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Formulating Linear Programming Models and Solution Concepts | MATH 251, Study notes of Operational Research

Material Type: Notes; Class: Prin of Operations Research; Subject: Mathematics; University: Saint Mary's College; Term: Unknown 1989;

Typology: Study notes

Pre 2010

Uploaded on 08/05/2009

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MATH 251 ACTIVITY 1: Formulating Linear Programming models, Solution concepts
WHY: Linear programming is the best-known and most commonly-used mathematical programming tools. The business
of writing software to implement solution methods is, itself, a million-dollar industry, and research for better
algorithms continues [Bell laboratories has copyrighted one]. The human part of this involves identifying and
setting up the models and interpreting the results of computations. Here we begin with some small examples.
VOCABULARY:
solution
feasible solution
feasible region
extreme point
boundary point
interior point
optimal solution
LEARNING OBJECTIVES:
1. Work as a team, using the team roles.
2. Be able to determine the variables and objective, and write the constraints, for a problem that fits the linear programming
model.
3. Understand the meanings of the feasible region for a linear programming problem
4. Understand why the optimal solution is at an extreme point of the feasible region
CRITERIA:
1. Success in completing the exercises.
2. Success in working as a team
3. Understanding of the material by all team members
RESOURCES:
1. Your text sections 2.2 – 2.3 [Setup example and graphical solution]
2. The two worked examples in the document “Setup and graphical solution of Linear Programming Problems [2-variables]”
available on the Blackboard site
3. 50 minutes
PLAN:
1. Select roles, if you have not already done so, and decide how you will carry out steps 2 and 3 (5 minutes)
2. Work through the exercises given below – you will submit one (team) copy of the work, with the usual reports [see the
syllabus]
3. Assess the team's work and roles performances and prepare the Reflector's and Recorder's reports including team grade (5
minutes).
4 Be prepared to discuss your results.
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MATH 251 ACTIVITY 1: Formulating Linear Programming models, Solution concepts WHY: Linear programming is the best-known and most commonly-used mathematical programming tools. The business of writing software to implement solution methods is, itself, a million-dollar industry, and research for better algorithms continues [Bell laboratories has copyrighted one]. The human part of this involves identifying and setting up the models and interpreting the results of computations. Here we begin with some small examples. VOCABULARY: solution feasible solution feasible region extreme point boundary point interior point optimal solution LEARNING OBJECTIVES:

  1. Work as a team, using the team roles.
  2. Be able to determine the variables and objective, and write the constraints, for a problem that fits the linear programming model.
  3. Understand the meanings of the feasible region for a linear programming problem
  4. Understand why the optimal solution is at an extreme point of the feasible region CRITERIA:
  5. Success in completing the exercises.
  6. Success in working as a team
  7. Understanding of the material by all team members RESOURCES:
  8. Your text sections 2.2 – 2.3 [Setup example and graphical solution]
  9. The two worked examples in the document “Setup and graphical solution of Linear Programming Problems [2-variables]” available on the Blackboard site
  10. 50 minutes PLAN:
  11. Select roles, if you have not already done so, and decide how you will carry out steps 2 and 3 (5 minutes)
  12. Work through the exercises given below – you will submit one (team) copy of the work, with the usual reports [see the syllabus]
  13. Assess the team's work and roles performances and prepare the Reflector's and Recorder's reports including team grade ( minutes). 4 Be prepared to discuss your results.

EXERCISES:

  1. Create the Linear Programming model for Problem #4 on p. 102 in your text. [Define variables, write the objective, determine “Max or min”, write the constraints. You need to calculate the profit per keyboard for each type – but you can check your result (you should show how you get it) by reading part a of the problem.]. Do not solve this problem
  2. Problem # 7 on p. 103 of your text [including data in part c] leads to the LP model given here. A.) graph the feasible region B.) By graphing the appropriate lines, show that a total (daily) profit of $4000 is possible – in fact, there are many feasible combinations that give this profit - and a total daily profit of $7000 is infeasible (impossible). C.) Use the graphical method to find the optimal solution – the most profitable feasible combination. What is the best possible profit?. X 1 = dozens of baseballs produced X 2 = dozens of softballs produced Objective: [maximize] z = 7 X 1 + 6 X 2 Subject to constraints 5 X 1 + 6 X 2 ≤ 3600 [cowhide] X 1 + 2 X 2 ≤ 960 [time available] X 1 , X 2 ≥ 0 Since this is “production per day” fractional values make sense – production of a dozen balls can be carried over & finished the next day. CRITICAL THINKING QUESTIONS: (answer individually in your journal)
    1. If Wilson Manufacturing (exercise 2) has learns that they can only ship up to 700 dozen balls (total – both types) a day, will this affect their best production plan? What if they learn they can only ship up to 600 dozen balls a day? How can you tell from the graph?
    2. What advantages are there in having different roles within the teams. What disadvantages?
    3. What advantages are there to having the roles rotate through the teams? Disadvantages? SKILL EXERCISES: (This is the assignment due next class) Text p. 101: #2(a only), 3, 10, 11(but read available cloth as 4200 linear feet, or problem can’t be set up]