Mastering Managerial Decision Analytics: Simplex & MLE Techniques
School
City University of Hong Kong**We aren't endorsed by this school
Course
MS 3128
Subject
Management
Date
Dec 11, 2024
Pages
6
Uploaded by DukeMusicKudu31
Assignment 2MS3128 Managerial Decision AnalyticsQ1. a.) Convert the following linear program to standard form of Simplex method. [10]Maximize z=3x1+2x2+4x3Subject to:2x1+x2−x3≤83x1+4x2+x3≥10x1−2x2+3x3=6x1isfree ,1≤ x2≤5,x3≥2Ans:Minimize z'=−3x'1+3x' '1−2x'2−4x'3Subject to:2x'1−2x' '1+x'2−x'3+s1=93x'1−3x' '1+4x'2+x'3+e1=4x'1−x''1−2x'2+3x'3+e2=2x'2+s2=5x'1, x' '1,x '2,x '3,s1,s2,e1,e2≥0b.) Find all basic solutions of x1 and x2for the following linear program. [10]Maximize z=5x1+4x2Subject to:x1+2x2≤83x1+2x2≤12x1≥0, x2≥0Ans:1.Set x1= 0, x2= 0: s1 = 8, s2 = 12, x1= 0, x2= 02.Set x1= 0, s1= 0: x2 = 4, s2 = 4, x1= 0, x2= 43.Set x1= 0, s2= 0: x2 = 6, s1 = -4, x1= 0, x2= 6 (infeasible)4.Set x2= 0, s1= 0: x1 = 8, s2 = -12, x1= 8, x2= 0 (infeasible)5.Set x2= 0, s2= 0: x1 = 4, s1 = 4, x1= 4, x2= 06.Set s1= 0, s2= 0: x1 = 2, x2 = 3, x1= 2, x2= 3
Q2. Given the function f(x , y)=x2+xy+y2+3x+2ya.)determine if it is convex. [10]Ans:fxx=∂2f∂ x2=2, fyy=∂2f∂ y2=2, fxy=∂2f∂x ∂ y=1, fyx=∂2f∂ y ∂x=1The Hessian matrix H is H=[2112]Convexity TestzT∇2f(x, y)z≥0for all z;zT∇2f(x, y)z=(z1z2)(2112)(z1z2)¿(z1z2)(2112)(z1z2)¿2(z12+z22+z1z2)≥0for all zf(x , y)isconvexb.)Shows the first 3 iterations of gradient descent for minimizing f(x , y)by initializing(xo, yo)=0,0and learning rate = 0.1. [10]Ans:∇f(x , y)=(2x+y+3,x+2y+2)Iteration1: ∇f(0,0)=(2(0)+0+3,0+2(0)+2)=(3,2)x1=0+0.1(3)=−0.3y1=0+0.1(2)=−0.2Iteration2: ∇f(−0.3,−0.2)=(2(−0.3)+(−0.2)+3,−0.3+2(−0.2)+2)=(2.2,1.3)x2=−0.3+0.1(2.2)=−0.52y2=−0.2+0.1(1.3)=−0.33Iteration13: ∇f(−0.52,−0.33)=(2(−0.52)+(−0.33)+3,−0.52+2(−0.33)+2)=(1.63,0.82)x2=−0.52+0.1(1.63)=−0.683y2=−0.33+0.1(0.82)=−0.412
Q3. Suppose you have a dataset consisting of the following five observed values: 2, 3, 3, 4, and 5. Assume these values are drawn from a normal distribution with unknown mean μand known variance σ2=1.a.)Write the likelihood function for the dataset.[8]Ans: L(μ)=∏i=1n1√2πe−(x¿¿i−μ)22¿Given the data point x1=2,x2=3,x3=3, x4=4, x5=5, the likelihood function becomes:L(μ)=(1√2π)5e−(2−μ)22e−(3−μ)22e−(3−μ)22e−(4−μ)22e−(5−μ)22b.)Derive the maximum likelihood estimator (MLE) for the mean μ. [10]Ans: lnL(μ)=−52ln(2π)−12∑i=15(xi−μ)2Differentiate the log-likelihood with respect to μand set the derivative to zero:ddμlnL(μ)=∑i=15(xi−μ)=0Solving for μ, we get:μ=15∑i=15xic.)Calculate the MLE using the observed data. [2]Ans: μ=2+3+3+4+55=175=3.4
Q4.You are given a dataset with the following observations:Feature 1Feature 2Feature 3ClassAHigh1YesALow2NoBHigh1NoBLow3NoBHigh2YesAHigh3Yesa.)Construct a decision tree based on the best split determined by Gini impurity. [10]Ans: Determine root node:Gini impurity of Feature 1: 36[1−(23)2−(13)2]+36[1−(13)2−(23)2]=0.445Gini impurity of Feature 2: 46[1−(34)2−(14)2]+26[1−(02)2−(22)2]=0.25Gini impurity of Feature 3: 26[1−(12)2−(12)2]+26[1−(12)2−(12)2]+26[1−(12)2−(12)2]=0.5The root node is Feature 2:Gini impurity of Feature 1 given Feature 2 is root node:24[1−(22)2−(02)2]+24[1−(12)2−(12)2]=0.25Gini impurity of Feature 3 given Feature 2 is root node:24[1−(12)2−(12)2]+14[1−(11)2−(01)2]++14[1−(11)2−(01)2]=0.25Or
b.)Construct a Neural network with 1 hidden layer with 3 neurons and train the model by using excel. [10]Ans: See Excel.
Q5. Two tech companies, AlphaTech and BetaCorp, are deciding whether to invest in developing a new AI technology. They can either collaborate or work independently. The payoff matrix is as follows:BetaCorp: CollaborateBetaCorp: IndependentAlphaTech: Collaborate8 , 82 , 6AlphaTech: Independent6 , 24 , 4a)Determine the Nash Equilibrium of the game. [8]Ans: Collaborate, Collaborate (8,8): Both companies collaborate, achieving the highest payoff for both, Neither can benefit by changing their strategy unilaterally. Independent, Independent (4,4): Both work independently. Again, neither can improve their outcomes by changing their strategy alone.b)Discuss the strategic implications for both companies if they choose to collaborate versus work independently. [6]Ans:Collaboration: Leads to the highest mutual benefit, fostering innovation and shared resources.Independent: Proviodes a safe, moderate payoff, reducing risks associated with dependency on the other company.c)If you were the CEO of AlphaTech, which strategy would you prefer and why? Consider both short-term and long-term impacts. [6]Ans:Preferred Strategy: Collaborate, due to the higher payoff(8) compared to independence(4)Short-term: Immediate benefits from shared development costs and resources.Long-term: Potential for stronger market positioning and innovation leadership.