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Econ 7322 Assessment

Creation date：2024-05-31 17:59:08

Belonging category：经济Economics

Tag Econ 7322

Assessment

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Econ 7322

Assessment
Q1
a.
TC=FC+VC
=18000+12000*0.9=$28800
TR=12000*$3.2=$38400
P=TR - TC= 38400-28800=$9600
b.
TC=TR
TFC+TVC=Price per cake*No.of cake
$18000+$0.9*No.of cake=$3.2*No.of cake
$1800=$2.3*No.of cake
So, No.of cake =18000/2.3=7826
c.
The revenue: 3.2x=y
Cost: 18000+0.9x=y
The x is number of cupcake and y is money So we can see the cross point is the
break-even
d.
7826/12000*100%=65.22%
So the maximum capacity break-even is 65.22%
e.
$18000+$0.9*No.of cake=$2.75*No.of cake
$1800=$1.85*No.of cake
So, No.of cake =18000/1.85=9730
Q2. a.
1. Decision variables: x1 = necklaces x2 = bracelets
2. Objective: Maximize profits
3. Constraints: Gold used ≤ Gold available
Platinum used ≤Platinum available
All variables ≥ 0, x2≤4
Max Z =300x1 + 400x2
s.t. 3x1 + 2x2 ≤ 18
2x1 + 4x2 ≤ 200
x1 ≥ 0, 4≤x2 ≥ 0
b.
c.
If x2=4, so 3x1 + 2x2 ≤ 18
2x1 + 4x2 ≤ 200 x1=2
Plug in Z’: 300x1 + 400x2 = 300*2 + 400*4
Z’=2200
So, will not the maximum demand for bracelets be met, will miss 200 unit.
d. If no bracelets being produced, so x2=0
The gold available is 18 units, platinum is 20 units
One necklaces need 3 units gold and 2 units platinum, so 18/3=6
20/2=10
6/3=2
10/2=5
So the max produce is make 6 necklaces, so profit =2*$300=$600
e.
So we can see, if increasing the profit on a bracelet from $400 to $600, the profit is $3000,
and if changing the platinum requirement for a necklace from 2 units to 3 units, the profit
is $2000.
Q3.
a. Decision variables: x1 = mine1 x2 = mine2
Objective: Minimize profits
Constraints: mine1 coasts ≥ mine1 coasts available
Mine2 coasts≥ mine2 coasts available
All variables ≥ 0
Min Z =12x1 + 8x2+24x3
s.t. 6x1 + 2x2 ≥12
2x1 +2x2 ≥8
4x1 +12x2 ≥24
x1 ≥ 0, x2 ≥ 0
b.
Q4.
a. Decision variables: x1 = permanent x2 = temporary
Objective: Minimize profits
Constraints: permanent coasts ≥ permanent coasts available
Temporary coasts≥ temporary coasts available
All variables ≥ 0
Min Z =64x1 + 42x2
s.t. 16x1 + 12x2 ≥450
0.5x1 + 1.4x2 ≤ 25
x1 + x2 ≤ 40
x1 ≥ 0, x2 ≥ 0
b.
c.
We can see the data is decreasing.
d.
The data shows more decreasing on objective.
e.
f.
Part3
Q1
a. Max g’y
Subject to B’y ≤ d
y ≥ 0
The LP problem:
Min d’x
Subject to Bx ≥ g
X ≥ 0
b.
Intuitive proof:
Primal is Unbounded: If the original problem is unbounded, this means that the objective
function can grow without bound while still satisfying all constraints.
Maximize g′y
subject to B′y≤d
y≥0
Being unbounded means there exists a feasible y such that g′y=∞
No Corresponding Dual Feasibility: If the primal target is unbounded, then there is no
limit in the dual to how far the primal target can grow. Therefore, the dual cannot find a
feasible solution corresponding to the infinite growth of the primordial.
minimize d′x
subject to Bx≥g
x≥0
If the dual is feasible, there exists x≥0 such that Bx≥g.
Contradiction: Using the fact that g′y=∞, multiply the dual constraint by y:
y′Bx≥y′g
x′By≥g′y=∞
But since B′y≤d and x≥0, we have:
x′B′y≤x′d
c.
Dual is Unbounded: If the dual problem is unbounded, this means that the dual objective
function can be reduced without limit while still satisfying all constraints.
minimize d′x
subject to
Bx≥g
x≥0
Being unbounded means there exists a feasible x such that d′x=-∞
No Corresponding Primal Feasibility: This means that there is no limit in the primal object
to how much the dual object's target can be reduced. Therefore, the primal cannot find a
feasible solution corresponding to the dual infinite reduction.
maximize g′y
subject to B′y≤d
y≥0

Contradiction: Using the fact that d′x=−∞, multiply the primal constraint by x:
x′B′y≤x′d=−∞
But since Bx≥g and y≥0, we have:
x′B′y≥x′g′
d.
Primal Problem:
maximize c′x
subject to Ax≤b
x≥0
Dual Problem:
minimize b′y
subject to A′y≥c
y≥0
Proof:
Primal Feasible and Dual Feasible: Since x0 and y0 are feasible for the primal and dual, we
have that:
Ax0≤b and x0≥0
A′y0≥c and y0≥0
Weak Duality: By weak duality, we know that for any feasible x and y:
c′x ≤ b′y
Given Condition: We are given that:
b′y0=c′x0
Putting It All Together: By combining the given condition with weak duality, we have:
c′x0=b′y0
Since c′x≤b′y for any feasible x and y, x0 and y0must be optimal.

Q2
a.
b.
c.
We can see from the sensitivity form data, the shadow price show more higher, so can
earn for each additional of that profit
d.
From question a we can see the process 2 is not produced in the optimal solution, need
to increase the profit for that product by the absolute value of the reduced cost in order
for it to be produced in the optimal solution.
Q3
1. Decision variables: x1 = midnight -4:00am
x2 =4:00am-8:00am
X3=start 8:00am
X4=start noon
X5=start 4:00pm
X6=8:00pm-midnught
2. Objective: Minimize profits
3. Constraints: midnight -4:00am ≥ 10
4:00am-8:00am ≥12
8:00am-noon ≥20
Noon-4:00pm ≥25
4:00pm -8:00pm ≥32
8:00pm-midnight ≥18
All variables ≥ 0
Min Z =80x1 + 500x2+420x3+300x4+270x5+210x6
s.t. x1≥10
x1 + x2 ≥ 12
X2 + x3 ≥20
X3+x4≥25
X4+x5≥32
X5+x6≥18
x123456 ≥ 0

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LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 FINC5090 MATH1052 SciComp Elec4622 FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 MAT 237 BISM2201 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SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum

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