Extended Examples of Best Matching

Abstract

This chapter presents four extended best matching case studies associated with distributed manufacturing , supply, social, and service networks. Detailed definitions, mathematical models , solution approaches, numerical analysis and discussions are presented for each case. The purpose is to demonstrate “A–Z” of modeling, optimizing, and controlling best matching processes via four comprehensive case studies.

The original version of this chapter was revised: For detailed information please see erratum. The erratum to this chapter is available at 10.1007/978-3-319-46070-3_9

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-319-46070-3_9

Appendix 1: Notation

Note: The notations are listed separately for each example, because some are repeated in different examples with different definitions.

E ₁ : Collaborative Supply Networks

Indices

$i$ :

Customer ($i \in I$)

$j,j^{{\prime }}$ :

Supplier ($j,j^{{\prime }} \in J$)

$t$ :

Period ($t \in T$)

Parameters

$\alpha_{i}$ :

Quantity of order from customer $i$

$b_{jt}$ :

Backordering cost of supplier $j$ in period $t$ ($/unit)

$B_{j,0}$ :

Initial backorder level of supplier $j$ at the beginning of planning horizon (units)

$D_{it}$ :

Demand of customer $i$ in period $t$ (units)

$\delta_{i}$ :

Due date of order from customer $i$

$f_{{jj^{{\prime }} t}}$ :

Fixed cost of collaboration between suppliers $j$ and $j^{{\prime }}$ in period $t$ ($)

$h_{jt}$ :

Inventory holding cost of supplier $j$ in period $t$ ($/unit)

$H_{j,0}$ :

Initial inventory level of supplier $j$ at the beginning of planning horizon (units)

$K_{jt}$ :

Capacity limit of supplier $j$ in period $t$ (units)

$o_{i}$ :

Order from customer $i$

$p_{jt}$ :

Production cost of supplier $j$ in period $t$ ($/unit)

$\pi$ :

Priority

$\tau_{i}$ :

Processing time of order from customer i

$v_{{jj^{{\prime }} t}}$ :

Cost of transshipping products suppliers $j$ and $j^{{\prime }}$ in period $t$ ($/unit)

M :

A sufficiently large positive number

Variables

$B_{jt}$ :

Backorder level of supplier j at the end of period t (units)

$H_{jt}$ :

Inventory level of supplier j at the end of period t (units)

$Q_{jt}$ :

Production level of supplier j in period t (units)

$S_{{jj^{{\prime }} t}}$ :

Transshipment level between suppliers $j$ and $j^{{\prime }}$ in period $t$ (units)

$\chi_{ijt}$ :

1, if customer $i$ is matched to supplier $j$ in period $t$; 0, otherwise

$\lambda_{{jj^{{\prime }} t}}$ :

1, if capacity sharing proposal of supplier $j$ is matched to demand sharing proposal of supplier $j^{{\prime }}$ in period $t$; 0, otherwise

E ₂ : Collaborative Assembly Lines

Indices

$i,i^{{\prime }}$ :

Workstation ($i,i^{{\prime }} \in I$)

$j,j^{{\prime }}$ :

Task ($j,j^{{\prime }} \in J$)

Parameters

$\alpha_{i} (t)$ :

Progress rate of workstation $i$ at time $t$

$\bar{C}$ :

Cycle time upper bound

$C^{B}$ :

Cycle time of balanceable line

D :

Demand

$\delta$ :

Deviation of cycle time from $C^{B}$

$e_{{ii^{{\prime }} }}$ :

CE for tool sharing from $i$ to $i^{{\prime }}$

$f_{{ii^{{\prime }} }}$ :

Fixed tool sharing cost from $i$ to $i^{{\prime }}$

$Pd_{j}$ :

Set of immediate predecessors of task $i$

$O_{k} (t)$ :

Set of non-target workstations with higher workload than $k$’s targets at time $t$

$\rho_{jt}$ :

Processing time of task $j$ at time/period $t$

$A_{t}$ :

Available production time in period $t$

$\overline{W}$ :

Upper bound for the number of workstations

$\underline{W}$ :

Lower bound for the number of workstations

$w_{i} (t)$ :

Workload of workstation $i$ at time $t$

Variables

$C$ :

Cycle time

$S_{{ii^{{\prime }} t}}$ :

Tool sharing between workstations $i$ and $i^{{\prime }}$ in period $t$

$W$ :

Number of workstations

$\chi_{jit}$ :

1, if task $i$ is matched to workstation $i$ in period $t$; 0, otherwise

$\psi_{{ii^{{\prime }} }}$ :

1, if workstation $i$ shares tools with workstation $i '$ in period $t$; 0, otherwise

$Z_{1}$ :

Objective 1: Number of workstations

$Z_{2}$ :

Objective 2: Cycle time

$Z_{3}$ :

Objective 3: Total collaboration cost

E ₃ : Clustering with Interdependent Preferences

Indices

i, k :

Set $I$

$j,l$ :

Set $J$

Parameters

$\alpha_{{ii^{{\prime }} }}$ :

Influence of $i^{{\prime }}$ on the preferences of $i$

${\mathbf{c}}$ :

Chromosome (solution set)

${\mathbf{c}}^{\text{m}}$ :

Modified chromosome

$M_{j}$ :

Capacity of $j$

$p_{i \to j}$ :

Preference of $i$ for $j$

$P_{ij}$ :

Mutual preference of $i$ and $j$

$\hat{P}_{ij}$ :

IP of $i$ and $j$

Variables

$\chi_{ij}$ :

1, if $i$ and $j$ are matched; 0, otherwise

E ₄ : Collaborative Service Enterprises

Indices

$o$ :

Organizations ($o \in O$)

$R_{t}$ :

Set of eligible resources for processing task $t$

$r$ :

Resources ($r \in R$)

$t$ :

Tasks ($t \in T$)

$T_{o}$ :

Set of tasks associated with organization $o$

$TL$ :

Tabu list

Parameters

$AR_{t}$ :

Average resource requirement of task $t$

$C_{rt}$ :

Capacity required by task $t$ if processed by resource $r$

$D_{o}$ :

Deviation between power and gain of organization $o$

$\delta$ :

Minimum deviation between the current and best fitness values

$ER_{o}$ :

Estimated (average) overall resource requirement of organization $o$

$G_{o}$ :

Gain of organization $o$ (total resources assigned to $o$)

$IT;IT_{ \hbox{max} }$ :

Number of iterations; Stopping criterion

$L_{r}$ :

Capacity limit of resource $r$

$\gamma_{0}$ :

Exploration/exploitation probability in neighborhood search

${\mathbf{M}}$ :

Encoding matrix

$P_{o}$ :

Power of organization $o$ (average requirements of tasks from $o$)

$\theta$ :

Tabu time function coefficient

$\tau (o,r,t)$ :

Tabu time of triplet $(o,r,t)$

$w_{FR}$ :

Weight of the FR objective

$w_{CR}$ :

Weight of the CR objective

Variables

$\alpha_{rt}$ :

Amounts of resources $r$ consumed by task $t$

$CR$ :

Collaboration rate

$F$ :

Fitness value

$FR_{t}$ :

Fulfillment rate of task $t$

$\chi_{ort}$ :

1, if task $t$, resource $r$, and organization $o$ are matched; 0, otherwise

Problems

• 7.1. Beyond the resource sharing cases discussed in this chapter, suggest other resource sharing applications and opportunities that can benefit from best matching.

• 7.2. What measurements would you apply to compare between resource sharing with best matching, and without it?

• 7.3. Distinguish between demand-capacity sharing by best matching for:

  • (a) Physical product supply.

  • (b) Digital product supply.

  • (c) Service delivery (e.g., transportation; food delivery; etc.).

• 7.4. Explain how you can benefit from the techniques of E₁ in the following cases of shared economy:

  • (a) Order delivery by shared drones .

  • (b) Repair of failed street lamps by shared maintenance-robots.

  • (c) Shared taxi service.

• 7.5. When would simulation be preferred to optimization heuristics in the solution of the four extended example problems in this chapter?

• 7.6. For the TAP described in Sect. 7.2.2, explain:

  • (a) What is the objective of applying a time-out?

  • (b) What are the disadvantages of applying time-out?

  • (c) Formulate an equation to quantify and compare the benefit and the loss due to the application of time-out.

  • (d) Can you suggest a protocol that can work effectively without a time-out?

• 7.7. For the TAP described in Sect. 7.2.2, develop three other protocol logic examples, as follows, and compare between them:

  • (a) Fixed priorities.

  • (b) Shortest Processing Time (SPT) first.

  • (c) An evolutionary (learning) logic.

• 7.8. Explain proposition 7.1 in your own words, and illustrate it with a numerical example.

• 7.9. Consider the negotiation function in Fig. 7.3, describe several alternative negotiation procedures for it, and define evaluation metrics for each of them.

• 7.10. What is the influence of the negotiation logic on measured impacts (as described in 7.2.3)?

• 7.11. In the workflow of the decentralized collaboration procedure shown in Fig. 7.4, explain the role of the best matching algorithms explained in Chaps. 5 and 6.

• 7.12. Repeat Problem 7.2 for the cases described in E₂, E₃, E₄.

• 7.13. Explain how best matching specifically impacts each of the four extended examples in this chapter. In your answer, focus on:

  • (a) The metrics of these impacts;

  • (b) How can these metrics be measured?

• 7.14. (a) Develop your own extended example, E₅, which is different from the four described in this chapter. (b) Use a table to compare the main modeling features of the five extended examples. (Hint: Use the PRISM taxonomy in your comparison.)