Sets and Parameters

T ={0,1, . . . , T} denotes the set of time steps. T⁰ =T \{T} and T⁻ =T \{0}. Table H.1 shows the time-related parameters (operation time and setup time) of the problem.

Further parameters describe the amount of raw material for the t^A-type opera-tions and thet^B-type operations and the number of desired nal products, see Table H.2.

H.2 Variables

The decision variables used in this model are all binary or integer. They are divided into two categories: stock variables and ow variables. Considering the production as a directed graph, the stock variabless are the nodes. They describe the amount of material available for production. The resources (machines) are the arcs, which correspond to the ow variables f. Both variables are time-indexed since they describe a dynamic setting that changes from time point to time point. The nal class of variables represents certain time points in the schedule, such as the end of the production. The details can be found in Tables H.3, H.4, and H.5.

H.3 Constraints

The constraints related to the operation execution are described in detail in the following subsections.

H.3.1 t

^A

-type operations

The t^A-type operations are executed by resources r₅ and r₆, using the raw material of t^A-type operations.

name index i description

/dur(t^X, Di) denotes the operation time of a t^X-type operation with a D_i-type resource,

following the notations introduced in Appendix B./

p^t_r^A_i 5&6 Shows the processing time of the r_i resource for each t^A-type operation. It is equal to dur(t^A, D₃).

p^t_r^B_i 14 Shows the processing time of the ri resource for executing a t^B-type operation. It is equal to dur(t^B, D₁) if i∈ {1,2,3}, and it is equal to dur(t^B, D₂) if i= 4.

p^t_r^C

i 14 Shows the processing time of the r_i resource for executing a t^C-type operation. It is equal to dur(t^C, D₂) if i= 4,

and it is equal to dur(t^C, D₁) if i∈ {1,2,3}.

p^t_r^D_i 7&8 Shows the processing time of r_i for each t^D-type operation.

It is equal todur(t^D, D₄).

u_ri 14 Shows the setup time for r_i for changing between a t^B-type operation and a t^C-type operation. It is equal to

setup(D₁, t^B, t^C) if i∈ {1,2,3}, and it is equal to setup(D₂, t^B, t^C) if i= 4.

Table H.1. Time-related parameters of the model.

name description

m^tA raw material input for thet^A-type operations m^tB raw material input for thet^B-type operations

n(p₁) the desired number of products of the rst type (p₁) n(p₂) the desired number of products of the second type (p₂)

Table H.2. Production input and output parameters.

name range description

Table H.3. Stock variables (associated with nodes in the production graph).

name i range description f_r^tA

i (t) 5&6 {0,1} r_i resource starts processing at time t

f_r^t_i^B(t) 1-4 {0,1} ri resource starts a t^B-type operation at time t

i (t) 7&8 {0,1} r_i resource starts nishing a t^D-type operation (an assembly task) at timet

Table H.4. Flow variables (associated with arcs in the production graph).

name i range description

C_max Z+ makespan

`_ri 7&8 Z+ time point of the end of manufacturing onr_i resource Table H.5. Time variables.

The initial conditions are:

s^tAraw(0) =m^tA, s^{tAf inished}(0) = 0,

which place all the raw material of t^A-type operations in front of r₅ and r₆ and assumes an empty stock of nished products of thet^A-type operations at time point 0. The boundary conditions enforce no production in the very last time pointT:

f_r^t₅^A(T) = f_r^t₆^A(T) = 0.

There are two nodes associated with the execution of t^A-type operations; both of them give rise to ow conservation constraints. The raw material of t^A-type operations of time point t is moved to the two resources r₅, r₆, and the remaining raw material is moved to the next time point:

s^tAraw(t) = s^tAraw(t+ 1) +f_r^tA

5 (t) +f_r^tA

6(t), ∀t∈ T⁰.

The ow conservation at the node at the other end of r₅, r₆ takes the nished materials and either stores them as a stock or moves them forward for the nal assembly: Note that here and in all the following equations, if the time index of a variable is not inT, the variable is replaced by the value0. In the above equation, it means that f_r^tA

5 (t−p^t_r^A

5) is replaced by 0, whenever t−p^t_r^A

5 ∈ T/ .

The production within r₅, r₆ takes a certain number of time steps, and only one item (raw material) can be processed within that time span:

X

∆∈{0,...,p^tA_ri−1}

f_r^tA

i (t−∆)≤1, ∀t∈ T, i= 5,6.

H.3.2 t

^B

- and t

^C

-type operations

First, a t^B-type operation is executed on the initial raw material by one of the re-sourcesr₁, . . . , r₄. Then, thet^B-type operation's resulted products are either moved for nal assembly, or they are further processed, and at^C-type operation is executed on them by one of the resourcesr₁, . . . , r₄. As an initial condition, all the initial raw material is placed in front of machinesr₁, . . . , r₄ at time point 0:

s^tBraw(0) =m^tB.

There is no possibility for executing a t^C-type operation at time point 0: f_r^t₁^C(0) =. . .=f_r^t₄^C(0) = 0.

There is an empty stock of products of executed t^B-type and t^C-type operations at time point 0:

s^{tBf inished}(0) =s^{tCf inished}(0) = 0.

As boundary conditions, it is required that no processing takes place in the very last time point T:

The raw material of at^B-type operation is either processed by one of the machines r₁, . . . , r₄, or it is moved to the next time point as raw material:

s^tBraw(t) =s^tBraw(t+ 1) +f_r^tB

1 (t) +. . .+f_r^tB

4 (t), ∀t∈ T⁰.

The products of nishedt^B-type operations are either stored or processed further by at^C-type operation on one of the machinesr₁, . . . , r₄, or moved for nal assembly to the machinesr₇, r₈:

s^{tBf inished}(t−1) +f_r^t₁^B(t−p^t_r^B₁) +. . .+f_r^t₄^B(t−p^t_r^B₄)

=s^{tBf inished}(t) +f_r^t₁^C(t) +. . .+f_r^t₄^C(t) +f_r^p₇¹(t) +f_r^p₈¹(t), ∀t ∈ T⁻.

The products of nished t^C-type operations are either stored or moved for nal assembly to the machines r₇, r₈:

s^{tCf inished}(t−1) +f_r^t₁^C(t−p^t_r^C₁) +. . .+f_r^t₄^C(t−p^t_r^C₄)

=s^{tCf inished}(t) +f_r^p₇²(t) +f_r^p₈²(t), ∀t ∈ T⁻.

Let it be either a t^B-type operation or a t^C-type operation within an r₁, . . . , r₄ resource, its execution takes a certain number of time steps. Moreover, only one

item (in case of a t^B-type operation, a raw material of a t^B-type operation; in case of a t^C-type operation, a product of a prior t^B-type operation) can be processed within that time span:

X

∆∈{0,...,p^tX_ri −1}

f_r^tX

i (t−∆) ≤1, ∀t∈ T, i= 1, . . . ,4, ∀t^X ∈ {t^B, t^C}.

When changing either from a t^B-type operation to a t^C-type operation or from at^C-type operation to a t^B-type operation, the setup time must be obeyed:

f_r^tX_i (t) +f_r^tY_i (t+ ∆)≤1, ∀∆∈ {0, . . . , p^t_r^X_i +u_ri−1}, i= 1, . . . ,4,

∀t^X, t^Y ∈ {t^B, t^C}, t^X 6=t^Y.

H.3.3 t

^D

-type operations

Two resources (r₇ and r₈) assemble the products in the nal step, resulting in prod-ucts of product typesp₁ and p₂.

As initial conditions, it is assumed that the stock of nal products of both p₁ and p₂ product types is empty:

s^p1(0) =s^p2(0) = 0.

The ow of the products of nished t^B-type, t^C-type, and t^A-type operations through resourcesr₇, r₈ is initially empty; moreover, there is no ow in the very last time point, too:

f_r^p1

i(t) = f_r^p2

i (t) =f_r^tD

i (t) = 0, i= 7,8, ∀t∈ {0, T}.

The nal stocks of the products of product type p₁ and the products of product typep₂ must meet the required demand:

s^pj(T) = n(p_j), ∀p_j ∈ {p₁, p₂}.

Both the stock ofp₁-type products and the stock ofp₂-type products are increased by adding nished products coming out of r₇, r₈ resources:

s^pj(t−1) +f_r^pj

7(t−p^t_r^D

7) +f_r^pj

8(t−p^t_r^D

8) =s^pj(t), ∀p_j ∈ {p₁, p₂}, ∀t ∈ T⁻. Only one product can be handled during the processing time while performing either at^A-type operation or at^B-type operation or a t^C-type operation by resource r7 orr8:

X

∆∈{0,...,p^tD_ri −1}

f_r^Y_i(t−∆)≤1, ∀t∈ T, ∀Y ∈ {t^D, p₁, p₂}, i= 7,8.

A product consists of an outcome of a t^A-type operation assembled with either an outcome of a t^C-type operation (on which a t^B-type operation was carried out

earlier) or an outcome of a t^B-type operation. The two parts have to be assembled at the same time:

f_r^tD

i (t) =f_r^p2

i(t) +f_r^p1

i (t), ∀t ∈ T, i= 7,8.

In document Planning of fault-tolerant solutions of operations research problems and multi-level heuristic scheduling (Pldal 132-137)