MINLPLib
A Library of Mixed-Integer and Continuous Nonlinear Programming Instances
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Instance tanksize
We discuss a tank design problem for a multi product plant, in which the optimal cycle time and the optimal campaign size are unknown. A mixed integer nonlinear programming formulation is presented, where non-convexities are due to the tank investment cost, storage cost, campaign setup cost and variable production rates. The objective of the optimization model is to minimize the sum of the production cost per ton per product produced. A continuous-time mathematical programming formulation for the problem is implemented with a fixed number of event points.
| Formatsⓘ | ams gms mod nl osil py |
| Primal Bounds (infeas ≤ 1e-08)ⓘ | |
| Other points (infeas > 1e-08)ⓘ | |
| Dual Boundsⓘ | 1.26864370 (ANTIGONE) 1.26864375 (BARON) 1.26864375 (COUENNE) 1.26864372 (LINDO) 1.26864373 (SCIP) 0.00000000 (SHOT) |
| Referencesⓘ | Rebennack, Steffen, Kallrath, Josef, and Pardalos, Panos M, Optimal Storage Design for a Multi-Product Plant: A Non-Convex MINLP Formulation, Computers and Chemical Engineering, 35:2, 2011, 255-271. |
| Sourceⓘ | GAMS Model Library model tanksize |
| Applicationⓘ | Tank Size Design |
| Added to libraryⓘ | 28 Feb 2014 |
| Problem typeⓘ | MBNLP |
| #Variablesⓘ | 47 |
| #Binary Variablesⓘ | 9 |
| #Integer Variablesⓘ | 0 |
| #Nonlinear Variablesⓘ | 27 |
| #Nonlinear Binary Variablesⓘ | 9 |
| #Nonlinear Integer Variablesⓘ | 0 |
| Objective Senseⓘ | min |
| Objective typeⓘ | linear |
| Objective curvatureⓘ | linear |
| #Nonzeros in Objectiveⓘ | 1 |
| #Nonlinear Nonzeros in Objectiveⓘ | 0 |
| #Constraintsⓘ | 74 |
| #Linear Constraintsⓘ | 53 |
| #Quadratic Constraintsⓘ | 20 |
| #Polynomial Constraintsⓘ | 0 |
| #Signomial Constraintsⓘ | 1 |
| #General Nonlinear Constraintsⓘ | 0 |
| Operands in Gen. Nonlin. Functionsⓘ | |
| Constraints curvatureⓘ | indefinite |
| #Nonzeros in Jacobianⓘ | 291 |
| #Nonlinear Nonzeros in Jacobianⓘ | 63 |
| #Nonzeros in (Upper-Left) Hessian of Lagrangianⓘ | 97 |
| #Nonzeros in Diagonal of Hessian of Lagrangianⓘ | 3 |
| #Blocks in Hessian of Lagrangianⓘ | 7 |
| Minimal blocksize in Hessian of Lagrangianⓘ | 1 |
| Maximal blocksize in Hessian of Lagrangianⓘ | 7 |
| Average blocksize in Hessian of Lagrangianⓘ | 3.857143 |
| #Semicontinuitiesⓘ | 0 |
| #Nonlinear Semicontinuitiesⓘ | 0 |
| #SOS type 1ⓘ | 0 |
| #SOS type 2ⓘ | 0 |
| Minimal coefficientⓘ | 5.1590e-02 |
| Maximal coefficientⓘ | 5.0000e+01 |
| Infeasibility of initial pointⓘ | 53.53 |
| Sparsity Jacobianⓘ |  |
| Sparsity Hessian of Lagrangianⓘ |  |
$offlisting
*
* Equation counts
* Total E G L N X C B
* 74 23 27 24 0 0 0 0
*
* Variable counts
* x b i s1s s2s sc si
* Total cont binary integer sos1 sos2 scont sint
* 47 38 9 0 0 0 0 0
* FX 5
*
* Nonzero counts
* Total const NL DLL
* 291 228 63 0
*
* Solve m using MINLP minimizing objvar;
Variables x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19
,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36
,x37,b38,b39,b40,b41,b42,b43,b44,b45,b46,objvar;
Positive Variables x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x25,x26,x27,x28,x29
,x30,x31,x32,x33,x34,x35,x36,x37;
Binary Variables b38,b39,b40,b41,b42,b43,b44,b45,b46;
Equations e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12,e13,e14,e15,e16,e17,e18,e19
,e20,e21,e22,e23,e24,e25,e26,e27,e28,e29,e30,e31,e32,e33,e34,e35,e36
,e37,e38,e39,e40,e41,e42,e43,e44,e45,e46,e47,e48,e49,e50,e51,e52,e53
,e54,e55,e56,e57,e58,e59,e60,e61,e62,e63,e64,e65,e66,e67,e68,e69,e70
,e71,e72,e73,e74;
e1.. x1 + x2 + x3 - x37 + 0.4*b38 + 0.4*b39 + 0.4*b40 + 0.2*b41 + 0.2*b42
+ 0.2*b43 + 0.1*b44 + 0.1*b45 + 0.1*b46 =E= 0;
e2.. b38 + b41 + b44 =L= 1;
e3.. b39 + b42 + b45 =L= 1;
e4.. b40 + b43 + b46 =L= 1;
e5.. 11.4794520547945*x1 - x4 - x13 + x14 + 4.59178082191781*b38
+ 2.2958904109589*b41 + 1.14794520547945*b44 =E= 0;
e6.. 11.4794520547945*x2 - x5 - x14 + x15 + 4.59178082191781*b39
+ 2.2958904109589*b42 + 1.14794520547945*b45 =E= 0;
e7.. 11.4794520547945*x3 - x6 + x13 - x15 + 4.59178082191781*b40
+ 2.2958904109589*b43 + 1.14794520547945*b46 =E= 0;
e8.. 9.56712328767123*x1 - x7 - x16 + x17 + 3.82684931506849*b38
+ 1.91342465753425*b41 + 0.956712328767123*b44 =E= 0;
e9.. 9.56712328767123*x2 - x8 - x17 + x18 + 3.82684931506849*b39
+ 1.91342465753425*b42 + 0.956712328767123*b45 =E= 0;
e10.. 9.56712328767123*x3 - x9 + x16 - x18 + 3.82684931506849*b40
+ 1.91342465753425*b43 + 0.956712328767123*b46 =E= 0;
e11.. 3.82739726027397*x1 - x10 - x19 + x20 + 1.53095890410959*b38
+ 0.765479452054795*b41 + 0.382739726027397*b44 =E= 0;
e12.. 3.82739726027397*x2 - x11 - x20 + x21 + 1.53095890410959*b39
+ 0.765479452054795*b42 + 0.382739726027397*b45 =E= 0;
e13.. 3.82739726027397*x3 - x12 + x19 - x21 + 1.53095890410959*b40
+ 0.765479452054795*b43 + 0.382739726027397*b46 =E= 0;
e14.. x13 - x22 =L= 0;
e15.. x14 - x22 =L= 0;
e16.. x15 - x22 =L= 0;
e17.. x16 - x23 =L= 0;
e18.. x17 - x23 =L= 0;
e19.. x18 - x23 =L= 0;
e20.. x19 - x24 =L= 0;
e21.. x20 - x24 =L= 0;
e22.. x21 - x24 =L= 0;
e23.. -50*x1*b38 + x4 =L= 0;
e24.. -50*x2*b39 + x5 =L= 0;
e25.. -50*x3*b40 + x6 =L= 0;
e26.. -50*x1*b41 + x7 =L= 0;
e27.. -50*x2*b42 + x8 =L= 0;
e28.. -50*x3*b43 + x9 =L= 0;
e29.. -50*x1*b44 + x10 =L= 0;
e30.. -50*x2*b45 + x11 =L= 0;
e31.. -50*x3*b46 + x12 =L= 0;
e32.. -15*x1*b38 + x4 =G= 0;
e33.. -15*x2*b39 + x5 =G= 0;
e34.. -15*x3*b40 + x6 =G= 0;
e35.. -15*x1*b41 + x7 =G= 0;
e36.. -15*x2*b42 + x8 =G= 0;
e37.. -15*x3*b43 + x9 =G= 0;
e38.. -7*x1*b44 + x10 =G= 0;
e39.. -7*x2*b45 + x11 =G= 0;
e40.. -7*x3*b46 + x12 =G= 0;
e41.. x1 - 40*b38 - 40*b41 - 40*b44 =L= 0;
e42.. x2 - 40*b39 - 40*b42 - 40*b45 =L= 0;
e43.. x3 - 40*b40 - 40*b43 - 40*b46 =L= 0;
e44.. x1 - b38 - b41 - b44 =G= 0;
e45.. x2 - b39 - b42 - b45 =G= 0;
e46.. x3 - b40 - b43 - b46 =G= 0;
e47.. x35 - 10*b38 - 10*b39 - 10*b40 - 20*b41 - 20*b42 - 20*b43 - 30*b44
- 30*b45 - 30*b46 =E= 0;
e48.. -0.3271*(sqrt(x22) + sqrt(x23) + sqrt(x24)) + x34 =E= 0;
e49.. -(0.0515901369863014*x25*(x1 + 0.4*b38 + 0.2*b41 + 0.1*b44) +
0.0515901369863014*x26*(x2 + 0.4*b39 + 0.2*b42 + 0.1*b45) +
0.0515901369863014*x27*(x3 + 0.4*b40 + 0.2*b43 + 0.1*b46) +
0.0528586301369863*x28*(x1 + 0.4*b38 + 0.2*b41 + 0.1*b44) +
0.0528586301369863*x29*(x2 + 0.4*b39 + 0.2*b42 + 0.1*b45) +
0.0528586301369863*x30*(x3 + 0.4*b40 + 0.2*b43 + 0.1*b46) +
0.0541268493150685*x31*(x1 + 0.4*b38 + 0.2*b41 + 0.1*b44) +
0.0541268493150685*x32*(x2 + 0.4*b39 + 0.2*b42 + 0.1*b45) +
0.0541268493150685*x33*(x3 + 0.4*b40 + 0.2*b43 + 0.1*b46)) + x36 =E= 0;
e50.. - 0.5*x13 - 0.5*x14 + x25 =E= -643;
e51.. - 0.5*x14 - 0.5*x15 + x26 =E= -643;
e52.. - 0.5*x13 - 0.5*x15 + x27 =E= -643;
e53.. - 0.5*x16 - 0.5*x17 + x28 =E= -536;
e54.. - 0.5*x17 - 0.5*x18 + x29 =E= -536;
e55.. - 0.5*x16 - 0.5*x18 + x30 =E= -536;
e56.. - 0.5*x19 - 0.5*x20 + x31 =E= -214;
e57.. - 0.5*x20 - 0.5*x21 + x32 =E= -214;
e58.. - 0.5*x19 - 0.5*x21 + x33 =E= -214;
e59.. (24.8739726027397*objvar - x34)*x37 - x35 - x36 =E= 0;
e60.. - x1 + 40*b38 + 40*b41 + 40*b44 =G= 0;
e61.. - x2 + 40*b39 + 40*b42 + 40*b45 =G= 0;
e62.. - x3 + 40*b40 + 40*b43 + 40*b46 =G= 0;
e63.. - b38 - b39 =G= -1;
e64.. - b39 - b40 =G= -1;
e65.. - b40 =G= -1;
e66.. - b41 - b42 =G= -1;
e67.. - b42 - b43 =G= -1;
e68.. - b43 =G= -1;
e69.. - b44 - b45 =G= -1;
e70.. - b45 - b46 =G= -1;
e71.. - b46 =G= -1;
e72.. b38 - b39 + b41 - b42 + b44 - b45 =G= 0;
e73.. b39 - b40 + b42 - b43 + b45 - b46 =G= 0;
e74.. b40 + b43 + b46 =G= 0;
* set non-default bounds
x13.fx = 643;
x14.lo = 643; x14.up = 4018.36;
x15.lo = 643; x15.up = 4018.36;
x16.lo = 536; x16.up = 3348.63;
x17.lo = 536; x17.up = 3348.63;
x18.lo = 536; x18.up = 3348.63;
x19.lo = 214; x19.up = 1339.45;
x20.lo = 214; x20.up = 1339.45;
x21.lo = 214; x21.up = 1339.45;
x22.lo = 643; x22.up = 4018.36;
x23.lo = 536; x23.up = 3348.63;
x24.lo = 214; x24.up = 1339.45;
b38.fx = 1;
b39.fx = 0;
b41.fx = 0;
b44.fx = 0;
* set non-default levels
b40.l = 0.550375356;
b42.l = 0.292212117;
b43.l = 0.224052867;
b45.l = 0.856270347;
b46.l = 0.067113723;
Model m / all /;
m.limrow=0; m.limcol=0;
m.tolproj=0.0;
$if NOT '%gams.u1%' == '' $include '%gams.u1%'
$if not set MINLP $set MINLP MINLP
Solve m using %MINLP% minimizing objvar;
Last updated: 2024-08-26 Git hash: 6cc1607f