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Due-date quotation model for manufacturing system scheduling under uncertainty

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Abstract

This paper studies the scheduling problem for the manufacturing systems with uncertain job duration, and the possibility of planning due-date quotations for critical manufacturing tasks given a fixed contingency budget. We propose a due-date quotation model to measure the risk of delay in the manufacturing process in terms of the allocated contingency budget. The risk of delay is measured in the same unit as its corresponding milestone factor such that the decision makers could directly visualize and quantify the level of risks in units of hours or days. In addition, the proposed model possesses various great properties required by a convex risk measure and it represents a minimized certainty equivalent of the overall expected risk in achieving the manufacturing due-dates. Extensive computational experiments are conducted to evaluate the model performance. The results show that our proposed model, compared to various existing methods, provides a much more balanced performance in terms of success rate of due-date achievement, due-date quotation shortfall, as well as, robustness against uncertainties. The practical applicability of the proposed models are also tested with the job scheduling problem in a real stamping industry application.

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Acknowledgements

This research was supported by a grant from the National Research Foundation, Prime Minister Office, Singapore under its Energy and Environmental Sustainability for Megacities (E2S2), Campus of Research Excellence and Technological Enterprise (CREATE) programme.

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Correspondence to Chee Khiang Pang.

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This article belongs to the Topical Collection: Smart Manufacturing - A New DES Frontier

Guest Editors: Rong Su and Bengt Lennartson

This research is supported by the National Research Foundation, Prime Minister’s Office, Singapore under its Energy and Environmental Sustainability for Megacities (E2S2), Campus for Research Excellence and Technological Enterprise (CREATE) programme.

Appendix

Appendix

1.1 A.1 Proof of Theorem 1

Proof

Each of the properties will be proved in detail here.

The proposed model in Eq. 1 can be equivalently written as:

$$ \psi_{\tau}(\boldsymbol{\tilde{z}}) := \left\{ \inf\limits_{\alpha_{i}, \tau_{i}}\left( \sum\limits_{i \in \mathcal{M}}^{}\omega_{i}\alpha_{i}+\tau\right) \ \Big| \ \sum\limits_{i \in \mathcal{M}}^{}\omega_{i} \tau_{i} = \tau, \ \forall i \in \mathcal{M}: \mathbb{E}[\tilde{z}_{i}-\alpha_{i}]^{+} \leq \tau_{i} , \tau_{i} \geq 0 \right\}. $$
(11)
  1. 1.

    Normalization: ψτ(0) = 0.

    By definition,

    $$ \begin{array}{@{}rcl@{}} \psi_{\tau}(\boldsymbol{0}):=\left\{\inf\limits_{\boldsymbol{\alpha}, \boldsymbol{\tau}}\left\{\sum\limits_{i=1}^{m}\omega_{i}\alpha_{i}+\tau\right\}: \sum\limits_{i=1}^{m}\omega_{i}\tau_{i}=\tau, \right.\ \forall i &\in& \mathcal{M}:\\ \mathbb{E}[0-\alpha_{i}]^{+} &\leq& \tau_{i},\\ \tau_{i} &\geq& 0 \left.\vphantom{\sum\limits_{i=1}^{m}}\right\}. \end{array} $$

    It is clear that αi = −τi is the smallest feasible value for the constraint.

    As such \(\psi _{\tau }(0)=\min \limits _{\boldsymbol {\alpha }, \boldsymbol {\tau }}\{{\sum }_{i=1}^{m}\omega _{i}(-\tau _{i})+\tau \}=(-\tau +\tau )=0\).

  2. 2.

    Monotonicity: If \(\boldsymbol {\tilde {z}_{1}} \leq \boldsymbol {\tilde {z}_{2}}\), then \(\psi _{\tau }(\boldsymbol {\tilde {z}_{1}}) \leq \psi _{\tau }(\boldsymbol {\tilde {z}_{2}}).\)

    By the definition of Eq. 1,

    Model in the case of \(\boldsymbol {\tilde {z}_{1}}\):

    $$ \begin{array}{@{}rcl@{}} \psi_{\tau}(\boldsymbol{\tilde{z}_{1}})&=&\inf\limits_{\boldsymbol{\alpha}_{1}, \boldsymbol{\tau}}\left\{\sum\limits_{i=1}^{m}\omega_{i}\alpha_{1,i}+\tau\right\}\\ &s.t. \quad &\mathbb{E}[\tilde{z}_{1,i}-\alpha_{1,i}]^{+} \leq \tau_{i}, \\ &&\sum\limits_{i=1}^{m}\omega_{i}\tau_{i}=\tau,\\ &&\tau_{i} \geq 0, \ \forall i. \end{array} $$

    Model in the case of \(\boldsymbol {\tilde {z}_{2}}\):

    $$ \begin{array}{@{}rcl@{}} \psi_{\tau}(\boldsymbol{\tilde{z}_{2}})&=&\inf\limits_{\boldsymbol{\alpha}_{2}, \boldsymbol{\tau}}\left\{\sum\limits_{i=1}^{m}\omega_{i}\alpha_{2,i}+\tau\right\}\\ &s.t. \quad &\mathbb{E}[\tilde{z}_{2,i}-\alpha_{2,i}]^{+} \leq \tau_{i}, \\ &&\sum\limits_{i=1}^{m}\omega_{i}\tau_{i}=\tau,\\ &&\tau_{i} \geq 0, \ \forall i. \end{array} $$

    Suppose the optimal α values are \(\boldsymbol {\alpha }_{1}^{*}\) and \(\boldsymbol {\alpha }_{2}^{*}\) respectively. As such, \(\psi _{\tau }(\boldsymbol {\tilde {z}_{1}})={\sum }_{i=1}^{m}\omega _{i}\alpha _{1,i}^{*}+\tau \) and \(\psi _{\tau }(\boldsymbol {\tilde {z}_{2}})={\sum }_{i=1}^{m}\omega _{i}\alpha _{2,i}^{*}+\tau \). For the same τ, the feasible values satisfy \(\alpha _{1,i}^{*} \leq \alpha _{2,i}^{*}\) since function \(\mathbb {E}[\tilde {z}_{i}-\alpha _{i}]^{+}\) is non-decreasing function w.r.t \(\tilde {z}_{i}\). As such, \(\psi _{\tau }(\boldsymbol {\tilde {z}_{1}}) \leq \psi _{\tau }(\boldsymbol {\tilde {z}_{2}})\).

  3. 3.

    Convexity: \(\forall \boldsymbol {\tilde {z}_{1}}, \boldsymbol {\tilde {z}_{2}} \in \boldsymbol {\mathcal {Z}}\) and λ ∈ [0, 1], \(\psi _{\tau }(\lambda \boldsymbol {\tilde {z}_{1}}+(1-\lambda )\boldsymbol {\tilde {z}_{2}}) \leq \lambda \psi _{\tau }(\boldsymbol {\tilde {z}_{1}})+(1-\lambda )\psi _{\tau }(\boldsymbol {\tilde {z}_{2}})\).

    The model is defined for \(\boldsymbol {\tilde {z}_{1}}\) as:

    $$ \begin{array}{@{}rcl@{}} &&\psi_{\tau}(\boldsymbol{\tilde{z}_{1}})= \sum\limits_{i=1}^{m}\omega_{i} \alpha_{1,i}^{*} +\tau, \\ &&\sum\limits_{i=1}^{m}\omega_{i} \tau_{1,i} =\tau, \end{array} $$

    and for \(\boldsymbol {\tilde {z}_{1}}\) as:

    $$ \begin{array}{@{}rcl@{}} &&\psi_{\tau}(\boldsymbol{\tilde{z}_{2}})= \sum\limits_{i=1}^{m}\omega_{i} \alpha_{2,i}^{*} +\tau, \\ &&\sum\limits_{i=1}^{m}\omega_{i} \tau_{2,i} =\tau. \end{array} $$

    It is also known that

    $$ \begin{array}{@{}rcl@{}} \psi_{\tau}(\lambda \boldsymbol{\tilde{z}_{1}}+(1-\lambda)\boldsymbol{\tilde{z}_{2}}) = \sum\limits_{i=1}^{m}\omega_{i} \alpha_{0,i}^{*} +\tau. \end{array} $$

    Proving the convexity is equivalent to proving that

    $$ \begin{array}{@{}rcl@{}} &&\sum\limits_{i=1}^{m}\omega_{i} \alpha_{0,i}^{*} +\tau \leq \lambda \left( \sum\limits_{i=1}^{m}\omega_{i} \alpha_{1,i}^{*} +\tau\right) + (1-\lambda) \left( \sum\limits_{i=1}^{m}\omega_{i} \alpha_{2,i}^{*} +\tau\right), \end{array} $$

    which can be further simplified as \({\sum }_{i=1}^{m}\omega _{i} \alpha _{0,i}^{*} \leq \lambda {\sum }_{i=1}^{m}\omega _{i} \alpha _{1,i}^{*} + (1-\lambda ){\sum }_{i=1}^{m}\omega _{i} \alpha _{2,i}^{*}\).

    Let \(\hat {\alpha _{i}}=\lambda \alpha _{1,i}^{*} + (1-\lambda )\alpha _{2,i}^{*}\). The function \(f(\tilde {z})=[\tilde {z}]^{+}=\max \limits \{0, \tilde {z}\}\) can be proved to be convex.

    $$ \begin{array}{@{}rcl@{}} (\lambda \tilde{z_{1}}+(1-\lambda) \tilde{z_{2}})^{+} &=&\max\{\lambda \tilde{z_{1}}+(1-\lambda) \tilde{z_{2}}, 0\} \\ &\leq& \max\{\lambda \tilde{z_{1}}, 0\} + \max\{(1-\lambda) \tilde{z_{2}}, 0\}\\ &=& \lambda \max\{\tilde{z_{1}}, 0\} + (1-\lambda) \max\{\tilde{z_{2}}, 0\}\\ &=& \lambda \{\tilde{z_{1}}\}^{+} + (1-\lambda) \{\tilde{z_{2}}\}^{+}.\\ \end{array} $$

    As such, \(\mathbb {E}(\tilde {z})^{+}\) is also convex. Then the following holds:

    $$ \begin{array}{@{}rcl@{}} &&\mathbb{E}[\lambda \tilde{z}_{1,i}+(1-\lambda) \tilde{z}_{2,i} - \hat{\alpha_{i}}]^{+}\\ &=&\mathbb{E}[\lambda \tilde{z}_{1,i}+(1-\lambda) \tilde{z}_{2,i} - \lambda \alpha_{1,i}^{*} - (1-\lambda)\alpha_{2,i}^{*}]^{+}\\ &\leq& \lambda \mathbb{E}[\tilde{z}_{1,i}-\alpha_{1,i}^{*}]^{+} + (1-\lambda) \mathbb{E}[\tilde{z}_{2,i}-\alpha_{2,i}^{*}]^{+} \\ &\leq& \lambda \tau_{1,i} + (1-\lambda) \tau_{2,i} \end{array} $$

    Since both τ1,i,τ2,i ∈Γ and set Γ is convex, λτ1,i + (1 − λ)τ2,i ∈Γ for any λ ∈ [0, 1].

    $$ \begin{array}{@{}rcl@{}} &&\sum\limits_{i=1}^{m}\omega_{i} \{\lambda \tau_{1,i} + (1-\lambda) \tau_{2,i}\}\\ &=&\lambda \sum\limits_{i=1}^{m}\omega_{i} \tau_{1,i}+ (1-\lambda)\sum\limits_{i=1}^{m}\omega_{i} \tau_{2,i}\\ &=&\lambda \tau + (1-\lambda) \tau\\ &=&\tau. \end{array} $$

    As a result, \(\hat {\alpha _{i}}\) is a feasible threshold value for \(\psi _{\tau }(\lambda \tilde {z}_{1,i}+(1-\lambda )\tilde {z}_{2,i})\). Nevertheless, \(\alpha _{0,i}^{*}\) is the optimal value that minimizes the objective function among all the feasible solutions. As a result, it is clear to conclude that \(\alpha _{0,i}^{*} \leq \hat {\alpha _{i}} \ \forall i=1,2,\dots ,m\). Further, it is not difficult to see that \({\sum }_{i=1}^{m}\omega _{i} \alpha _{0,i}^{*} \leq \lambda {\sum }_{i=1}^{m}\omega _{i} \alpha _{1,i}^{*} + (1-\lambda ){\sum }_{i=1}^{m}\omega _{i} \alpha _{2,i}^{*}\).

    In conclusion, \(\psi _{\tau }(\lambda \boldsymbol {\tilde {z}_{1}}+(1-\lambda )\boldsymbol {\tilde {z}_{2}}) \leq \lambda \psi _{\tau }(\boldsymbol {\tilde {z}_{1}})+(1-\lambda )\psi _{\tau }(\boldsymbol {\tilde {z}_{2}})\) and the convexity of ψ is proved.

  4. 4.

    Translation Invariance: If \(\boldsymbol {\delta }=(\delta _{1}, \delta _{2}, \dots , \delta _{m}) \in \mathbb {R}^{m}\), then \(\psi _{\tau }(\boldsymbol {\tilde {z}}-\boldsymbol {\delta })=\psi _{\tau }(\boldsymbol {\tilde {z}})-{\sum }_{i=1}^{m}\omega _{i} \delta _{i}\).

    By definition,

    $$ \begin{array}{@{}rcl@{}} \psi_{\tau}(\boldsymbol{\tilde{z}}-\boldsymbol{\delta})&=&\min\limits_{\boldsymbol{\alpha}, \boldsymbol{\tau}}\left\{\sum\limits_{i=1}^{m}\omega_{i}\alpha_{i} + \tau\right\}\\ &s.t. \quad &\mathbb{E}[\tilde{z}_{i}-\delta_{i}-\alpha_{i}]^{+} \leq \tau_{i},\\ &&\sum\limits_{i=1}^{m}\omega_{i}\tau_{i}=\tau. \end{array} $$

    For the objective function,

    $$ \begin{array}{@{}rcl@{}} \psi_{\tau}(\boldsymbol{\tilde{z}}-\boldsymbol{\delta})&=&\min\limits_{\boldsymbol{\alpha}, \boldsymbol{\tau}}\left\{\sum\limits_{i=1}^{m}\omega_{i}\alpha_{i} + \tau\right\}\\ &=&\min\limits_{\boldsymbol{\alpha}, \boldsymbol{\tau}}\left\{\sum\limits_{i=1}^{m}\omega_{i}\alpha_{i} + \sum\limits_{i=1}^{m}\omega_{i} \delta_{i}+ \tau\right\} - \sum\limits_{i=1}^{m}\omega_{i} \delta_{i} \\ &=&\min\limits_{\boldsymbol{\alpha}, \boldsymbol{\tau}}\left\{\sum\limits_{i=1}^{m}\omega_{i}(\alpha_{i} + \delta_{i})+ \tau\right\} - \sum\limits_{i=1}^{m}\omega_{i} \delta_{i} \\ &=&\min\limits_{\boldsymbol{\alpha}, \boldsymbol{\tau}}\left\{\sum\limits_{i=1}^{m}\omega_{i} \hat{\alpha_{i}}+ \tau\right\} - \sum\limits_{i=1}^{m}\omega_{i} \delta_{i}. \end{array} $$

    For the constraint,

    $$ \begin{array}{@{}rcl@{}} \mathbb{E}[\tilde{z}_{i}-\delta_{i}-\alpha_{i}]^{+} &\leq& \tau_{i}\\ \mathbb{E}[\tilde{z}_{i}-(\delta_{i}+\alpha_{i})]^{+} &\leq& \tau_{i}\\ \mathbb{E}[\tilde{z}_{i}-\hat{\alpha_{i}}]^{+} &\leq& \tau_{i} \end{array} $$

    As a result, the model can be written as:

    $$ \begin{array}{@{}rcl@{}} &&\min\limits_{\boldsymbol{\tilde{\alpha}}, \boldsymbol{\tau}}\left\{\sum\limits_{i=1}^{m}\omega_{i}\hat{\alpha_{i}} + \tau\right\} - \sum\limits_{i=1}^{m}\omega_{i} \delta_{i} = \psi(\boldsymbol{\tilde{z}})-{\sum}_{i=1}^{m}\omega_{i} \delta_{i}\\ &&s.t. \ \ \mathbb{E}[\tilde{z}_{i}-\hat{\alpha_{i}}]^{+} \leq \tau_{i}. \end{array} $$
  5. 5.

    Contingency Sensitivity: if τ1τ2, then \(\psi _{\tau _{2}}(\boldsymbol {\tilde {z}}) \leq \psi _{\tau _{1}}(\boldsymbol {\tilde {z}})\).

    Since τ1τ2, suppose τ2 = τ1 + Δττ ≥ 0 without loss of generality.

    1. (P1)
      $$ \begin{array}{@{}rcl@{}} \psi_{\tau_{1}}(\boldsymbol{\tilde{z}})&=&\min\limits_{\boldsymbol{\alpha}, \boldsymbol{\tau}}\left\{\sum\limits_{i=1}^{m}\omega_{i}\alpha_{1,i} + \tau_{1}\right\}\\ &s.t. \quad &\mathbb{E}[\tilde{z}_{i}-\alpha_{1,i}]^{+} \leq \tau_{1,i},\\ &&\sum\limits_{i=1}^{m}\omega_{i}\tau_{1,i}=\tau_{1}, \\ &&\tau_{1,i} \geq 0, \forall i. \end{array} $$
    2. (P2)
      $$ \begin{array}{@{}rcl@{}} \psi_{\tau_{2}}(\boldsymbol{\tilde{z}})&=&\min\limits_{\boldsymbol{\alpha}, \boldsymbol{\tau}}\left\{\sum\limits_{i=1}^{m}\omega_{i}\alpha_{2,i} + \tau_{2}\right\}\\ &s.t. \quad&\mathbb{E}[\tilde{z}_{i}-\alpha_{2,i}]^{+} \leq \tau_{2,i},\\ &&\sum\limits_{i=1}^{m}\omega_{i}\tau_{2,i}=\tau_{2}, \\ &&\tau_{2,i} \geq 0, \forall i. \end{array} $$

Suppose that \(\alpha _{1,i}^{*}, \tau _{1,i}^{*}\) are optimal solutions to (P1). The idea is to construct one feasible solution to (P2) using expressions of the optimal solutions to (P1). Constructing the following:

$$ \left\{ \begin{array}{ll} \hat{\tau_{i}}=\tau_{1,i}^{*} &, \forall i \neq j.\\ \hat{\tau_{j}}=\tau_{1,j}^{*}+\frac{\Delta \tau}{\omega_{j}} &, j \in \{1,2,\dots, n\}. \end{array} \right. $$

Since \(\alpha _{1,i}^{*}, \tau _{1,i}^{*}\) are optimal solutions to (P1), it is true that

$$ \begin{array}{@{}rcl@{}} \mathbb{E}[\tilde{z}_{j}-\alpha_{1,j}^{*}]^{+} & \leq& \tau_{1,j}^{*}\\ \rightarrow \mathbb{E}\left[\tilde{z}_{j}-\alpha_{1,j}^{*}+\frac{\Delta \tau}{\omega_{j}}-\frac{\Delta \tau}{\omega_{j}}\right]^{+} & \leq& \tau_{1,j}^{*}\\ \rightarrow \mathbb{E}\left[\tilde{z}_{j}-\left( \alpha_{1,j}^{*}-\frac{\Delta\tau}{\omega_{j}}\right)-\frac{\Delta\tau}{\omega_{j}}\right]^{+} & \leq& \tau_{1,j}^{*}\\ \rightarrow \mathbb{E}\left[\tilde{z}_{j}-\left( \alpha_{1,j}^{*}-\frac{\Delta\tau}{\omega_{j}}\right)\right]^{+}-\frac{\Delta\tau}{\omega_{j}} & \leq& \tau_{1,j}^{*}\\ \rightarrow \mathbb{E}\left[\tilde{z}_{j}-\left( \alpha_{1,j}^{*}-\frac{\Delta\tau}{\omega_{j}}\right)\right]^{+}& \leq& \tau_{1,j}^{*} + \frac{\Delta\tau}{\omega_{j}} \\ \rightarrow \mathbb{E}\left[\tilde{z}_{j}-\left( \alpha_{1,j}^{*}-\frac{\Delta\tau}{\omega_{j}}\right)\right]^{+}& \leq& \hat{\tau_{j}} \end{array} $$

For the case that ij,

$$ \begin{array}{@{}rcl@{}} \mathbb{E}[\tilde{z}_{i}-\alpha_{1,i}^{*}]^{+} & \leq \tau_{1,i}^{*}\\ \rightarrow \mathbb{E}[\tilde{z}_{i}-\alpha_{1,i}^{*}]^{+} & \leq \hat{\tau_{i}}\\ \end{array} $$

As such, \(\{\alpha _{1,j}^{*}-\frac {\Delta \tau }{\omega _{j}}, \alpha _{1,i}^{*}\}\) is one feasible solution to (P2). In this case, the value of the objective function of (P2) can be calculated as:

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{i \neq j}^{}\omega_{i}\alpha_{1,i}^{*}+\omega_{j}\left( \alpha_{1,j}^{*}-\frac{\Delta\tau}{\omega_{j}}\right) + \tau_{1}+{\Delta} \tau\\ &=&\sum\limits_{i \neq j}^{}\omega_{i}\alpha_{1,i}^{*}+\omega_{j}\alpha_{1,j}^{*}-{\Delta}\tau+\tau_{1}+{\Delta}\tau\\ &=&\sum\limits_{\forall i}^{}\omega_{i}\alpha_{1,i}^{*} + \tau_{1}\\ &=&\psi^{*}(\tau_{1}). \end{array} $$

In other words, the minimal solution of (P1) is one feasible solution of (P2). As such, \(\psi _{\tau _{2}}(\boldsymbol {\tilde {z}}) \leq \psi _{\tau _{1}}(\boldsymbol {\tilde {z}})\). This completes the proof. □

1.2 A.2 Single due-date Quotation Model (Theorem 1 under m = 1)

In the special case that m = 1 in the model (1), i.e. there exists only a single milestone variable, the single milestone quotation model can be formulated as

$$ \begin{array}{@{}rcl@{}} \rho(\tau)&=&\min\limits_{\alpha}\{\alpha+\tau\}\\ & \text{s.t.}\quad &\mathbb{E}[\tilde{z}-\alpha]^{+} \leq \tau, \\ &&\tau \geq 0, \end{array} $$
(12)

where \([\tilde {z}-\alpha ]^{+}\) is defined as a positive function:

$$ [\tilde{z}-\alpha]^{+}=\max\{\tilde{z}-\alpha,0\}=\left\{ \begin{array}{ll} \tilde{z}-\alpha &, \tilde{z}\geq \alpha.\\ 0 &, \tilde{z} < \alpha. \end{array} \right. $$

Theorem 2

Single milestone quotation model (12) possesses the following properties:

  1. 1.

    Normalization: ρ(0) = 0.

  2. 2.

    Monotonicity: If z1z2, then ρ(z1) ≤ ρ(z2).

  3. 3.

    Convexity: \(\forall z_{1}, z_{2} \in \mathcal {Z}\) and λ ∈ [0, 1], ρ(λz1 + (1 − λ)z2) ≤ λρ(z1) + (1 − λ)ρ(z2).

  4. 4.

    Translation Invariance: If \(m \in \mathbb {R}\), then ρ(zm) = ρ(z) − m.

  5. 5.

    Contingency Sensitivity: If τ1τ2, then ρ(τ1) ≥ ρ(τ2).

The first four properties in Theorem 2 together proves that the proposed milestone quotation model in Eq. 12 is a convex risk measure defined in Follmer and Schied (2002). The fifth property reveals that this proposed milestone quotation model possesses is contingency budget sensitive. In other words, the proposed risk measure will return a smaller risk value when there is a larger amount of contingency budget and vice versa, which coincides with the decision maker’s intuition.

Proof

Each of the properties will be proved in detail here.

  1. 1.

    Normalization: ρ(0) = 0.

    By definition,

    $$ \begin{array}{@{}rcl@{}} &&\rho(0)=\min\limits_{\alpha}\{\alpha+\tau\}\\ &s.t. \ \ &\mathbb{E}[0-\alpha]^{+} \leq \tau. \end{array} $$

    It is clear that α = −τ is the smallest feasible value for the constraint.

    As such \(\rho (0)=\min \limits _{\alpha }\{\alpha +\tau \}=(-\tau +\tau )=0\).

  2. 2.

    Monotonicity: If \(\tilde {z}_{1} \leq \tilde {z}_{2}\), then \(\rho (\tilde {z}_{1}) \leq \rho (\tilde {z}_{2}).\)

    By the definition of Eq. 12,

    Model in the case of \(\tilde {z}_{1}\):

    $$ \begin{array}{@{}rcl@{}} &&\rho(\tilde{z}_{1})=\min\limits_{\alpha_{1}}\{\alpha_{1}+\tau\}\\ &s.t. \quad &\mathbb{E}[\tilde{z}_{1}-\alpha_{1}]^{+} \leq \tau. \end{array} $$

    Model in the case of \(\tilde {z}_{2}\):

    $$ \begin{array}{@{}rcl@{}} &&\rho(\tilde{z}_{2})=\min\limits_{\alpha_{2}}\{\alpha_{2}+\tau\}\\ &s.t. \quad &\mathbb{E}[\tilde{z}_{2}-\alpha_{2}]^{+} \leq \tau. \end{array} $$

    Define sets of feasible α1,α2 values that the 2 models can take as \(C_{\alpha _{1}}\) and \(C_{\alpha _{2}}\).

    $$ \begin{array}{@{}rcl@{}} && C_{\alpha_{1}}=\{\alpha_{1} | \mathbb{E}[\tilde{z}_{1}-\alpha_{1}]^{+} \leq \tau\},\\ && C_{\alpha_{2}}=\{\alpha_{2} | \mathbb{E}[\tilde{z}_{2}-\alpha_{2}]^{+} \leq \tau\}. \end{array} $$

    Suppose the optimal α values are \(\alpha _{1}^{*}\) and \(\alpha _{2}^{*}\) respectively. As such, \(\rho (\tilde {z}_{1})=\alpha _{1}^{*}+\tau \) and \(\rho (\tilde {z}_{2})=\alpha _{2}^{*}+\tau \). For the same τ, the feasible values satisfy \(\alpha _{1}^{*} \leq \alpha _{2}^{*}\) since function \(E[\tilde {z}-\alpha ]^{+}\) is non-decreasing function w.r.t z. As such, \(\rho (\tilde {z}_{1}) \leq \rho (\tilde {z}_{2})\).

  3. 3.

    Convexity: \(\forall \tilde {z}_{1}, \tilde {z}_{2} \in \mathcal {Z}\) and λ ∈ [0, 1], \(\rho (\lambda \tilde {z}_{1}+(1-\lambda )\tilde {z}_{2}) \leq \lambda \rho (\tilde {z}_{1})+(1-\lambda )\rho (\tilde {z}_{2})\).

    Firstly, the function \(f(\tilde {z})=[\tilde {z}]^{+}=\max \limits \{0, \tilde {z}\}\) can be proved to be convex.

    $$ \begin{array}{@{}rcl@{}} (\lambda \tilde{z_{1}}+(1-\lambda) \tilde{z_{2}})^{+} &=&\max\{\lambda \tilde{z_{1}}+(1-\lambda) \tilde{z_{2}}, 0\} \\ &\leq& \max\{\lambda \tilde{z_{1}}, 0\} + \max\{(1-\lambda) \tilde{z_{2}}, 0\}\\ &=& \lambda \max\{\tilde{z_{1}}, 0\} + (1-\lambda) \max\{\tilde{z_{2}}, 0\}\\ &= &\lambda \{\tilde{z_{1}}\}^{+} + (1-\lambda) \{\tilde{z_{2}}\}^{+}.\\ \end{array} $$

    Then any function \(\mathbb {E}(\tilde {z})^{+}\) is also convex.

    Now suppose that:

    $$ \begin{array}{@{}rcl@{}} \rho(\lambda \tilde{z}_{1}+(1-\lambda)\tilde{z}_{2})&=&\alpha^{*}+\tau, \\ \rho(\tilde{z}_{1})&=& \alpha_{1}^{*}+\tau, \\ \rho(\tilde{z}_{2})&=& \alpha_{2}^{*}+\tau . \end{array} $$

    Proving the convexity is equivalent to proving that \(\alpha ^{*}+\tau \leq \lambda (\alpha _{1}^{*}+\tau ) + (1-\lambda ) (\alpha _{2}^{*}+\tau ) \), which can be further simplified as \(\alpha ^{*} \leq \lambda \alpha _{1}^{*} + (1-\lambda )\alpha _{2}^{*}\).

    Let \(\hat {\alpha }=\lambda \alpha _{1}^{*} + (1-\lambda )\alpha _{2}^{*}\). Then the following holds:

    $$ \begin{array}{@{}rcl@{}} &&\mathbb{E}[\lambda \tilde{z_{1}}+(1-\lambda) \tilde{z_{2}} - \hat{\alpha}]^{+}\\ &=&\mathbb{E}[\lambda \tilde{z_{1}}+(1-\lambda) \tilde{z_{2}} - \lambda \alpha_{1}^{*} - (1-\lambda)\alpha_{2}^{*}]^{+}\\ &\leq& \lambda \mathbb{E}[\tilde{z_{1}}-\alpha_{1}^{*}]^{+} + (1-\lambda) \mathbb{E}[\tilde{z_{2}}-\alpha_{2}^{*}]^{+} \\ &\leq& \lambda \tau + (1-\lambda) \tau \\ &\leq& \tau \end{array} $$

    As a result, \(\hat {\alpha }\) is a feasible threshold value for \(\rho (\lambda \tilde {z}_{1}+(1-\lambda )\tilde {z}_{2})\). Nevertheless, α is the optimal value that minimizes the objective function among all the feasible solutions. As a result, it is clear to conclude that \(\alpha ^{*} \leq \hat {\alpha }\) which in return will prove that the convexity holds for ρ.

  4. 4.

    Translation Invariance: If \(\delta \in \mathbb {R}\), then \(\rho (\tilde {z}-\delta )=\rho (\tilde {z})-\delta \).

    By definition,

    $$ \begin{array}{@{}rcl@{}} &&\rho(\tilde{z}-\delta)=\min\limits_{\alpha}\{\alpha + \tau\}\\ s.t. \quad &&\mathbb{E}[\tilde{z}-\delta-\alpha]^{+} \leq \tau. \end{array} $$

    For the objective function,

    $$ \begin{array}{@{}rcl@{}} \rho(\tilde{z}-\delta)&=&\min\limits_{\alpha}\{\alpha + \tau\}\\ &=&\min\limits_{\alpha}\{\alpha + \delta + \tau\}-\delta \\ &=&\min\limits_{\hat{\alpha}}\{\tilde{\alpha} + \tau\}-\delta. \end{array} $$

    For the constraint,

    $$ \begin{array}{@{}rcl@{}} \mathbb{E}[\tilde{z}-\delta-\alpha]^{+} &\leq& \tau\\ \mathbb{E}[\tilde{z}-(\delta+\alpha)]^{+} &\leq& \tau\\ \mathbb{E}[\tilde{z}-\hat{\alpha}]^{+} &\leq& \tau. \end{array} $$

    As a result, the model can be written as:

    $$ \begin{array}{@{}rcl@{}} \min\limits_{\hat{\alpha}}\{\tilde{\alpha} + \tau\} - \delta &=& \rho(\tilde{z})-\delta\\ s.t. \quad \mathbb{E}[\tilde{z}-\hat{\alpha}]^{+} &\leq& \tau. \end{array} $$
  5. 5.

    Contingency Sensitivity: If τ1τ2, then ρ(τ1) ≥ ρ(τ2).

    Suppose

    $$ \begin{array}{@{}rcl@{}} &&\rho(\tau_{1})= \alpha_{1}^{*}+\tau_{1}, \\ &&\rho(\tau_{2})= \alpha_{2}^{*}+\tau_{2}. \end{array} $$

    In order to prove that ρ(τ1) ≥ ρ(τ2), the following condition should hold:

    $$ \begin{array}{@{}rcl@{}} &&\alpha_{1}^{*}+\tau_{1} \geq \alpha_{2}^{*}+\tau_{2}. \end{array} $$

    Let \(\tilde {\alpha }=\alpha _{1}^{*}+\tau _{1}-\tau _{2}\),

    $$ \begin{array}{@{}rcl@{}} \mathbb{E}[\tilde{z}-\tilde{\alpha}]^{+}&=&\mathbb{E}[\tilde{z}-(\alpha_{1}^{*}+\tau_{1}-\tau_{2})]^{+}\\ &\leq& \mathbb{E}[\tilde{z}-\alpha_{1}^{*}]^{+} + \mathbb{E}[\tau_{2}-\tau_{1}]^{+}\\ &\leq& \tau_{1} + \tau_{2}-\tau_{1}\\ &=&\tau_{2}. \end{array} $$

    Hence, \(\tilde {\alpha }\) is a feasible solution of ρ(τ2). Besides, \(\alpha _{2}^{*}\) is the optimal (minimum) value for ρ(τ2). As a result, \(\alpha _{2}^{*} \leq \tilde {\alpha }\) and further it is proved that ρ(τ1) ≥ ρ(τ2). This completes the proof.

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Wang, Z., Ng, T.S. & Pang, C.K. Due-date quotation model for manufacturing system scheduling under uncertainty. Discrete Event Dyn Syst 31, 271–293 (2021). https://doi.org/10.1007/s10626-020-00332-y

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