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Question: How to solve Non linear objective function with inequality constraints using NLp solve?

Question:How to solve Non linear objective function with inequality constraints using NLp solve?

MANUTTM 30 Maple 2019
syntax plotting optimization constraints
March 01 2023
1

I have to maximize my non linear objective function TBCI (given in attached worksheet) with 4 inequality constraints. The decision variables are i and q.The range of i is 0 to 1 and of q is 0 to 76.

When i tru to solve it using NLP solve it is showing error-"Error, (in Optimization:-NLPSolve) constraints must be specified as a set or list of equalities and inequalities".

Please help me to solve it. please find attached the maple file

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restart
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with(plots); with(DEtools); with(LinearAlgebra); with(Student[VectorCalculus]); with(linalg); with(Optimization); with(student)
> 

TC := proc (Q) options operator, arrow; O1*(1-beta)*D/Q+(1/2)*h1*Q+(1/2)*h2*(gamma-beta)*Qr/gamma+O2*beta*D/Qr+r*beta*D-q*beta*D-P*(1-i)*beta*D end proc

proc (Q) options operator, arrow; Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(O1, Student:-VectorCalculus:-`+`(1, Student:-VectorCalculus:-`-`(beta))), Student:-VectorCalculus:-D), Q^Student:-VectorCalculus:-`-`(1)), Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(h1, Q), 2^Student:-VectorCalculus:-`-`(1))), Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(h2, Student:-VectorCalculus:-`+`(gamma, Student:-VectorCalculus:-`-`(beta))), Qr), Student:-VectorCalculus:-`*`(2, gamma)^Student:-VectorCalculus:-`-`(1))), Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(O2, beta), Student:-VectorCalculus:-D), Qr^Student:-VectorCalculus:-`-`(1))), Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(r, beta), Student:-VectorCalculus:-D)), Student:-VectorCalculus:-`-`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(q, beta), Student:-VectorCalculus:-D))), Student:-VectorCalculus:-`-`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(P, Student:-VectorCalculus:-`+`(1, Student:-VectorCalculus:-`-`(i))), beta), Student:-VectorCalculus:-D))) end proc

 (1)
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diff(TC(Q), Q)

-O1*(1-beta)*Student:-VectorCalculus:-D/Q^2+(1/2)*h1

 (2)
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simplify(isolate(%, Q))

Q = RootOf(2*Student:-VectorCalculus:-D*O1*beta+_Z^2*h1-2*Student:-VectorCalculus:-D*O1)

 (3)
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diff(TC(Q), Qr)

(1/2)*h2*(gamma-beta)/gamma-O2*beta*Student:-VectorCalculus:-D/Qr^2

 (4)
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simplify(isolate(%, Qr))

Qr = RootOf((beta*h2-gamma*h2)*_Z^2+2*gamma*O2*beta*Student:-VectorCalculus:-D)

 (5)
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diff(TC(Q), `$`(Q, 2))

2*O1*(1-beta)*Student:-VectorCalculus:-D/Q^3

 (6)
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diff(TC(Q), `$`(Qr, 2))

2*O2*beta*Student:-VectorCalculus:-D/Qr^3

 (7)
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TC1 := proc (Q, Qr) options operator, arrow; O1*(-beta*i+1)*D/Q+(1/2)*h1*Q+(1/2)*h2*(-beta*i+gamma)*Qr/gamma+O2*i*beta*D/Qr+r*i*beta*D-q*i*beta*D-P*(1-i)*i*beta*D end proc

proc (Q, Qr) options operator, arrow; Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(O1, Student:-VectorCalculus:-`+`(1, Student:-VectorCalculus:-`-`(Student:-VectorCalculus:-`*`(i, beta)))), Student:-VectorCalculus:-D), Q^Student:-VectorCalculus:-`-`(1)), Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(h1, Q), 2^Student:-VectorCalculus:-`-`(1))), Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(h2, Student:-VectorCalculus:-`+`(gamma, Student:-VectorCalculus:-`-`(Student:-VectorCalculus:-`*`(i, beta)))), Qr), Student:-VectorCalculus:-`*`(2, gamma)^Student:-VectorCalculus:-`-`(1))), Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(O2, i), beta), Student:-VectorCalculus:-D), Qr^Student:-VectorCalculus:-`-`(1))), Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(r, i), beta), Student:-VectorCalculus:-D)), Student:-VectorCalculus:-`-`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(q, i), beta), Student:-VectorCalculus:-D))), Student:-VectorCalculus:-`-`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(P, Student:-VectorCalculus:-`+`(1, Student:-VectorCalculus:-`-`(i))), i), beta), Student:-VectorCalculus:-D))) end proc

 (8)
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diff(TC1(Q, Qr), Q)

-O1*(-beta*i+1)*Student:-VectorCalculus:-D/Q^2+(1/2)*h1

 (9)
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a1 := isolate(-O1*(-beta*i+1)*D/Q^2+(1/2)*h1, Q)

Q = RootOf(2*Student:-VectorCalculus:-D*O1*beta*i+_Z^2*h1-2*Student:-VectorCalculus:-D*O1)

 (10)
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diff(TC1(Q, Qr), Qr)

(1/2)*h2*(-beta*i+gamma)/gamma-O2*i*beta*Student:-VectorCalculus:-D/Qr^2

 (11)
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a2 := isolate(diff(TC1(Q, Qr), Qr), Qr)

Qr = RootOf((beta*h2*i-gamma*h2)*_Z^2+2*gamma*O2*i*beta*Student:-VectorCalculus:-D)

 (12)
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TC2 := proc (Q) options operator, arrow; O1*(1-gamma)*D/Q+(1/2)*h1*Q+r*gamma*D-q*gamma*D-P*(1-i)*gamma*D end proc

proc (Q) options operator, arrow; Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`+`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(O1, Student:-VectorCalculus:-`+`(1, Student:-VectorCalculus:-`-`(gamma))), Student:-VectorCalculus:-D), Q^Student:-VectorCalculus:-`-`(1)), Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(h1, Q), 2^Student:-VectorCalculus:-`-`(1))), Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(r, gamma), Student:-VectorCalculus:-D)), Student:-VectorCalculus:-`-`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(q, gamma), Student:-VectorCalculus:-D))), Student:-VectorCalculus:-`-`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(Student:-VectorCalculus:-`*`(P, Student:-VectorCalculus:-`+`(1, Student:-VectorCalculus:-`-`(i))), gamma), Student:-VectorCalculus:-D))) end proc

 (13)
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b1 := isolate(diff(TC2(Q), Q), Q)

Q = RootOf(2*Student:-VectorCalculus:-D*O1*gamma+_Z^2*h1-2*Student:-VectorCalculus:-D*O1)

 (14)
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E1 := eval(a1, [P = 765, D = 347933, O1 = 30, O2 = 10, h1 = 76.5, h2 = 38.25, beta = 0.5e-1, gamma = 0.4e-1, q = 7.65, r = 382.5, i = .5])

Q = RootOf(76.5*_Z^2-20354080.50)

 (15)
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simplify(E1)

Q = 515.8162578

 (16)
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E2 := eval(a2, [P = 765, D = 347933, O1 = 30, O2 = 10, h1 = 76.5, h2 = 38.25, beta = 0.5e-1, gamma = 0.4e-1, q = 7.65, r = 382.5, i = .5])

Qr = RootOf(.57375*_Z^2-6958.66000)

 (17)
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simplify(E2)

Qr = 110.1289401

 (18)
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Totcost := eval(TC1(Q, Qr), [Q = 516, Qr = 110, P = 765, D = 347933, O1 = 30, O2 = 10, h1 = 76.5, h2 = 38.25, beta = 0.5e-1, gamma = 0.4e-1, q = 7.65, r = 382.5, i = .5])

-25502.577

 (19)
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TB1 := B1*D*P*beta*i-D*beta*i*q

B1*Student:-VectorCalculus:-D*P*beta*i-Student:-VectorCalculus:-D*beta*i*q

 (20)
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TBC := eval(TB1, [B1 = 3.36, P = 765, D = 347933, O1 = 30, O2 = 10, h1 = 76.5, h2 = 38.25, beta = 0.5e-1, gamma = 0.4e-1, r = 382.5])

-17396.65*q*i+44716349.16*i

 (21)
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Totincent := P*i^2*beta*D

P*i^2*beta*Student:-VectorCalculus:-D

 (22)
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TI := eval(Totincent, [B1 = 3.36, P = 765, D = 347933, O1 = 30, O2 = 10, h1 = 76.5, h2 = 38.25, beta = 0.5e-1, gamma = 0.4e-1, r = 382.5])

13308437.25*i^2

 (23)
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NULL
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simplify(subs(P = 765, D = 347933, O1 = 30, O2 = 10, h1 = 76.5, h2 = 38.25, gamma = 0.4e-1, q = 7.65, r = 382.5, beta = 0.5e-1, Q = sqrt((2*(-beta*i+1))*O1*D/h1))); simplify(subs(P = 765, D = 347933, O1 = 30, O2 = 10, h1 = 76.5, h2 = 38.25, gamma = 0.4e-1, q = 7.65, r = 382.5, beta = 0.5e-1, Qr = sqrt(2*i*beta*gamma*O2*D/((gamma-beta)*h2))))

Q = 3230.787830*(-0.1307189542e-2*i+0.2614379084e-1)^(1/2)

  

Qr = 190.7489196*(-i)^(1/2)

 (24)
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TCo := eval(TC1(Q, Qr), [Q = 3230.787830*sqrt(-0.1307189542e-2*i+0.2614379084e-1), Qr = 190.7489196*sqrt(-i), P = 765, D = 347933, O1 = 30, O2 = 10, h1 = 76.5, h2 = 38.25, gamma = 0.4e-1, q = 7.65, r = 382.5, beta = 0.5e-1])

3230.787829*(-0.5e-1*i+1)/(-0.1307189542e-2*i+0.2614379084e-1)^(1/2)+123577.6345*(-0.1307189542e-2*i+0.2614379084e-1)^(1/2)+91201.82720*(-0.5e-1*i+0.4e-1)*(-i)^(1/2)+912.0182718*i/(-i)^(1/2)+6521134.252*i-13308437.25*(1-i)*i

 (25)
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eq1 := TCo <= 0

3230.787829*(-0.5e-1*i+1)/(-0.1307189542e-2*i+0.2614379084e-1)^(1/2)+123577.6345*(-0.1307189542e-2*i+0.2614379084e-1)^(1/2)+91201.82720*(-0.5e-1*i+0.4e-1)*(-i)^(1/2)+912.0182718*i/(-i)^(1/2)+6521134.252*i-13308437.25*(1-i)*i <= 0

 (26)
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eq2 := 0 <= q/P and q/P <= 1

0 <= q/P and q/P <= 1

 (27)
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eq3 := 0 <= i and i <= gamma/beta

0 <= i and i <= gamma/beta

 (28)
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eq4 := TBC >= 0

0 <= -17396.65*q*i+44716349.16*i

 (29)
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``
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TBCI := TBC-TCo+TI

-17396.65*q*i+38195214.91*i-3230.787829*(-0.5e-1*i+1)/(-0.1307189542e-2*i+0.2614379084e-1)^(1/2)-123577.6345*(-0.1307189542e-2*i+0.2614379084e-1)^(1/2)-91201.82720*(-0.5e-1*i+0.4e-1)*(-i)^(1/2)-912.0182718*i/(-i)^(1/2)+13308437.25*(1-i)*i+13308437.25*i^2

 (30)
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NLPSolve(TBCI, {eq1, eq2, eq3, eq4}, assume = nonnegative)

Error, (in Optimization:-NLPSolve) constraints must be specified as a set or list of equalities and inequalities

  
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``
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