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AIMMS Tutorial for Professionals - Model Description

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AIMMS Tutorial for Professionals - Model DescriptionThis file contains only one chapter of the book. For a free download of thecomplete book in pdf format, please visit www.aimms.comAimms 3.11cCopyright 1993–2010 by Paragon Decision Technology B.V. All rights reserved.Paragon Decision Technology B.V. Paragon Decision Technology Inc. Paragon Decision Technology Pte.Schipholweg 1 500 108th Avenue NE Ltd.2034 LS Haarlem Ste. # 1085 80 Raffles PlaceThe Netherlands Bellevue, WA 98004 UOB Plaza 1, Level 36-01Tel.: +31 23 5511512 USA Singapore 048624Fax: +31 23 5511517 Tel.: +1 425 458 4024 Tel.: +65 9640 4182Fax: +1 425 458 4025Email: info@aimms.comWWW: www.aimms.comAimms is a registered trademark of Paragon Decision Technology B.V. IBM ILOG CPLEX and sc CPLEX isa registered trademark of IBM Corporation. GUROBI is a registered trademark of Gurobi Optimization,Inc. KNITRO is a registered trademark of Ziena Optimization, Inc. XPRESS-MP is a registered trademarkof FICO Fair Isaac Corporation. Mosek is a registered trademark of Mosek ApS. Windows and Excel areA Aregistered trademarks of Microsoft Corporation. T X, LT X, andA S-LT X are trademarks of the AmericanME E EMathematical Society. Lucida is a registered trademark of Bigelow & Holmes Inc. Acrobat is a registeredtrademark of Adobe Systems Inc. Other brands and their products are trademarks of their respectiveholders.Information in this document is subject to change without notice and does not represent a ...
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AIMMS Tutorial for Professionals - Model Description
This file contains only one chapter of the book.For a free download of the complete book in pdf format, please visitwww.aimms.com
Aimms3.11
Copyright c1993–2010 by Paragon Decision Technology B.V. All rights reserved.
Paragon Decision Technology B.V. Schipholweg 1 2034 LS Haarlem The Netherlands Tel.: +3123 5511512 Fax: +3123 5511517
Email: info@aimms.com WWW:www.aimms.com
Paragon Decision Technology Inc. 500 108th Avenue NE Ste. # 1085 Bellevue, WA 98004 USA Tel.: +1425 458 4024 Fax: +1425 458 4025
Paragon Decision Technology Pte. Ltd. 80 Raffles Place UOB Plaza 1, Level 36-01 Singapore 048624 Tel.: +659640 4182
Aimmsis a registered trademark of Paragon Decision Technology B.V.IBM ILOG CPLEXand sc CPLEX is a registered trademark of IBM Corporation.GUROBIis a registered trademark of Gurobi Optimization, Inc.KNITROis a registered trademark of Ziena Optimization, Inc.XPRESS-MPis a registered trademark of FICO Fair Isaac Corporation.Mosekis a registered trademark of Mosek ApS.WindowsandExcelare registered trademarks of Microsoft Corporation. T X, LT X, andA S-LT X are trademarks of the American A A E EME Mathematical Society.Lucidais a registered trademark of Bigelow & Holmes Inc.Acrobatis a registered trademark of Adobe Systems Inc.Other brands and their products are trademarks of their respective holders.
Information in this document is subject to change without notice and does not represent a commitment on the part of Paragon Decision Technology B.V. The software described in this document is furnished under a license agreement and may only be used and copied in accordance with the terms of the agreement. The documentation may not, in whole or in part, be copied, photocopied, reproduced, translated, or reduced to any electronic medium or machine-readable form without prior consent, in writing, from Paragon Decision Technology B.V.
Paragon Decision Technology B.V. makes no representation or warranty with respect to the adequacy of this documentation or the programs which it describes for any particular purpose or with respect to its adequacy to produce any particular result.In no event shall Paragon Decision Technology B.V., its employees, its contractors or the authors of this documentation be liable for special, direct, indirect or consequential damages, losses, costs, charges, claims, demands, or claims for lost profits, fees or expenses of any nature or kind.
In addition to the foregoing, users should recognize that all complex software systems and their doc-umentation contain errors and omissions. The authors, Paragon Decision Technology B.V. and its em-ployees, and its contractors shall not be responsible under any circumstances for providing information or corrections to errors and omissions discovered at any time in this book or the software it describes, whether or not they are aware of the errors or omissions. The authors, Paragon Decision Technology B.V. and its employees, and its contractors do not recommend the use of the software described in this book for applications in which errors or omissions could threaten life, injury or significant loss.
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Chapter 3
Model Description
In this chapter you will find a description of the mathematical program corre-This chapter sponding to the problem description of the previous chapter.
3.1 Productflow
The following indices capture the dimensions of the problem, and are usedIndices throughout this chapter. Indices: llocations ffactorieslocations cdistribution centerslocations pproduction lines ttime periods sdemand scenarios
The following product flow decision variables determine the levels of produc-tion, distribution and storage.
Variables: qf t uf pt x f cts ylts
total factory production [hl (hectoliter)] binary to indicate that production line is in use transport [TL (truckload)] stock [hl]
Note that the production variables are identical for all demand scenarios, while the distribution and storage variables can vary for each scenario.Note also that both hectoliters and truckloads are used to measure the quantities of soft drinks. In this tutorial a truckload is defined as 12 cubic meters.
The following product flow related parameters are used in this chapter.
Parameters: Dcts Lt
demand [hl] actual period length [day]
Decision variables
Parameters. . .
Qf p Mf pt Vf t F A f pt X f Yl Y l
Chapter 3.Model Description
production at full operation [hl/day] binary to indicate that production line is in maintenance binary to indicate a vacation period drop in workforce during vacation periods (fraction) potential production [hl] number of available truckloads [TL] maximum stock level [hl] minimum stock level [hl]
The parameters related to production line capacity, demand and vacations will be read from external data sources. The maintenance parameter will be deter-mined as part of the rolling horizon solution process.
. . .and their data source
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The potential production of a production line,Af pt, is dependent on the main-Potential tenance and vacation parameters, and is defined as follows.production determination Af pt=Lt(1Mf pt)(1FVf t)Qf p,(f , p, t) Note that nonzero values of parametersMf pt,FandVf tresult in the potential production,Af pt, being less than the production level at full operationQf p.
The following stock balance constraint relates stock to previous stock, produc-tion, distribution and demand.
  ylts=yl,t1,s+qlt+xf ltsxlctsDlts,(l, t, s) c f yY ],[Y ,(l, t, s) lts ll
Note that this balance constraint is used for all locations (thus both factories and distribution centers), and that particular terms inside this constraint must on some occasions be interpreted as non-existent.For instance, the production term is non-existent for distribution centers, while the demand term is non-existent for factories. InAimmsyou can specify a global index domain for each identifier, and the system will automatically restrict all identifier references to such an index domain.
Using the potential production parameterAf ptas defined previously, it is now straightforward to determine the total weekly production at each of the facto-ries. q=A u ,(f , t) f tf ptf pt p
Balance constraint
Domain restrictions
Factory production
Chapter 3.Model Description
It is also straightforward to model the restriction that the number of truck-loads to be moved from a factory during a particular week is limited by the number of trucks available at that factory. xf ctsXf,(f , t, s) c Note that the above planning constraint is, in practice, a simplification of the detailed transport capacity scheduling limitations.In scheduling applications the routing of vehicles, the distances to be traveled, plus the time-windows for the drivers would all be key factors in the determination of a final schedule. These factors are considered to be less important for the current one-year plan.
3.2 Modeswitches
The following variable is needed to register the mode switches,
Variable: v f pt
binary to register a mode switch
The registration of mode switches seems tricky at first, but becomes straight-forward with some additional explanation. Consider the following two inequal-ities.
vf ptuf ptuf p,t1,(f , p, t) vf ptuf p,t1uf pt,(f , p, t) Whenever a production line switches from being used to not being used, or vice versa, the switch-registration variablevwill be greater than or equal to unity. The penalty term in the objective discussed in the next section will ensure that this variable remains as small as possible. Thus, without a switch in the use of a production line, the variablevwill be zero.
Consider a production line in use.Whenever such a line needs to be main-tained, its production drops to zero.Immediately following the maintenance week, its production is likely to restart.In this case, the change in produc-tion is not considered to be a mode switch.The definition of the potential production parameter,Af pt, in the previous section is consistent with this ob-servation. The maintenance parameter,Mf pt, is set to one when maintenance is planned, which forces the potential production parameter,Af pt, to be zero for that week. The penalty term in the objective function, however, will cause theuvariable to remain at level one, thus avoiding the unwanted mode switch. A similar argument applies to maintenance while a line is not in use.
Transport limitation
Additional notation
Mode switch registration
Effect on maintenance
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3.3 Objective
Chapter 3.Model Description
The following parameters and variables are needed to specify the objective function of the mathematical program.
Parameters: q C f y C l x C f c v C Ps
Variables: rs z
unit production cost [$/hl] unit stock cost [$/hl] unit transport cost [$/TL] penalty cost due to mode switch [$] demand scenario probability
demand scenario cost [$] total cost [$]
Additional notation
The cost per single demand scenario is the sum of the production costs, theCost per scenario-specific storage and distribution costs, plus a penalty term to reflectscenario the costs associated with mode switching.    q y x v ys+ rs=C qf t+Cl ltC xf cts+C vf pt,s f fc f tlt fct fpt
The total cost to be minimized is simply the weighted sum of the scenario costs. Minimize: z=Psrs s
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Minimize total cost
3.4 Modelsummary
Chapter 3.Model Description
The full mathematical description of the optimization model can now be sum-marized as follows.
Minimize:
z=Psrs s
Subject to:   ylts=yl,t1,s+qlt+xf ltsxlctsDlts(l, t, s) c f qf t=Af ptuf pt(f , t) p xX(f , t, s) f ctsf c
vf ptuf ptuf p,t1 vuu f ptf p,t1f pt   q y q+C y rs=Cf tlts+ f l f tlt   x v C xf cts+C vf pt f c f ctf pt
uf pt∈ {0,1} xf cts0 [Y ylts l, Yl] vf pt0
(f , p, t) (f , p, t)
s
(f , p, t) (f , c, t, s) (l, t, s) (f , p, t)
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