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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

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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 ffactories⊂locations cdistribution centers⊂locations 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(1−Mf pt)(1−F∙Vf 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,t−1,s+qlt+xf lts−xlcts−Dlts,∀(l, t, s) c f y∈Y ],[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 cts≤Xf,∀(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 pt≥uf pt−uf p,t−1,∀(f , p, t) vf pt≥uf p,t−1−uf 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,t−1,s+qlt+xf lts−xlcts−Dlts∀(l, t, s) c f qf t=Af ptuf pt∀(f , t) p x≤X∀(f , t, s) f ctsf c

vf pt≥uf pt−uf p,t−1 v≥u−u f ptf p,t−1f 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 cts≥0 ∈[Y ylts l, Yl] vf pt≥0

∀(f , p, t) ∀(f , p, t)

∀s

∀(f , p, t) ∀(f , c, t, s) ∀(l, t, s) ∀(f , p, t)

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