# Uncertainty of transportation time due to stochastic degradation

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## Computational results: Extended Problem

This section provides the results of implementing the mathematical models of the Extended Problem (see Sections 3.3.1 and 3.3.2) on the real industrial case and solving them by the proposed multi-objective differential algorithm (MODE) in Section A3.5. The results of the proposed BMILP_EP and TMINLP_EP models are provided in the following Sections 4.3.1 and 4.3.2, respectively.

BMILP_EP model

This section provides the results of the proposed BMILP_EP model. First, the Pareto frontier of the BMILP_EP model is extracted only for MI-and-CI strategy. The real industrial case was solved using the proposed MODE algorithm and the objective function values for the Pareto solutions have been listed in Table 4.4 for both deterministic and uncertain models. In the uncertain model, it is considered that all parameters are uncertain. In Table 4.4, first column shows the number of Pareto solution. Second and third columns represent the values of the first and second objective functions for deterministic model. Similarly, fourth and fifth columns show the values of the first and second objective functions for uncertain model. Accordingly, twenty one and six Pareto solutions were obtained for deterministic and uncertain parameters, respectively.
The results of Table 4.4 have been illustrated in Figure 4.13 wherein dash and solid lines represent the Pareto frontier of deterministic and uncertain models, respectively.
The inspection plans for different six Pareto solutions of the uncertain model have been depicted in Figure 4.14. As it can be seen, solutions with lower value of the total internal cost, e.g., solutions number 1 and 2, include inspection plans with less number of MI and CI inspections. This is because of that the total internal cost is decreased by decreasing the cost of inspections, while lower inspection cost deals with less numbers of inspections. On the other hand, solutions with lower values of the warranty cost relate to those inspection plans wherein the minimum nonconforming items reach the customers. Accordingly, more numbers of inspections are performed in these plans. These different plans show the conflict of the total internal and warranty costs and highlight the applicability and validity of the proposed BMILP_EP.
The contribution of each uncertain parameter in the increase of objective functions has been depicted as Figure 4.15. As it can be seen, misadjustment and dispersion have the highest effect on the both objective functions. In addition, misadjustment has higher effect on objective function 1 rather than objective function 2 and vice versa for dispersion. Therefore, companies who eager to minimize manufacturing cost should determine exact value for misadjustment and try to omit variation in it as much as possible. On the other hand, companies who attempt to keep their customers satisfied should control both misadjustment and dispersion and determine exact value for them in their plans.
Respecting the validation of our results, Adragna et al. (2010) and Thornton (2004) have also proved the significant impact of misadjustment and dispersion in calculating inertial tolerancing and process capability index.

### TMINLP_EP model

This section provides the results of the proposed TMINLP_EP model. In order to apply the proposed TMINLP_EP model on the real industrial case, there are other parameters besides to those in the BMILP_EP model that need to be provided. Due to limited access to the industrial party, the new parameters are randomly generated based on expert’s idea and near to real settings. In this section, the same multistage production system as the real industrial case (see Section 4.1.1) is studied with four products (i.e., P1 to P4), three choices for machines at each stage, and three choices for inspection tools for inspecting each quality characteristic. There are 10, 12, 8, and 15 quality characteristics to be inspected for P1 to P4, respectively. The characteristics utilize the same machine and inspection tools. In generating the data, following logical rules are respected:
• Machines with higher cost have higher process capability and consequently have lower failure rate.
• Machines with higher fixed cost have lower unit production time as well as lower unit production cost.
• Machines with higher cost have lower breakdown rate, lower retrieve time, and higher production rate.
• Machines with higher cost have higher fixed and variable monitoring inspection (i.e., MI) cost.
• Machines with higher cost have higher production capacities.
• Inspection tools with higher cost have higher failure detection and consequently have lower values of error type I and II for CI.
• Inspection tools with higher fixed cost have lower unit inspection time as well as lower unit inspection cost for both MI and CI.
• Inspection tools with higher cost have lower breakdown rate, lower retrieve time, and higher inspection rate for both MI and CI.
• Inspection tools with higher cost have higher inspection capacities for both MI and CI.
After generating new data for the real industrial case, the proposed TMINLP_EP model is solved using the proposed MODE algorithm. It should be noted that all parameters are considered under uncertainty and the same robust approach as BMILP_EP model is utilized. After solving the model, fifteen and ten Pareto solutions have been obtained for deterministic and uncertain models, respectively, as Table 4.5. The first column shows the number of Pareto solutions. The second and third columns show the first, second and third objective function values for the deterministic problem. Besides, fourth and fifth columns represent the first, second and third objective function values for the uncertain problem.
It is obvious that the objective function values are increased by proposing the robust solutions. The mean increases are 8%, 12%, and 8% for the first, second, and third objective functions, respectively. These values show that the second objective function is more sensitive to the uncertainty of the parameters.
The inspection plan for the Pareto solution number 5 of the uncertain model has been depicted in Figure 4.16. For example in product number 4 (i.e., P4), quality characteristic number 6 needs MI after operation number 6. In addition, quality characteristics number 7, 8, 11 and 15 need MI after operation number 15. Besides, quality characteristics number 7, 8, and 11 to 15 need CI after operation number 15.
The information regarding to the machines and inspection tools selection has been presented in Table 4.6. It can be seen that all three types of machines and inspection tools have been utilized through manufacturing these four products. For instance, the operation number 1 is performed by the machine number 2 for P1, P2, and P2, and the same operation is performed by machine number 3 in P4. In addition, the MI for quality characteristic number 8 is performed by inspection tools number 1, 1, 3, and 3, for P1, P2, P3, and P4, respectively. Furthermore, the CI for quality characteristic number 8 is performed by inspection tools number 3, 1, and 1, for P1, P3, and P4, respectively. It is noteworthy that the inspections of those quality characteristics that their corresponding operations are performed on the machines with lower capability are performed by inspection tools with higher capability as well as lower errors type I and II. For instance, the CI of the quality characteristic number 8 in P1 is performed by inspection tool number 3 that has higher capability. This issue is due to that the corresponding operation which realizes the quality characteristic number 8 in P1, is performed on machine number 1 that has lower capability regarding to the machines with higher numbering index.
In the production system, the usage percentages of machines number 1, 2, and 3 are 53%, 38% and 5%, respectively. On the another hand, the usage percentages of the monitoring inspection tools number 1, 2, and 3 are 52%, 32% and 16%, respectively. These percentages are 25%, 30% and 45% for the conformity inspection tools number 1, 2, and 3, respectively.
Among P1 to P4, the maximum manufacturing time (i.e., 3 − ) is equal to 5132 seconds and this value belongs to P4 with 15 quality characteristics. From this value, 3747 seconds (73%) is spent for production and the rest 1385 seconds (27%) is spent for the waiting time of the items to be processed or be inspected.

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#### Sensitivity analysis

This section investigates the sensitivity of the objective functions in the TMINLP_EP model regarding to the input parameters such as misadjustment, dispersion, capacity of machines and inspection tools, production time, production and inspection rates as well as breakdown rate and retrieve time. In all of the following analyses, the sensitivity of the objective functions has been calculated based on the based deterministic scenario.
Figure 4.17 represents the sensitivity of the objective functions to the increase in misadjustment value. As it can be seen, although all the objective functions increase once the misadjustment increases, but the second objective function is more sensitive to the misadjustment variation. In addition, the third objective function is less sensitive to the misadjustment variation. Figure 4.17 highlights this issue that manufacturers, who are customer satisfaction oriented, need to control the variation of the misadjustment.
Figure 5.18 illustrates the sensitivity of the objective functions to the increase in dispersion of the production processes. Similar to the results of Figure 5.17, all the objective functions are increased once the dispersion is increased, wherein the highest and lowest sensitivities belong to the second and the third objective functions. These results also impose more attention to the uncertainty of the dispersion and the need to control this variation.
Figure 4.19 investigates the effect of increasing the capacity of machines and inspection tools on the objective function values. As it was previously assumed in Section 4.3.2 (i.e., logical rules), machines and inspection tools with higher capacity have also have higher service rate as well as lower failure rate. Accordingly, increasing the capacity increases the first objective function, while decreases the second and the third objectives. It is obvious that the third objective function is more sensitive to capacity of machines and inspection tools. This sensitivity can be considered as the effect of the waiting time on the total manufacturing time. By the other words, increasing the capacity as well as increasing the service rate of machines and inspection tools will definitively lead to lower waiting time and lower total manufacturing time.

Chapter I: General Introduction
1.2. Chapter purpose and outline
1.2. Introduction
1.2. Problem statement .
1.2. Scope and objectives ….
1.2. Research methodology
1.2. Thesis contribution
1.2. Organization of the thesis
Chapter II: Literature Review
2.0. Chapter purpose and outline
2.1. Introduction
2.1.1. Production system characteristics
2.1.2. Methodology
2.2. Separate optimization
2.3. Simultaneous optimization
2.4. Conclusion and gap analysis
Chapter III: Mathematical Formulations & Solution Approaches
3.0. Chapter purpose and outline
3.1. Problem statement
3.2. Main Problem (MP)
3.2.1. Assumption
3.2.2. Notations
3.2.3. Mathematical formulation
3.2.3.1. Formulation of MI-or-CI strategy
3.2.3.2. Formulation of MI-and-CI strategy
3.2.4. Robust Optimization
3.3. Extended Problem (EP)
3.3.1. Bi-objective model
3.3.1.1. Notations
3.3.1.2. Mathematical formulation
3.3.1.3. Robust Optimization
3.3.2. Multi-objective model
3.3.2.1. Assumptions
3.3.2.2. Notations
3.3.2.3. Mathematical formulation
3.3.2.4. Robust Optimization
3.4. Solution algorithms: Meta-heuristics
3.4.1. Solution representation: Main Problem
3.5. Discussion and Summary
Chapter VI: Experimental Results
4.0. Chapter purpose and outline
4.1. Experiments
4.1.1. Case study
4.1.2. Experimental setup
4.2. Computational results: Main Problem
4.2.1. Main Problem results
4.3. Computational results: Extended Problem
4.3.1. BMILP_EP model
4.3.2. TMINLP_EP model
4.3.3. Sensitivity analysis
4.3.4. Computational time complexity
4.3.5. Model uncertainty
4.4. Summary
Chapter V: Hub Location Problem
5.0. Chapter purpose and outline
5.1. Introduction
5.2. Literature review
5.3. The reliability hub location problem
5.3.1. Partial disruptions in hubs and links
5.3.2. Uncertainty of transportation time due to stochastic degradation
5.3.3. Stochastic disruption of hubs’ service rates
5.4. Mathematical formulation
5.5. Linearization of the model: A piecewise function
5.6. Approximated BOCRHLP (AMIP)
5.7. Multi-objective lower bound approach (MOLB)
5.8. Multi-objective meta-heuristic algorithm
5.8.1. Solution representation scheme
5.8.1.1. Hub location (Phase 1)
5.8.1.2. Hub capacity level (Phase 2)
5.8.1.3. Spoke allocation (Phase 3)
5.8.1.4. Transportation mode (Phase 4)
5.8.2. The Proposed algorithm
5.8.2.2. SGV-II algorithm’s framework
5.8.2.3. Intensification using VNS
5.8.2.4. Termination criteria
5.9. Experiments
5.9.1. Test problems
5.9.2. Experimental setup
5.9.3. Performance metrics
5.10. Results and discussion
5.11. Case study
5.11.1. Sensitivity analysis
5.12. Summery
Chapter VI: Conclusion & Future Research
6.0. Chapter purpose and outline