or minimize the objective function.
Problem:
Roscoe Davis, Chairman of a college’s business department, has decided to apply a new method in criterion for judging who should teach each course, Professor Davis reviews the past two years teaching evaluations (which were filled out by students). Since each of the four professors taught each of the four courses at one time or another during the two-year period, Davis is able to record a course rating for each instructor. These ratings are shown in the table at top of next column. Find the best assignment of professors to courses to maximize the overall teaching rating.
Professor
Statistics
Management
Finance
Economics
Anderson
90
65
95
40
Sweeney
70
60
80
75
Williams
85
40
80
60
Mckinney
55
80
65
55
Solution:
Process to formulate a Linear Programming problem
Let us look at the steps of defining a Linear Programming problem generically:
For a problem to be a linear programming problem, the decision variables, objective function and constraints all have to be linear functions.
If the all the three conditions are satisfied, it is called a Linear Programming Problem.
Step:1 Identify the decision variables.
Decision variables are Professors here. The decision variables are the variables which will decide my output.
Professor
Statistics
Management
Finance
Economics
Anderson
90
65
95
40
Sweeney
70
60
80
75
Williams
85
40
80
60
McKinney
55
80
65
55
Step: 2
Objective function is Z. It is defined as the objective of making decisions. In the above example, the Chaiman wishes to increase the overall technical rating represented by Z. So, overall technical rating is my objective function.
Step:3
Constraints are given below.
Statistics
Management
Finance
Economics
90
65
95
40
70
60
80
75
85
40
80
60
55
80
65
55
Step: 4 Explicit state non-negativity restriction
Professors are having more than the rating expected and greater than or equal to zero rating.
Here cost(C0, C1, C2, C3) are considered as Rating Factors of each professor is considered as below
C0
C1
C2
C3
50
30
20
15
Constraint matrix is given below.
Statistics
Management
Finance
Economics
90
65
95
40
70
60
80
75
85
40
80
60
55
80
65
55
Limit is more than one, that is, each of the four professors taught each of the four courses at one time or another during the two-year period
Cost factor(Rating)
Statistics
Management
Finance
Economics
0
90
65
95
40
0
70
60
80
75
0
85
40
80
60
0
55
80
65
55
Zj-Cj
-50
-30
-20
-15
Cost factor(Rating)
Statistics
Management
Finance
Economics
50
1
0.7
1
0.4
0
0
9
6
44
0
0
21
10
0
0
0
40
7
0
Zj-Cj
0
6
33
0
Result:
Optimal finite solution is found. Best assignment of professors to courses to maximize the overall teaching rating based on Anderson professor’s rating in Statistics subject. Chairman can increase the rating based on the Anderson Professors progress for last two years.
Professor
Statistics
Management
Finance
Economics
Anderson
90
65
95
40
Sweeney
70
60
80
75
Williams
85
40
80
60
Mckinney
55
80
65
55
Explanation / Answer
Formulate problem 9-26 as a linear programming problem and solve by computer
istance Wtal distance with the dis- alite Touhd in Problem 9-24 of a college's business de- method in partment, has decided to apply a new assigning professors to courses next semester. As a criterion for judging who should teach each course, Professor Davis reviews the past two years' teaching evaluations (which were filled out by students). Since each of the four professors taught each of the four courses at one time or another during the two-year poe- riod, Davis is able to record a course rating for each instructor. These ratings are shown in the table at top of next column. Find the best assignment of professors to courses to maximize the overall teaching rating. 2:9-26 Ros