A GRASP heuristic for the hot strip mill scheduling problem under consideration of energy consumption

Abstract

Hot strip mill rolling is an energy intensive production process in the steel industry. It converts steel slabs at high temperatures into steel strips. In this paper we address the related planning problem, i.e. the hot strip mill scheduling problem. The task is to determine the production sequence of production orders within a schedule. The involved energy consumption for heating individual slabs is explicitly considered in a new mixed integer problem formulation. The model is solved using a greedy randomized adaptive search procedure. In a numerical case study based on real world data the applicability and performance of the proposed heuristic is analyzed. The solution approach is able to find optimal solutions for small problem instances. Moreover, it solves industry size problem instances within reasonable time and outperforms the rule based planning approach prevalent in praxis.

Appendix

The formulated problem contains two kinds of nonlinearities. First, products of binary and continuous decision variables are embedded in constraints (17), (24) and (25). Second, absolute values are used in constraints (15), (22), and (23). In the following a linearization approach is given for both kinds of nonlinearities.

First, the product of a binary variable a and a continuous variable b is regarded. It conveys the following statement: If a = 1 the product has the value b. If a = 0 the product has the value 0. The term a · b can be linearized by introducing an additional continuous variable x that is restricted by four linear constraints:

$$\left( {a - 1} \right) \cdot M + b \le x$$

(35)

$$x \le - \left( {a - 1} \right) \cdot M + b$$

(36)

$$- a \cdot M \le x$$

(37)

$$x \le a \cdot M$$

(38)

On the one hand the constraints (35) and (36) demand that if a = 1 then x = b. If a = 0 these constraints are not binding. On the other hand the constraints (37) and (38) demand that if a = 0 then x = 0. If a = 1 these constraints are not binding. In consequence, the term a · b conveys the same statement as the new variable x with the constraints (35) to (38), and thus, can be replaced by the latter.

Considering the model in Sect. 3.3, the term x _hj  · [∑  ^k _l=j  ∑  ⁿ _i=1 x _il  · Swd _hi  · lg _i ] in constraint (17) can be interpreted as product of the binary variable x _hj and the continuous variable ∑  ^k _l=j  ∑  ⁿ _i=1 x _il  · Swd _hi  · lg _i . The term is replaced by the continuous variable u _hjk which leads to the following reformulation:

$$- \mathop \sum \limits_{l = j + 1}^{k} \left( {\alpha_{l} + \beta_{l} } \right) \cdot M + \mathop \sum \limits_{h = 1}^{n} u_{hjk} \le lg_{Swd}^{max}\quad \quad \forall j, k = 1 \ldots n, j < k$$

(39)

$$(x_{hj} - 1) \cdot M + \left[ {\mathop \sum \limits_{l = j}^{k} \mathop \sum \limits_{i = 1}^{n} x_{il} \cdot Swd_{hi} \cdot lg_{i} } \right] \le u_{hjk}\quad \quad \forall h,j,k = 1 \ldots n, j < k$$

(40)

$$u_{hjk} \le - (x_{hj} - 1) \cdot M + \left[ {\mathop \sum \limits_{l = j}^{k} \mathop \sum \limits_{i = 1}^{n} x_{il} \cdot Swd_{hi} \cdot lg_{i} } \right]\quad \quad \forall h,j,k = 1 \ldots n, j < k$$

(41)

$$- x_{hj} \cdot M \le u_{hjk}\quad \quad \forall h,j,k = 1 \ldots n, j < k$$

(42)

$$u_{hjk} \le x_{hj} \cdot M\quad \quad \forall h,j,k = 1 \ldots n, j < k$$

(43)

The same approach is applicable to constraints (24) and (25).

Second, the linearization of absolute values is presented by regarding the term |a| ≤ b. It says that a is bounded by − b ≤ a ≤ b. The same statement is conveyed by the linear constraints (44) and (45)

$$a \le b$$

(44)

$$- b \le a$$

(45)

Considering the model in Sect. 3.3, the according transformation of constraint (22) leads to the constraints (46) and (47). The same approach is applicable to constraints (15) and (23).

$$\mathop \sum \limits_{i = 1}^{n} x_{ij} {\cdot}hd_{i} - \mathop \sum \limits_{i = 1}^{n} x_{i,j - 1} {\cdot}hd_{i} \le \Delta hd^{max} + y_{j} \cdot M\quad \quad \forall j = 2 \ldots n$$

(46)

$$- \left( {\Delta hd^{max} + y_{j} \cdot M} \right) \le \mathop \sum \limits_{i = 1}^{n} x_{ij} {\cdot}hd_{i} - \mathop \sum \limits_{i = 1}^{n} x_{i,j - 1} {\cdot}hd_{i}\quad \quad \forall j = 2 \ldots n$$

(47)