AIMMS Tutorial for Professionals - Model Description

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This documentation was typeset by Paragon Decision Technology B.V. using LATEX and the Lucida font family. 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 Product ﬂow

The following indices capture the dimensions of the problem, and are used Indices throughout this chapter.

Indices: l locations f factories ⊂ locations c distribution centers ⊂ locations p production lines t time periods s demand scenarios

The following product ﬂow decision variables determine the levels of produc- Decision tion, distribution and storage. variables

Variables:

qf t total factory production [hl (hectoliter)] uf pt binary to indicate that production line is in use xfcts transport [TL (truckload)] ylts 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 ...

Parameters:

Dcts demand [hl] Lt actual period length [day] Chapter 3. Model Description 11

Qf p production at full operation [hl/day] Mf pt binary to indicate that production line is in maintenance Vf t binary to indicate a vacation period F drop in workforce during vacation periods (fraction)

Af pt potential production [hl] Xf number of available truckloads [TL] Y l maximum stock level [hl]

Y l minimum stock level [hl]

The parameters related to production line capacity, demand and vacations will ... and their be read from external data sources. The maintenance parameter will be deter- data source mined as part of the rolling horizon solution process.

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 parameters Mf pt, F and Vf t result in the potential production, Af pt, being less than the production level at full operation Qf p.

The following stock balance constraint relates stock to previous stock, produc- Balance tion, distribution and demand. constraint

ylts = yl,t−1,s + qlt + X xflts − X xlcts − Dlts , ∀(l, t, s) f c

ylts ∈ [Y l, Y l], ∀(l, t, s)

Note that this balance constraint is used for all locations (thus both factories Domain and distribution centers), and that particular terms inside this constraint must restrictions 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. In Aimms you 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 parameter Af pt as deﬁned previously, it is now Factory straightforward to determine the total weekly production at each of the facto- production ries.

qf t = X Af pt uf pt, ∀(f , t) p Chapter 3. Model Description 12

It is also straightforward to model the restriction that the number of truck- Transport loads to be moved from a factory during a particular week is limited by the limitation number of trucks available at that factory.

X xfcts ≤ 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 Mode switches

The following variable is needed to register the mode switches, Additional notation Variable:

vf pt binary to register a mode switch

The registration of mode switches seems tricky at ﬁrst, but becomes straight- Mode switch forward with some additional explanation. Consider the following two inequal- registration ities.

vf pt ≥ uf pt − ufp,t−1, ∀(f,p,t)

vf pt ≥ ufp,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 variable v will 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 variable v will be zero.

Consider a production line in use. Whenever such a line needs to be main- Eﬀect on tained, its production drops to zero. Immediately following the maintenance maintenance week, its production is likely to restart. In this case, the change in production 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 the u variable to remain at level one, thus avoiding the unwanted mode switch. A similar argument applies to maintenance while a line is not in use. Chapter 3. Model Description 13

3.3 Objective

The following parameters and variables are needed to specify the objective Additional function of the mathematical program. notation

Parameters: q Cf unit production cost [$/hl] y Cl unit stock cost [$/hl] x Cf c unit transport cost [$/TL] Cv penalty cost due to mode switch [$]

Ps demand scenario probability Variables:

rs demand scenario cost [$] z total cost [$]

The cost per single demand scenario is the sum of the production costs, the Cost per scenario-speciﬁc storage and distribution costs, plus a penalty term to reﬂect scenario the costs associated with mode switching.

q y x v rs = X Cf qf t + X Cl ylts + X Cf c xfcts + X C vf pt, ∀s f t lt f ct f pt

The total cost to be minimized is simply the weighted sum of the scenario Minimize total costs. cost

Minimize:

z = X Psrs s Chapter 3. Model Description 14

3.4 Model summary

The full mathematical description of the optimization model can now be sum- marized as follows.

Minimize:

z = X Psrs s

Subject to:

ylts = yl,t−1,s + qlt + X xflts − X xlcts − Dlts ∀(l, t, s) f c

qf t = X Af pt uf pt ∀(f , t) p

X xfcts ≤ Xf ∀(f,t,s) c

vf pt ≥ uf pt − ufp,t−1 ∀(f,p,t)

vf pt ≥ ufp,t−1 − uf pt ∀(f,p,t)

q y rs = X Cf qf t + X Cl ylts + f t lt

x v X Cf c xfcts + X C vf pt ∀s f ct f pt

uf pt ∈{0, 1} ∀(f,p,t)

xfcts ≥ 0 ∀(f,c,t,s)

ylts ∈ [Y l, Y l] ∀(l, t, s)

vf pt ≥ 0 ∀(f,p,t)