IE003+lab

Course Code: Course Title: Section: Members:

Laboratory Exercise No 1 Basic Linear Programming Program: Date Performed: Date Submitted: Instructor:

1. Objective(s):

The activity aims to formulate linear programming maximization problems using Lindo Software. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: 2.1 identify requirements in solving maximization problems of LP. 2.2 List down the procedures/steps in soling LP maximization problem using Lindo software. 2.3 interpret the results provided by Lindo software. 3. Discussion:

The word “linear” implies direct proportionality of relationship of variable. “Programming” means making schedules or plans of activities to undertake in the future. “Linear Programming” therefore is planning by the use of linear relationship of variables involved. It makes use of certain mathematical techniques to get the best possible solution to a problem involving limited resources. There are two ways of solving a linear programming problem: By the graphical and by the simplex method. The graphical method can only be used if the problem has two or three variable, since there are only two coordinate axis in a plan and three coordinates in space. The simplex method can handle a problem having any number of variables.

4. Resources:

Lindo Software Textbooks 1

5. Procedure:

Problem 1: Assume that a small machine shop manufactures two models, standard and deluxe. Each standard model requires two hours of grinding and four hours of polishing; each deluxe module requires five hours of grinding and two hours of polishing. The manufacturer has three grinders and two polishers. Therefore in 40 hours week there are 120 hours of grinding capacity and 80 hours of polishing capacity. There is a contribution a contribution margin of $3 on each standard model and $4 on each deluxe model. To maximize the total contribution margin, the management must decide on: 1.) the allocation of the available production capacity to standard and deluxe models 2.) the number of units of each model to produce. Problem 2: (Production allocation problem) Four different typeof metals, namely, iron, copper, zinc and manganese are required to produce commodities A, B and C. To produce one unit of A, 40kg iron, 30kg copper, 7kg zinc and 4kg manganese are needed. Similarly, to produce one unit of B, 70kg iron, 14kg copper and 9kg manganese are needed and for producing one unit of C, 50kg iron, 18kg copper and 8kg zinc are required. The total available quantities of metals are 1 metric ton iron, 5 quintals copper, 2 quintals of zinc and manganese each. The profits are Rs 300, Rs 200 and Rs 100 by selling one unit of A, B and C respectively. Formulate the problem mathematically and solve it using Lindo software. Procedure: 1. Identify all the given information from the problem. 2. Create a first draft of the LP program by determining its objective function and constraints. 3. Open the Lindo Application 4. Input the objective function and the given constraints in the problem in the worksheet. 5. In order to solve the objective function and constraints click “SOLVE” in the menu bar . The following figure will appear. 6. Click the “YES” in “Do range (sensitivity) Analysis” dialog box. 7. The solution will be shown in a separate window. 8. Interpret the result. 9. Draw conclusions

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6. Data and Results:

7. Data Analysis and Conclusion:

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8. Assessment (Rubric for Laboratory Performance):

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Course Code: Course Title: Section: Members:

Laboratory Exercise No 2 Linear Programming - Maximization Problem Program: Date Performed: Date Submitted: Instructor:

1. Objective(s):

The activity aims to introduce the basic linear programming including different cost and non-cost variables related to manufacturing setting 2. Intended Learning Outcomes (ILOs):

The students shall be able to: 2.1 Solve basic linear programming particularly maximization problems using Lindo software. 3. Discussion:

Linear programming is not a programming language like C++, Java, or Visual Basic. Linear programming can be defined as: “A mathematical method to allocate scarce resources to competing activities  in an optimal manner when the problem can be expressed using a linear objective function and linear inequality constraints.”  A linear program consists of a set of variables, a linear objective function indicating the contribution of each variable to the desired outcome, and a set of linear constraints describing the limits on the values of the variables. The “answer” to a linear program is a set of values for the problem variables that results in the best —  largest or smallest —  value of the objective function and yet is consistent with all the constraints. Formulation is the process of translating a real-world problem into a linear program. Once a problem has been formulated as a linear program, a computer program can be used to solve the problem. In this regard, solving a linear program is relatively easy. The hardest part about applying linear programming is formulating the problem and interpreting the solution.

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4. Resources:

Lindo Software Textbooks 5. Procedure:

Practice Problem 1: A farmer has 10 acres to plant in wheat and rye. He has to plant at least 7 acres. However, he has only $1200 to spend and each acre of wheat costs $200 to plant and each acre of rye costs $100 to plant. Moreover, the farmer has to get the planting done in 12 hours and it takes an hour to plant an acre of wheat and 2 hours to plant an acre of rye. If the profit is $500 per acre of wheat and $300 per acre of rye how many acres of each should be planted to maximize profits? Practice Problem 2: A gold processor has two sources of gold ore, source A and source B. In order to kep his plant running, at least three tons of ore must be processed each day. Ore from source A costs $20 per ton to process, and ore from source B costs $10 per ton to process. Costs must be kept to less than $80 per day. Moreover, Federal Regulations require that the amount of ore from source B cannot exceed twice the amount of ore from source A. If ore from source A yields 2 oz. of gold per ton, and ore from source B yields 3 oz. of gold per ton, how many tons of ore from both sources must be processed each day to maximize the amount of gold extracted subject to the above constraints? 1. Identify all the given information from the problem. 2. Create a first draft of the LP program by determining its objective function and constraints. 3. Open the Lindo Application. 4. Input the objective function and the given constraints in the problem in the worksheet. 5. In order to solve the objective function and constraints click “SOLVE” in the menu bar  6. Click the “YES” in “Do range (sensitivity) Analysis” dialog box. 7. The solution will be shown in a separate window. 8. Interpret the result. 9. Draw conclusions.

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6. Data and Results:

7. Data Analysis and Conclusion:

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8. Assessment (Rubric for Laboratory Performance):

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Course Code: Course Title: Section: Members:

Laboratory Exercise No 3 Linear Programming - Minimization Problem Program: Date Performed: Date Submitted: Instructor:

1. Objective(s):

The activity aims to formulate linear programming minimization problems using Lindo Software. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: 2.4 identify requirements in solving minimization problems of LP. 2.5 interpret the results provided by Lindo software. 3. Discussion:

The word “linear” implies direct proportionality of relationship of variable. “Programming” means making schedules or plans of activities to undertake in the future. “Linear Programming” therefore is planning by the use of linear relationship of variables involved. It makes use of certain mathematical techniques to get the best possible solution to a problem involving limited resources. There are two ways of solving a linear programming problem: By the graphical and by the simplex method. The graphical method can only be used if the problem has two or three variable, since there are only two coordinate axis in a plan and three coordinates in space. The simplex method can handle a problem having any number of variables.

4. Resources:

Lindo Software Textbooks 9

5. Procedure:

Problem 1: Assume that a pharmaceutical firm is to produce exactly 40 gallons of mixture in which the basic ingredients, x and y, cost $8 per gallon and $15 per gallon, respectively, No more than 12 gallons of x can be used, and at least 10 gallons of y must be used. The firm wants to minimize cost. Problem 2: A patient needs daily 5mg, 20mg and 15mg of vitamins A, B and C respectively. The vitamins available from a mango are 0.5mg of A, 1mg of B, 1mg of C, that from an orange is 2mg of B, 3mg of C and that from an apple is 0.5mg of A, 3mg of B, 1mg of C. Ifthe cost of a mango, an orange and an apple be Rs 0.50, Rs 0.25 and Rs 0.40respectively, find the minimum cost of buying the fruits so that the dailyrequirement of the patient be met. Formulate the problem mathematically and solve it using Lindo. Procedure: 1. Identify all the given information from the problem. 2. Create a first draft of the LP program by determining its objective function and constraints. 3. Open the Lindo Application. 4. Input the objective function and the given constraints in the problem in the worksheet. 5. In order to solve the objective function and constraints click “SOLVE” in the menu bar . The following figure will appear. 6. Click the “YES” in “Do range (sensitivity) Analysis” dialog box. 7. The solution will be shown in a separate window. 8. Interpret the result. 9. Draw conclusions.

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6. Data and Results:

7. Data Analysis and Conclusion:

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8. Assessment (Rubric for Laboratory Performance):

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Course Code: Course Title: Section: Members:

Laboratory Exercise No. 4 Linear Programming - Simplex Minimization Problem Program: Date Performed: Date Submitted: Instructor:

1. Objective(s):

The activity aims to formulate linear programming, simplex minimization problems using Lindo Software. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: 2.6 identify requirements in solving simplex minimization problems of LP. 2.7 interpret the results provided by Lindo software. 3. Discussion:

 Although >= and = symbols are occasionally used in constraints of maximization problems, these are more common among minimization problems. This is how to change these constraints with >= and = symbols to equations. Subtraction to slack variables is permitted in minimization, but not in maximization, because if we intend to minimize, it is but logical to subtract, but if we intend to maximize, it is otherwise.

4. Resources:

Lindo Software Textbooks

5. Procedure:

Problem 1: The owner of a shop producing automobile trailers wishes to determine the best mix for his three products: at-bed trailers, economy trailers, and luxury trailers. His shop is limited to working 24 days per month on metalworking and 60 days per month on woodworking for these products. The following table indicates the production data for the trailers. 13

Problem 2: A small petroleum company owns two refineries. Refinery 1 costs $25,000 per day to operate, and it can produce 300 barrels of high-grade oil, 200 barrels of medium-grade oil, and 150 barrels of lowgrade oil each day. Refinery 2 is newer and more modern. It costs $30,000 per day to operate, and it can produce 300 barrels of high-grade oil, 250 barrels of medium-grade oil, and 400 barrels of low-grade oil each day. The company has orders of 35,000 barrels of high-grade oil, 30,000 barrels of medium-grade oil, and 40,000 barrels of low-grade oil. How many days should the company run each refinery to minimize its costs and still meet its orders? Procedure: 1. Identify the requirements of the problem. 2. Create a first draft of the LP program by determining its objective function and constraints 3. Open the Lindo Application. 4. Input the objective function and the given constraints in the problem in the worksheet. 5. In order to solve the objective function and constraints click “SOLVE” in the menu bar . The following figure will appear. 6. Click the “YES” in “Do range (sensitivity) Analysis” dialog box. 7. The solution will be shown in a separate window 8. Interpret the result. 9. Draw conclusions.

6. Data and Results:

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7. Data Analysis and Conclusion:

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8. Assessment (Rubric for Laboratory Performance):

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Laboratory Exercise No. 5 Linear Programming - Simplex Minimization Problem Involving Constraints With Pure Greater than/Equal Signs Course Code: Program: Course Title: Date Performed: Section: Date Submitted: Members: Instructor:

1. Objective(s):

The activity aims to formulate linear programming simplex maximization problems using Lingo Software. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: 

Identify requirements in solving minimization problems of LP whose constraints involve pure greater than/equal sign.



Interpret the results provided by Lingo software.

3. Discussion:

Minimization problems whose constraints involve pure greater than/equal sign are concerned with selecting variables from surplus to artificial. The objective is to minimize cost. Repetitive procedures are used in going from one table to another, to come up with an optimum solution. This is solved using the Lingo Software.

4. Resources:

Lingo Software Textbooks

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5. Procedure:

Problem 1: A small jewelry manufacturing company employs a person who is a highly skilled gem cutter, and it wishes to use this person at least 6 hours per day for this purpose. On the other hand, the polishing facilities can be used in any at least 10 hours per day. The company specializes in three kinds of semiprecious gemstones, J, K, and L. Relevant cutting, polishing, and cost requirements are listed in the table. How many gemstones of each type should be processed each day to minimize the cost of the finished stones? What is the minimum cost?

J

K

L

Cutting

1hr

1hr

1hr

Polishing

2hr

1 hr

2hr

Cost per stone

$30

$30

$10

Problem 2: Livestock Nutrition Co. produces specially blended feed supplements. LNC currently has an order for at least 200 kgs of its mixture. This consists of two ingredients X1 ( a protein source ) X2 ( a carbohydrate source ) The first ingredient, X1 costs $ 3 a kg. The second ingredient, X2 costs $ 8 a kg. The mixture must be at least 40% X1 and it must be at least 30% X2. LNC’s problem is to determine how much of each ingredient to use to minimize cost. Procedure: 1. Identify the requirements of the problem. 2. Create a first draft of the LP program by determining its objective function and constraints 3.  Augment the relevant surplus and artificial variables. Add the artificial variables while deduct the surplus variables. 4. Open the Lingo Software. 5. Input the objective function and the given constraints in the problem in the worksheet. 6. In order to solve the objective function and constraints click “SOLVE” in the menu bar. The 18

following figure will appear. 7. The solution will be shown in a separate window. 8. Interpret the result. 9. Draw conclusions. 6. Data and Results:

7. Data Analysis and Conclusion:

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8. Assessment (Rubric for Laboratory Performance):

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Laboratory Exercise No. 6 Linear Programming - Simplex Minimization Problem Involving Constraints With Equal Sign and Greater Than/Equal Signs Course Code: Program: Course Title: Date Performed: Section: Date Submitted: Members: Instructor:

1. Objective(s):

The activity aims to formulate linear programming simplex maximization problems using Lingo Software. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: 

Identify requirements in solving minimization problems of LP whose constraints involve pure greater than/equal sign.



Interpret the results provided by Lingo software.

3. Discussion:

 Although >= and = symbols are occasionally used in constraints of maximization problems, these are more common among minimization problems. This is how to change these constraints with >= and = symbols to equations. Subtraction to slack variables is permitted in minimization, but not in maximization, because if we intend to minimize, it is but logical to subtract, but if we intend to maximize, it is otherwise.

4. Resources:

Lingo Software Textbooks

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5. Procedure:

Problem 1: A Furniture Ltd., wants to determine the least expensive combination of products to manufacture. The Furniture Ltd., makes two products, tables and chairs, which must be processed through assembly and finishing departments. Assembly operates in exactly 60 hours; Finishing can handle at least 48 hours of work. Manufacturing one table requires 4 hours in assembly and 2 hours in finishing. Each chair requires 2 hours in assembly and 4 hours in finishing. Cost is $8 per table and $6 per chair.

Problem 2: Suppose a manufacturer of printed circuits has a minimum quantity of stock of 120 transistors and an exact quantity of stock of 200 resistors and is required to produce 2 types of circuits. Type A requires 20 resistors and 10 transistors. Type B requires 10 resistors and 20 transistors. If the cost on Type A circuits is $ 5 and that of Type B circuits is $ 12. How many of each circuit should be produced in order to minimize the cost?

Procedure: 1. Identify the requirements of the problem. 2. Create a first draft of the LP program by determining its objective function and constraints 3.  Augment the relevant surplus and artificial variables. Add the artificial variables while deduct the surplus variables. 4. Open the Lingo Software. 5. Input the objective function and the given constraints in the problem in the worksheet. 6. In order to solve the objective function and constraints click “SOLVE” in  the menu bar. The 22

following figure will appear. 7. The solution will be shown in a separate window. 8. Interpret the result. Draw conclusions. 6. Data and Results:

7. Data Analysis and Conclusion:

8. Assessment (Rubric for Laboratory Performance):

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Laboratory Exercise No. 7 Linear Programming Simplex Maximization Problems Involving Constraints with Less than or Equal sign Course Code: Program: Course Title: Date Performed: Section: Date Submitted: Members: Instructor:

1. Objective(s):

The activity aims to formulate and solve maximization problems by the Lingo Software. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: Identify requirements in solving maximization problems of LP whose constraints involve less than or equal sign. Interpret the results provided by Lingo software. 3. Discussion:

Maximization problems are concerned with selecting variables from slack to surplus to artificial. The objective is to maximize cost. Repetitive procedures are used in going from one table to another, to come up with an optimum solution. This is solved using the Lingo Software. 4. Resources:

Lingo Software Textbooks 5. Procedure:

Problem 1: High Tech industries import components for production of two different models of personal computers, called deskpro and portable. High Tech’s management is currently interested in developing a weekly production schedule for both products and maximize profit. The deskpro generates a profit contribution of $50/unit, and portable generates a profit contribution of $40/unit. For next week’s production, a max of 150 hours of assembly time is available. Each unit of deskpro requires 3 hours of assembly time. And each unit of portable requires 5 hours of assembly time. High Tech currently has only 20 portable display components in inventory; thus no more than 20 units of 25

portable may be assembled. Only 300 sq. feet of warehouse space can be made available for new production. Assembly of each Deskpro requires 8 sq. ft. of warehouse space, and each Portable requires 5 sq. ft. of warehouse space. Problem 2: A company produces golf equipment and decided to move into the market for standard and deluxe golf bags. Each golf bag requires the following operations: Cutting and dyeing the material, Sewing, Finishing (inserting umbrella holder, club separators etc.), Inspection and packaging. Each standard golf-bag will require 7/10 hr. in the cutting and dyeing department, 1/2 hr. in the sewing department, 1 hr. in the finishing department and 1/10 hr. in the inspection & packaging department. Deluxe model will require 1 hr. in the cutting and dyeing department, 5/6 hr. for sewing, 2/3 hr. for finishing and 1/4 hr. for inspection and packaging The profit contribution for every standard bag is 10 MU and for every deluxe bag is 9 MU. In addition the total hours available during the next 3 months are as follows: Cutting & dyeing dept

630 hrs

Sewing dept

600 hrs

Finishing

708 hrs

Inspection & packaging

135 hrs

The company’s problem is to determine how many standard and deluxe bags should be produced in the next 3 months to maximize profit? Procedure: 10. Identify the requirements of the problem. 11. Create first a draft of the LP program by determining its objective function and constraints 12.  Augment the necessary variables variab les to the model. 13. Open the Lingo Software. 14. Input the objective function and the given constraints in the problem in the worksheet. 15. In order to solve the objective function and constraints click “SOLVE” in the menu bar . The 26

following figure will appear. 16. The solution will be shown in a separate window. 17. Interpret the result. 18. Draw conclusions.

6. Data and Results:

7. Data Analysis and Conclusion:

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8. Assessment (Rubric for Laboratory Performance):

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Laboratory Exercise No. 8 Linear Programming Simplex Maximization Problems Involving Constraints with Less than or Equal sign and Equal sign Course Code: Program: Course Title: Date Performed: Section: Date Submitted: Members: Instructor:

1. Objective(s):

The activity aims to formulate linear programming, simplex maximization problems using Lingo Software. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: 2.8 identify requirements in solving simplex maximization problems of LP. 2.9 interpret the results provided by Lingo Software 3. Discussion:

 Although