Linear
Programming: Sections: Optimizing linear systems, Setting up word problems
The warehouse on the east side of town has eighty sheets in stock; the westside warehouse has fortyfive sheets in stock. Delivery costs per sheet are as follows: $0.50 from the eastern warehouse to Customer A, $0.60 from the eastern warehouse to Customer B, $0.40 from the western warehouse to Customer A, and $0.55 from the western warehouse to Customer B. Find the shipping arrangement which minimizes costs. Hmm... I've got four things to consider: east warehouse to Customer A But I only have two variables. How can I handle this? The variables obviously need to stand for the number of sheets being shipped, but I have four different sets of sheets. This calls for subscripts and explicit labelling: shipped from east warehouse to Customer
A: A_{e} Since Customer A wants 50 sheets and Customer B wants 70 sheets, then: A_{e} + A_{w}
= 50, so A_{w}
= 50 – A_{e} (I'll
call this Equation
I) Since the eastern warehouse can ship no more than eighty sheets and the western warehouse can ship no more than fortyfive sheets, then: 0 < A_{e}
+ B_{e} < 80 And the optimization equation will be the shipping cost: C = 0.5A_{e} + 0.6B_{e} + 0.4Aw + 0.55Bw Where does this leave me? The equations (labelled as Equations I and II above) allow me to substitute and get rid of two of the variables in the second inequality above, so: 0 < A_{e}
+ B_{e} < 80 Simplifying the second inequality above gives me the new system: 0 < A_{e}
+ B_{e} < 80 Multiplying through by –1 (thereby flipping the inequality signs) and adding 120 to all three "sides" of the second inequality, I get the new system: 0 < A_{e}
+ B_{e} < 80 Since A_{e}
+ B_{e} is no less
than 75 and is no more than 80, then these two inequalities reduce to
one: 75 < A_{e} + B_{e} < 80 I can also simplify the optimization equation: C = 0.5A_{e}
+ 0.6B_{e} + 0.4Aw + 0.55Bw Due to the size of the orders, I have: 0 < A_{e}
< 50 Since I have only two variables now, and since I'll be graphing with x and y, I'll rename the variables: Copyright © Elizabeth Stapel 20062011 All Rights Reserved x = A_{e} Then entire system is as follows:
The feasibility region graphs as: When you test the corner points, (5, 70), (10, 70), (50, 30), and (50, 25), you should get the minimum cost when you ship as follows: 5
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