P r q p r q p r q p z p q r are called decision

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P R Q P R Q P R Q P Z P , Q , R are called decision variables Z is the objective function to be maximized The set of inequalities are constraints
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16 Additional Constraints Non-negativity or lower bound Upper bound Each type of product has a maximum market demand 0 , 0 , 0 R Q P 60 40 100 R Q P
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17 Complete Linear Programming Model 0 , 0 , 0 60 , 40 , 100 2400 15 10 2400 16 6 15 2400 16 28 12 2400 10 10 20 to subjec 50 60 45 Maximize + + + + + + + + + + = R Q P R Q P Q P R Q P R Q P R Q P R Q P Z
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18 Important Assumptions in Linear Programming Divisibility (continuous model) Decision variables can be real numbers Proportionality (linear model) Objective functions and constraints are linear functions of decision variables , 2 2 1 1 n n x a x a x a + + + Where a i are fixed parameters in the problem, and x i are decision variables. Also written as: . 1 = n i i i x a
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19 General Form of LP = = n j j j x c 1 Z minimize or Maximize n j l x n j u x m i b x a j j j j i j n j ij ,..., 1 , ,..., 1 , ,..., 1 , subject to 1 = = = = = The number of variables is n The number of constraints is m (excluding lower and upper bounds)
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20 Optimal Solutions Optimal solution: P =81.28, Q =16.36, R =60 Optimal objective function value: Z = 7664 This solution is infeasible if production levels have to be integers!
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21 Business Meanings 60 60 : R Product 40 36 . 16 : Q Product 100 82 . 81 : P Product : bounds upper with Comparing 60 , 36 . 16 , 82 . 81 : Solution Optimal = = < = < = = = = R Q P R Q P Observation : Product R reaches its upper bound, implying that we should produce as many R as possible. Product Q is far from its upper bound, implying that it is the least profitable to produce Q
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22 Business Meanings Question 1: The company plans to make some advertisement to increase potential market demand. Which type of product should the company invest to advertise? Question 2: If we can increase the maximum market demand of product R, how much should we increase it? A new LP model: drop the constraint: R <=60 New optimal solution: P=78.42, Q=10.69, R=72.48, with total profit=7794.
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