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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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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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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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.


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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:


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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 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:


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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:



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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.



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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 <= and = symbols are occasionally used in constraints of maximization probl ems, these are more common in complex problems. This is how to change these constraints with <= and = symbols to equations. Just augment slack variables because if we intend to maximize, it is but logical to add, but if we intend to maximize, it is otherwise.

4. Resources:


5. Procedure:

Problem 1:

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Each day there are at most 30 hours of machine time available and exactly 60 hours of craftsman time. The profit on each type A shed is P60 and on each type B shed is P84 Problem 2: A furniture company makes 2 products: tables and chairs, which must be processed through assembly and finishing departments. Assembly department is available for at most 60 hours in every production period, while the finishing department is available for exactly 48 hours of work. Manufacturing one table requires 4 hours in the assembly and 2 hours in the finishing. Each chair requires 2 hours in the assembly and 4 hours in the finishing. One table contributes P180 to profit, while a chair contributes P100. The problem is to determine the number of tables and chairs to make per production period in order to maximize the 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. Augment the necessary variables to the model. 12. Open the Lingo Software. 13. Input the objective function and the given constraints in the problem in the worksheet. 14. In order to solve the objective function and constraints click “SOLVE” in the menu bar . 15. The solution will be shown in a separate window 16. Interpret the result. 17. Draw conclusions.


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Laboratory Exercise No. 9 Linear Programming - Simplex Maximization Problem Involving Constraints With Less Than / Equal Sign, With Equal Sign and Greater than/Equal Signs Course Code: Program: Course Title: Date Performed: Section: Date Submitted: Members: Instructor:

1. Objective(s):


The students shall be able to: 2.1 identify requirements in solving simplex maximization problems of LP. 2.2 interpret the results provided by Lingo software. 3. Discussion:

The simplex method of linear programming was developed by George B. Dantzig of Stanford University. It is a repetitive optimizing technique. It repeats the process of mathematically moving from an extreme point (in the graphical method) until an optimal solution is reached. Unlike the graphical, the simplex can handle infinite number of variables.

4. Resources:

Lingo Software Textbooks 5. Procedure:

Problem 1:

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The total amount of raw material available per day for both products is at least 1,575 lbs. The total storage space for all products is exactly 1,500 sq. ft., and a maximum of 7 hours per day can be used for production. Maximize profit.

Problem 2: Suppose a manufacturer of printed circuits has a stock of exactly 200 resistors, at least 120 transistors and maximum of 150 capacitors,

Procedure: 35

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 Lingo 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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Laboratory Exercise No. 10 Linear Programming - Simplex Maximization Problem Program: Date Performed: Date Submitted: Instructor:

1. Objective(s):

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

The students shall be able to: 2.3 identify requirements in solving simplex maximization problems of LP. 2.4 interpret the results provided by Excel Solver. 3. Discussion:


4. Resources:

Excel Solver Textbooks

5. Procedure:

Problem 1: A farmer has 20 hectares for growing barley and Swedes. The farmer has to decide how much of each to grow. The cost per hectare for barley is P30 and for Swedes is P20. The farmer has budgeted P480. 39

Barley requires 1 man-day per hectare and Swedes require 2 man-days per hectare. There are 36 mandays available. The profit on barley is P100 per hectare and on Swedes is P120 per hectare. Find the number of hectares of each crop the farmer should sow to maximize profit. Problem 2: A manufacturer produces three types of plastic fixtures. The time required for molding, trimming, and packaging is given in Table 9.1. (Times are given in hours per dozen fixtures.) Process Type A Type B Type C Total time available Molding 1 2 3/2 12,000 Trimming 2/3 2/3 1 4,600 Packaging ½ Profit $11

1/3 $16

½ $15

2,400 —

How many dozen of each type of fixture should be produced to obtain a maximum profit?

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

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1. Objective(s):

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



4. Resources:


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 44

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: 9. Identify the requirements of the problem. 10. Create a first draft of the LP program by determining its objective function and constraints 11. Open the Excel Solver. 12. Input the objective function and the given constraints in the problem in the worksheet. 13. In order to solve the objective function and constraints click “SOLVE” in the menu bar . The following figure will appear. 14. The solution will be shown in a separate window. 15. Interpret the result. 45

16. Draw conclusions. 6. Data and Results:

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1. Objective(s):

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



4. Resources:


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 50

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: 17. Identify the requirements of the problem. 18. Create a first draft of the LP program by determining its objective function and constraints 19. Open the Excel Solver. 20. Input the objective function and the given constraints in the problem in the worksheet. 21. In order to solve the objective function and constraints click “SOLVE” in the menu bar. The following figure will appear. 22. The solution will be shown in a separate window. 23. Interpret the result. 24. Draw conclusions. 51


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Laboratory Exercise No. 12 Linear Programming – Graphical Method Program: Date Performed: Date Submitted: Instructor:

1. Objective(s):

The activity aims to solve linear programming simplex maximization problems using Graphical Method. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: 2.9 identify requirements in solving simplex maximization problems graphically. 3. Discussion:

The simplex method of linear programming was developed by George B. Dantzig of Stanford University. It is a repetitive optimizing technique. It repeats the process of mathematically moving from an extreme point (in the graphical method) until an optimal solution is reached. The graphical method handles finite number of variables.

4. Resources:

TORA Textbooks 5. Procedure:

Problem 1:

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The total amount of raw material available per day for both products is at least 1,575 lbs. The total storage space for all products is exactly 1,500 sq. ft., and a maximum of 7 hours per day can be used for production. Maximize profit.

Problem 2: Suppose a manufacturer of printed circuits has a stock of exactly 200 resistors, at least 120 transistors and maximum of 150 capacitors,

Procedure: 10. Identify the requirements of the problem. 11. Create a first draft of the LP program by determining its objective function and constraints 57

12. Graph the first constraint by the intercept method. 13. Graph the last constraints by the intercept method. 14. Determine the corner-point feasible solutions in the graph, including the point of intersection of the constraint lines. 15. Compute for the total contribution of each corner-point feasible solution using the objective function. 16. Choose the highest value for maximization problems. Choose the lowest value for minimization problems. 17. Interpret the result. 18. Draw conclusions


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Laboratory Exercise No. 13 Linear Programming – Graphical Method Program: Date Performed: Date Submitted: Instructor:


1. Objective(s):

The activity aims to solve linear programming simplex minimization problems using Graphical Method. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: 2.10

identify requirements in solving simplex minimization problems graphically.

3. Discussion:

The simplex method of linear programming was developed by George B. Dantzig of Stanford University. It is a repetitive optimizing technique. It repeats the process of mathematically moving from an extreme point (in the graphical method) until an optimal solution is reached. The graphical method handles finite number of variables.

4. Resources:

TORA / Graph Website Textbooks 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.

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Problem 2: A small petroleum company owns two refineries. Refinery 1 costs $20,000 per day to operate, and it can produce 400 barrels of high-grade oil, 300 barrels of medium-grade oil, and 200 barrels of low-grade oil each day. Refinery 2 is newer and more modern. It costs $25,000 per day to operate, and it can produce 300 barrels of high-grade oil, 400 barrels of mediumgrade oil, and 500 barrels of low-grade oil each day. The company has orders totaling 25,000 barrels of high-grade oil, 27,000 barrels of medium-grade oil, and 30,000 barrels of low-grade oil. How many days should it run each refinery to minimize its costs and still refine enough oil to meet its orders?

Procedure: 19. Identify the requirements of the problem. 20. Create a first draft of the LP program by determining its objective function and constraints 21. Graph the first constraint by the intercept method. 22. Graph the last constraints by the intercept method. 23. Determine the corner-point feasible solutions in the graph, including the point of intersection of the constraint lines. 24. Compute for the total contribution of each corner-point feasible solution using the objective function. 25. Choose the highest value for maximization problems. Choose the lowest value for minimization problems. 26. Interpret the result. 27. Draw conclusions


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7. Data Analysis and Conclusion: 8. Assessment (Rubric for Laboratory Performance):

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Laboratory Exercise No. 14 Linear Programming – Transportation Problems Program: Date Performed: Date Submitted: Instructor:

1. Objective(s):

The activity aims to formulate linear programming transportation problems using Excel Solver. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: 2.1 identify requirements in solving transportation problems of LP. 2.2 interpret the results provided by Excel Solver. 3. Discussion:

Transportation problems are concerned with selecting routes from the source of supply to distribution outlets. The objective is either to minimize cost of transportation to maximize the contribution to profit. Repetitive procedures are used in going from one table to another, to come up with an optimum solution. In the minimization process, the table is said to be optimum if the improvement computations are all positive; while in the maximization process, the table is said to be optimum if the improvements are all negative. 4. Resources:

Excel Solver Textbooks 5. Procedure:

Problem 1: Wheat is harvested in the Midwest and stored in grain elevators in three different cities— Kansas City, Omaha, and Des Moines. These grain elevators supply three flour mills, located in Chicago, St. Louis, and Cincinnati. Grain is shipped to the mills in railroad cars, each of which is capable of holding one ton of wheat. Each grain elevator is able to supply the following number of tons (i.e., railroad cars) of wheat to the mills on a monthly basis:

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Each mill demands the following number of tons of wheat per month.

The cost of transporting one ton of wheat from each grain elevator (source) to each mill (destination) differs according to the distance and rail system. These costs are shown in the following table. For example, the cost of shipping one ton of wheat from the grain elevator at Omaha to the mill at Chicago is $7.

The problem is to determine how many tons of wheat to transport from each grain elevator to each mill on a monthly basis in order to minimize the total cost of transportation. Problem 2: National Foods Company has five plants where it processes and packages fruits and vegetables. It has suppliers in six cities in California, Texas, Alabama, and Florida. The company has owned and operated its own trucking system in the past for transporting fruits and vegetables from its suppliers to its plants. However, it is now considering outsourcing all its shipping to outside trucking firms and getting rid of its own trucks. It currently spends $245,000 per month to operate its own trucking system. It has determined monthly shipping costs (in $1,000s per ton) using outside shippers from each of its suppliers to each of its plants as shown in the following table:

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Should National Foods continue to operate its own shipping network or sell its trucks and out- source its shipping to independent trucking firms? 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 Excel Solver. 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. The solution will be shown in a separate window. 7. Interpret the result. 8. Draw conclusions.


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Laboratory Exercise No. 15 Network Problems – Shortest Route Algorithm Program: Date Performed: Date Submitted: Instructor:

1. Objective(s):

The activity aims to formulate and solve network problems by the Shortest Route using Solver. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: Identify requirements in solving network problems of LP. Interpret the results provided by Solver. 3. Discussion:

Network problems are concerned with selecting routes from the source of supply to distribution outlets. The objective is either to minimize cost of routing. Repetitive procedures are used in going from one table to another, to come up with an optimum solution. This is solved by the Shortest Route. 4. Resources:

Solver Textbooks 5. Procedure:

Problem 1: A park has recently been set aside for a limited amount of sightseeing and backpack hiking. Cars are not allowed into the park, but there is a narrow, winding road system for trams and for jeepneys driven by the park rangers. This road system is shown below, where location O is the entrance to the park; other letters designate the locations of ranger stations and facilities. The numbers give the distances of these winding roads in miles. The park contains a scenic wonder at station T. A small number of trams are used to transport sightseers from the park entrance to station T and back. The park management currently faces a problem to determine which route from the park entrance to station T has the smallest total distance for the operation 70

of the trams.

A 2

7 2

T

5

5

4

O

D

B

7 3

1

1 4 E

C 4

Problem 2: Suppose the park renovated the layout to a new one as follows. Determine which route from the park entrance to station T has the smallest total distance for the operation of the trams.

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A 7 4

D

1

6 5

6 O

T

B 1

5

8

4

2

E

C 5

Procedure: 9. Identify the requirements of the problem. 10. Create a first draft of the LP program by determining its objective function and constraints 11. Open the Solver in Open Office Calculator. 12. Input the objective function and the given constraints in the problem in the worksheet. 13. In order to solve the objective function and constraints click “SOLVE” in the menu bar . The following figure will appear. 14. The solution will be shown in a separate window. 15. Interpret the result. 16. Draw conclusions.

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Laboratory Exercise No. 16 Network Problems – Maximal Flow Algorithm Program: Date Performed: Date Submitted: Instructor:

1. Objective(s):

The activity aims to formulate and solve network problems by the Maximal Flow Algorithm using Solver. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: Identify requirements in solving network problems of LP. Interpret the results provided by Solver. 3. Discussion:

Network problems are concerned with selecting routes from the source of supply to distribution outlets. The objective is either to minimize cost of routing. Repetitive procedures are used in going from one table to another, to come up with an optimum solution. This is solved by the Maximal Flow Algorithm. 4. Resources:

Solver Textbooks 5. Procedure:


of the trams.

A 2

7 2

T

5

5

4

O

D

B

7 3

1

1 4 E

C 4


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A 7 4

D

1

6 5

6 O

T

B 1

5

8

4

2

E

C 5

Procedure: 19. Identify the requirements of the problem. 20. Create a first draft of the LP program by determining its objective function and constraints 21. Open the Solver. 22. Input the objective function and the given constraints in the problem in the worksheet. 23. In order to solve the objective function and constraints click “SOLVE” in the menu bar . The following figure will appear. 24. The solution will be shown in a separate window. 25. Interpret the result. 26. Draw conclusions.

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Laboratory Exercise No. 17 Network Problems – Shortest Route Algorithm Program: Date Performed: Date Submitted: Instructor:

1. Objective(s):

The activity aims to formulate and solve network problems by the Shortest Route using Lingo. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: Identify requirements in solving network problems of LP. Interpret the results provided by Lingo. 3. Discussion:

Network problems are concerned with selecting routes from the source of supply to distribution outlets. The objective is either to minimize cost of routing. Repetitive procedures are used in going from one table to another, to come up with an optimum solution. This is solved by the Shortest Route. 4. Resources:

Lingo Textbooks 5. Procedure:


of the trams.

A 2

7 2

T

5

5

4

O

D

B

7 3

1

1 4 E

C 4


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A 7 4

D

1

6 5

6 O

T

B 1

2

5

8

4

E

C 5

Procedure: 17. Identify the requirements of the problem. 18. Create a first draft of the LP program by determining its objective function and constraints 19. Open the software Lingo. 20. Input the objective function and the given constraints in the problem in the worksheet. 21. In order to solve the objective function and constraints click “SOLVE” in the menu bar . The following figure will appear. 22. The solution will be shown in a separate window. 23. Interpret the result. 24. Draw conclusions.

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Laboratory Exercise No. 18 Network Problems – Maximal Flow Algorithm Program: Date Performed: Date Submitted: Instructor:

1. Objective(s):

The activity aims to formulate and solve network problems by the Maximal Flow Algorithm using Lingo. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: Identify requirements in solving network problems of LP. Interpret the results provided by Lingo. 3. Discussion:

Network problems are concerned with selecting routes from the source of supply to distribution outlets. The objective is either to minimize cost of routing. Repetitive procedures are used in going from one table to another, to come up with an optimum solution. This is solved by the Maximal Flow Algorithm. 4. Resources:

Lingo Textbooks 5. Procedure:


of the trams.

A 2

7 2

T

5

5

4

O

D

B

7 3

1

1 4 E

C 4


86

A 7 4

D

1

6 5

6 O

T

B 1

2

5

8

4

E

C 5

Procedure: 27. Identify the requirements of the problem. 28. Create a first draft of the LP program by determining its objective function and constraints 29. Open the software Lingo. 30. Input the objective function and the given constraints in the problem in the worksheet. 31. In order to solve the objective function and constraints click “SOLVE” in the menu bar . The following figure will appear. 32. The solution will be shown in a separate window. 33. Interpret the result. 34. Draw conclusions.

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Laboratory Exercise No. 19 Sensitivity Analysis Program: Date Performed: Date Submitted: Instructor:

1. Objective(s):

The activity aims to apply several techniques of Operations Research in an actual operation of an existing company. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: 2.2 Analyze the existing operations of the selected company. 2.3 Apply Operations Research tools and techniques to the identified problem(s) or goal(s). 2.4 Apply design of experiment structured methodologies in the chosen study. 2.5 Communicate problems and solutions through written and oral presentation. 3. Discussion:

1. The students will choose a practical problem for their final project. They will be expected in each case to define a problem, collect data and present recommendations. 2. Students will apply design of experiment structured methodologies in the conduct of the study. 3. Final projects will be presented in written and oral presentations. 4. Prepare a report paper in 15-20 pages double space. 4. Resources:

5. Procedure:

1. 2. 3. 4. 5.

Choose a company and select an area of study Identifies problems/goals Collect pertinent data Use appropriate modern tools in Operations Research to solve the identified problems/goals. Discuss in written and oral presentations the detailed measurement of the identified system. 90



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Laboratory Exercise No 20 Final Project – Case Study Company Program: Date Performed: Date Submitted: Instructor:

1. Objective(s):

The activity aims to apply several techniques of Operations Research in an actual operation of an existing company. 2. Intended Learning Outcomes (ILOs):

The students shall be able to: 2.6 Analyze the existing operations of the selected company. 2.7 Apply Operations Research tools and techniques to the identified problem(s) or goal(s). 2.8 Apply design of experiment structured methodologies in the chosen study. 2.9 Communicate problems and solutions through written and oral presentation. 3. Discussion:

5. The students will choose a practical problem for their final project. They will be expected in each case to define a problem, collect data and present recommendations. 6. Students will apply design of experiment structured methodologies in the conduct of the study. 7. Final projects will be presented in written and oral presentations. 8. Prepare a report paper in 15-20 pages double space. 4. Resources:

5. Procedure:

6. 7. 8. 9. 10.

Choose a company and select an area of study Identifies problems/goals Collect pertinent data Use appropriate modern tools in Operations Research to solve the identified problems/goals. Discuss in written and oral presentations the detailed measurement of the identified system. 93



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IE003+lab

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