1.1 For the problems below, indicate the degrees of freedom and the prob-

CHAPTER

ONE Introduction 1.1 For the problems below, indicate the degrees of freedom and the prob- lem type (LP, NLP, IP, etc.) a) maxf(x, y) = 3x+ 4y s.t. x+ 4y−z≤10 y+z≥6 x−y≤3

Solution: Assuming all the variables are real. This is an LP as the objective

function and constraints are linear. There are three variables, and three inequality constraints, so the degrees of freedom are three.b) minf(x, y) = 3∙x 2

• 4sin(y∙z)

s.t. x+ 4y≤10 y+z= 6 +π x−y≤3 z∈ {0, π/2, π}

Solution: Assuming all the variables are real. This is an NLP as the ob-

jective function is nonlinear and constraints are linear. There are three variables, one equality and two inequality constraints, so the degrees of freedom are two.c) min 4.35∙x 2 ∙y1+ 1.74∙x∙z∙y2−2.5∙k∙y3 s.t. x−z+k≤10 1 (Introduction to Applied Optimization, 1e Urmila M. Diwekar) (Solution Manual, For Complete File, Download link at the end of this File) 1 / 4

2Introduction to Applied Optimization y1+y2≤1 y2≤y3 x≤8 k≤7 x, k≥0 y1, y2, y3∈ {0,1}

Solution: This is an an MINLP as the objective function is nonlinear, there

are binary (integer) variables in the problem and constraints are linear. There are three continuous variables, three binary variables, and three inequality constraints (2 bounds), so the degrees of freedom are six.d) minσ 2 RB = Z 1

(RB− RB)

2 dF s.t.RB= Z 1

RB(θ, x, u)dF CA= CAi

• +k

A ∙e

−EA/RT

∙τ CB= CBi +k

A ∙e

−EA/RT

∙τ∙CA

• +k

B ∙e

−EB/RT

∙τ −rA=k

A∙e

−EA/RT

−rB=k

B∙e

−EB/RT

−k

A∙e

−EA/RT

Q=F ρCp∙(T−Ti) +V∙(rAHRA+rBHRB)

τ=V/F

RB=rb∙V

Solution: In this problem the objective function, variance is an integral

over uncertainty (u) surface. The problem is NLP under un- certainty (Stochastic Nonlinear Programming) as the objective function and constraints are nonlinear. The variables areCA, CAi ,CB,CBi ,rA,rB,Q,F,V,RB,T,τ,Ti, and (k

A ,k

B ,EA, EB,R,HRA,HRB). There are 20 variables (not including the 2 / 4

Introduction3 objective functionσ) and 7 equations, so degrees of freedom are

• In general, the 7 variables in the brackets () are related to

the reactions constants and normally specified, if you consider that then the degrees of freedom will reduce to 6.

1.2 Indicate whether the problem below is an NLP or an LP. What meth- ods do you expect to be most effective for solving this problem?maxf(x, y, z, m) =x−3y+ 1.25z−2∙log(m) s.t. m∙exp(y)≥10 log(m)−x+ 4z≥6 x−3y≤9

Solution: The problem in the present form is an NLP as the objective function

and two of the constraints are nonlinear. However, the problem can be converted to LP by transforming the variables as given below.maxf(x, y, z, n) =x−3y+ 1.25z−2∙n s.t. y≥log(10)−n n−x+ 4z≥6 x−3y≤9 n=log(m). It is easier to solve the LP formulation than the original NLP. This is also illustrated in the GAMS solutions. GAMS could not find solution for the original NLP but LP could be solved very easily. 3 / 4

4Introduction to Applied Optimization

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