# Optimizing Process-Improvement Efforts for Supply Chain Operations under Disruptions: New Structural Results

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## Abstract

**:**

## 1. Background and Brief Literature Review

## 2. The Generic Model Formulation

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**1:**

## 3. Optimality and Structural Results for Different Production Functions

**Proposition**

**2.**

- (a)
- Under the labor-reliant (single input) policy ($\widehat{j}=1$), the (least-cost) additional capital input is $\widehat{{K}^{\prime}}(\Delta |\widehat{j})=0$, the additional labor input is $\widehat{{L}^{\prime}}(\Delta |\widehat{j})={(\Delta /A)}^{1/{\beta}_{L}}{K}_{0}{}^{-{\beta}_{K}/{\beta}_{L}}-{L}_{0}$ and the corresponding minimal cost of fixed cost reduction is ${\Gamma}_{1}^{\ast}(\Delta )={\Psi}_{1}{\Delta}^{1/{\beta}_{L}}-{\psi}_{1}$ where ${\Psi}_{1}={c}_{L}{(\Delta /A)}^{1/{\beta}_{L}}{K}_{0}{}^{-{\beta}_{K}/{\beta}_{L}}$ and ${\psi}_{1}={c}_{L}{L}_{0}$.
- (b)
- Under the capital-reliant (single input) policy ($\widehat{j}=2$), the (least-cost) additional labor input is $\widehat{{L}^{\prime}}(\Delta |\widehat{j})=0$, the additional capital input is $\widehat{{K}^{\prime}}(\Delta |\widehat{j})={(\Delta /A)}^{1/{\beta}_{K}}{L}_{0}{}^{-{\beta}_{L}/{\beta}_{K}}-{K}_{0}$ and the corresponding minimal cost of fixed cost reduction ${\Gamma}_{2}^{\ast}(\Delta )={\Psi}_{2}{\Delta}^{1/{\beta}_{K}}-{\psi}_{2}$ where ${\Psi}_{2}={c}_{K}{(\Delta /A)}^{1/{\beta}_{K}}{L}_{0}{}^{-{\beta}_{L}/{\beta}_{K}}$ and ${\psi}_{1}={c}_{K}{K}_{0}$.
- (c)
- Under the dual input policy ($\widehat{j}=3$), the least-cost additional labor input $\widehat{{L}^{\prime}}(\Delta |\widehat{j})$, the least-cost additional capital input $\widehat{{K}^{\prime}}(\Delta |\widehat{j})$ and the corresponding minimal cost of fixed cost reduction ${\Gamma}_{3}^{\ast}(\Delta )$ follow.$$\begin{array}{c}{\widehat{K}}^{\prime}*(\Delta |\widehat{j}*)={({\Omega}^{-{\beta}_{L}}\Delta /A)}^{1/\rho}-{K}_{0},\\ {\widehat{L}}^{\prime}*(\Delta |\widehat{j}*)={({\Omega}^{{\beta}_{K}}\Delta /A)}^{1/\rho}-{L}_{0},\\ {\widehat{\Gamma}}_{3}^{}(\Delta )={\Psi}_{3}{\Delta}^{1/\rho}-{\psi}_{3},\end{array}$$

**Proof**

**of**

**Proposition**

**2.**

**Corollary**

**1.**

- (a)
- If $\Delta >\mathrm{max}\left\{A{\Omega}^{{\beta}_{L}}{K}_{0}{}^{\rho},A{\Omega}^{-{\beta}_{K}}{L}_{0}{}^{\rho}\right\}$, the best input usage policy achieving the least cost for $\Delta $ is the dual input usage policy (${\widehat{j}}^{\ast}(\Delta )=3$) and ${\Gamma}^{\ast}(\Delta )={\widehat{\Gamma}}_{3}(\Delta )$.
- (b)
- If $\Delta \le \mathrm{max}\left\{A{\Omega}^{{\beta}_{L}}{K}_{0}{}^{\rho},A{\Omega}^{-{\beta}_{K}}{L}_{0}{}^{\rho}\right\}$, ${c}_{L}{(\Delta /A)}^{1/{\beta}_{L}}{K}_{0}{}^{-{\beta}_{K}/{\beta}_{L}}-{c}_{L}{L}_{0}\le {c}_{K}{(\Delta /A)}^{1/{\beta}_{K}}{L}_{0}{}^{-{\beta}_{L}/{\beta}_{K}}-{c}_{K}{K}_{0}$, then the best input-usage policy achieving the least cost for $\Delta $ is the labor-reliant single input-usage policy (${\widehat{j}}^{\ast}(\Delta )=1$) and ${\Gamma}^{\ast}(\Delta )={\widehat{\Gamma}}_{1}(\Delta )$.
- (c)
- If $\Delta \le \mathrm{max}\left\{A{\Omega}^{{\beta}_{L}}{K}_{0}{}^{\rho},A{\Omega}^{-{\beta}_{K}}{L}_{0}{}^{\rho}\right\}$, ${c}_{L}{(\Delta /A)}^{1/{\beta}_{L}}{K}_{0}{}^{-{\beta}_{K}/{\beta}_{L}}-{c}_{L}{L}_{0}>{c}_{K}{(\Delta /A)}^{1/{\beta}_{K}}{L}_{0}{}^{-{\beta}_{L}/{\beta}_{K}}-{c}_{K}{K}_{0}$ then, the best input-usage policy achieving the least cost for $\Delta $ is the capital-reliant single-input-usage policy (${\widehat{j}}^{\ast}(\Delta )=2$) and ${\Gamma}^{\ast}(\Delta )={\widehat{\Gamma}}_{2}(\Delta )$.
- (d)
- If the initial capital and labor endowments ${K}_{0}$ and ${L}_{0}$ have been chosen according to the marginal cost condition ${c}_{K}\partial \Gamma (\Delta )/\partial {L}_{0}={c}_{L}\partial \Gamma (\Delta )/\partial {K}_{0}$, then it is always optimal to use the dual-input policy.

**Proof**

**of**

**Corollary**

**1.**

**Proposition**

**3.**

- (a)
- Under the labor-reliant (single input) policy ($\widehat{j}=1$), the (least-cost) additional capital input is $\widehat{{K}^{\prime}}(\Delta |\widehat{j})=0$, the additional labor input is $\widehat{{L}^{\prime}}(\Delta |\widehat{j})={[(1/{\alpha}_{L}){\Delta}^{-\beta}-({\alpha}_{K}/{\alpha}_{L}){K}_{0}{}^{\beta}]}^{1/\beta}-{L}_{0}$ and the corresponding minimal cost of setup reduction is ${\widehat{\Gamma}}_{1}^{}(\Delta )={c}_{L}{[(1/{\alpha}_{L}){\Delta}^{-\beta}-({\alpha}_{K}/{\alpha}_{L}){K}_{0}{}^{\beta}]}^{1/\beta}-{c}_{L}{L}_{0}$.
- (b)
- Under the capital-reliant (single input) policy ($\widehat{j}=2$), the (least-cost) additional labor input is $\widehat{{L}^{\prime}}(\Delta |\widehat{j})=0$, the additional capital input is $\widehat{{K}^{\prime}}(\Delta |\widehat{j})={[(1/{\alpha}_{K}){\Delta}^{-\beta}-({\alpha}_{L}/{\alpha}_{K}){L}_{0}{}^{\beta}]}^{1/\beta}-{K}_{0}$ and the corresponding minimal cost of setup reduction is ${\widehat{\Gamma}}_{2}^{}(\Delta )={c}_{K}{[(1/{\alpha}_{K}){\Delta}^{-\beta}-({\alpha}_{L}/{\alpha}_{K}){L}_{0}{}^{\beta}]}^{1/\beta}-{c}_{K}{K}_{0}$.
- (c)
- Under the dual-input policy ($\widehat{j}=3$), the least-cost additional labor input $\widehat{{L}^{\prime}}(\Delta |\widehat{j})$, the least-cost additional capital input $\widehat{{K}^{\prime}}(\Delta |\widehat{j})$ and the corresponding minimal cost of fixed-cost reduction ${\widehat{\Gamma}}_{3}^{}(\Delta )$ follow:$$\begin{array}{c}(\widehat{{K}^{\prime}}(\Delta |\widehat{j})+{K}_{0})/(\widehat{{L}^{\prime}}(\Delta |\widehat{j})+{L}_{0})={\left({c}_{L}{\alpha}_{K}/{c}_{K}{\alpha}_{L}\right)}^{1/(\beta -1)}\\ \widehat{{K}^{\prime}}(\Delta |\widehat{j})={\Delta}^{-1}{\left[{\alpha}_{K}+{\alpha}_{L}{\left({c}_{L}{\alpha}_{K}/{c}_{K}{\alpha}_{L}\right)}^{\beta /(\beta -1)}\right]}^{-1/\beta}-{K}_{0},\\ \widehat{{L}^{\prime}}(\Delta |\widehat{j})={\Delta}^{-1}{\left[{\alpha}_{L}+{\alpha}_{K}{\left({c}_{K}{\alpha}_{L}/{c}_{L}{\alpha}_{K}\right)}^{\beta /(\beta -1)}\right]}^{-1/\beta}-{L}_{0},\\ {\Gamma}_{3}^{\ast}(\Delta )={\Psi}_{3}{\Delta}^{-1}-{\psi}_{3},\end{array}$$

**Proof**

**of**

**Proposition**

**3.**

**Corollary**

**2.**

- (a)
- $\Delta >max\left\{{K}_{0}{\left[{\alpha}_{K}+{\alpha}_{L}{\left({c}_{L}{\alpha}_{K}/{c}_{K}{\alpha}_{L}\right)}^{\beta /(\beta -1)}\right]}^{1/\beta},{L}_{0}{\left[{\alpha}_{L}+{\alpha}_{K}{\left({c}_{K}{\alpha}_{L}/{c}_{L}{\alpha}_{K}\right)}^{\beta /(\beta -1)}\right]}^{1/\beta}\right\}$, or
- (b)
- the initial capital and labor endowments ${K}_{0}$ and ${L}_{0}$ have been chosen according to the marginal cost condition ${K}_{0}/{L}_{0}={\left({c}_{L}{\alpha}_{K}/{c}_{K}{\alpha}_{L}\right)}^{1/(\beta -1)}$.

**Proof**

**of**

**Corollary**

**2.**

**Proposition**

**4.**

**Proof**

**(sketch)**

**of**

**Proposition**

**4.**

**Proposition**

**5.**

**Proof**

**of**

**Proposition**

**5.**

**Proposition**

**6.**

**Proof**

**(sketch)**

**of**

**Proposition**

**6.**

**Corollary**

**3.**

**Proposition**

**7.**

**Proof**

**of**

**Proposition**

**7.**

**Corollary**

**4.**

## 4. Some Insights on Process Improvements for Sustainable Supply Chains

#### 4.1. Resource Planning and Allocation

#### 4.2. Impact of Disruptions

#### 4.3. Incorporation of Carbon Emissions Penalties

## 5. Numerical Examples

## 6. Concluding Remarks and Future Research

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Notations | |

$D$ | Mean demand rate (units per unit time, say, per year) |

${\mu}_{DL}$ | Mean demand during lead time |

${\sigma}_{DL}$ | The standard deviation of demand during lead time |

$S$ | Fixed cost per setup (or order) |

$H$ | Inventory cost rate per unit per unit time held in stock |

$\pi $ | Backordering cost rate per unit |

$\eta $ | Desired service level (fraction of immediately unmet demand per year) |

$P$ | Unit acquisition cost |

$Q$ | Lot size |

$r$ | Reorder point |

$z$ | Safety stock factor |

$\mathcal{L}[z]$ | The loss function of a standard normal variable |

$\Phi (z)$ | Cumulative distribution function (cdf) of a standard normal variable |

$OC(Q,r)$ | Expected total operating cost/unit time for lot size-reorder $(Q,r)$ policy |

${S}_{0}$ | Setup level before reduction efforts |

$\Delta $ | A nonnegative scalar $(\le 1)$ denoting the reduction effort |

$\Gamma (\Delta )$ | Cost of achieving a $(1-\Delta )\times 100\%$ fixed-replenishment cost reduction |

${\Gamma}^{\ast}(\Delta )$ | Least-cost of achieving a reduction of $\Delta $ in fixed-replenishment cost |

$K$ | Capital input level devoted to the reduction activity |

$L$ | Labor input level devoted to the reduction activity |

${K}_{0}$ | Inherent technical capital stock |

${L}_{0}$ | Inherent human know-how |

${K}^{\prime}$ | Additional capital investment |

${L}^{\prime}$ | Additional labor investment |

${K}^{\prime}*({\Delta}^{\ast})$ | Optimal allocation of capital input to achieve that reduction |

${L}^{\prime}*({\Delta}^{\ast})$ | Optimal allocation of labor input to achieve that reduction |

$\theta (K,L)$ | Production function describing reduction activity |

$A$ | Productivity or technological efficiency for the overall conversion process |

${A}_{K}$ | Capital productivity or technological efficiency in Leontief function |

${A}_{L}$ | Labor productivity or technological efficiency in Leontief function |

${\beta}_{K}$ | Capital input elasticity in Leontief production function ($<0$) |

${\beta}_{L}$ | Labor input elasticity in Leontief production function ($<0$) |

$\rho $ | The elasticity of scale for Cobb–Douglas production function |

$\widehat{j}$ | Solution type when the least cost is achieved |

${\widehat{j}}^{\ast}(\Delta )$ | Optimal input usage policy |

## References

- Porteus, E.L. Optimal lot sizing, process quality improvement and setup cost reduction. Oper. Res.
**1986**, 34, 137–144. [Google Scholar] [CrossRef] - Li, G.; Rajagopalan, S. Process improvement, quality, and learning effects. Manag. Sci.
**1998**, 44, 1517–1532. [Google Scholar] [CrossRef] - Xiao, W.; Gaimon, C.; Subramanian, R.; Biehl, M. Investment in environmental process improvement. Prod. Oper. Manag.
**2019**, 28, 407–420. [Google Scholar] [CrossRef] - Malmberg, F.; Marklund, J. Evaluation and control of inventory distribution systems with quantity based shipment consolidation. Nav. Res. Logist. (NRL)
**2023**, 70, 205–227. [Google Scholar] [CrossRef] - Porteus, E.L. Investing in reduced setups in the EOQ model. Manag. Sci.
**1985**, 31, 998–1010. [Google Scholar] [CrossRef] - Billington, P.J. The classic economic production quantity model with setup cost as a function of capital expenditure. Decis. Sci.
**1987**, 18, 25–42. [Google Scholar] [CrossRef] - Kim, S.L.; Hayya, J.C.; Hong, J.D. Setup reduction in the economic production quantity model. Decis. Sci.
**1992**, 23, 500–508. [Google Scholar] [CrossRef] - Trevino, J.; Hurley, B.J.; Friedrich, W. A mathematical model for the economic justification of setup time reduction. Int. J. Prod. Res.
**1993**, 31, 191–202. [Google Scholar] [CrossRef] - Hall, R.W. Zero Inventories; Dow Jones-Irwin: Homewood, IL, USA, 1983. [Google Scholar]
- Keller, G.; Noori, H. Justifying new technology acquisition through its impact on the cost of running an inventory policy. IIE Trans.
**1988**, 20, 284–291. [Google Scholar] [CrossRef] - Nasri, F.; Paknejad, M.J. Flexibility improvement in continuous-review (s, Q) systems. Omega
**1992**, 20, 408–410. [Google Scholar] [CrossRef] - Paknejad, M.J.; Nasri, F.A.; Affisco, J.F. Defective units in a continuous review (s, Q) system. Int. J. Prod. Res.
**1995**, 33, 2767–2777. [Google Scholar] [CrossRef] - Hill, A.V.; Khosla, I.S. Models for optimal lead time reduction. Prod. Oper. Manag.
**1992**, 1, 185–197. [Google Scholar] [CrossRef] - Ouyang, L.Y.; Chang, H.C. Mixture inventory model involving setup cost reduction with a service level constraint. Opsearch
**2000**, 37, 327–339. [Google Scholar] [CrossRef] - Gallego, G.; Moon, I. The distribution free newsboy problem: Review and extensions. J. Oper. Res. Soc.
**1993**, 44, 825–834. [Google Scholar] [CrossRef] - Ouyang, L.Y.; Chang, H.C. Lot size reorder point inventory model with controllable lead time and setup cost. Int. J. Syst. Sci.
**2002**, 33, 635–642. [Google Scholar] [CrossRef] - Ouyang, L.Y.; Chuang, B.R.; Chang, H.C. Setup cost and lead time reductions on stochastic inventory models with a service level constraint. J. Oper. Res. Soc. Jpn.
**2002**, 45, 113–122. [Google Scholar] [CrossRef] - Lin, L.C.; Hou, K.L. Optimal ordering policies and capital investment in setup reduction for continuous review inventory system with random yields. J. Stat. Manag. Syst.
**2003**, 6, 65–77. [Google Scholar] [CrossRef] - Uthayakumar, R.; Parvathi, P. A continuous review inventory model with controllable backorder rate and investments. Int. J. Syst. Sci.
**2009**, 40, 245–254. [Google Scholar] [CrossRef] - Annadurai, K.; Uthayakumar, R. Controlling setup cost in (Q, r, L) inventory model with defective items. Appl. Math. Model.
**2010**, 34, 1418–1427. [Google Scholar] [CrossRef] - Cheng, T.L.; Huang, C.K.; Chen, K.C. Inventory model involving lead time and setup cost as decision variables. J. Stat. Manag. Syst.
**2004**, 7, 131–141. [Google Scholar] [CrossRef] - Uthayakumar, R.; Parvathi, P. Inventory models with mixture of backorders involving reducible lead time and setup cost. Opsearch
**2008**, 45, 12–33. [Google Scholar] [CrossRef] - Ben-Daya, M.; Hariga, M. Lead-time reduction in a stochastic inventory system with learning consideration. Int. J. Prod. Res.
**2003**, 41, 571–579. [Google Scholar] [CrossRef] - Kim, J.S.; Benton, W.C. Lot size dependent lead times in a Q, R inventory system. Int. J. Prod. Res.
**1995**, 33, 41–58. [Google Scholar] [CrossRef] - Mefford, R.N. The economic value of a sustainable supply chain. Bus. Soc. Rev.
**2011**, 116, 109–143. [Google Scholar] [CrossRef] - Khan, S.A.R.; Yu, Z.; Golpira, H.; Sharif, A.; Mardani, A. A state-of-the-art review and meta-analysis on sustainable supply chain management: Future research directions. J. Clean. Prod.
**2021**, 278, 123357. [Google Scholar] [CrossRef] - Ülkü, M.A.; Engau, A. Sustainable supply chain analytics. In Industry, Innovation and Infrastructure. Encyclopedia of the UN Sustainable Development Goals; Leal Filho, W., Azul, A.M., Brandli, L., Lange Salvia, A., Wall, T., Eds.; Springer: Berlin/Heidelberg, Germany, 2021; pp. 1123–1134. [Google Scholar]
- Kleindorfer, P.R.; Saad, G.H. Managing disruption risks in supply chains. Prod. Oper. Manag.
**2005**, 14, 53–68. [Google Scholar] [CrossRef] - Sodhi, M.S.; Tang, C.S.; Willenson, E.T. Research opportunities in preparing supply chains of essential goods for future pandemics. Int. J. Prod. Res.
**2023**, 61, 2416–2431. [Google Scholar] [CrossRef] - Altay, N.; Pal, R. Coping in supply chains: A conceptual framework for disruption management. Int. J. Logist. Manag.
**2023**, 34, 261–279. [Google Scholar] [CrossRef] - Goodarzian, F.; Ghasemi, P.; Gunasekaren, A.; Taleizadeh, A.A.; Abraham, A. A sustainable-resilience healthcare network for handling COVID-19 pandemic. Ann. Oper. Res.
**2022**, 312, 761–825. [Google Scholar] [CrossRef] - Nabipour, M.; Ülkü, M.A. On deploying blockchain technologies in supply chain strategies and the COVID-19 pandemic: A systematic literature review and research outlook. Sustainability
**2021**, 13, 10566. [Google Scholar] [CrossRef] - Kamran, M.A.; Kia, R.; Goodarzian, F.; Ghasemi, P. A new vaccine supply chain network under COVID-19 conditions considering system dynamic: Artificial intelligence algorithms. Socio-Econ. Plan. Sci.
**2023**, 85, 101378. [Google Scholar] [CrossRef] - Oguntola, I.O.; Ülkü, M.A. Artificial intelligence for sustainable humanitarian logistics. In Encyclopedia of Data Science and Machine Learning; Wang, J., Ed.; IGI-Global: Hershey, PA, USA, 2023; pp. 2970–2983. [Google Scholar]
- Kabadurmus, O.; Kayikci, Y.; Demir, S.; Koc, B. A data-driven decision support system with smart packaging in grocery store supply chains during outbreaks. Socio-Econ. Plan.Sci.
**2023**, 85, 101417. [Google Scholar] [CrossRef] - Dai, T.; Tang, C.S. Frontiers in service science: Integrating ESG measures and supply chain management: Research opportunities in the postpandemic era. Serv. Sci.
**2022**, 14, 1–12. [Google Scholar] [CrossRef] - Shingo, S. Non-Stock Production: The Shingo System of Continuous Improvement; CRC Press: Boca Raton, FL, USA, 1988. [Google Scholar]
- Bookbinder, J.H.; Ülkü, M.A. Freight transport and lojistics in JIT systems. In International Encyclopedia of Transportation; Vickerman, R., Ed.; Elsevier Ltd.: Kidlington, UK, 2021; Volume 3, pp. 107–112. [Google Scholar]
- Agi, M.A.; Faramarzi-Oghani, S.; Hazır, Ö. Game theory-based models in green supply chain management: A review of the literature. Int. J. Prod. Res.
**2021**, 59, 4736–4755. [Google Scholar] [CrossRef] - Hua, G.; Cheng, T.C.E.; Wang, S. Managing carbon footprints in inventory management. Int. J. Prod. Econ.
**2011**, 132, 178–185. [Google Scholar] [CrossRef] - Benjaafar, S.; Li, Y.; Daskin, M. Carbon footprint and the management of supply chains: Insights from simple models. IEEE Trans. Autom. Sci. Eng.
**2013**, 10, 99–116. [Google Scholar] [CrossRef] - Chen, X.; Benjaafar, S.; Elomri, A. The carbon-constrained EOQ. Oper. Res. Lett.
**2013**, 41, 172–179. [Google Scholar] [CrossRef] - Toptal, A.; Ozlu, H.; Konur, D. Joint decisions on inventory replenishment and emission reduction investment under different emission regulations. Int. J. Prod. Res.
**2014**, 52, 243–269. [Google Scholar] [CrossRef] - Heathfield, D.F.; Wibe, S. An Introduction to Cost and Production Functions; Humanities Press International: Trenton, NJ, USA, 1987. [Google Scholar]
- Cobb, C.W.; Douglas, P.H. A Theory of Production. Am. Econ. Rev.
**1928**, 18, 139–165. [Google Scholar] - Komiya, R. Technological Progress and the Production Function in the United States Steam Power Industry. Rev. Econ. Stat.
**1962**, 44, 156–166. [Google Scholar] [CrossRef] - Wei, T. Impact of energy efficiency gains on output and energy use with Cobb-Douglas production function. Energy Policy
**2007**, 35, 2023–2030. [Google Scholar] [CrossRef] - Ozaki, I. Industrial structure and employment. Dev. Econ.
**1976**, 14, 341–365. [Google Scholar] [CrossRef] - Shadbegian, R.; Gray, W.B. Pollution abatement expenditures and plant-level productivity: A production function approach. Ecol. Econ.
**2005**, 54, 196–208. [Google Scholar] [CrossRef] - Hatirli, S.A.; Ozkan, B.; Fert, C. Energy inputs and crop yield relationship in greenhouse tomato production. Renew. Econ.
**2006**, 31, 427–438. [Google Scholar] [CrossRef] - Kogan, K.; Tapiero, C.S. Optimal co-investment in supply chain infrastructure. Eur. J. Oper. Res.
**2009**, 192, 265–276. [Google Scholar] [CrossRef] - Yuan, C.; Liu, S.; Wu, J. Research on energy-saving effect of technological progress based on Cobb-Douglas production function. Energy Policy
**2009**, 37, 2842–2846. [Google Scholar] [CrossRef] - Shin, S.; Eksioglu, B. An empirical study of RFID productivity in the US retail supply chain. Int. J. Prod. Econ.
**2015**, 163, 89–96. [Google Scholar] [CrossRef] - Arrow, K.J.; Chenery, H.B.; Minhas, B.S.; Solow, R.M. Capital-labor substitution and economic efficiency. Rev. Econ. Stat.
**1961**, 43, 225–250. [Google Scholar] [CrossRef] - Derymes, P.J.; Kurz, M. Technology and scale in electricity generation. Econometrica
**1964**, 32, 287–315. [Google Scholar] [CrossRef] - Acemoglu, D.; Restrepo, P. Modeling automation. AEA Pap. Proc.
**2018**, 108, 48–53. [Google Scholar] [CrossRef] - Leontief, W. Introduction to a theory of the internal structure of functional relationship. Econometrica
**1947**, 15, 361–373. [Google Scholar] [CrossRef] - Haldi, J.; Whitcomb, D. Economies of scale in industrial plants. J. Political Econ.
**1967**, 75, 373–385. [Google Scholar] [CrossRef] - Nakamura, S. A nonhomothetic generalized Leontief cost function based on pooled data. Rev. Econ. Stat.
**1990**, 72, 649–656. [Google Scholar] [CrossRef] - Alizadeh, F.; Goldfarb, D. Second-order cone programming. Math. Program.
**2003**, 95, 3–51. [Google Scholar] [CrossRef] - Kian, R.; Berk, E.; Gürler, Ü. Minimal conic quadratic reformulations and an optimization model. Oper. Res. Lett.
**2019**, 47, 489–493. [Google Scholar] [CrossRef] - Saberi, S.; Kouhizadeh, M.; Sarkis, J.; Shen, L. Blockchain technology and its relationships to sustainable supply chain management. Int. J. Prod. Res.
**2019**, 57, 2117–2135. [Google Scholar] [CrossRef]

**Figure 1.**Comparison of setup cost-reduction functions with data in Trevino et al. [8].

**Table 1.**Optimal lot size ${Q}^{\ast}$, optimal safety stock factor ${z}^{\ast}$ and optimal setup reduction ${\Delta}^{\ast}$ in response to disruptive demand regime changes ($\lambda =0.25$, $K=\mathrm{32,500}$, $\phi =\mathrm{32,500}$, ${D}_{0}=\mathrm{100,000}$, $D={D}_{0}$, ${\sigma}_{DL}^{0}={D}_{0}/3$, $H=10$, ${S}_{0}=1000$).

$\mathit{\eta}$ | ${\mathit{\sigma}}_{\mathit{D}\mathit{L}}/{\mathit{\sigma}}_{\mathit{D}\mathit{L}}^{0}$ | ${\mathit{Q}}^{\ast}$ | ${\mathit{z}}^{\ast}$ | ${\Delta}^{\ast}$ |
---|---|---|---|---|

0.01 | 0.05 | 7628.2 | 1.29 | 0.68 |

0.1 | 8748.3 | 1.54 | 0.76 | |

0.2 | 11,041.7 | 1.74 | 0.92 | |

0.4 | 11,128.7 | 1.98 | 1.00 | |

0.02 | 0.05 | 7825.4 | 0.93 | 0.70 |

0.1 | 9082.2 | 1. | 0.78 | |

0.2 | 11,522.0 | 1.42 | 0.95 | |

0.4 | 11,978.4 | 1.68 | 1.00 | |

0.05 | 0.05 | 8257.9 | 0.35 | 0.73 |

0.1 | 9702.5 | 0.68 | 0.83 | |

0.2 | 8927.7 | 1.11 | 1.00 | |

0.4 | 14,073.2 | 1.22 | 1.00 |

**Table 2.**Optimal lot size ${Q}^{\ast}$, optimal safety stock factor ${z}^{\ast}$ and optimal setup reduction ${\Delta}^{\ast}$ in response to disruptive demand regime changes ($\lambda =0.25$, $K=\mathrm{32,500}$, $\phi =\mathrm{32,500}$, ${D}_{0}=\mathrm{100,000}$, $D=0.9{D}_{0}$, ${\sigma}_{DL}^{0}={D}_{0}/3$, $H=10$, ${S}_{0}=1000$).

$\mathit{\eta}$ | ${\mathit{\sigma}}_{\mathit{D}\mathit{L}}/{\mathit{\sigma}}_{\mathit{D}\mathit{L}}^{0}$ | ${\mathit{Q}}^{\ast}$ | ${\mathit{z}}^{\ast}$ | ${\Delta}^{\ast}$ |
---|---|---|---|---|

0.01 | 0.05 | 7523.9 | 1.30 | 0.73 |

0.1 | 8664.1 | 1.55 | 0.82 | |

0.2 | 10,849.5 | 1.74 | 0.98 | |

0.4 | 11,280.1 | 1.99 | 1.00 | |

0.02 | 0.05 | 7717.2 | 0.94 | 0.75 |

0.1 | 8924.1 | 1.21 | 0.84 | |

0.2 | 7760.2 | 1.58 | 1.00 | |

0.4 | 12,189.2 | 1.69 | 1.00 | |

0.05 | 0.05 | 8141.0 | 0.36 | 0.78 |

0.1 | 9603.1 | 0.69 | 0.89 | |

0.2 | 8683.5 | 1.12 | 1.00 | |

0.4 | 13,351.0 | 1.22 | 1.00 |

**Table 3.**Optimal lot size ${Q}^{\ast}$, optimal safety stock factor ${z}^{\ast}$ and optimal setup reduction ${\Delta}^{\ast}$ in response to disruptive demand regime changes ($\lambda =0.25$, $K=\mathrm{32,500}$, $\phi =\mathrm{32,500}$, ${D}_{0}=\mathrm{100,000}$, $D=0.8{D}_{0}$, ${\sigma}_{DL}^{0}={D}_{0}/3$, $H=10$, ${S}_{0}=1000$).

$\mathit{\eta}$ | ${\mathit{\sigma}}_{\mathit{D}\mathit{L}}/{\mathit{\sigma}}_{\mathit{D}\mathit{L}}^{0}$ | ${\mathit{Q}}^{\ast}$ | ${\mathit{z}}^{\ast}$ | ${\Delta}^{\ast}$ |
---|---|---|---|---|

0.01 | 0.05 | 7406.4 | 1.31 | 0.80 |

0.1 | 8495.6 | 1.55 | 0.89 | |

0.2 | 7260.3 | 1.90 | 1.00 | |

0.4 | 10,635.0 | 1.99 | 1.00 | |

0.02 | 0.05 | 7595.5 | 0.95 | 0.81 |

0.1 | 8819.6 | 1.22 | 0.92 | |

0.2 | 7697.1 | 1.60 | 1.00 | |

0.4 | 11,492.1 | 1.69 | 1.00 | |

0.05 | 0.05 | 8009.8 | 0.37 | 0.85 |

0.1 | 9487.1 | 0.70 | 0.97 | |

0.2 | 8409.7 | 1.13 | 1.00 | |

0.4 | 13,680.3 | 1.23 | 1.00 |

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**MDPI and ACS Style**

Berk, E.; Ayas, O.; Ülkü, M.A.
Optimizing Process-Improvement Efforts for Supply Chain Operations under Disruptions: New Structural Results. *Sustainability* **2023**, *15*, 13117.
https://doi.org/10.3390/su151713117

**AMA Style**

Berk E, Ayas O, Ülkü MA.
Optimizing Process-Improvement Efforts for Supply Chain Operations under Disruptions: New Structural Results. *Sustainability*. 2023; 15(17):13117.
https://doi.org/10.3390/su151713117

**Chicago/Turabian Style**

Berk, Emre, Onurcan Ayas, and M. Ali Ülkü.
2023. "Optimizing Process-Improvement Efforts for Supply Chain Operations under Disruptions: New Structural Results" *Sustainability* 15, no. 17: 13117.
https://doi.org/10.3390/su151713117