Other
Exercises decision science
- Course
- Institution
Explanation of all exercises
[Show more]
1 review
By: svenkolk18 • 2 year ago
Seller
Follow
elisabertels
Reviews received
Content preview
Linear Programming problem (p.35-49):Example 1.1 of exercises and computer practicals:
1. Define the decision variable
Xb and Xc
2. Define the objective function
w = 60Xb + 10Xc
3. Add constraints & restrictions
Xb + 2Xc = 1500
4Xb + Xc = 2000
Xb + Xc = 1000
Non-negativity constraint = Xb ≥ 0 and Xc ≥ 0
4. Define the feasible are
0Xb + 2Xc = 1500 → Xc = 750
1Xb + 0Xc = 1500 → Xb = 1500
(1500,750)
0Xb + 1Xc = 2000 → Xc = 2000
4Xb + 0Xc = 2000 → Xb = 500
(500,2000)
0Xb + 1Xc = 1000 → Xc = 1000
1Xb + 0Xc = 1000 → Xb = 1000
(1000,1000)
5. Determine the objective coefficient and the iso-profit lines
𝑋𝑏 60 𝑋𝑏
w = 60Xb + 10Xc → c = (60,10) ( 𝑋𝑐 ) → c = ( 10 ) and x = ( 𝑋𝑐 )
6. Optimal solution
Xb* = 500
Xc* = 0
w = 60 x 500 + 10 x 0 = 30.000
Example 1.2
1. Decide the decision variables
X1 = kilorad of direction 1
X2 = kilorad of direction 2
2. Define the objective function
min(0.2X1 + 0.1X2)
3. Add constraints and restrictions
0.3X1 + 0.2X2 ≤ 1.2
0.4X1 + 0.6X2 = 2.4
0.6X1 + 0.3X2 ≥ 1.8
, Non-negativity constraint = X1 ≥ 0 and X2 ≥ 0
Example 2.1
a) 1) −x1 + 2x2 = − 1 → x1 = 0, x2 = -0.5 → (0,-5). x2 = 0, x1 = 1 → (1,0)
(2) 3x1 + 3x2 = 3 → x1 = 0, x2 = 1 → (0,1). x2 = 0, x1 = 1 → (1,0)
(3) −x1 + x2 = 2 → x1 = 0, x2 = 2 → (0,2). x2 = 0, x1 = -2 → (-2,0)
(4) x1 + x2 = 4 → x1 = 0, x2 = 4 (0,4). x2 =0, x1 = 4 → (4,0)
(5) 2x1 + 3x2 = 12 → x1 = 0, x2 = 4 → (0,4). x2 = 0 , x1 = 6 → (6,0)
b) Unbounded region = V1 is unbounded because the feasible area has infinite solutions.
Empty region = V3 set of feasible solutions is not possible. Because 1-4 lies within the
feasible area of V2 but 5 says it should be ≥ 12. (can’t be in two feasible areas at the same
time).
c)
V1 V2 V3
W1 X1* = 0, X2* = 2 X1* = 0, X2* = 2 infeasible
W* = -0 + 0,2 = 0,2 W* = 0x2
W2 Half line Line segments infeasible
W* = 102 𝑋1 0 1
X =( 𝑋2 ) = λ1+ ( 2 ) + λ2( 3 )
𝑋1 0 1
X = ( 𝑋2 ) = ( 2 ) + λ ( 1 ) λ1+λ2 ≥ 0
λ ≥0 w* = 102
W3 Solution is unbounded X1* = 1, X2* = 3 infeasible
W*3 = 8
V1, W1 V1, W2
Example 1.3
1. Decide the decision variables
Xa = price of A
Xb = price of B
, 2. Define the objective function
Max ❵ w= 1000Xa + 1500Xb
3. Add constraints and restrictions
15-Xa ≥ 10 - Xb → 15-Xa + Xb ≥ 10
25-Xb ≥ 20 - Xa → 25-Xa + Xb ≥ 20
Xa ≤ 15
Xb ≤ 25
Xa, Xb ≥ 0
Example 2.2
Sensitivity analysis:
3x1 + x2 = 9 → x1 = 0, x2 = 9 → (0,9). x2 = 0, x1 = 3 → (3,0)
x1 + x2 = 5 → x1 = 0, x2 = 5 → (0,5). x2 =0, x1 = 5 → (5,0)
Right hand side ranging
α X W
-infinity ≤ α≤ 1 X1* = 0, X2* = 5 W* = 5
1 ≤α ≤ 3 X1* = 2, X2* = 3 W* = 2α + 3
3 ≤α ≤ +infinity X1* = 3, X2* = 0 W* = 3α
How to find 1? → for which value do you find the yellow point instead of the blue point? → the
gradient needs to be perpendicual to the line:
αx1 + x2
x1 + x2
→ α/1 = 1/1 → α = 1
Example 2.3
3x1 + x2 = 9 → x1 = 0, x2 = 9 → (0,9). x2 = 0, x1 = 3 → (3,0)
x1 + x2 = 𝛃
, 𝛃 X W
-infinity ≤ 𝛃 ≤ 0 infeasible infeasible
0≤𝛃≤3 X1* = 𝛃, X2* = 0 W* = 2𝛃
3≤𝛃≤9 X1* = (-3+9)/2+𝛃 W* = 2X1 + X2
X2* = (3𝛃 -9)/2 W* = 2((-3𝛃+9)/(2+𝛃)) + (3𝛃-9)/2
9 ≤ 𝛃 ≤ infinity X1*= 0, X2* = 9 W* = 9
Exercise 1.4
Xc = # commercials
Xn = # newspapers
Xm = # magazines
Max ( np = 2Xc + Xn + 0.5Xm)
25Xc + 10Xn + 5Xm ≤ 2000
Xc + 0.6Xn + 0.4Xm ≥ 100
Xc,Xn,Xm ≥ 0
Excercise 3.1
min{w = 4x1 + 5x2 + 6x3}
x1 − 2x2 + 3x3 ≤ 6
−2x1 + 3x2 − x3 ≥ 7
x1 + x2 + x3 = −3
x1 0, x2 0, x3 free.
max{w = -4x1 - 5x2 - 6x3}
x1 − 2x2 + 3x3 + Y1 = 6
−2x1 + 3x2 − X3 - Y2 = 7
-x1 - x2 - x3 = 3
max{w = -4x1 + 5x2* - 6x3^+ + 6X3^-}
x1 + 2x2* + 3x3^+ - 3X3^- + Y1 = 6
−2x1 - 3x2* − X3^+ + X3^- - Y2 = 7
-X1 + X2* - X3^+ + X3^- = 3
Exercise 3.2
x1 + x2 + x3 = 1
−x1 + 2x3 + x4 = 4
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller elisabertels. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $4.32. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
56326 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now