Optimal control strategies for single-machine family scheduling with sequence-dependent batch setup and controllable processing times

Abstract

The problem of scheduling jobs on an unreliable single machine is considered in this paper. The scheduling problem is characterized by the following features: jobs are grouped into classes of equivalent jobs; the generalized due date model is adopted for each class of jobs; it is possible to reduce the processing time of a job, at the price of the payment of an extra cost; a costly setup is required when switching between jobs of different classes. The scheduling problem is solved from a perspective which is different from the traditional determination of an optimal sequence of jobs; in fact, the objective of the paper is to determine optimal control strategies (functions of the system state) which allow generating the optimal decisions during the evolution of the system, taking into account the actual system state. In this way, optimal decisions can be promptly taken also in the presence of perturbations which affect the single machine (such as breakdowns and slowdowns). To this aim, a specific optimal control problem is stated and solved in the paper.

Appendices

Appendix 1: Algorithm providing time instants $\omega _{1}, \ldots , \omega _{|B |}$

Algorithm 1

Determination of the time instant $\omega _{j}$, $j = 1, \ldots , |B |$, at which, in case $\omega _{j} < \infty$, the function $x^{\circ }(t)$ jumps in an upward direction

SECTION A – INITIALIZATION

SECTION B – FIRST LOOP

SECTION C – SECOND LOOP

Appendix 2: sketches of the proofs of Lemmas 1 and 2

1.1 Lemma 1

The function $x^{\circ }(t)$ can be obtained by analyzing the shape of the function $f(x+t) + g(x)$ in the interval $[x_{1},x_{2}]$, with t moving from $-\infty$ to $+\infty$.

Several cases must be considered, mainly depending on the width of the intervals between abscissae $\gamma _{a_{q}}$ and $\gamma _{b_{q}}$, or abscissae $\gamma _{b_{q}}$ and $\gamma _{a_{q+1}}$, with respect to the width of interval $[x_{1},x_{2}]$. The essential case is the one in which all the intervals between these abscissae are wider than the width of $[x_{1},x_{2}]$, that is, when $\gamma _{b_{j}} - \gamma _{a_{j}} > x_{2}-x_{1}$ $\forall \, j = 1, \ldots , |B |$, $|B |> 0$, and $\gamma _{a_{j+1}} - \gamma _{b_{j}} > x_{2}-x_{1}$ $\forall j = 1, \ldots , |A |- 1$, $|A |> 1$. In this case, the function $x^{\circ }(t)$ has the following behavior.

• 1. When t is such that the slopes of $f(x+t)$ in the interval $[x_{1},x_{2}]$ are less than $\nu$, that is, $\forall \, t < \gamma _{a_{1}} - x_{2}$, $\forall \, t \in [ \gamma _{b_{j}} - x_{1} , \gamma _{a_{j+1}} - x_{2})$, $j = 1, \ldots , |A |- 1$, and $\forall \, t \ge \gamma _{b_{|B |}} - x_{1}$ if $|A |= |B |$, the minimum of the function $f(x+t) + g(x)$, with respect to $[x_{1},x_{2}]$, is obtained at $x_{2}$, since $f(x+t) + g(x)$ is strictly decreasing in $[x_{1},x_{2}]$.

• 2. When t is such that $\gamma _{a_{j}} - t \in [x_{1},x_{2}]$, $j = 1, \ldots , |A |$, that is, $\forall \, t \in [ \gamma _{a_{j}} - x_{2} , \gamma _{a_{j}} - x_{1} )$, $j = 1, \ldots , |A |$, the function $f(x+t) + g(x)$ is strictly decreasing in $[x_{1},\gamma _{a_{j}} - t]$ and nondecreasing in $[\gamma _{a_{j}} - t , x_{2}]$; then, it has a minimum, with respect to $[x_{1},x_{2}]$, in $\gamma _{a_{j}} - t$; when t increases in the interval $[\gamma _{a_{j}} - x_{2} , \gamma _{a_{j}} - x_{1})$, the minimum decreases (with unitary speed) from $x_{2}$ to $x_{1}$.

• 3. When t is such that the slope of $f(x+t)$ in the interval $[x_{1},x_{2}]$ is grater than or equal to $\nu$, that is, $\forall \, t \in [ \gamma _{a_{j}} - x_{1} , \gamma _{b_{j}} - x_{2})$, $j = 1, \ldots , |B |$, and $\forall \, t \ge \gamma _{a_{|A |}} - x_{1}$ if $|A |> |B |$, the minimum of the function $f(x+t) + g(x)$, with respect to $[x_{1},x_{2}]$, is obtained at $x_{1}$, since $f(x+t) + g(x)$ is nondecreasing in $[x_{1},x_{2}]$.

• 4. When t is such that $\gamma _{b_{j}} - t \in [x_{1},x_{2}]$, $j = 1, \ldots , |B |$, that is, $\forall \, t \in [ \gamma _{b_{j}} - x_{2} , \gamma _{b_{j}} - x_{1} )$, $j = 1, \ldots , |B |$, the function $f(x+t) + g(x)$ is nondecreasing in $[x_{1},\gamma _{b_{j}} - t]$ and strictly decreasing in $[\gamma _{b_{j}} - t , x_{2}]$; then, it has a maximum, with respect to $[x_{1},x_{2}]$, in $x = \gamma _{b_{j}} - t$, and the minimum is obtained either at $x_{1}$ or $x_{2}$, depending on the values $f(x_{1}+t) + g(x_{1})$ and $f(x_{2}+t) + g(x_{2})$, $t \in [ \gamma _{b_{j}} - x_{2} , \gamma _{b_{j}} - x_{1} )$ (the minimum is obtained at $x^{\circ } = x_{1}$ if $f(x_{1}+t) + g(x_{1}) < f(x_{2}+t) + g(x_{2})$ and at $x^{\circ } = x_{2}$ otherwise). In this connection, note that

  • when $t = \gamma _{b_{j}} - x_{2}$, it is certainly $f(x_{1}+t) + g(x_{1}) \le f(x_{2}+t) + g(x_{2})$;

  • when t increases in the interval $(\gamma _{b_{j}} - x_{2} , \gamma _{b_{j}} - x_{1})$, the value of $f(x_{1}+t) + g(x_{1})$ increases or remains constant and the value of $f(x_{2}+t) + g(x_{2})$ decreases;

  • when $t = \gamma _{b_{j}} - x_{1}$, it is certainly $f(x_{1}+t) + g(x_{1}) > f(x_{2}+t) + g(x_{2})$.

  This means that it certainly exists $\omega _{j} \in [ \gamma _{b_{j}} - x_{2} , \gamma _{b_{j}} - x_{1} )$ such that $f(x_{1}+t) + g(x_{1}) \le f(x_{2}+t) + g(x_{2})$ $\forall \, t \in [ \gamma _{b_{j}} - x_{2}, \omega _{j})$, $f(x_{1}+\omega _{j}) + g(x_{1}) = f(x_{2}+\omega _{j}) + g(x_{2})$, and $f(x_{1}+t) + g(x_{1}) > f(x_{2}+t) + g(x_{2})$ $\forall \, t \in ( \omega _{j} , \gamma _{b_{j}} - x_{1} )$; then, the minimum is obtained at $x_{1}$ $\forall \, t \in [ \gamma _{b_{j}} - x_{2}, \omega _{j})$, “jumps” from $x_{1}$ to $x_{2}$ at $\omega _{j}$, and is obtained at $x_{2}$ $\forall \, t \in [ \omega _{j} , \gamma _{b_{j}} - x_{1} )$.

Thus, according to the previous four “rules”, since $\mu _{0} = 0$, the function $x^{\circ }(t)$ is $x_{2}$ at the beginning (rule 1), decreases with slope $-1$ in the interval $[ \gamma _{a_{1}} - x_{2} , \gamma _{a_{1}} - x_{1} )$ (rule 2), and is equal to $x_{1}$ from $\gamma _{a_{1}} - x_{1}$ to $\omega _{1}$, at which it jumps to $x_{2}$ (rules 3 and 4); $x^{\circ }(t)$ remains equal to $x_{2}$ from $\omega _{1}$ up to $\gamma _{a_{2}} - x_{2}$ (rules 4 and 1), then it decreases with slope $-1$ in the interval $[ \gamma _{a_{2}} - x_{2} , \gamma _{a_{2}} - x_{1} )$ (rule 2), is equal to $x_{1}$ from $\gamma _{a_{2}} - x_{1}$ to $\omega _{2}$, at which jumps to $x_{2}$ (rules 3 and 4), and so on. In its last part, the function $x^{\circ }(t)$ is $x_{1}$ if $|A |> |B |$ or $x_{2}$ if $|A |= |B |$, in accordance with rules 3 and 1. Note that, in this case, Algorithm 1 provides a finite value $\omega _{j}$ for each abscissa $\gamma _{b_{j}}-t$, $j = 1, \ldots , |B |$; then, $T = \varOmega$, being $\varOmega = \{ \omega _{1} , \ldots , \omega _{j}, \ldots , \omega _{|B |} \}$. Moreover, $Q = |B |$, $t^{\star }_{q} = \omega _{q}$, and $l(q) = q$, $\forall \, q = 1, \ldots , Q$. Thus, it is possible to write $t^{\star }_{q} \in [\gamma _{b_{l(q)}} - x_{2}, \gamma _{b_{l(q)}} - x_{1})$, $\gamma _{b_{l(q)}} - \gamma _{a_{l(q)}} > x_{2}-x_{1}$ $\forall \, q = 1, \ldots , Q$, and $\gamma _{a_{l(q)+1}} - \gamma _{b_{l(q)}} > x_{2}-x_{1}$ $\forall \, q = 1, \ldots , Q$ (if $l(Q) < |A |$) or $\forall \, q = 1, \ldots , Q-1$ (if $l(Q) = |A |$), which imply $t^{\star }_{q} > \gamma _{a_{l(q)}} - x_{1}$ $\forall \, q = 1, \ldots , Q$ and $t^{\star }_{q} < \gamma _{a_{l(q)+1}} - x_{2}$ $\forall \, q = 1, \ldots , Q$ (if $l(Q) < |A |$) or $\forall \, q = 1, \ldots , Q-1$ (if $l(Q) = |A |$). Then, $x^{\circ }(t)$ has the structure provided by (6b) or (6c), with $x_{\mathrm {s}}(t)$ provided by (7a), $x_{q}(t)$, $q = 1, \ldots , Q-1$, $Q > 1$, provided by (8a), and $x_{\mathrm {e}}(t)$ provided by (9a) (if $l(Q) < |A |$) or (9c) (if $l(Q) = |A |$).

In presence of narrower intervals, some of the abscissae $\gamma _{b_{j}} - t$, $j = 1, \ldots , |B |$, may not cause a jump discontinuity in $x^{\circ }(t)$. Then, in general, $x^{\circ }(t)$ has a number of discontinuities (at which it jumps in an upward direction) equal to $Q \le |B |$. In accordance with the notation adopted in Algorithm 1, values $\omega _{j} < +\infty$, $j \in \{ 1, \ldots , |B |\}$, are those actually corresponding to jump discontinuities. These Q values are denoted by $t^{\star }_{q}$, $q = 1, \ldots , Q$. The link between values $\omega _{j}$ and $t^{\star }_{q}$ is represented by the mapping function $l(q) = j$, $j \in \{ 1, \ldots , |B |\}$, $q = 1, \ldots , Q$ (that is, $l(q) = j \Leftrightarrow \omega _{j} = t^{\star }_{q}$). Another consequence of having narrower intervals is that some of the discontinuities are such that function $x^{\circ }(t)$ does not jump from $x_{1}$ to $x_{2}$ but jumps from a value greater than $x_{1}$ and/or jumps to a value less than $x_{2}$. In these cases, $x_{\mathrm {s}}(t)$ in (6b) and (6c) has the structure of (7b), $x_{q}(t)$ in (6c) has the structure of (8c), (8c), or (8d), and $x_{\mathrm {e}}(t)$ in (6b) and (6c) has the structure of (9b).

1.2 Lemma 2

When $x = x_{2}$, it is $g(x) = 0$; then, when $x^{\circ }(t) = x_{2}$, it turns out $h(t) = f(x_{2}+t)$. Instead, when $x = x_{1}$, it is $g(x) = \nu (x_{2} - x_{1})$; then, when $x^{\circ }(t) = x_{1}$, it turns out $h(t) = f(x_{1}+t) + \nu (x_{2} - x_{1})$.

When $\gamma _{a_{1}} - x_{2} < t < \gamma _{a_{1}} - x_{1}$, with regard to (7a), $x^{\circ }(t)$ passes linearly (with unitary speed) from the value $x_{2}$ at $t = \gamma _{a_{1}} - x_{2}$ to the value $x_{1}$ at $t = \gamma _{a_{1}} - x_{1}$; then, in the same interval, function h(t) passes, with the same dynamics (that is, linearly), from the value $f(x_{2} + \gamma _{a_{1}} - x_{2}) = f(\gamma _{a_{1}})$ (at $t = \gamma _{a_{1}} - x_{2}$) to the value $f(x_{1} + \gamma _{a_{1}} - x_{1}) + \nu ( x_{2} - x_{1} ) = f(\gamma _{a_{1}}) + \nu ( x_{2} - x_{1} )$ (at $t = \gamma _{a_{1}} - x_{1}$); the segment which joins such values belongs to the line $\nu t + [ f(\gamma _{a_{1}}) - \nu (\gamma _{a_{1}} - x_{2}) ]$. In the same way, when $\gamma _{a_{1}} - x_{2} < t < t^{\star }_{1}$, with regard to (7b), $x^{\circ }(t)$ passes linearly (with unitary speed) from the value $x_{2}$ at $t = \gamma _{a_{1}} - x_{2}$ to the value $-t^{\star }_{1}+\gamma _{a_{1}}$ at $t = t^{\star }_{1}$; then, in the same interval, function h(t) passes linearly from the value $f(\gamma _{a_{1}})$ (at $t = \gamma _{a_{1}} - x_{2}$) to the value $f(\gamma _{a_{1}}) + \nu ( x_{2} + t^{\star }_{1} - \gamma _{a_{1}} )$ (at $t = t^{\star }_{1}$); the segment which joins such values belongs again to the line $\nu t + [ f(\gamma _{a_{1}}) - \nu (\gamma _{a_{1}} - x_{2}) ]$. This proves that, when $t : x^{\circ }(t) \ne \{ x_{1} , x_{2} \}$ in (11a), and when $t < t^{\star }_{1} : x^{\circ }(t) \ne \{ x_{1} , x_{2} \}$ in (11b) and (11c), function $h(t) = \nu t + [ f(\gamma _{a_{1}}) - \nu (\gamma _{a_{1}} - x_{2}) ]$.

In analogous way, it is possible to prove that, when $t \in [ t^{\star }_{q} , t^{\star }_{q+1} ) : x^{\circ }(t) \ne \{ x_{1} , x_{2} \}$, $q = 1, \ldots , Q-1$, in (11c), and when $t \ge t^{\star }_{Q} : x^{\circ }(t) \ne \{ x_{1} , x_{2} \}$ in (11b) and (11c), it turns out $h(t) = \nu t \ + \ [ f(\gamma _{a_{l(q)+1}}) \ - \ \nu (\gamma _{a_{l(q)+1}} - x_{2}) ]$, $q = 1, \ldots , Q$.

Appendix 3: optimal control strategies for the numerical example

1.1 Strategies used to obtain the nominal optimal solution

• stage 0 – state $[0 \; 0 \; 0 \; t_{0}]^{T}$

  $$\begin{aligned} \delta _{1}^{\circ } (0,0,0, t_{0})= & {} \left\{ \begin{array}{ll} 1 &{} t_{0} < -20.5\\ 0 &{} t_{0} \ge -20.5 \end{array} \right. \\ \delta _{2}^{\circ } (0,0,0, t_{0})= & {} \left\{ \begin{array}{ll} 0 &{} t_{0} < -20.5\\ 1 &{} t_{0} \ge -20.5 \end{array} \right. \\ \tau ^{\circ } (0, 0, 0, t_{0})= & {} \left\{ \begin{array}{ll} 8 &{} t_{0} < -21\\ -t_{0} - 13 &{} -21 \le t_{0} < -20.5\\ 6 &{} -20.5 \le t_{0} < 2.5\\ -t_{0} + 8.5 &{} 2.5 \le t_{0} < 4.5\\ 4 &{} t_{0} \ge 4.5 \end{array} \right. \end{aligned}$$

• stage 1 – state $[0 \; 1 \; 2 \; t_{1}]^{T}$

  $$\begin{aligned} \delta _{1}^{\circ } (0,1,2, t_{1})= & {} \left\{ \begin{array}{ll} 0 &{} t_{1} < 15\\ 1 &{} 15 \le t_{1} < 17.5\\ 0 &{} t_{1} \ge 17.5 \end{array} \right. \\ \delta _{2}^{\circ } (0,1,2, t_{1})= & {} \left\{ \begin{array}{ll} 1 &{} t_{1} < 15\\ 0 &{} 15 \le t_{1} < 17.5\\ 1 &{} t_{1} \ge 17.5 \end{array} \right. \\ \tau ^{\circ } (0,1,2, t_{1})= & {} \left\{ \begin{array}{ll} 6 &{} t_{1} < 8.5\\ -t_{1} + 14.5 &{} 8.5 \le t_{1} < 10.5\\ 4 &{} t_{1} \ge 10.5 \end{array} \right. \end{aligned}$$

• stage 2 – state $[0 \; 2 \; 2 \; t_{2}]^{T}$

  $$\begin{aligned} \delta _{1}^{\circ } (0,2,2, t_{2})= & {} \left\{ \begin{array}{ll} 0 &{} t_{2} < 0\\ 1 &{} 0 \le t_{2} < 29.25\\ 0 &{} t_{2} \ge 29.25 \end{array} \right. \\ \delta _{2}^{\circ } (0,2,2, t_{2})= & {} \left\{ \begin{array}{ll} 1 &{} t_{2} < 0\\ 0 &{} 0 \le t_{2} < 29.25\\ 1 &{} t_{2} \ge 29.25 \end{array} \right. \\ \tau ^{\circ } (0,2,2, t_{2})= & {} \left\{ \begin{array}{ll} 6 &{} t_{2} < 0\\ 8 &{} 0 \le t_{2} < 4.5\\ -t_{2} + 12.5 &{} 4.5 \le t_{2} < 8.5\\ 4 &{} t_{2} \ge 8.5 \end{array} \right. \end{aligned}$$

• stage 3 – state $[1 \; 2 \; 1 \; t_{3}]^{T}$

  $$\begin{aligned} \delta _{1}^{\circ } (1,2,1, t_{3})= & {} 1 \quad \forall \, t_{3}\\ \delta _{2}^{\circ } (1,2,1, t_{3})= & {} 0 \quad \forall \, t_{3}\\ \tau ^{\circ } (1,2,1, t_{3})= & {} \left\{ \begin{array}{ll} 8 &{} t_{3} < 7\\ -t_{3} + 15 &{} 7 \le t_{3} < 8\\ 8 &{} 8 \le t_{3} < 13\\ -t_{3} + 21 &{} 13 \le t_{3} < 17\\ 4 &{} t_{3} \ge 17 \end{array} \right. \end{aligned}$$

• stage 4 – state $[2 \; 2 \; 1 \; t_{4}]^{T}$

  $$\begin{aligned} \delta _{1}^{\circ } (2,2,1, t_{4})= & {} 1 \quad \forall \, t_{4}\\ \delta _{2}^{\circ } (2,2,1, t_{4})= & {} 0 \quad \forall \, t_{4}\\ \tau ^{\circ } (2,2,1, t_{4})= & {} \left\{ \begin{array}{ll} 8 &{} t_{4} < 15\\ -t_{4} + 23 &{} 15 \le t_{4} < 16\\ 8 &{} 16 \le t_{4} < 21\\ -t_{4} + 29 &{} 21 \le t_{4} < 25\\ 4 &{} t_{4} \ge 25 \end{array} \right. \end{aligned}$$

• stage 5 – state $[3 \; 2 \; 1 \; t_{5}]^{T}$

  $$\begin{aligned} \delta _{1}^{\circ } (3,2,1, t_{5})= & {} \left\{ \begin{array}{ll} 1 &{} t_{5} < 24\\ 0 &{} t_{5} \ge 24 \end{array} \right. \\ \delta _{2}^{\circ } (3,2,1, t_{5})= & {} \left\{ \begin{array}{ll} 0 &{} t_{5} < 24\\ 1 &{} t_{5} \ge 24 \end{array} \right. \\ \tau ^{\circ } (3,2,1, t_{5})= & {} \left\{ \begin{array}{ll} 8 &{} t_{5} < 23\\ -t_{5} + 31 &{} 23 \le t_{5} < 24\\ 6 &{} 24 \le t_{5} < 31\\ -t_{5} + 37 &{} 31 \le t_{5} < 33\\ 4 &{} t_{5} \ge 33 \end{array} \right. \end{aligned}$$

• stage 6 – state $[3 \; 3 \; 2 \; t_{6}]^{T}$

  $$\begin{aligned} \delta _{1}^{\circ } (3,3,2, t_{6})= & {} 1 \quad \forall \, t_{6}\\ \delta _{2}^{\circ } (3,3,2, t_{6})= & {} 0 \quad \forall \, t_{6}\\ \tau ^{\circ } (3,3,2, t_{6})= & {} 8 \quad \forall \, t_{6} \end{aligned}$$

1.2 Strategies used to obtain the optimal solution in the perturbed case #1

• stage 0 – state $[0 \; 0 \; 0 \; t_{0}]^{T}$ (see previous page)

• stage 1 – state $[0 \; 1 \; 2 \; t_{1}]^{T}$ (see previous page)

• stage 2 – state $[1 \; 1 \; 1 \; t_{2}]^{T}$

  $$\begin{aligned} \delta _{1}^{\circ } (1,1,1, t_{2})= & {} \left\{ \begin{array}{ll} 1 &{} t_{2} < -6\\ 0 &{} -6 \le t_{2} < 1\\ 1 &{} 1 \le t_{2} < 15.5\\ 0 &{} 15.5 \le t_{2} < 17.25\\ 1 &{} t_{2} \ge 17.25 \end{array} \right. \\ \delta _{2}^{\circ } (1,1,1, t_{2})= & {} \left\{ \begin{array}{ll} 0 &{} t_{2} < -6\\ 1 &{} -6 \le t_{2} < 1\\ 0 &{} 1 \le t_{2} < 15.5\\ 1 &{} 15.5 \le t_{2} < 17.25\\ 0 &{} t_{2} \ge 17.25 \end{array} \right. \\ \tau ^{\circ } (1,1,1, t_{2})= & {} \left\{ \begin{array}{ll} 8 &{} t_{2} < -7\\ -t_{2} + 1 &{} -7 \le t_{2} < -6\\ 6 &{} -6 \le t_{2} < 1\\ -t_{2} + 9 &{} 1 \le t_{2} < 3\\ 8 &{} 3 \le t_{2} < 5.5\\ -t_{2} +13.5 &{} 5.5 \le t_{2} < 9.5\\ 4 &{} t_{2} \ge 9.5 \end{array} \right. \end{aligned}$$

• stage 3 – state $[2 \; 1 \; 1 \; t_{3}]^{T}$

  $$\begin{aligned} \delta _{1}^{\circ } (2,1,1, t_{3})= & {} \left\{ \begin{array}{ll} 1 &{} t_{3} < 2\\ 0 &{} 2 \le t_{3} < 9\\ 1 &{} 9 \le t_{3} < 11\\ 0 &{} 11 \le t_{3} < 20.\overline{6}\\ 1 &{} t_{3} \ge 20.\overline{6} \end{array} \right. \\ \delta _{2}^{\circ } (2,1,1, t_{3})= & {} \left\{ \begin{array}{ll} 0 &{} t_{3} < 2\\ 1 &{} 2 \le t_{3} < 9\\ 0 &{} 9 \le t_{3} < 11\\ 1 &{} 11 \le t_{3} < 20.\overline{6}\\ 0 &{} t_{3} \ge 20.\overline{6} \end{array} \right. \\ \tau ^{\circ } (2,1,1, t_{3})= & {} \left\{ \begin{array}{ll} 8 &{} t_{3} < 1\\ -t_{3} + 9 &{} 1 \le t_{3} < 2\\ 6 &{} 2 \le t_{3} < 9\\ -t_{3} + 17 &{} 9 \le t_{3} < 11\\ 6 &{} 11 \le t_{3} < 17\\ -t_{3} + 23 &{} 17 \le t_{3} < 19\\ 4 &{} t_{3} \ge 19 \end{array} \right. \end{aligned}$$

• stage 4 – state $[3 \; 1 \; 1 \; t_{4}]^{T}$

  $$\begin{aligned} \delta _{1}^{\circ } (3,1,1, t_{4})= & {} \left\{ \begin{array}{ll} 1 &{} t_{4} < 10\\ 0 &{} t_{4} \ge 10 \end{array} \right. \\ \delta _{2}^{\circ } (3,1,1, t_{4})= & {} \left\{ \begin{array}{ll} 0 &{} t_{4} < 10\\ 1 &{} t_{4} \ge 10 \end{array} \right. \\ \tau ^{\circ } (3,1,1, t_{4})= & {} \left\{ \begin{array}{ll} 8 &{} t_{4} < 9\\ -t_{4} + 17 &{} 9 \le t_{4} < 24.5\\ 6 &{} 10 \le t_{4} < 19.5\\ -t_{4} + 25.5 &{} 19.5 \le t_{4} < 21.5\\ 4 &{} t_{4} \ge 21.5 \end{array} \right. \end{aligned}$$

• stage 5 – state $[3 \; 2 \; 2 \; t_{5}]^{T}$

  $$\begin{aligned} \delta _{1}^{\circ } (3,2,2, t_{5})= & {} 0 \quad \forall \, t_{5}\\ \delta _{2}^{\circ } (3,2,2, t_{5})= & {} 1 \quad \forall \, t_{5}\\ \tau ^{\circ } (3,2,2, t_{5})= & {} \left\{ \begin{array}{ll} 6 &{} t_{5} < 32\\ -t_{5} + 38 &{} 32 \le t_{5} < 34\\ 4 &{} t_{4} \ge 34 \end{array} \right. \end{aligned}$$

• stage 6 – state $[3 \; 3 \; 2 \; t_{6}]^{T}$ (see above)

1.3 Strategies used to obtain the optimal solution in the perturbed case #2

• stage 0 – state $[0 \; 0 \; 0 \; t_{0}]^{T}$ (see previous page)

• stage 1 – state $[0 \; 1 \; 2 \; t_{1}]^{T}$ (see previous page)

• stage 2 – state $[0 \; 2 \; 2 \; t_{2}]^{T}$ (see previous page)

• stage 3 – state $[1 \; 2 \; 1 \; t_{3}]^{T}$ (see previous page)

• stage 4 – state $[2 \; 2 \; 1 \; t_{4}]^{T}$ (see above)

• stage 5 – state $[3 \; 2 \; 1 \; t_{5}]^{T}$ (see above)

• stage 6 – state $[3 \; 3 \; 2 \; t_{6}]^{T}$ (see above)