Please show work, thanks. Problem 2 A trucking company routinely transports toma
ID: 367349 • Letter: P
Question
Please show work, thanks.
Problem 2 A trucking company routinely transports tomatoes from Devis and Bakersfield to San Francisco and Los Angeles, the demand and supply in each location are shown below: Supply Demand 400 A. Davis 1. SF 800 B. Bakersfield LA Let XA1 be the quantity sh Davis to SF CA1-$3 (thousands of dollars) [Hint for simplicity, use $3 in calculation) CA: $12 (thousands of dollars) CB1-$9 (thousands of dollars) CB2 $9 (thousands of dollars) The goal of the company is to minimize total shipping cost in order to meet demand subject to supply availability 1) Fommulate the Primal LP 2) What is the ecceomic meaning of the shadow price of each constraint in Primal LPYExplanation / Answer
(2)
Primal
min.Z = 3XA1 + 12XA2 + 9XB1 + 9XB2
Subject to,
XA1 + XA2 400 OR, -XA1 - XA2 -400
XB1 + XB2 800 OR, -XB1 - XB2 -800
XA1 + XB1 300
XA2 + XB2 900
XA1, XA2, XB1, XB2 0
Shadow price stands for the rate of change of the objective function value for one unit change of capacity of a constraint (i.e. its RHS).
(3)
Dual
max. W = -400YA - 800YB + 300V1 + 900V2
Subject to,
-YA + 0YB + V1 + 0V2 3
-YA + 0YB + 0V1 + V2 12
0YA - YB + V1 + 0V2 9
0YA - YB + 0V1 + V2 9
YA, YB, V1, V2 0
(4)
Given that XA1 has a positive (non-zero) optimal solution, the first constraint of the dual will be binding i.e. we can write
-yA + 0yB + v1 + 0v2 3 as -yA + 0yB + v1 + 0v2 = 3
or, -yA + v1 = 3 -------------(1)
Given that v1 = 3, from (1), we can say that yA = 0
Again note the second constraint of the dual. it is -
-yA + 0yB + 0v1 + v2 12
or, simply, yA + v2 12
Since yA = 0, v2 12
As this is a binding constraint (as XA2 has a positive optimal value of 100), we will replace the '' with '=' sign and confirm that v2 = 12
(5)
The demand of 1 was 300 which is already fulfilled by A as XA1=300, So, no more flow from B to 1 i.e. XB1=0.
(6)
Note that the cost incurred in flow of one unit from A to 1 is $3 and for B to 1 it is $9. So, there is a marginal loss of $9 - $3 = $6 in selecting location B for demand center 1.