Introduction to Linear Programming

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Linear Programming sounds really difficult, but it’s just a neat way to use math to find out the best way to do things – for example, how many things to make or buy.  It usually involves a system of linear inequalities, called constraints, but in the end, we want to either maximize something (like profit) or minimize something (like cost).   Whatever we’re maximizing or minimizing is called the objective function.

Linear programming was developed during the second World War for solving military logistic problems.  It is used extensively today in business to minimize costs and maximize profits.

Before we start Linear Programming, let’s review Graphing Linear Inequalities with Two Variables.

Review of Inequalities

Let’s go back and revisit graphing linear inequalities on the coordinate system.

To graph inequalities on the coordinate system, we need to get “y” by itself on the left hand side, so it’s best to use the slope-intercept or “y = mx + b” formula.  This is because we’ll easily know which way to shade the graph, and we’ll make fewer errors doing this.  The shaded areas will be where the equation “works”; in other words, where the solutions are.

When we have “y  < ”, we always shade in under the line that we draw (or to the left, if we have a vertical line).  So think of “less than” as “raining down” from the graph.

When we have “y  > ”, we always shade above the line that we draw (or to the right, if we have a vertical line).  So think of “greater than” as “raining up” from the graph.  (I know – it doesn’t really “rain up”, but I still like to explain the graphs that way.)

Note that you can always plug in an (x, y) ordered pair to see if it shows up in the shaded areas (which means it’s a solution), or the unshaded areas (which means it’s not a solution.)  For an example of this, see the first inequality below.

With “<” and “>” inequalities, we draw a dashed (or dotted) line to indicate that we’re not  including that line (but everything up to it), whereas with “Less Than or Equal To” and “Greater Than or Equal”, we draw a regular line, to indicate that we are including it in the solution.  To remember this, I think about the fact that “<” and “>” have less pencil marks than “Less Than or Equal To” and “Greater Than or Equal”, so there is less pencil used when you draw the lines on the graph.  You can also remember this by thinking the line under the “Less Than or Equal To” and “Greater Than or Equal” means you draw a solid line on the graph.

Note that the last example is a “Compound Inequality” since it involves more than one inequality.  The solution set is the ordered pairs that satisfy both inequalities; it is indicated by the darker shading.
Here are some examples:

Graphing Inequalities

Again, note that the last example is a “Compound Inequality” since it involves more than one inequality.  The solution set is the ordered pairs that satisfy both inequalities; it is indicated by the darker shading.

Bounded and Unbounded Regions

With our Linear Programming examples, we’ll have a set of compound inequalities, and they will be bounded inequalities, meaning the inequalities will have both maximum and minimum values.  (We’ll show examples below, but think of bounded meaning that you could draw a “circle” around the feasible region, which is the solution set to the inequalities).

Here are what some typical Systems of Linear Inequalities might look like in Linear Programming:

Bounded and Unbounded Inequalities

Inequality Word Problem:

Linear Programming problems are typically word problems – not cool.   But most will fit in the same mold: for these beginning problems, they will have two types of unknowns or variables, like earrings and necklaces, and they will involve inequalities.

You’ll just put the first variable on the x axis and the second on the y axis.  We will be solving for the number of each type that either gives the maximum profit, or minimum cost.

Jewelry

Let’s first see an example of just graphing an inequality in a real-life situation:

Lisa has an online jewelry shop where she sells earrings and necklaces.  She sells earrings for $30 and necklaces for $40.  To make a profit, she must sell at least $1200 of jewelry per month.

Define the variables, write an inequality for this situation, and graph the solutions to the inequality.

It’s best to make the first item (pairs of earrings) the x, and the second item (necklaces) y.  (Usually we can look at what the problem is asking to get these variables).   So let’s plug in some real numbers to try to get our equations.  If she sells 1 pair of earrings and 1 necklace, she’ll only make $70, which is way below her goal for a profit.  How about 20 pairs of earrings and 15 necklaces?  Do you see what we’re doing here?   The number of pairs of earrings she sells times $30 plus the number of necklaces she sells times $40 has to be greater than $1200.

Also, she can’t sell less than 0 pairs of earrings or necklaces, so we need to include those inequalities, too.

So here are the inequalities, and we’ll also draw a graph:

Jewelry Inequality Problem

Linear Programming Terms

Again, the linear programming problems we’ll be working with have the first variable on the x axis and the second on the y axis.   In our example, x is the number of pairs of earrings and y is the number of necklaces. Typically you can look at what the problem is asking to determine what the variables are.

Typically, we will be solving for the number of each type that either gives the maximum profit, or minimum cost.  The maximum profit or minimum cost expression is called the objective function.  We’ll do an example where we want to maximize profit, so our objective function will be “Maximize 30x + 40y”.

The inequalities of the problem are called the constraints, since we are limiting what we have, such as time or resources.  We’ll do an example where we are limiting our time to make jewelry.  We also will always have our non-negative constraints, where the x and y values have to be greater than or equal to 0.  Some constraints will involve greater than inequalities, for example, if a certain number of things need to be sold.  Usually there will be a sentence or phrase in the word problem for each constraint.  And match units when coming up with inequality constraints; for example, one may have to do with money, and another with hours.

We have to make sure our inequality is bounded, in order to find a maximum profit (for minimizing cost, it doesn’t have to be bounded, but it usually is).  Again, the bounded region (solutions to the system of inequalities) is called the feasible region, which will be the double-shaded region..

After we graph the inequalities, we’ll want to find the corner points.  The corner points are the vertices of the feasible region, which are the intersections of the lines of the feasible region.   The solution to the linear programming will occur at one of the corner points.  (There is something called a Corner Point Theorem that proves this, but we won’t have to worry about it).

Then we’ll substitute our corner points into the objective function to see which point yields the largest (for maximizing profit) or smallest (for minimizing cost) value.  We can do this with a table, as we’ll see later.

Linear Programming Word Problems

Now let’s put it all together and solve “real” linear programming problems!

Problem:

Lisa has an online jewelry shop where she sells earrings and necklaces.  She sells earrings for $30 and necklaces for $40.  It takes 30 minutes to make a pair of earrings and 1 hour to make a necklace, and, since Lisa is a math tutor, she only has 10 hours a week to make jewelry.  In addition, she only has enough materials to make 15 total jewelry items per week.  She makes a profit of $15 on each pair of earrings and $20 on each necklace.  How many pairs of earrings and necklaces should Lisa make each week in order to maximize her profit, assuming she sells all her jewelry?

Define the variables, write an inequality for this situation, and graph the solutions to the inequality.

Solution:

Variables:  Let’s look at what the problem is asking: how many pairs of earrings and how many necklaces should Lisa make to make a profit?  So let’s let x = the number of pairs of earrings, and y = the number of necklaces.

Objective Function:  Since we are maximizing profit, this will be a maximum, and it will be total dollars.  So the objective function is Maximize 15x + 20y.

Hint:  Usually the objective function is a money function.

Constraints:   Lisa’s constraints are based on her time, and also her materials.  Always make sure all the units match; we had to change 30 minutes into .5 hours.  Also note that we didn’t need to know what the jewelry sells for; sometimes they will give you extra information!

Hint:  To figure out the constraint inequalities, match units.  For example, one inequality has to do with hours, the other with number of items.

Let’s set up a separate constraint for each sentence, and first put it in a table:

Constraint Table

Let’s turn these into inequalities, and also add the non-negative constraints:

Jewelry Graph

Corner Points:  Think of the corner points as on the outside of the shaded area, but where there is a turn in the graph (the vertices).  In this case, we can easily see what they are from the graph; if we can’t, we’ll have to solve a system of equations (see next example).

Then take these points and plug in the x and y values into the objective function.  Here is what we get:

Corner Point Table

So, in order to maximize her profits, Lisa should make 10 pair of earrings and 5 necklaces per week, and her weekly profit is $250.

Problem:

Anchor

Bountiful Boats has to produce at least 5000 cabin cruisers and 12,000 pontoons each year; they can produce at most 30,000 jet skis in a year.  The company has two factories:  one in Michigan, and one in Wisconsin; each factory is open for a maximum of 240 days per year.  The Michigan factory makes 20 cabin cruisers, 40 pontoons, and 60 jet skis per day.  The Wisconsin factory makes 10 cruisers, 30 pontoons, and 50 jet skis per day.  The cost to run the Michigan factory per day is $960,000; the cost to run the Wisconsin factory per day is $750,000.  How many days of the year should each factory run in order to meet the boat production, yet do so at a minimum cost?

Solution:

Variables:  Let’s look at what the problem is asking: how many days of the year should each factory run?  So let’s let x = the number days per year that the Michigan factory should run, and y = the number days per year that the Wisconsin factory should run.

Objective Function:  Since we are minimizing cost per day, this will be a minimum, and it will be total dollars.  So the objective function is Minimize 960000x + 750000y.

Constraints:   Bountiful Boats’ constraints are based on how much they need to produce (both less than and greater than inequalities), how many days the factories are open (less than inequality), along with the non-negative constraints.

If you can’t find the constraints from sentences in the problem, we can look for another set of items (like cabin cruisers, pontoons, and jet skis), and these will usually each represent an inequality.  Let’s set up a separate constraint for each of the boats, and put them in a table.  Note that the first two inequalities are less than (“at least”), and the last one is greater than (“at most”).

Boat Problem Table

Let’s turn these into inequalities, and also add the factory days open and non-negative constraints.   Notice how we use compound inequalities for the number of days open.

Boat Linear Programming Graph

Corner Points:  Think of the corner points as on the outside of the shaded area, but where there is a turn in the graph (the vertices).  In this case, we can easily see what they are from the graph; if we can’t, we’ll have to solve a system of equations.

Let’s say we couldn’t see the exact intersection of Equations to get Intersection from the graphWe could turn the equations into equalities, and use Linear Combination of Systems to solve:

Linear Combination

Then take these points and plug in the x and y values into the objective function.  Here is what we get:

Corner Point Table Boats

Note that you can see that the point (240, 240) won’t provide a minimum since the point (240, 80) will provide a smaller amount.  So, technically, you wouldn’t even have to plug this point into the objective function.

So, in order to minimize their costs, Bountiful Boats should make boats 240 days a year at their Michigan plant, and 80 days a year at their Wisconsin plant.  This will yield a cost of $290,400,000.  So you can see that your answer may be surprising!

Learn these rules, and practice, practice, practice!

On to Rational Functions and Equations  – you are ready!

234 thoughts on “Introduction to Linear Programming

  1. I’m teaching the lesson during my student teaching. This was a helpful and simple outline of this topic. I appreciate the help.
    Kyle

    • Kyle,
      Thanks so much for taking the time to write me! Please spread the word about She Loves Math and let me know if you find any errors, etc.
      Thanks again!
      Lisa

      • Hello Lisa!
        My apologies for butting in. I dont know how I could ask your help regarding my two LPP. Could you please help me out of this?
        These are the problems:
        1) A candy manufacturer makes two types of special candy, say A and B. Candy A consists of equal parts of dark chocolate and caramel and Candy B consists of two parts of dark chocolate and one part of walnut. The company has in stock 430 kilograms of caramel, 360 kilograms of dark chocolate, and 210 kilograms of walnuts. The company sells Candy A for P285 and Candy B for P260 per kilograms. How much of each candy should the manufacturer produce to maximize profit?

        2) A factory manufactures two products each requiring the use of three machines. The machine A can be used at most 70 hours; the machine B at most 50 hours; and the machine C at most 90 hours. The first product requires 2 hours on machine A, 4 hours on machine B, and 3 hours on machine C; the second product requires 5 hours on machine A, 1 hour on machine B, and 4 hours on machine C. If the profit is P2,200 per unit for the first product and P2,700 unit for the second product, how many units of each product should be manufactured to maximize profit?
        Would you please show me the objective function and the constraints too. I need to see the graph as well for the feasible region.
        Thank you Lisa!
        Glenn

    • I find many math-help websites overwhelming and confusing but this was exactly what I was looking for and right to the point. SOOO helpful! I will bookmark this site for future reference. 🙂

      • Thanks so much for using She Loves Math, and for writing! Your comment really makes me want to work harder on it and finish it 😉 Lisa

  2. This is great! I really needed some good examples broken down into steps for my teaching of this subject. This was my first year teaching it. I did it at university but am teaching it at high school so it has to be scaled back a bit. Your examples are awesome! Thank you so much for sharing: Websites like yours contribute so much to the teaching profession.

    • Sarah,
      Thank you so much for taking the time to comment on She Loves Math! I really enjoy writing it and I’m glad it’s helping 😉 Keep reading, and let me know if you see any errors or problems 😉
      Thanks again,
      Lisa

  3. Doing these objective functions for linear programming in my Calculus class… your table for the corner points made things so much simpler to understand!!! Thanks 🙂

  4. I’ve been sitting and trying to make sense of this because my textbook doesn’t exactly do so properly. This was perfect! Thanks a million!

  5. A marchant plans to sell two models of home computers at costs of 250 dollar and 400 dollar,respectively.The 250 dollar model yields a profit of 45 dollar and the 400 dollar yields a profit of 50 dollar.The merchant estimates that the total monthly demand will not exceed 250 units.Find the number of units of each model that should be stocked in order to maximum profit.Assumee that the merchant does not want to invest more than 700000 in computer inventory

    • Thanks for writing! Here’s how I would set this up: x = number of units that cost $250, y = number of units that cost $400. We want to maximize 45x + 50y (objective function) such that: x + y < = 250, x >= 0, y >= 0, 250x + 400y <= 70000 (I think it's supposed to be 70000, not 700000). Graph all the inequalities, find the boundary points, and plug them into the objective function to see what combination will give you the maximum. When I did this, I got the boundary points are (0,0), (0,165), (265,0), (195,50). Plugging these points into the objective function, I got the point that maximizes the profit is (265,0) at a profit of $11925. Does that make sense? Lisa

      • Hello! I worked more on this problem and updated my answer: Here’s how I would set this up: x = number of units that cost $250, y = number of units that cost $400. We want to maximize 45x + 50y (objective function) such that: x + y < = 250, x >= 0, y >= 0, 250x + 400y <= 70000 (I think it’s supposed to be 70000, not 700000). Graph all the inequalities, find the boundary points, and plug them into the objective function to see what combination will give you the maximum. When I did this, I got the boundary points are (0,0), (0,165), (265,0), (195,50). Plugging these points into the objective function, I got the point that maximizes the profit is (265,0) at a profit of $11925. Does that make sense? Lisa

  6. A company has budgeted a maximum of 600000 dollars for advertising a certain product nationally.Each minute of television time costs 60000 dollar and each one page newspaper ad cost 15000 dollars.Each television ad is expected to be viewed by 15 millon viewers, and each newpaper is expected to ne seen by 3 million readers . The company market research department advises the company to use 90% of the advertising budget on television ads.how should the advertising budget be allocated to maximize the total audience.

    • Thanks for writing! Here’s how I’d set this up: max 15000000t + 3000000n, such that 60000t + 15000n < = 600000, t >= 0, n >= 0,
      60000t <= .9(600000t + 15000n) (since this is the budget and we want 90% of it). Does that make sense? Than you can graph and find the critical points and plug in these to the objective function. When I graphed, I got the number of television ads to be 4.74 television ads (at one minute each) and about 21 newspaper ads. This would be a budget of $284400 for television ads and $315750 for newspaper ads. I'm not sure if this is right - can you check it? Lisa

  7. I really appreciate this website. Please continue to share knowledge. I understand all the topics more by means of this. Thank you. 🙂

  8. A small refinery produces only two products: lubricants and sealants. These are produced by processing crude oil through 3 processors; a cracker, a splitter and a separator. These processors have limited capacities. For the cracker, at most 1000hours; for the splitter; at most 4200hours; and the separator has at most 2400hours per week. Similarly, there is a limit on the supply of crude oil; at most 700 barrels per week. To produce 1 barrel of lubricant, we need 1 hour at the cracker, 6 hours at the splitter and 3 hours at the separator.
    Under these conditions, what product combinations of lubricants and sealants are feasible? If a barrel of lubricants nets $2000 and a barrel of sealants nets $2500, which product combination will maximize combined profits?
    THANK YOU

    • Good problem!
      Here’s the full original problem that I’m using to set this up:
      A small refinery produces only two products: lubricants and sealants. These are produced by processing crude oil through 3 processors: a cracker, a splitter, and a separator. These processors have limited capacities. For the cracker, at most 1000 hours; for the splitter, at most 4200 hours; and for the separator, at most 2400 hours per week. Similarly, there is a limit on the supply of crude oil: at most 700 barrels per week.

      To produce one barrel of lubricant, we need one hour at the cracker, 6 hours at the splitter, and 4 hours at the separator. To produce a barrel of sealant, we need 2 hours at the cracker, 7 hours at the splitter, and 3 hours at the separator.

      Under these conditions, what product combinations of lubricants and sealants are feasible? If a barrel of lubricant nets P2000 and a barrel of sealant nets P2500, which product combination will maximize combined profits?

      I set up the following constraints: x + 2y < = 1000, 6x + 7y <= 4200, 4x + 3y <= 2400, x + y <= 700, and x, y >= 0. The objective function would be max 2000x + 2500y. When I did the graph, I got points (600, 0), (0, 500), (420, 240), and (280, 360) with the maximum profits at (280, 360) at $1460000. Not sure if this is correct, but you can check the work. Thanks! Lisa

      • Nice work Lisa,

        But how did you get the coefficients for all the ys in the inequalities? Nothing talks about sealants in the word problem or am I missing something? I can’t seem to wrap my head around that.

        • Thanks so much for writing. You are absolutely correct; I used more information than what the original Comment supplied, and I’ve updated my response. It should be updated here in the Comments section; let me know if that doesn’t make sense. Lisa

          • Yes! It makes absolute sense now. Thank you very much. This is a nice way of brushing up my linear programming as a grad student.

            Nice work Lisa.

          • I’m so glad, and I’m glad you found the site. This makes want to keep writing 🙂 Lisa

  9. Pingback: December 8,9 | Integrated Math III

  10. Your club has decided to volunteer to clean up your school grounds and plant some bushes and
    trees. You determine that the bushes you want to plant average $15 each and each tree $25.
    You will definitely buy both bushes and trees. You realize that you cannot plant more than 18
    trees. Your school says that you should plant at least 12 plants but no more than 30. The
    number of trees must be at least ½ the number of bushes.

    • Thanks for writing! Let x = number of bushes, and y = number of trees to plant. So you probably want to minimize the cost of the plants, or 15x + 25y. Then your constraints will be $latex x\ge 1,y\ge 1,y\le 18,x+y\ge 12,x+y\le 30,y\ge .5x$. Then you would graph all the inequalities and test the corner points to see which will give the minimum cost. Does that make sense? If you want to me try to graph and solve, let me know. Thanks! Lisa (The inequalities weren’t showing up correctly, so I fixed it – sorry)

  11. Am soooo much overwhelmed of this website. I’ve difficulties in operations research which deals with linear programming topics

  12. Hi Lisa,
    Bill’s Grill is a popular college restaurant that is famous for its hamburgers. The owner of the restaurant, Bill, mixes fresh ground beef and pork with a secret ingredient to make delicious quarter-pound hamburgers that are advertised as having no more than 25% fat. Bill can buy beef containing 80% meat and 20% fat at $0.85 per pound. He can buy pork containing 70% meat and 30% fat at $0.65 per pound. Bill wants to determine the minimum cost way to blend the beef and pork to make hamburgers that have no more than 25% fat.
    Formulate an LP model for this problem. (Hint: The decision variables for this problem represent the percentage of beef and the percentage of pork to combine.)

  13. A gardener has a square vegetable garden. The length of the garden’s diagonal is 25 feet. Sketch a picture of this situation and find the area and perimeter of the garden. Round answer to two decimal places. PLEASE HELP!!!

    • Thanks for writing! Since the length of the garden’s diagonal is 25, each side is 25/sqrt(2), because of a 45-45-90 triangle (have you had this?). Then the area of the garden is 25/sqrt(2) squared, or 625 / 2 = 312.5 ft squared. You can also get this by knowing that the area of a square is 1/2 times diagonal times diagonal = 312.5 ft squared. Hope this helps! lisa

  14. can you help me with my project about linear programming ?
    Grubby Stakes Mining Company is establishing a production plan for the current week at its Bonstock Lode, which has three main veins of varying characteristics. The net yields per ton for each of the veins is provided below.

    Ore Mining Veins
    Eastern Northern Tom’s Lucky
    Gold .2 oz .3 oz .4 oz
    Silver 30 oz 20 oz 30 oz
    Copper 50 lb 20 lb 25 lb

    Gold presently sells for $150 per ounce, silver sells for $5 per ounce, and copper sells for $2 per pound. Eastern is the most accessible vein, requiring 1 worker-hour per ton of ore. Northern and Tom’s Lucky veins are more remote and require 2 worker-hours per ton. Only 300 workers-hours are available, and all labor costs are fixed. At least 100 tons must be mined from the Northern vein this week, so that it can be reshored next week, there are no tonnage limitations for the other tunnels. The company must also yield at least 5,000 pounds of copper to meet contractual commitments.
    (a.) Formulate Grubby’s linear program to determine how many tons must be mined from each vein to maximize total revenue.
    (b.) Find the optimal solution.

    • Thanks for writing! Here’s what I got for the equations for this one – not sure if it is correct 😉
      Let x = # tons from Eastern vein
      y = # tons from Northern vein
      z = # tons from Tom’s Lucky vein

      For constraints, we have 1x + 2y + 2z < = 300, y >= 100,
      50x + 20y + 25z >= 5000
      We want to maximize 150(.2x + .3y + .4z) + 5(30z + 20y + 30z) + 2(50x + 20y + 25z), which is 280x + 185y + 260z.
      Does this look correct? If so, we can go on to (b).
      Lisa

  15. hi there,

    I am clueless when it comes to this, please try and explain this question to me.

    A manufacturer of leather articles produces boots and jackets. The manufacturing process consists of two activities, – Making and finishing
    There are 800 hours available for making the articles and 1200 hours available for finishing them. It takes 4 hour to make and 3 hours to finish a pair of boots, and 2 hours to make and 4hours to finish a jacket. Market experience requires the production of boots to be a minimum of 150 pair per month . Write down a system of linear inequalities that describe the appropriate constraints if x is the number of pairs of boots and y the number of jackets manufactured.

    • Thanks for writing! Here’s how I would set this up:
      Making: 4x + 2y less than or equal to 800
      Finishing: 3x + 4y less than or equal to 1200
      Production: x greater than or equal to 150

      (When I graphed the 3 lines and found the boundary points, I got (150, 100), (150, 0), and (200, 0)).
      Does that make sense?
      Lisa

  16. a collection of weights (22, 7, 13, 10, 20, 17), Construct an LP model that would result in partitioning them into two groups such that the difference in the sums is as small as possible.

    • Thanks for writing! I’m not sure how to do this one, sorry. I think it’s beyond the scope of this web site for now. I’ll keep looking for a solution though, and maybe someone else can solve it. Sorry again, Lisa

  17. Thank you so much for the reply.
    can you do the working for me please? I just need to understand how you got to that,

    • Can you check my response again to see if it makes sense? For some reason, when I type in symbols for greater than and less than, it doesn’t work. I wrote it out in English. Basically we have an inequality for Making, one for Finishing, and one for Making the Boots (minimum of 150 per month). And if x equals the number of boots, 4x will equal the number of hours to make x boots, since it takes 4 hours to make one pair. See if the equations make sense now. Thanks! Lisa

  18. hi there, another battle, please help.

    how many units must be produced to maximise a profit defined by the function:
    2y=-4x²+16x-12

    • I think you just need to find the vertex of this function: y = -2x^2 + 8x – 6. The x (units) part of the vertex would be -b/2a or 2 units.

  19. The land of milkandhoney produced only three products–machinery, steel, and automobiles. All other goods were imported. The minister of the economy held the responsibility for economic planning, and he felt that the welfare of the country could best be served by maximizing the net dollar value of exports (that is, the value of the imported to produce those exports). Milkandhoney could sell all the steel, automobiles, or machinery it could produce on the world market at prices of $500 per unit for steel, $1,500 per unit for automobiles, and $2,500 per unit for machinery.
      In order to produce on unit of steel, it took 0.05 units of machinery, 0.01 of automobiles, 2 units of ore purchased on the world market for $100 per unit, and other imported materials costing $50. In addition, it took one half man-years of labor to produce each unit of steel. Mildandhoney’s steel mills had a rated capacity of 100,000 units per year.
      To produce one unit of automobiles. It took one unit of steel, 0.1 units of machinery, and one man-year of labor. In addition, it took $300 worth of imported materials to produce each unit of automobiles. Automobile capacity was 700,000 units per year.
      To produce one unit of machinery required 0.01 units of automobiles, 0.5 units of steel, and 2 man-years of labor, in addition to $100 for items imported from outside. The capacity of the machinery plants was 50,000 units per year.
      The total manpower available for labor in Milkandhoney was 800,000 persons per year.
      
     1. formulate a linear-programming modeling to determine the production mix that will maximize net exports (dollars). Be careful to define all variable and to state exactly all relationships among variables.

    • Thanks for writing! I’m so sorry; I’m not sure how to do this, since it involves an input-output model. I think you’d have x less than or equal to 100000, y less than or equal to 700000, z less than or equal to 50000, .5x + 1y + 2x less than or equal to 800000. Then I would max x((500-250) – (.05z + .01y)) + y((1500-30) – (x + .1z)) + z((2500-100) – (.01y + .5x)), but I think there’s a better way to do this. Lisa

      • hi lisa, you said that there may be another answer for this. i would just like to ask if this is the correct alternative? thank you 🙂

        ES= Steel production for export
        EA=Automobile production for export
        EM= Machinery production for export
        PS= Total steel production
        PA= Total automobile production
        PM= Total machinery production

        Step 2: Formulation of the Objective Function and Constraints

        Maximize z = 500ES + 1500EA + 2500EM – 250PS – 300PA – 100PM

        Subject to:

        -ES + PS – PA – O.5PM = 0
        -EA – 0.01PS + PA – 0.0.1PM = 0
        -EM – 0.05PS – 0.1PA + PM = 0
        PS ≤ 100,000
        PA≤ 700,000
        PM ≤ 50,000
        0.5PS + PA + 2PM ≤ 800,000
        i made a table but it wont show correctly in the coment box 🙂

        IS THIS CORRECT? THANK YOU SO MUCH 🙂 and with the answer you gave me, what is the variables? thank you 🙂

        • My variables were just total steel, automobile, and machinery units. Yours looks good, but I’m not sure if they wanted 6 variables. Can you check with someone else? Lisa

  20. HI! 🙁 can you help me about linear programming?? pleasee. I need answers :(((

    Investment Advisor Inc. is a brokerage firm that manages stock portfolios for a number of clients. A new client has requested that the firm handle an $80,000 investment portfolio. As an initial investment strategy the client would like to restrict the portfolio to a mix of the following stocks:

    STOCK PRICE/SHARE ESTIMATED ANNUAL RETURN/SHARE RISK INDEX / SHARE
    US Oil $ 25 $3 .50
    Hub Properties $ 50 $5 .25

    The risk index for the stock is a rating of the relative risk of the two investment alternatives. For the data given, US Oil is judge to be the more risky investment. By constraining the total risk for the portfolio, the investment firm avoids placing excessive amounts of the portfolio in potentially high-return but also high risk investment. For the current portfolio an upper limit of 700 has been set for the total risk index of all investments. In addition, the firm has set an upper limit of 1000 shares for the more risky US Oil stock. How many shares of each stock should be purchased in order to maximize the total annual return?

    GOD BLESSxx

  21. hi. can you help me with my project about linear programming?
    The Butterfield Company makes a variety of hunting knives. Each knife is processed on four machines. The processing times required are as follows. Machine capacities (in hours) are 1500 for machine 1; 1400 for machine 2; 1600 for machine 3; and 1500 for machine 4.
    Processing Time (hr)
    Knife Machine 1 Machine 2 Machine 3 Machine 4
    A 0.05 0.10 0.15 0.05
    B 0.15 0.10 0.05 0.05
    C 0.20 0.05 0.10 0.20
    D 0.15 0.10 0.10 0.10
    E 0.05 0.10 0.10 0.05

    Each product contains a different amount of two basic raw materials. Raw material 1 costs $0.50 per ounce, and raw material 2 costs $1.50 per ounce. There are 75000 ounces of raw material 1 and 100000 ounces of raw material 2 available.
    Requirements (oz/unit)
    Knife Raw Material 1 Raw Material 2 Selling Price ($/unit)
    A 4 2 15.00
    B 6 8 25.50
    C 1 3 14.00
    D 2 5 19.50
    E 6 10 27.00

    a. If the objective is to maximize profit, specify the objective function and constraints for the problem. Assume that labor costs are negligible.
    b. Solve the problem with a computer package.

    thank you so much 🙂

  22. Hi, can you help me answering this problem please? I really have no clue on how to do it. This is our project so I really badly need your help. So here’s how it goes..

    All-American Meat Processors is mixing the ingredients for a batch of German and Italian sausages. The following cuts are to be used:
    (MEAT) – (COST PER POUND) – (AVAILABLE QUANTITY (lbs)
    Beef rib ———– $1.00 —- ——- 200
    Beef shank ——- 1.50 ———- 500
    Beef tongue —— 1.00 ———– 700
    Pork ————– .90 —————– 1,000
    Lamb ————- 1.20 ————– 800

    All-American wants to determine the minimum cost mixture that will provide at least 500 pounds of German sausage and 300 pounds of Italian. This must be done so that at least 60% of the German sausage is beef and at least 80% of the Italian sausage is beef. Furthermore, the German sausage can contain no more than 10% lamb, while the Italian sausage cannot have more than 20% pork.Using double-subscripted variables throughout, formulate All-American’s problem as a linear program.

    • Thanks for writing! I’m sorry, but this question is a little too complicated – I can try to look at it later, but I have no time now ;( Lisa

  23. Hi can you please help me? I got this homework and i cant solve it

    Hart Venture Capital (HVC) specializes in providing venture capital for software development and Internet applications. Currently HVC has two investment opportunities: (1) Security Systems, a firm that needs additional capital to develop an Internet security software package.
    (2) Market Analysis, a market research company that needs additional capital to develop a software package for conducting customer satisfaction surveys.
    In exchange for Security Systems stock, the firm asked HVC to provide $600,000 in year 1, $600,000 in year 2, and $250,000 in year 3 over the coming three-year period. In exchange for their stock, Market Analysis asked HVC to provide $500,000 in year 1, $350,000 in year 2, and $400,000 in year 3 over the same three-year period. HVC believes that both investment opportunities are worth pursuing. However, because of other investments, they are willing to commit at most $800,000 for both projects in the first year, at most $700,000 in the second year, and $500,000 in the third year.
    HVC’s financial analysis team reviewed both project and recommended that the company’s objective should be maximize the net present value of the total investment in Security System and Market Analysis. The present value takes into account the estimated value of the stock at the end of the three-year period as well as the capital outflows that are necessary during each of the three years. Using he 8% required rate of return, HVC’s financial team estimates that 100% funding of the Security Systems has a net present value of 1,800,000, and 100% funding of the Market Analysis project has a net present value of 1,600,000.
    HVC has the option to fund any percentage of the Security Systems and Market Analysis projects. For example, if HVC decides to fund 40% of the Security Systems projects, investment of 0.4(600,000)=250,000 would be required in year 1, 0.4(600,000)=250,000 would be required in year 2, and 0.4(250,000)=100,000 would be required in year 3. In this case, the net present value of the social security projects would be 0.4(1,800,000)=720,000. The investment amounts and the net present value for partial funding of the market analysis would be computed in the same manner.

    please help me in this question… i dont know where to start

    • Thanks for writing! I’m sorry, but this question is a little too complicated – I can try to look at it later, but I have no time now ;( Lisa

  24. Hello, I can’t figure out how to answer this problem using linear programming. Can you please help me?

    In addition to its line of bicycles, Hot Wheels, Inc. manufactures three types of kiddie tricycles: a model known as Fat Wheel, a model called the Toad, and their ever-popular model, the Ridge Runner. Hot Wheels manufactures these tricycle models on special order or whenever slack time is available and would like to determine the optimal number of each type of tricycle to produce in order to maximize the total number of tricycles produced. Because of the popularity of the Ridge Runner model, Hot Wheels would like the number of Ridge Runners to be at least twice the number of Fat Wheels. In addition, the number of Ridge Runners should also be at least twice the number of Toads. In terms of manufacturing time each Fat Wheel and each Toad tricycle requires 10 minutes, where as each Ridge Runner tricycle requires 4 minutes. In addition, Fat Wheels require 8 minutes of assembly time, Toads require 6 minutes, and Ridge Runners require 4 minutes. There are 40 hours of manufacturing time and 20 hours of assembly time available. The warehouse has capacity to store a maximum of 150 tricycles. What should Hot Wheels do? Consider the possibility of alternate optimal solutions. What flexibility does this provide for Hot Wheels?

    • Thanks for writing. If x = Fat Wheels, y = Toads and z = RR, I get z greater than or equal 2x, z greater than or equal to 2y, 1/6x + 1/6y + 4z less than or equal to 40, 8x + 6y + 4z less than or equal to 20, and x + y + z less than or equal to 150. Hope this is a start! Lisa

      • Oh my God! THANK YOU SO MUCH! This is truly helpful. Thank you so much for taking your time in answering my questions! More power to you! Btw, you just saved me from failing this class. ♥

  25. Hi Lisa! Good day. I would like to ask for help with my BA project if you don’t mind. I am really hopeless right now and I don’t think I’d be able to pass this on monday because I really don’t have any idea how to answer the problem assigned to me because unfortunately I have not learned anything from my professor. Please help me, it would really mean a lot and I’d really owe you big time.

    Here is my problem:
    8-5 (Ballplayer selection problem) The Dubuque Sackers, a class D baseball team, face a tough four-game road trip against league rivals in Des Moines, Davenport, Omaha, and Peoria. Manager “Red” Revelle faces the task of scheduling his four starting pitchers for appropriate games. Because the games are to be played back to back in less than one week, Revelle cannot count on any pitcher to start in more than one game.

    Revelle knows the strengths and weaknesses not only of his pitchers, but also of his opponents. He has developed a performance rating for each of his starting pitchers against each of these teams. The ratings are listed in the table on this page. What pitching rotation should manager Revelle set to provide the highest total of the performance using LP.

    (a) Formulate this problem using LP.
    (b) Solve the problem.

    OPPONENT
    STARTING PITCHER DES MOINES DAVENPORT OMAHA PEORIA
    Dead-Arm Jones 0.60 0.80 0.50 0.40
    SpitBall Baker 0.70 0.40 0.80 0.30
    Ace Parker 0.90 0.80 0.70 0.80
    Gutter Wilson 0.50 0.30 0.40 0.20

    Thank you so much, this would really mean a lot. Hoping for your response.

    • Thanks for writing! I’m sorry, but this question is a little too complicated – I can try to look at it later, but I have no time now ;( Lisa

  26. please help me with this

    Carolina Health Services Inc., is a clinic specializing in four types of patient are: cosmetic surgery, dermatology, orthopedic surgery, and neurosurgery. It has been determined from past records that a patient in each of these specialists contributes to the profit of the clinic as follows: cosmetic $200, dermatology $150, orthopedic $150, and neurosurgery $250. The physicians are convinced that patients are not being processed in an optimal manner. The clinic has contracted with you to provide a weekly patient processing system. You have been able to determine the following time requirements and limitations :

    specialty Lab X-ray Therapy Surgery Physicians
    cosmetic 5 2 1 4 10
    dermatolory 5 8 10 8 14
    orthopedic 2 1 0 16 8
    neurosurgery 4 5 8 10 12
    total hours available per week 200 140 110 240 320

    The physicians have access to as many of each type of patient as they wish. Additional they have limited their cosmetic and orthopedic practice to a combined total of more than 120 hours a week. set up the objective function and constraints to the optimal patient mix on a weekly basis.
    Thanks :*

    • Thanks for writing! I’m sorry, but this question is a little too complicated – I can try to look at it later, but I have no time now ;( Lisa

  27. hi can you please help me about this problem? 🙁
    Cinergy Corporation manufactures and distributes electricity for customers located in
    Indiana, Kentucky, and Ohio. The company spends $725 to $750 million each year for the
    fuel needed to operate its coal-fired and gas-fired power plants; 92% to 95% of the fuel used
    is coal. Cinergy uses 10 coal-burning generating plants: five located inland and five located
    on the Ohio River. Some plants have more than one generating unit. As the seventh-largest
    coal-burning utility in the United States, Cinergy uses 28-29 million tons of coal per year at
    a cost of approximately $2 million every day.
    The company purchases coal using fixed-tonnage or variable-tonnage contracts from mines
    in Indiana (49%), West Virginia (20%), Ohio (12%), Kentucky (11%), Illinois (5%), and
    Pennsylvania (3%). The company must purchase all of the coal contracted for on fixedtonnage
    contracts, but on variable-tonnage contracts it can purchase varying amounts up to
    the limit specified in the contract. The coal is shipped from the mines to Cinergy’s
    generating facilities in Ohio, Kentucky, and Indiana. The cost of coal varies from $19 to $35
    dollars per ton and transportation/delivery charges range from $1.50 to $5.00 per ton.
    A model is used to determine the megawatt hours (mWh) of electricity that each generating
    unit is expected to produce and to provide a measure of each generating unit’s efficiency,
    referred to as the heat rate. The heat rate is the total BTUs required to produce 1-kilowatt
    hour (kWh) of electrical power

    then, there are tables like this

    https://brainmass.com/file/
    YTo0OntzOjk6InVzZXJfdHlwZSI7aTozO3M6NzoidXNl
    cl9pZCI7aTowO3M6NzoiZmlsZV9pZCI7czo2OiI4MDY0OTYiO3M
    6OToic2lnbmF0dXJlIjtzOjQwOiI0YmEzOGZlZmE3YjU5ZjZhNDU
    4YzhjZDBmOTgwZWY2YjY5MjFlMDM1Ijt9

    • Thanks for writing! I’m sorry, but this question is a little too complicated – I can try to look at it later, but I have no time now ;( Lisa

  28. Hi Lisa,
    I am working on this problem, and have been killing my brain for over a week now. Linear programming is just so confusing to me!!!! here is what I am trying to figure out and make sense of.
    Scott owns a manufacturing company that produces two models of entertainment centers. The Athens requires 4 feet of fancy molding and takes 4 hours to manufacture. The Barcelona needs 15 needs 15 feet of molding and 3 hours to manufacture. In a given week, there are 120 hours of labor available and the company has 360 feet of molding to use for the entertainment centers. The company makes a profit of $9 on the Athens and $12 on the Barcelona. How many of each model should the company manufacture to maximize its profit?

    I have to state the constraints. so would that be: 4x + 15y < 360 and 4x + 3 y < 120 ?
    and the objective function…is that f= 9x + 12y ?
    identify whether or not I need to find a maximum or minimum in order to solve…I think it is maximum.

    The following is the part that LOSES me. and all the words and letters seem to blend together and make my head mush!!!
    I have to solve the problem by graphing and indicating the feasible region and identify the corners/vertices of the feasible regions. and state how many of each model should the company manufacture to maximize its profit

    please help…I am trying sooo hard to make sense of this!!!

  29. Hi Lisa,

    I am doing a project for Quantitative Techniques in Decision Making. I need to do a problem regarding our work and this is the scenario.

    Our company receives numerous request for a techical service called WEA, request totalling 300 per year. We have 6 senior staff and 8 junior staff to do the requests. Our turn around time for each request is 30 days. The senior staff must have more requests to be served than the junior staff. What method in Quantitative Techniqes can I use to know how many request must be assigned to each staff and to reach our target of 300/year? Can linear programming be used in this situation? Is it possible to shorten the turn-around time? I really don’t know how to come up with an appropriate problem that would require Quantitative Techniques Method in Decision Making with these scenario. Please help. I need this badly and asap. Thank you in advance.

    Kelly

    • Kelly,
      Thanks for writing! Here’s how I would do it: Let s = number requests assigned to each senior staff for a year, j = number of requests assigned to each junior staff for a year. Then s > j, s > 0, j > 0, 6s + 8j >= 300, and 30s <= 365, 30j <= 365. Does that make sense? I'm not sure if this is right 😉 Lisa

  30. Hi Lisa,

    Thank you so much for your immediate reply. I really needed it. I’ll try to come up with the solutions and hope I will be able to make it on time. I commend your work and dedication in helping people like us who are having difficulties in Math.

    Best regards and more power,
    Kelly

  31. Plz Help!!!

    I do not know if it is Law of SInes or Cosines!!

    The dimensions of a triangular lot are 100 feet by 50 feet by
    75 feet. If the price of such land is $3 per square foot, how much does the lot cost?

  32. Also this one!! Plz and thanks!!

    An airplane flies 165 miles from point A in the direction 130º and then
    travels in the direction 245ºfor 80 miles. Approximately how far is the airplane from A?

    • I would use Law of Cosines on this one – we can see that the angle between the 165 miles and 80 miles is 65 degrees (draw it) and then we’ll have x squared = 80 squared + 165 squared – 2 * 80 * 165 * cos(65) to get the answer. Does this make sense? Lisa

  33. Three ships are within site of each other. The Maverick spots the two other ships, the Northern-Star and the Throw-Rug, and measures the angle between them to be 72º. The
    distance between the Maverick and the Northern Star is 1527 meters. The Northern Star measures an angle of 59º between the Maverick and the Throw-Rug. What is the distance between the Northern Star
    and the Throw-Rug?

    • We have a Law of Sines here, since we have A-S-A. The third angle (angle from TR) is 49 (180 – 59 – 72), so we have sin(72)/x = sin(49)/1527. Solve for x to get 1924.268 meters. Hope that helps, Lisa

  34. Hey!! I forgot how to do this!! Help

    Find the area of a triangle with sides 10 cm
    and 3 cm, with an included angle of 120º.

    • Thanks for writing! This would use the Area of Triangle formula with the sin in it: A = 1/2(10)(3)(sin120) = 12.99 cm squared. Does that make sense? Lisa

  35. A businessman wishes to buy a triangular lot
    whose sides measure 125 feet, 280 feet, &
    315 feet. Find the area of the lot.
    This is so confusing to me!!

  36. Hey Lisa, I want to know how to write this.

    Problem: Write a power function to model the situation.

    The surface area S of a sphere varies directly as the square root of the pendulum’s length L.

    Problem: Write a power function to model the situation.

    The surface area of a sphere varies directly as the square of the radius r.

    Problem: The time it takes for a group of volunteers to build a house varies inversely with the # of volunteers v.

    a.) Write a power function to model this situation
    b.) If 20 volunteers can build a house in 62.5 working hours, how many volunteers would be needed to build a house in 50 working hours. (hint: find the value of k first)

    • Hi! For the first one, I get S = k*sqrt(L), second one I get S = k*r^2, and the third one (a) t = k/v (b) 62.5 = k/20, k = 1250, then 50 = 1250/v; v = 25. (Notice that 62.5 * 20 = 50 * 25 = k.)
      Hope that helps, Lisa

  37. I do NOT know the answer to this question. Please help!!!

    Which of the following exhibit inverse variation?

    a. The distance traveled as a function of speed
    b. The total cost as a function of the number of items purchased
    c. The area of a circular swimming pool as a function of its radius
    d. The number of posts in a 20ft fence as a function of distance between posts

  38. An electronics store sells an average of 60 entertainment systems per month at an average of $800 more than the cost price. For every $20 increase in the selling price, the store sells one fewer system. What amount over the cost price will maximize profit?

    • Here’s how you would set up: maximize (60 – x)(800 + 20x) (x = number of fewer systems sold). So x = 5. So they should sell 5 * 20 more than $800, so $900 more than cost price. Does that make sense? Lisa

  39. A ball is kicked into the air and follows a path described by h(t) 5 -4.9t 2 +6t + 0.6, where t is the time, in seconds, and h is the height, in metres, above the ground. Determine the maximum height of the ball, to the nearest tenth of a metre.

  40. Arnold has 24 m of fencing to surround a garden, bounded on one side by the wall of his house. What are the dimensions of the largest rectangular garden that he can enclose?

    • Here’s how you can set up: Let x = width of garden, 24 – 2x length. Then you want to maximize y = x(24 – 2x), which would be x = 6. So dimensions would be 6 by 12. Does that make sense? Lisa

    • You could find the vertex of the quadratic y = x(10-x), since that will give you the maximum. So you’d get x = 5, and the maximum product is 25. Hope that helps 🙂 Lisa

  41. I need help to write this!! Please

    The volume of a box has a width of (x-2) inches. The volume is expressed as a product of the length of its dimensions and is expressed by V(x)= x^3+2x^3-5x-6. Use synthetic division and the given width to completely factor V(x). Put the dimensions in the blanks.

    The dimensions of the box are (x-2), _________, and ______________ inches.

  42. I did not learn this and my teacher is making me do it!! Helppp PLEASE!!!!!!!!!

    #1) If -7-8i is a root of a polynomial equation, what does the Imaginary Root Theorem tell you?

    #2.) If 4+ the square root of 7 is a root of a polynomial equation, what does the Irrational Root Theorem tell you?

    • The irrational root theorem tells you that the conjugate of an irrational root is also an irrational root. So for 1) -7+8i is also a root and for 2)4-sqrt(7) is also a root. Does that makes sense? Lisa

  43. Using Descartes’s Rule Of Signs, list the possible number of positive and negative roots. Show your work.

    f(x)= x^5+4x^4+x-6=0

    • You would take the factors of 6 (constant) over the factors of 1 (coefficient of x^5) (plus and minus): +-6, +-3, +-2, +-1. Does this make sense? Lisa

    • You would just go 1 to the right and 16 up (like point (1, 16). To get the distance, use the distance formula: sqrt(1^2 + 16^2) = sqrt(17). Lisa

  44. hi there, I really need help with this two questions please,
    1. How many units must be produced to maximise a profit defined by the function: 2y=-4x²+16x-12
    2.Find the derivative of the function: G(x) =x (x²-4√x+4)

    • For 1), you’ll want to get the vertex to find the maximum point: y = -2x^2+8x-6. The vertex is (-b/2a, f(-b/2a)), which is (2, 2). So look at the x, which is 2.
      For 2), I get 3x^2 – 6x^(1/2) + 4. Lisa

  45. I get a different answer for the 2nd problem. But, while I’m here, I’d like to show 2 ways to do each problem.

    Problem 1 (Method 1) Same as Lisa’s approach
    Rewrite problem as: y = -4x^2 + 16x – 12
    Negative coefficient of x^2 ==> sad parabola ==> opens downward
    Therefore, max at vertex, when x = -b/2a = -16/(2*-4) = 16/8 = 2

    Problem 1 (Method 2)
    Find the x intercepts. The line of symmetry will be in the middle. Vertex (max) on LOS.
    Set y = zero to find the x intercepts.
    0 = -2x^2 + 8x -6
    0 = x^2 -4x -3
    0 = (x-1)(x-3) ==> zeros at x = 1, 3
    Line of symmetry in middle, at x=2. This is where the max occurs.

    Problem 2 (Method 1)
    To avoid using the product rule, multiply the two functions.
    G(x) = x^3 – 4x^(3/2) + 4x
    G'(x) = 3x^2 – 6x^(1/2) + 4

    Problem 2 (Method 2)
    Use the product rule: G'(x) = f'(x)*p(x) + f(x)*p'(x)
    G(x) = x(x^2 – 4x^(1/2) + 4)
    G'(x) = [1](x^2 – 4x^(1/2) + 4) + (x)[2x – 2x^(-1/2) ]
    G'(x) = x^2 – 4x^(1/2) + 4 + 2x^2 – 2x^(1/2)
    G'(x) = 3x^2 – 6x^(1/2) + 4

    I hope this helps! Kathy Paulk (www.PaulkUSA.com)

  46. A third method to solve Problem #1. This method used calculus to solve it.

    Background: The derivative of a function tells you the rate of change at a particular point. In other words, the derivative of a function tells you the slope of a line, tangent to a function, at a particular point. If you find the derivative of a function, then set the derivative = 0, you will find the places where the slope of the tangent line is zero. This is where maximums or minimums of the original function occur.

    Problem 1 (Method 3) Using Calculus
    Rewrite problem as: y = -4x^2 + 16x – 12
    y’ = -8x + 16
    y” = -8 ==> negative 2nd derivative ==> concave down ==> opens downward
    Set the first derivative = 0 to find maximum or minimum points.
    Since the 2nd derivative is always negative, the 1st derivative is zero at a maximum.
    y’ = -8x + 16 (from above)
    0 = -8x + 16
    x = 2 ==> The maximum of the function occurs when x = 2.

  47. hi Lisa, I really need help with this question, can u please solve it, thanks

    A UAE company produces commodities at 3 factories. The commodities are shipped from the factories to the 2 warehouses and it is also possible directly to the 2 clients. The commodities are distributed to the 2 clients from the warehouses. The distribution is also possible between the 3 factories between the two warehouses and between the 2 clients as shown by the distribution network. It is constrained that the shipment between any 2 network nodes does not exceed 150 units. The maximum capacity of the 3 factories and the demand from the 2 clients and the shipping costs are given below. The objective is to minimise the shipping costs between various network nodes.

    Shipping costs between nodes
    To Node
    From Node FA FB FC W1 W2 Client 1 Client 2
    FA 10 6 10 10 40 20
    FB 18 18 2 2 16 30
    FC 0.8 16 2 1 20 24
    W1 2.4 4 24
    W2 0.8 4 24
    Client 1 2
    Client 2 14

    The capacity of the factories and the demand of the clients are given below

    Capacity (# of units)
    Factory A 400
    Factory B 500
    Factory C 200

    Demand (# of units)
    Client 1 550
    Client 2 400

    1) Write the Integer Linear Program
    2) Solve the Integer Linear Program

    • Thanks for writing! I’m so sorry, but this problem is beyond the scope of my knowledge! You might look elsewhere on the web for it. Lisa

    • If you want to see a short video about factoring a 4th degree polynomial, given one zero, you may want to watch a video on my website (www.PaulkUSA.com). Click on the “Math Videos” tab. Then click on the last video on that page.

      The polynomial in the video is different than the polynomial in your question. I agree with Lisa’s answer to your question (the zeros).

  48. the texas technical institute for turtle tutelage offers classes in language, mathematics, and physics. language courses are worth 3 credit , math courses worth 4 credit , and physics are worth 5 credit. the prise to take a language course is $15, math course is $12, physics course is $18. to graduate, sydney the turtle must take at least 35 classes and accumulate a minimum of 130 credits. in addition, the college requires that a minimum of 5 each type of course be taken. how many courses of each type should sydney take if she wants to minimize the cost of her education?

    • Thanks for writing! Here’s how I’d do this problem: Let l = language courses, m = math courses, p = physics courses. Then we want to minimize 15l + 12m + 18p such that l >= 5, m >= 5, p >= 5, 3l + 4m + 5p >= 130, l + m + p >=35. Does this make sense? Let me know if you want me to graph and find the answer. Lisa

  49. A company budgeted a maximum of 600000 for advertising a certain product nationally. Each minute of television time costs 60000 and each one-page newspaper ad costs 15000. Each television ad is expected to be viewed by 15 million viewers. And each newspaper ad is expected to be seen by 3 million readers. The company’s market research department advises the company to use at most 90% of the advertising budget on television ads. How should the advertising budget be allocated to maximize the total audience?

    • Thanks for writing! Here’s how I’d set this up: max 15000000t + 3000000n, such that 60000t + 15000n < = 600000, t >= 0, n >= 0,
      60000t <= .9(600000t + 15000n) (since this is the budget and we want 90% of it). Does that make sense? Than you can graph and find the critical points and plug in these to the objective function. When I graphed, I got the number of television ads to be 4.74 television ads (at one minute each) and about 21 newspaper ads. This would be a budget of $284400 for television ads and $315750 for newspaper ads. I'm not sure if this is right - can you check it? Lisa

  50. 5. A merchant plans to sell two models of home computers at cost of $250 and $400, respectively. The $250 model yields a profit of $45 and the $400 model yields a profit of $50. The merchant estimates that the total monthly demand will not exceed 250 units. Find the number of units of each model that should be stocked in order to maximize profit. Assume that the merchant does not want to invest more than $70000 in computer inventory.

    • Here’s how I’d set this up: maximize 45x + 50y, such that x >= 0, y >= 0, x + y <= 250, 250x + 400y <= 70000. Does that make sense? Lisa

  51. Hi Lisa! I’m having a hard time solving this problem. Ca you kindly show me how to answer this? Please and thank you so much..

    mildred’s tool and die shop must provide exactly 10 experimental bits to a pneumatic drill company. The bits can be shaped either by forging or by machining. Both procedures involve a final milling stage, but the forged bits require more milling because they are not as smooth initially. In either case, only one bit can be shaped at a time using either process, and the order must be filled within two working days. The table on page 297 summarizes the restrictions.

    A. Assuming that the proprietor, Ms. Mildred Riveter, wants to maximize profits, formulate a linear program specifying the number of bits that should be shaped using each process.

    B. Solve this problem graphically.

  52. I’m trying to do this linear programming problem:
    Minimize Z=2x+3y subject to constraints
    x+2greater than equal to 2
    x+y less than equal to 50
    2x+3y less than equal to 6

    • Thanks for writing! I would graph this and look at the corner points – and put those x/y values in the objective function. So when I graph, I get the corner points are (0,0), (3,0), and (0,2). So the best point will be (0, 0). Are you sure the numbers are correct? lisa

  53. Metropolitan Police Patrol
    The Metropolitan Police Department was recently criticized in the local media for not responding to police calls in the downtown area rapidly enough. In several recent cases, alarms had sounded for break-ins, but by the time the police car arrived, the perpetrators had left, and in one instance a store owner had been shot. Sergeant Joe Davis was assigned by the chief as head of a task force to find a way to determine the optimal patrol area (dimensions) for their cars that would minimize the average time it took to respond to a call in the downtown area.
    Sergeant Davis solicited help from Angela Maris, an analyst in the operations area for the police department. Together they began to work through the problem.
    Joe noted to Angela that normal patrol sectors are laid out in rectangles, with each rectangle including a number of city blocks. For illustrative purposes he defined the dimensions of the sector as x in the horizontal direction and as y in the vertical direction. He explained to Angela that cars traveled in straight lines either horizontally or vertically and turned at right angles. Travel in a horizontal direction must be accompanied by travel in a vertical direction, and the total distance traveled is the sum of the horizontal and vertical segments. He further noted that past research on police patrolling in urban areas had shown that the average distance traveled by a patrol car responding to a call in either direction was one-third of the dimensions of the sector, or x / 3 and y /3 . He also explained that the travel time it took to respond to a call (assuming that a car left immediately upon receiving the call) is simply the average distance traveled divided by the average travel speed.
    Angela told Joe that now that she understood how aver- age travel time to a call was determined, she could see that it was closely related to the size of the patrol area. She asked Joe if there were any restrictions on the size of the area sectors that cars patrolled. He responded that for their city, the department believed that the perimeter of a patrol sector should not be less than 5 miles or exceed 12 miles. He noted several policy issues and staffing constraints that required these specifications. Angela wanted to know if any additional restrictions existed, and Joe indicated that the distance in the vertical direction must be at least 50% more than the horizontal distance for the sector. He explained that laying out sectors in that manner meant that the patrol areas would have a greater tendency to overlap different residential, income, and retail areas than if they ran the other way. He said that these areas were layered from north to south in the city, so if a sector area was laid out east to west, all of it would tend to be in one demographic layer.
    Angela indicated that she had almost enough information to develop a model; except that she also needed to know the average travel speed the patrol cars could travel. Joe told her that cars moving vertically traveled an average of 15 miles per hour, whereas cars traveled horizontally an average of 20 miles per hour. He said that the difference was due to different traffic flows.
    Develop a linear programming model for this problem and solve it by using the graphical method.

  54. Question
    AngelaandBobRaykeepalargegardeninwhichtheygrowcabbage,tomatoes,andonionsto make two kinds of relish—chow-chow and tomato. The chow-chow is made primarily of cab- bage, whereas the tomato relish has more tomatoes than does the chow-chow. Both relishes include onions, and negligible amounts of bell peppers and spices. A jar of chow-chow contains 8 ounces of cabbage, 3 ounces of tomatoes, and 3 ounces of onions, whereas a jar of tomato relish contains 6 ounces of tomatoes, 6 ounces of cabbage, and 2 ounces of onions. The Rays grow 120 pounds of cabbage, 90 pounds of tomatoes, and 45 pounds of onions each summer. The Rays can produce no more than 24 dozen jars of relish. They make $2.25 in profit from a jar of chow-chow and $1.95 in profit from a jar of tomato relish. The Rays want to know how many jars of each kind of relish to produce to generate the most profit.
    a. Formulate a linear programming model for this problem.
    b. Solve this model graphically.

    • I’m sorry – this linear programming section is more directed to high school pre-calculus, where they don’t use slack variables. I will look into adding that at a later time. Lisa

  55. Hi, I am taking a math class and my professor is not being any help at all. How do I set up and solve this problem?

    It takes 4 days for an auto body paint shop to paint and dry a compact sedan, and 5 days to paint and dry a sport-utility vehicle (SUV). There are 28 days for the paint shop to do work before it closes for system upgrade. What combination of sedans and SUVs can be inducted for painting and drying with the guarantee of being finished before the shop closes?

    a. Express the answer as a linear inequality

    Answer: __________________________________

    b. Draw its graph, showing the solution set (Use following grid, if desired)

    • Here’s how I’d do this one: Let c = number of compact sedans, s = number of SUVs. Then we have 4c + 5s < = 28. We also have s and c greater than or equal to 0. When we graph we get a triangle that start with the origin, goes into the first quadrant, and ends with the line 4c + 5s = 28 (a line with x-intercept (4, 0) and y-intercept (0, 28/5)). Then you'd have to take only the integer solutions for x (sedans) and y (SUVs), in that area, since you can't have part of a car. Here's a graph: Does that make sense? Lisa

  56. As well as this one. As i said my professor is not helping at all.

    Your cybersecurity firm is submitting a bid for a government contract. For a work shift at the cyber defense operations center, you determine that:

    • The “computer network defense (CND) analysis” task requires 8 work-hours network monitoring and 2 labor-hours of “active remediation of unauthorized activities”.
    • The “CND infrastructure support” task requires 2 work-hours for network monitoring and 1 work-hour for “active remediation of unauthorized activities”.

    The MAXIMUM work-hours specified per shift for network monitoring is 48 and for “active remediation” is 16. If x is the number of CND analysis tasks and y is the number of CND infrastructure support tasks performed per shift:

    a. Write a system of linear inequalities indicating appropriate restraints on x and y.

    • Here’s how I’d do this one: 8x + 2y <= 48 and 2x + 1y <= 16. Also x and y have to be greater than or equal to 0. Then you can graph the two equations to see what the feasible region is. Lisa

  57. Not sure what the objective function of this problem is…

    A calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100 scientific and 80 graphing each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing can be made daily. To satisfy a shipping contract, a total of at least 200 calculators must be shipped each day.

    If each scientific sold results in a $2.00 loss, but the graphing produces a $5.00 profit, ho many of each type should be made daily to maximize net profits?

    Pretty sure I got the inequalities right. Just not sure about the objective function.

    Thank you!!

  58. Having some problems trying to figure out this problem.
    Zach is planning to invest up to $50,000 in corporate and municipal bonds. The least he will invest in corporate bonds is $6000 and he does not want to invest more than $27,000 in corporate bonds. He also does not want to invest more than $34,650 in municipal bonds. The interest is 8.5% on corporate bonds and 6.8% on municipal bonds. This is simple interest for one year. What is the maximum income?
    Write the objective Function:
    Write the Constraints:
    Graph and find the region of feasible solutions:
    Identify all vertices:
    Find the value of the objective function at each vertex and record them in the box next to the graph:
    Solution:What is the benefit to using this method over another method like trial and error?

  59. What a great site! I am on the lookout for some linear programming problems involving history. For example I would love to tie it to the idea of linear programming during WWII. Any ideas or resources?

  60. I am trying to help my son with this problem and need help understanding it so that I can lead him to the answer but not give it to him ;-). So here’s the problem and what I have run across.

    Bob builds tool sheds. He uses 10 sheets of drywall and 15 studs for a small shed and 15 sheets of drywall and 45 studs for a large shed. He has available 60 sheets of drywall and 135 studs. If Bob makes $390 profit on a small shed and $520 on a large shed, how many of each type of building should Bob build to maximize his profit?

    When putting in the constraints and looking at the counter points, most of these points are giving decimal points (such as 23.67 small building – just an example, not from the problem). Since this is not possible, have I done something wrong? I used constraints of 10x + 15y less than or equal to 60; and 15x + 45y is less than or equal to 135

    Help?

  61. Pendell Home Products produces decorative wood frame doors (D) and windows (W). Each item produced goes through three manufacturing processes: cutting, sanding, and finishing. Each door (D) produced requires 1 hour in cutting, 30 minutes in sanding, and 30 minutes in finishing. Each window (W) requires 30 minutes in cutting, 45 minutes in sanding, and 1 hour in finishing. Next week, Pendell Home Products has 40 hours of cutting capacity available, 40 hours of sanding capacity, and 60 hours of finishing capacity. Assume all doors (D) produced can be sold for a profit of $500 and all windows (W) can be sold for a profit of $400.

    a. Formulate an LP model for this problem.
    b. Sketch the feasible region for this problem. Label all axes, constraints, extreme point(s) and the feasible region.
    c. Indicate the optimal objective function value at the extreme point.

    Ive tried to work through other problems on the site but I am still so confused

    • Thanks for writing! Here’s how I’d set up this problem: Maximize 500D + 400W such that: 1D + .5W <= 40, .5D + .75W <= 40, .5D + 1W <= 60. Does that make sense? Let me know if you want me to try to graph and solve it. Lisa

  62. Hi!

    Could you explain this question? I am stuck on how to correctly set up the constraint equations.

    HiRise Construction can bid on two 1-year projects. The following table provides the quarterly cash flow (in millions of dollars) for the two projects.
    Cash flow (in millions of $) at 1/1/08 4/1/08 7/1/08 10/1/08 for Projects 1 and 2 are/; -1.0 -3.0 -3.1 -2.5 -1.5 1.5 1.8 1.8 12/31/08 5.0 2.8
    HiRise has cash funds $1 million at the beginning of each quarter and may borrow at most $1 million at a 10% nominal annual interest rate. Any borrowed money must be returned at the end of the quarter. Surplus cash can earn quarterly interest at an 8% nominal annual rate. Net accumulation at the end of one quarter is invested in the next quarter. (i) Assume that HiRise is allowed partial or full participation in the two projects. Determine the level of participation that will maximize the net cash accumulated on 12/31/2008. Is it possible in any quarter to borrow money and simultaneously end up with surplus funds?

    I appreciate your assistance!

  63. A small nation has shifted its economy from farming to the production of food products, yarn (woven from wool or cotton) and clothes. The nation exports the excess of its production of these goods over the amount it consumes. In working out the plan, assume that world market prices are $3 per unit for food, $10 per unit for yarn and $25 per unit for clothes. The volumes the nation exports have a negligible impact on market prices. It wishes to work out its production plan for the next year such that net dollar value of exports is maximized, given that there are constraints on production, labour and available land.

    To produce each unit of food the nation must import $0.50 worth of goods (farm machinery and fertilizer), consume 0.2 units of food (e.g. animal feed), consume 0.5 units of labour and use 0.9 units of land. To produce each unit of yarn, the nation must import $1.25 worth of goods and consume 1 unit of labour and 1.5 units of land. To produce each unit of clothes, the nation must import $5.00 worth of goods and consume 1 unit of yarn and 4 units of labour.

    The population consumption is food (billions of units) 11.5, yarn (billions of units) 0.6, Clothes (billions of units) 1.2. The nation has 65 billion units of labour, 27 billion units of land available for farming and has the capacity to produce 9.6 billion units of clothes annually. The economic policy calls for maximizing the net dollar value of exports for the coming year, exports being the excess of production over domestic consumption.

    How would I formulate this in an LP problem?

  64. A shopkeeper orders packets of soap powder. The cost price of a large packet is 180 dollars and that of a small packet is 80 dollar. She is prepared to spend up to 4000 dollars altogether and needs twice as many large packets as small packets with a minimum of 10 large and 20 small packets. What is the greatest number of packets she can buy? The profit is 20 dollars on a large packet and 10 dollars on a small packet. Which arrangement gives the greatest profit? What is that profit?

  65. A farmer has a garden of 10 hecors to plant rice and wheat. He has to plant at least 7 hectors. However he has at most $1200 to spend, and each hector of wheat cost $200 to plant and each hector of rice costs $100 o plant. More over the farmer has to get the work done in a time not more than 12 hours. It takes 1 hour to plant each hector of wheat and takes 2 hours to plant each hector of rice. If the profit is $500 per hector of wheat and $300 per hector of rice, how many hectors of each kind should be planted to maximize the profit?

  66. Hey there can you please help me to solve the word problem you really know how to simplify things

    small petroleum company owns two refineries.Refinery 1 costs R20000 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.the company has orders totalling 25000 barrels of high grade oil, 27000 barrels of medium grade oil and 30000 barrels of low grade oil.The company wants to determine how many days should it run each refinery to minimize its costs and still refine enough oil to meet its orders.formulate the above problem as an lp program and use simplex method to find the optimal solution

    • Thanks for writing! I’m sorry – we don’t cover the Simplex Method on this site; you’ll have to get help somewhere else. Thanks, Lisa

  67. Maximize: 2222x + 1545 = Z
    Subject to, 2x + 3y >=12, 4x + 5y >=20, x <=9, y=0 and y>=0
    Use the graphic method of linear programming to maximize the profit.

    Is there anybody who can send me the solution. Thanks for your nice cooperation.

    • Hi! Thanks for writing! What you do is graph the four inequalities and then find the points in the CORNERS of all the lines. Then use the x value and y value to plug in to the MAXIMIZE (objective) function to see which point makes that the largest. That’s the answer! Does that make sense? Lisa

  68. Pls Lisa am finding these questions confusing pls can u assist me.
    John must work at least 20hrs a week to supplement his income while attending school .in store 1,he can work between 5 and 12hrs a week, and in store 2 he is allowed between 6 and 10hrs. Both stores pay the same hourly wage. In deciding how many hours to work in each store, John wants to base his decision on work stress.based on interview with present employees, john estimates that, on an ascending scale of 1 to 10, the stress factors are 8 and 6 at store 1 and 2, respectively. Because stress mounts by the hour, he assumed that the total stress for each store at the end of the week proportional to the number of hours he works in the store. How many hours should john work in each store?

    • Thanks for writing. I found the answer to this on a weebly website. Does this answer make sense? Lisa
      Minimize Optimum solution Z = 8×1 + 6×2
      Subject to
      5 ≤ x1 ≤ 12
      6 ≤ x2 ≤ 10
      x1 + x2 ≤ 20
      x1, x2 ≥ 0
      (Using the graphical method as presented by 2.2A #4, however searching for minimization point; and neglecting (12,10)(infeasible))
      1. x1 = 12, x2 = 8; Z = 144
      2. x1 =10, x2 = 10; Z = 140  Optimum mix

  69. HELP

    Local restaurant would like to determine best way to allocate monthly advertising of $1,000.00 between newspaper and radio. 25% must be budgeted for each type. Marketing consultant has devised a index for advertising from 0-100. Higher values demonstrate more exposure. If the value for newspaper is 50 and advertising is 80, how should restaurant allocate advertising for maximum exposure.

  70. Hi Lisa, I really need your help to solve this question.

    *Write the linear programming model: objective function and its constraints.
    *Draw the feasible solution for the LP problem.
    The Janie Gioffre Drapery Company makes three types of draperies at two different locations. At location A, it can make 10 pairs of deluxe drapes, 20 pairs of better drapes and 13 pairs of standard drapes per day. At location B, it can make 20 pairs of deluxe, 50 pairs of better and 6 pairs of standard per day. The company has orders of 2000 pairs of deluxe drapes, 4200 pairs of better and 1200 pairs of standard. The daily costs are $500 per day at location A and $800 per day at location B. How many days should Janie schedule at each location in order to fill the orders at minimum cost? What is the minimum cost?

  71. I’m in a bus to school now and we are writing a semester test on linear programming, I’m so glad I bumped into this post before writing. It was very helpful, thank you!!

    • Thanks so much for the kind words, and I’m so glad the web page helped you! Have fun in math class, and please spread the word about shelovesmath.com 😉 Lisa

  72. Hi! Can you help me solve this problem?

    MSA Computer Corporation manufactures two models of minicomputers, the Alpha 4 and the Beta 5. The firm employs five technicians, working 160 hours each per month, on its assembly line. Management insists that full employment (i.e. all 160 hours of time) be maintained for each worker during next month’s operations. It requires 20 labor hours to assemble each Alpha 4 computer and 25 labor hours to assemble each Beta 5 model. MSA wants to see at least 10 Alpha 4s and at least 15 Beta 5s produced during the production period. Alpha 4s generate $1200 profit per unit, and Beta 5s yield $1800 each. Determine the most profitable number of each model of minicomputer to produce during the coming month.

  73. A Rancher’s Problem

    A rancher has 1,000 acres on which to grow corn or barley, or feed cattle. Information on
    the two crops under consideration is given below:

    Corn Barley
    Cash costs per acre 100 120
    Labor hours per acre 10 8
    Yield (bushel per acre) 120 100
    Selling price per bushel 4.25 5.25
    Purchase price per bushel 4.50 n/a

    The rancher can plant the land in any mix of the two crops; or the land can be used to
    raise young steers and feed them for a year. The steers cost $150 each and are sold for $800
    after feeding. Each steer requires 20 hours of labor, one half acre of land, and 80 bushels of
    corn. The corn for the steers can be that grown on the ranch if enough is available, or it can
    be purchased.

    The rancher can use either inexperienced labor, which costs $6 per hour, or experienced
    labor at a rate of $10 per hour. Each hour of inexperienced labor requires 0.15 hours of
    supervision, and each hour of experienced labor requires 0.05 hours of supervision. There
    are 2,000 hours of supervision available.

    The rancher is limited to funds totaling $200,000 for buying seed and other cash costs
    for crops, for purchasing steers, and for paying labor.

    Suppose that the rancher wants to maximize his profit for the season and is willing to
    undertake any land usage plan, as long as it is feasible.

  74. An investor has up to $250,000 to invest in three types of investments.
    Type A pays 8% annually and has a risk factor of
    0. Type B pays 10% annually and has a risk factor of 0.06.
    Type C pays 14% annually and has a risk factor of 0.10. To
    have a well-balanced portfolio, the investor imposes the following
    conditions. The average risk factor should be no
    greater than 0.05. Moreover, at least one-fourth of the total
    portfolio is to be allocated to Type A investments and at least
    one-fourth of the portfolio is to be allocated to Type B investments.
    How much should be allocated to each type of investment
    to obtain a maximum return?

  75. An accounting firm has 900 hours of staff time and 100 hours
    of reviewing time available each week. The firm charges
    $2000 for an audit and $300 for a tax return. Each audit
    requires 100 hours of staff time and 10 hours of review time,
    and each tax return requires 12.5 hours of staff time and 2.5
    hours of review time. What number of audits and tax returns
    will bring in a maximum revenue?

  76. A company has budgeted a maximum of $600,000 for advertising
    a certain product nationally. Each minute of television
    time costs $60,000 and each one-page newspaper ad costs
    $15,000. Each television ad is expected to be viewed by 15
    million viewers, and each newspaper ad is expected to be
    seen by 3 million readers. The company’s market research
    department advises the company to use at most 90% of the
    advertising budget on television ads. How should the
    advertising budget be allocated to maximize the total audience?

  77. Hi! I need help solving this problem…

    Lisa is making cookies to sell at the Annual Dirt Bike Competition. A dozen oatmeal
    cookies require 3 cups of flour and 2 eggs. A dozen sugar cookies require 4 cups of
    flour and 1 egg. She has 40 cups of flour and 20 eggs. She can make no more than 9
    dozen oatmeal cookies and no more than 7 dozen sugar cookies, and she earns $3 for
    each dozen oatmeal cookies and $2 for each dozen sugar cookies. How many dozens
    of each type of cookie should she make to maximize her profit?

    • Sorry I took so long to answer. Here’s how I’d set this up: Maximize 3x + 2y, such that 3x + 4y <= 40, 2x + 1y <= 20, 0 <= x <= 9, 0 <= y <= 7. Does that make sense? Lisa

  78. please help answer this
    A manufacturer will produce two products A and B, product A has a contribution of shs 3/unit and product B has a contribution of shs 4/unit. The manufacturer wishes to establish a weekly production plan which maximizes profit.

    Because of trade agreement, sale of product A are limited to a weekly maximum of 20 units and to honor an agreement with a well established customers, at least 10 units of part B must be sold per week. Develop a linear program and hence find the optimal solution.

    • Thanks for writing. I would set this up like this: Max 3x + 4y, such that 0 < = x <= 20, y >= 10. Then draw graphs, find corner points, and plug in to see which points yields the maximum profit. Lisa

  79. Hi,

    i have trouble to solve this question. Your help would be appreciated

    Howard Mechanical Corporation manufactures two products. Each product involves a certain amount of broaching, grinding and polishing as detailed below:
    Product A B
    Broaching 8 hours 10 hours
    Grinding 5 hours 4 hours
    Polishing 10 hours 6 hours

    For the coming month, the corporation has maximum available capacities of 600 hours for polishing, 800 hours for broaching and 400 hours for grinding.
    Product A is fabricated from 5 kg of aluminium; product B requires 2 kg of brass. For the coming month, 100 kg of brass and 350 kg of aluminium are available for use in the corporation.
    If the same profit, $800, is derived from the sale of each product,
    (i) What is the optimum combination of products for the coming month and what profit will be made?
    (ii) Two proposals have been submitted for your consideration. The first proposal recommends the purchase of additional supplies of aluminium for the coming month and the second proposal recommends that additional people should be employed on a casual basis for polishing. Is either proposal worth further investigation? Justify your answer by indicating the increased potential profit associated with each of the two proposals.

  80. Hey Lisa — great site. I am a long time teacher looking for a linear programming problem I can modify down for an Algebra 1 class to explore real world graphing of linear inequalities — that I can have the kiddies do by hand one day and as a Desmos exploration on a second day. Found lots of good ideas in the Q & A section of your blog. And thanks to you I won’t have to reinvent the wheel to produce the answer. Very rich site from multiple perspectives – much thanks!!!!

    • Thank you SO MUCH for your sweet note, and I’m glad you are enjoying the site. Please let me know if there’s anything I need to add or change, or there are any typos 😉 Please spread the word about SheLovesMath.com! Lisa

  81. Please help me on this… thank u..

    CASE 1 – APPLIED TECHNOLOGY INCORPORATED
    (Solve the case using linear programming graphical method)
    Applied Technology Inc. (ATI) produces bicycle frames using two fiberglass materials that improve the strength-to-weight ratio of the frames. The cost of the standard grade material is USD7.50 per meter and the cost of the professional grade material is USD9.00 per meter. The standard and professional grade materials contain different amount of fiberglass, carbon fiber, and Kevlar as shown in the following table.
    Material Standard Grade Professional Grade
    Fiberglass 84% 58%
    Carbon Fiber 10% 30%
    Kevlar 6% 12%
    The company signed a contract with a bicycle manufacturer to produce a new frame with a carbon fiber content of at least 20% and a Kevlar content of not greater than 10%. To meet the required weight specification, a total of 30 meters of material must be used for each frame.
    a. Formulate a linear program to determine the number of meters of each grade of fiberglass material that ATI should use in each frame in order to minimize total cost. Define the decision variables and indicate the purpose of each constraint.
    b. Use the graphical solution procedure to determine the feasible region. What are the coordinates of the extreme points?
    c. Compute the total cost at each extreme point. What is the optimal solution?
    d. The distributor of the fiberglass material is currently overstocked with the professional grade material. To reduce inventory, the distributor offered ATI the opportunity to purchase the professional grade for USD8.00 per meter. Will the optimal solution change?
    e. Suppose that the distributor further lowers the price of the professional grade material to USD7.40 per meter, will the optimal solution change? What effect would an even lower price for the professional grade material have on the optimal solution? Explain.

  82. please help me…

    CASE 2 – BLUEGRASS FARM CORPORATION
    (Solve the case using linear programming simplex method)

    The BlueGrass Farm, located in Alitagtag, Batangas, has been experimenting with special diet for its horses. The feed components available for the diet are a standard horse feed product, an enriched oat product, and a new vitamin and mineral feed additive. The nutritional values in units per kilo and the costs for the three feed components are summarized in the table.
    Feed Component Standard Enriched Oat Additive
    Ingredient A 0.8 0.2 0.0
    Ingredient B 1.0 1.5 3.0
    Ingredient C 0.1 0.6 2.0

    Cost per kilo 25 50 300

    The minimum daily diet requirements for each horse are 3 units of ingredient A, 6 units of ingredient B, and 4 units of ingredient C. In addition, to control the weight of horses, the total daily feed for a horse should not exceed 6 kilos.
    Determine the minimum cost mix that will satisfy the daily diet requirements of horses.

  83. Hi Lisa! Can you help me with this problem?
    I’m not gonna write the whole thing but this part is giving me a hard time…How would write this constraint?
    The company doesn’t want to depend on just one type of bread so they have mandated that at least 25% of its total bread production must be from both types (Baguette and Ciabatta). Also, the amount of Ciabatta produced should not be more than half of the total production.

    Thank you so much for your help!

    • Here’s how I might set this up: Let b = amount of baguette produced and c = amount of ciabatta produced. Do you have a variable for the total amount of bread produced? Then we may have b + c is greater than or equal to .25T and c is less than or equal to .5T. Does that help? Lisa

  84. Hi Lisa, please help me with this question.
    A car repair workshop uses large numbers of filters, costing #300 each, and plugs, costing #400 each. The workshop owner allows #30000 for these parts and needs at least twice as many filters as plugs. There must also be at least 50 filters and 20 plugs.

    a). What is the largest numbers of parts that can be bought, and in what way?
    b). If it is decided to buy as many plugs as conditions allow, how many of each part can be bought?.

    • Thanks for writing. Here’s how I’d do this problem: 300f + 400p less than or equal to 30000. f >= 50, p >= 20, 2p <= f. Maximize f + p. Graph, find corner points, and see which point allows the maximize number of parts. Does that make sense? Lisa

  85. Henry arrived at school late because her car broke down, and therefore, has only 45 min to complete a history exam. the exam has 2 open-ended question and 30 multi choice questions. Each correct open-ended question is worth 20 points, and each multi-choice question is worth 2 points. she knows it usually takes her 15 min to answer open ended question and only one min to answer a multi choice question. assume that for each question henry answers, she receives full credit. how many of each type of question should she answer to receive the maximum possible points? what is the maximum possible points that henry can receive?

  86. Angeline’s Handicraft manufactures two products in two departments. Product X contributes P26 and takes 6 hours in department 1, and 6 hours in department 2. Product Y contributes P50 and takes 8 hours in department 1 and 12 hours in department 2. Department 1 has a capacity of 38 hours and department 2 has a capacity of 42 hours. Indicate the maximum level in units and maximum profit contribution production level.

    • Here’s how I’d set this up:
      Max 26x + 50y
      such that
      6x + 8y <= 38 6x + 12y <= 42Draw graphs, find corner points, and plug in objective function to find maximum level and profit.

  87. Given the equivalent inequality which results from multiplying both sides of the given inequality by -2: -4x < 10
    There is a line under the sign, < but it won't let me put it

    • Thanks for your question, and sorry the editor doesn’t work well. Multiplying the equation by -2 yields 8x is greater than or equal to -20. Does that make sense? Lisa

  88. Hi Lisa please help me answer this question .
    After making various presentations to potential investors, MC has finally hit the jackpot. An investor wants to invest 100,000,000 pesos in MC.

    After doing all your staff work in the previous weeks, the two of you have been tasked to find the most efficient way to invest the money. This involves making the highest profit possible from the money you have invested. You have decided to consult your friend Olivia, who is also a financial investor.

    Olivia discusses that you need to use a mathematical method called Linear Programming to solve this problem. She explains that Linear Programming is a method that has to be set up very carefully. It involves creating a function to optimize and modelling various constraints using linear equations. She has decided to present you the profitability percentages of MC, given that she has already done prior work with MC.

    The startup arm is projected to return at least 12%. To minimize risk, you must invest no more than 30,000,000 pesos. The financial stocks return 3%, while the retail arm returns 5%.

    For tax reasons, you must invest at least 3 times more in the financial stocks than the retail arm.

    Your task is to set up the linear program. You have to define the following:
    The variables that will be used
    The optimization function that will be used
    What kind of optimization will be done: minimization or maximization?
    The constraints that the linear program will be subjected to

    The model will be presented to the MC CEO, who will verify if the model is satisfactory.

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