The formal description of the general production planning problem

The simple problem and the schedule in Figure B.1 are referred to during the intro-duction of the notations for better understanding.

Notations B.0.1. Input-related notations:

M ={m₁, ..., m_s}: the set of the workow types also called process models. In the example problem of Figure B.1, there are two process models: m1 and m2. W ={w₁, ..., w_n}: the set of the workows also called processes. The problem of Figure B.1 has three workows: two of them have workow type m₁, and the third one has workow type m₂.

T_i ={t_i,1, ..., t_i,m}: the set of the operations (also called tasks) of process w_i. The problem of Figure B.1 has 10 operations. The operations of process w₁ aret_1,1, t_1,2, and t_1,3, the operations of processw₂ aret_2,1, t_2,2, and t_2,3, nally, the operations of process w₃ are t_3,1, t_3,2, t_3,3, and t_3,4.

T =∪ⁿ_i=1T_i: the set of the operations of all processes. In the case of the problem of Figure B.1, T ={t_1,1, t_1,2, t_1,3, t_2,1, t_2,2, t_2,3, t_3,1, t_3,2, t_3,3, t_3,4}.

P re(t_i,j)⊂T_i: the subset of the operations of process w_i (it can even be an empty set) whose elements are direct prior operations to operation t_i,j in process wi. This set may have more than one element because of the possible parallel structure of the processes. In the case of the problem of Figure B.1, e.g., P re(t_3,2) =P re(t_1,1) =∅, P re(t_3,3) = {t_3,2}, and P re(t_3,4) ={t_3,1, t_3,3}. allP re(t_i,j)⊂T_i: the subset of the operations of process w_i (it can even be an empty set) whose elements are prior operations to operation t_i,j in process w_i. E.g., if t_i,b ∈P re(t_i,c) and t_i,a ∈P re(t_i,b), then t_i,a∈allP re(t_i,c). In the case of the problem of Figure B.1, allP re(t_3,1) = allP re(t_3,2) = ∅, allP re(t_3,3) = {t_3,2}, and allP re(t_3,4) ={t_3,1, t_3,2, t_3,3}, for example.

Figure B.1. A simple scheduling problem and its schedule. They are applied during the explanation of the introduced notations.

R={r₁, ..., r_u}: the set of the resources. In Figure B.1, there are 8 resources:

R={r₁, r₂, r₃, r₄, r₅, r₆, r₇, r₈}.

D={D1, ..., Dy}: the set of the resource types. In Figure B.1, there are 4 resource types: D={D₁, D₂, D₃, D₄}.

n(Dl)∈Z: the number of the resources of resource type Dl. As two examples, n(D₁) = 3 and n(D₂) = 1 in the problem of Figure B.1.

RT(r_k)∈D: the resource type of resource r_k. For example, RT(r₆) =D₃ in the problem of Figure B.1.

setup(D_l, t^S, t^U)∈N: the setup time required to set a resource with resource type D_l from its state that can perform a t^S-type operation to a new state that makes the resource capable of performing a t^U-type operation.

A={t^A, t^B, ..., t^Z}: the set of the operation types. In the problem of Figure B.1, there are 4 operation types: A={t^A, t^B, t^C, t^D}.

T T(t_i,j)∈A: the operation type of operation t_i,j. For example, T T(t_2,3) = t^D and T T(t_3,3) = t^C in the problem of Figure B.1.

capable:D×A→ {0,1}: a function that describes whether a resource with a certain resource type can carry out a certain operation type. As two examples,capable(D₁, t^A) = 0 and capable(D₁, t^B) = 1, considering the problem of Figure B.1.

dur(t^S, D_l)∈N: the duration of carrying out an operation with operation type t^S by a D_l-type resource. If capable(D_l, t^S) = 0, then dur(t^S, D_l) =∞. For example, in the problem of Figure B.1,dur(t^B, D₂) = 10 anddur(t^D, D₂) =∞. durRe(t^S, Dl)∈N: the duration; after that, a resource with resource type Dl

can be used again for performing a new operation after starting a t^S-type operation. If capable(D_l, t^S) = 0, then durRe(t^S, D_l) = ∞. This notion is introduced because of the existence of transportation devices. It is the sum of the operation time and a specialized setup time that follows the operation.

Because there is no transportation resource in the problem of Figure B.1, in the example, the values of functiondurRe()are equal to the values of functiondur() for each operation type - resource type pair.

P ={p1, ..., ps}: the set of the product types. In the problem of Figure B.1, there are two product items (I₁ and I₂) with product type p₁ and one product item (I₃) with product type p₂.

P T(I_v)∈P: the product type of product item I_v. For example, P T(I₃) =p₂ in the example of Figure B.1.

W F(I_v)∈W: the workow that produces product I_v. In the problem of Figure B.1, W F(I₁) =w₁, W F(I₂) =w₂, and W F(I₃) = w₃.

W M(w_v)∈M: the workow model of w_v workow. In the problem of Figure B.1, W M(w₁) =W M(w₂) = m₁, and W M(w₃) = m₂.

W M(p_l)∈M: the workow model that produces products with product type p_l. In the problem of Figure B.1,W M(p₁) = m₁, and W M(p₂) = m₂.

Notations B.0.2. Schedule-related variables:

X(t_i,j)∈R: the resource which is assigned to operationt_i,j by the scheduler.

For example, X(t_1,2) = r₁ in the schedule of Figure B.1.

start(t_i,j)∈N: the start time of operation t_i,j of process w_i. As an example, start(t_3,3) = 3 in Figure B.1.

end(t_i,j)∈N: the nish time of operation t_i,j of process w_i. As an example, end(t1,3) = 6, based on Figure B.1.

makespan(W) = max(end(t_i,j))−min(start(t_k,l)) for all t_i,j, t_k,l ∈T. In the example of Figure B.1 makespan(W) =end(t3,4)−start(t2,1) = 9−0 = 9. The input-related constraints are:

W F(I_v)≡W F(I_q)⇔v =q,∀ 1≤v, q≤n.

This constraint ensures that each product item has its own, separate workow that the item's production is based on.

W M(p_l)≡W M(p_k)⇔l=k,∀ 1≤l, k≤s.

This constraint ensures that each product type has its own, separate workow model that the production of the products with this product type is based on.

n(p_l) =|

n

[

i=1

I_i, where P T(I_i) =p_l|,∀1≤l ≤s.

The above constraint denes the calculation of the cardinality of the set of prod-ucts with a certain product type.

Based on the above 3 constraints, the following constraint can be gained:

s

X

l=1

n(p_l) = |W|.

The constraint prescribes that each workow produces exactly one product.

F or ∀ r_k,1≤k ≤u, ∃D_l,1≤l ≤y, such that RT(r_k) =D_l, and n(D_l) = |

u

[

i=1

r_i, where RT(r_i) =D_l|,∀ 1≤l ≤y.

The above constraint denes the calculation of the number of resources with a certain resource type.

dur(t^S, D_l)≤durRe(t^S, D_l), ∀t^S ∈A, D_l ∈D.

This constraint prescribes that a resource can not be ready to perform any next operation before its ongoing operation is nished. It is worth noting that practi-cally the dierentiation between functions dur() and durRe() is required because of the possible existing of transportation machines: after a transportation device transports a product to the destination, ensuring that the product is ready for the following operation, the machine still requires time to get back to its initial location for starting a new similar transportation operation.

The constraints for determining start(t_i,j) and end(t_i,j) for all t_i,j ∈ T are:

X(t_i,j)6=∅,∀t_i,j ∈T

This constraint means that each operation has to be carried out by a resource.

end(ti,j) = start(ti,j) +dur(T T(ti,j), RT(X(ti,j))), ∀ ti,j ∈T

The above constraint represents the connection between the start and the nish time of an operation regarding the related operation time.

start(tl,m)≥start(ti,j) +durRe(T T(ti,j), RT(X(ti,j))) +setup(X(t_i,j), T T(t_i,j), T T(t_l,m)) or start(t_l,m) +durRe(T T(t_l,m), RT(X(t_l,m))) +setup(X(t_l,m), T T(t_l,m), T T(t_i,j))≤start(t_i,j),

∀t_i,j, t_l,m ∈T, if t_i,j 6=t_l,m and X(t_i,j)≡X(t_l,m))

This constraint species that the operations allocated to the same resource must not overlap each other.

end(t_i,a) ≤ start(t_i,b), ∀ 1 ≤ i ≤ n, ∀ t_i,a, t_i,b ∈ T_i, if t_i,a ∈ allP re(t_i,b) The nal constraint describes that an operation of a process must not start be-fore its prior operations of the considered operation's process are not nished.

For the problem with the process set W, the solution with minimalmakespan(W) is intended to be found.

Appendix C