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Systems of Linear Inequalities and Introduction to Linear Programming<br />
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Fundamental theorem of linear programming<br />Over the closed convex polygons, the maximum and minimum values of any linear expansion occur at corner points or vertices of the region or at all points along one side.<br />The inequalities that determine the solution region form the set of feasible solutions. <br />Any solution satisfies the constraints or restrictions placed on the variables of the given problem. <br />An optimal feasible solution is any feasible solution which optimizes the subject function under consideration. <br />
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APPLICATIONS<br />In real situations, x and y often represent quantities or amounts, which cannot be negative.<br />In this case our graphs are restricted to the first quadrant, where x and y are both nonnegative.<br />
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EXAMPLE 1:<br />On a particular airline flight a passenger is allowed to check a piece of luggage, without additional cost, provided the sum of the 3 dimensions (length, width, height) does not exceed 62 in. Suppose a piece of luggage has a height of 20in., and the length is greater than the width, but not more than twice its width. <br />If the length is x inches and the width is y inches, what is the system of linear inequalities involving x and y?<br />Graph the region of permissible values of x and y.<br />
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EXAMPLE 1:<br />On a particular airline flight a passenger is allowed to check a piece of luggage, without additional cost, provided the sum of the 3 dimensions (length, width, height) does not exceed 62 in. Suppose a piece of luggage has a height of 20in., and the length is greater than the width, but not more than twice its width. <br />If the length is x inches and the width is y inches, what is the system of linear inequalities involving x and y?<br />
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b. Graph the region of permissible values of x and y.<br />
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Exercise 1:<br />Suppose, on the flight in Example #1, that a passenger has 2 pieces of luggage and that she is allowed to check the two pieces without additional cost provided their total dimension do not exceed 106 in. The sum of the 3 dimensions of the larger piece is 58 in. Furthermore, the smaller piece has a height of 12in. And the length greater than the width but not more than 10 in. greater.<br />If the smaller piece of luggage has a length of x inches and a width of y inches, what is the system of linear inequalities involving x and y?<br />Graph the region of permissible values of x and y.<br />
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Example 2:<br />A plastic toy factory manufactures two types of toy cars – type A and type B. Regular orders require a daily output of at least 40 pieces of type A and 30 pieces of type B. It takes 1.5 man hours to produce a type A car and 2 man hours to produce a type B car . The staff can work no more than 204 man hours per day.<br />A. Represent the given information in the form of inequalities and graph them to show the possible numbers of each type that can be produced each day.<br />B. If the profit on each type A car is P30 and on each type B car is P20, find the number of each type that should be produced to maximize profit granting that all toy cars can be sold. <br />
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Example 2:<br />A plastic toy factory manufactures two types of toy cars – type A and type B. Regular orders require a daily output of at least 40 pieces of type A and 30 pieces of type B. It takes 1.5 man hours to produce a type A car and 2 man hours to produce a type B car . The staff can work no more than 204 man hours per day.<br />A. Represent the given information in the form of inequalities and graph them to show the possible numbers of each type that can be produced each day.<br />Solution:<br />Let x be the number of type A toy cars produced per day.<br />Let ybe the number of type B toy cars produced per day.<br />
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Example 2:<br />A plastic toy factory manufactures two types of toy cars – type A and type B. Regular orders require a daily output of at least 40 pieces of type Aand 30 pieces of type B. It takes 1.5 man hours to produce a type A car and 2 man hours to produce a type B car. The staff can work no more than 204 man hours per day.<br />Solution:<br />Let x be the number of type A toy cars produced per day.<br />Let ybe the number of type B toy cars produced per day.<br />Constraints:<br />x ≥ 40<br />y ≥ 30<br />1.5x + 2y ≤ 204 <br />x, y ≥ 0<br />
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Example 2:<br />Graph the constraints to show the possible numbers of each type that can be produced each day. :<br />x ≥ 40<br />y ≥ 30<br />1.5x + 2y ≤ 204 <br />
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Example 2:<br />Graph the constraints to show the possible numbers of each type that can be produced each day. :<br />x ≥ 40<br />y ≥ 30<br />1.5x + 2y ≤ 204 <br />
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Example 2:<br />B. If the profit on each type A car is P30 and on each type B car is P20, find the number of each type that should be produced to maximize profit granting that all toy cars can be sold. <br />Objective: maximize profit, P<br />P = 30x +20y ( in pesos)<br />Evaluate the value of P at each vertex to find the maximum profit<br />P = 30x + 20y (Php)<br />
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Example 2:<br />Evaluate the value of P at each vertex to find the maximum profit.<br />P = 30x + 20y (Php)<br />A(40, 30)<br />P= 30(40) + 20(30) <br />P = P1,800<br />B(40, 72)<br />P = 30(40) + 20(72) <br />P = P2,640<br />96 type A and 30 type B toy cars must be produced per day to maximize profit. <br />C(96, 30)<br />P = 30(96) + 20(30) <br />P = P3, 480<br />
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Exercise 2:<br />Tommy assembles stereo equipment for resale in his shop. He offers two products, mp3 and DVD players. He makes a profit of $10 on each mp3 and $6 on each DVD. Both must go through two steps in his shop, assemble and bench checking. An mp3 player takes 12 hours to assemble and 4 hours to bench check. A DVD player takes 4 hours to assemble but 8 hours to bench check. Looking at this month’s schedule, Tommy sees that he has 60 assembly hours uncommitted and 40 hours of bench checking time available. Find the best possible combination of mp3 and DVD players to maximize profit.<br />
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Example 3:<br />An animal feed is to be a mixture of foodstuffs, each unit of which contains protein, fat, and carbohydrates in the number of grams:<br />Each bag of the resulting mixture is to contain at least 400g of protein, 21g of fat, and at most 1,400g of carbohydrates.<br />Find the mixtures that meet these requirements. If each gram of foodstuff A costs P100 and each gram of foodstuff B costs P200, how much of each foodstuff should be used to get the least cost? <br />
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Example 3:<br />Solution:<br />Let x be the number of units of foodstuff A<br />Let y be the number of units of foodstuff B<br />The following are the constraints, demands, or restrictions placed on the variables of our problem:<br />Protein: at least 400g<br />100x + 50 y ≥ 400<br />
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Example 3:<br />Solution:<br />The following are the constraints, demands, or restrictions placed on the variables of our problem:<br />Fats: at least 21g<br />x + 9y ≥ 21<br />Carbohydrates: at most 1,400g<br />100x + 300y ≤ 1400<br />Nonnegativity constraint<br />x ≥ 0<br />y ≥ 0<br />
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Example 3:<br />Solution:<br />The cost for each combination could be represented by<br />C = 100x + 200y in pesos.<br />Objective: minimize cost, C = 100x +200y.<br />
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Example 3:<br />Solution:<br />Graph the system of inequalities.<br />100x + 50 y ≥ 400<br />x + 9y ≥ 21<br />100x + 300y ≤ 1400<br />x ≥ 0<br />y ≥ 0<br />
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Example 3:<br />Evaluate the value of C at each vertex to find the minimum cost.<br />C = 100x + 200y (Php)<br />A(3, 2) <br />C = 100(3) + 200(2) C = P700<br />B(2, 4)<br />C = 100(2) + 200(4) <br />C = P1,000<br />C(21/2, 7/6)<br />C = 100(21/2) + 200(7/6) <br />C = P1,283<br />The least cost is P700, that is when 3 units of foodstuff A and 2 units of foodstuff B are taken.<br />
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EXERCISE 3:<br />A mixture of food A and food B is to be made so that it contains at least 45 oz of nutrient 1 and 40 oz of nutrient 2. The cost per pound of A is $4 and each pound of A contains 1 oz of nutrient 1 and 2 oz of nutrient 2. Food B costs $8 per pound and each pound of B contains 1.5 oz of nutrient 1 and 0.5 oz of nutrient 2. If the weight of the mixture must not exceed 40 lb., how many pounds of each food should be used so that the total cost is a minimum?<br />
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HOMEWORK #5:<br />A nut packager has on hand 121 kilos of peanuts and 49 kilos of cashew nuts. He can sell 2 kinds of mixtures of these nuts; a cheap mix is 80% peanuts and 20% cashew nuts and a party mix is 30% peanuts and 70% cashew nuts. He can sell the party mix at 160 pesos per kilo and the cheap mix at 100 pesos per kilo. How many kilos of each should he make in order to get the best revenues?<br />
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homework #5:<br />A canteen is minimizing the cost of its beef and pork menus. The average menu requires 1 kg of lean meat and 0.75 kg of fat meat per person per week. The beef which costs P150 per kg is 20% fat and 80% lean. The pork which costs P120 per kg is 60% fat and 40% lean. If the canteen has 300 customers on these menus and if it cannot purchase more than 625 kg. of meat per week because of refrigeration space, how many kilograms of beef and how many kilograms of pork should be purchased to keep the cost at a minimum?<br />
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HOMEWORK #5:<br />Machine A runs at a cost of P50 producing 100 bolts and 50 screws in 1 hour. Machine B runs for 1 hour at P56 producing 80 bolts and 60 screws. With a combined time of no more than 15 hours, how long should each of the machines run to produce an order of 100,000 bolts and 75,000 screws at the minimum operating cost? <br />
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HOMEWORK #5:<br />A company can advertise its product by using local radio and TV stations. Its budget limits the advertisement expenditures to $1000 a month. Each minute of radio advertisement costs $5 and each minute of TV advertisement costs $100. The company would like to use the radio at least twice as much as the TV. Past experience shows that each minute of TV advertisement will usually generate 25 times as many sales as each minute of radio advertisement. Determine the optimum allocation of the monthly budget to radio and TV advertisements.<br />
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HOMEWORK #5:<br />An electronic company manufactures 2 radio models, each on a separate production line. The daily capacity of the first line is 60 radios and that of the second is 75 radios. Each unit of the first model uses 10 pieces of a certain electronic component, whereas each unit of the second model requires 8 pieces of the same component. The maximum daily availability of the special component is 800 pieces. The profit per unit of models 1 and 2 is $30 and $20, respectively. Determine the optimum daily production of each model.<br />
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HOMEWORK #5:<br />A company produces two types of cowboy hats. Each hat of the first type requires twice as much labor time as does each hat the second type. If all hats are of the second type only, the company can produce a total of 500hats a day. The market limits daily sales of the first and second types to 150 and 200 hats. Assume that the profit per hat is $8 for type 1 and $5 for type 2. Determine the number of hats of each type to produce in order to maximize profit.<br />
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