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Unit 1 Linear programming

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Define: LINEAR PROGRAMMING – is a method for finding a minimum or maximum value of some quantity, given a set of constraints (Limits). Define: OBJECTIVE FUNCTION – the equation/quantity you are trying to maximize or minimize Define: FEASIBLE REGION – The area created by the system of inequalities (constraints)

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Example #1 Find the maximal and minimal value of z = 3x + 4y given the following constraints: Objective functions Feasible region constraints List vertices (intersecting points) (-1,-3) Z = 3(-1) +4(-3)= -15 (2,6) Z = 3(2) +4(6) = 30 (6,4) Z = 3(6) + 4(4) = 34 The maximum value occurs at point (6,4) & minimum value (-1, -3) GRAPH!

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Example #2 Given the following constraints, maximize and minimize the value of z = –0.4x + 3.2y List vertices (intersecting points) (0,2) = 6.4(5,2) = 4.4 (0,5) = 16(5,0) = -2 (1,6) = 18.8(4,0) = -1.6 GRAPH!

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Example #3 (2, -4) = 16(-3, -3) = 3 (0, -6) = 24(2,7) =-17 P = 2x – 3y

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linear programming Minimum for: C = 3x + 4y

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Extra practice Pg 160. #10-12, 22

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Linear programming Word problems

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You are making H-Dub T-shirts & Hats to sell for homecoming and under the following constraints. You have at most 20 hours to work You only have room to sell 60 items H-DUB HAT Takes 10 minutes to make Profit $6 H-DUB T-SHIRT Takes 30 minutes to make Profit $20 Time = 10x + 30y ≤ 1200 Amount= x + y ≤ 60 Real life= x ≥ 0 & y ≥ 0 Profit: P = 6x + 20y Constraints: x = hat y = shirt Objective function:

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Example #5 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 calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators much be shipped each day. If each scientific calculator sold results in a $2 loss, but each graphing calculator produces a $5 profit, how many of each type should be made daily to maximize net profits? Scientific = x Graphing = y Demand: x ≥ 100 y ≥ 80 Production: x ≤ 200 y ≤ 170 Shipping contract: x+y > 200 Objective function P = –2x + 5y, GRAPH!

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Example #6 You need to buy some filing cabinets. You know that Cabinet X costs $10 per unit, requires six square feet of floor space, and holds eight cubic feet of files. Cabinet Y costs $20 per unit, requires eight square feet of floor space, and holds twelve cubic feet of files. You have been given $140 for this purchase, though you don't have to spend that much. The office has room for no more than 72 square feet of cabinets. How many of which model should you buy, in order to maximize storage volume? cost: 10x + 20y < 140 space: 6x + 8y < 72 Real life: x ≥ 0 y ≥ 0 Objective function volume: V = 8x + 12y GRAPH!

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Extra practice # 13, 14, 16

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