# Solving a Fabrication Lot-Size and Shipping Frequency Problem with an Outsourcing Policy and Random Scrap

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Methods

#### Problem Statement and Mathematical Modeling

_{2π}.

- T
_{π}= the replenishment cycle time of the proposed system, - T = the replenishment cycle time for a production system without outsourcing,
- πQ = outsourcing quantity per replenishment cycle,
- K = in-house production setup cost per cycle,
- C = unit fabrication cost, which includes unit cost of inspection,
- h = holding cost per item per unit time,
- C
_{S}= unit disposal cost, - K
_{π}= fixed order (setup) cost of outsourcing items per cycle, - C
_{π}= unit purchasing cost of outsourcing items, - β
_{1}= the relating parameter between K_{π}and K, that is K_{π}= (1 + β_{1})K where 0 ≤ β_{1}≤ 1, - β
_{2}= the relating parameter between C_{π}and C, that is C_{π}= (1 + β_{2})C where β_{2}≥ 0, - H
_{1}= on-hand inventory in units at the time in-house production ends, - H = maximum level of on-hand inventory in units after receiving outsourcing items,
- t
_{1π}= production uptime for the proposed system, - t
_{2π}= time required for transporting all items, - t
_{1}= uptime of the conventional EPQ model, - t
_{2}= delivery time of the conventional EPQ model, - I(t) = on-hand inventory of perfect quality items at time t,
- I
_{d}(t) = on-hand inventory of scrap items at time t, - K
_{1}= fixed transportation cost per shipment, - C
_{T}= unit transportation cost, - n = number of fixed quantity installments of the finished batch to be delivered per cycle,
- t
_{n}= the fixed interval of time between each installment delivered during downtime t_{2π}, - h
_{2}= unit stock holding cost per unit time at the customer’s side, - I
_{c}(t) = on-hand inventory of stocks at the customer’s side at time t, - TC(Q,n) = total production-inventory-delivery cost per cycle for the proposed system,
- E[TCU(Q,n)] = the long-run average costs per unit time for the proposed system.

_{2π}, total holding cost is (see Figure 2) [18]:

_{1π}and downtime t

_{2π}, and stock holding cost at the customer’s side. Hence, TC(Q, n) is as follows:

_{π}and C

_{π}in Equation (12), we have:

## 4. Results

#### 4.1. The Convexity of E[TCU(Q, n)]

_{1}), K, n, K

_{1}, λ, [1 − E[x](1 − π)], and Q are all positive. Hence, E[TCU(Q, n)] is convex with respect to all Q different from zero.

_{n}

_{+1}= E[TCU(Q, n + 1)], δ

_{n}= E[TCU(Q, n)], and δ

_{n}

_{−}

_{1}= E[TCU(Q, n − 1)]. By applying Equation (14), we have (δ

_{n}

_{+1}− δ

_{n}) and (δ

_{n}

_{−1}− δ

_{n}) as follows:

_{2}− h) ≥ 0; also, based on the assumption EPQ model, annual production rate is much greater than annual demand rate, hence P[1 − E[x]] − λ ≥ 0. Therefore, for n ≥ 2, Equation (20) is resulting greater than or equal to zero. A computational verification is provided in Table A1 (refer to Appendix A for details).

#### 4.2. Deriving the Optimal Operating Policy

- (1)
- Let n = 1 initially, and apply Equations (14) and (21) to compute the values of Q, δ
_{n}, δ_{n}_{+1}, and (δ_{n}_{+1}− δ_{n}). - (2)
- Let n = n + 1, and calculate the values of Q, δ
_{n}, δ_{n}_{−}_{1}, δ_{n}_{+1}, (δ_{n}_{−}_{1}− δ_{n}), and (δ_{n}_{+1}− δ_{n}). - (3)
- If both (δ
_{n}_{−}_{1}− δ_{n}) ≥ 0 and (δ_{n}_{+1}− δ_{n}) ≥ 0, then go to step (4); otherwise, go to step (2). - (4)
- Stop. The optimal number of shipments n* and optimal lot size Q* are obtained.

## 5. Implications

#### 5.1. Numerical Example

_{π}= $30 (i.e., β

_{1}= −0.7) and unit outsourcing cost C

_{π}= $130 (i.e., β

_{2}= 0.3). The in-house production process may randomly produce an x portion of scrap items which follows a uniform distribution over the interval [0, 0.2]. Other values of system parameters include the following:

- h = $30 per item per year,
- C
_{S}= $20, disposal cost per scrap item, - K
_{1}= $800, fixed transportation cost per shipment, - C
_{T}= $0.5, transportation cost per item.

_{n}

_{−1}+ δ

_{n}

_{+1}) − 2(δ

_{n}) ≥ 0 (please refer to Table A1 in Appendix A for details). Then, for the optimal operating policy, we apply the searching procedure presented in Section 4.2 and obtain the optimal number of shipments n* = 3, optimal replenishment batch size per cycle Q* = 1229, and the long-run average system cost per unit time E[TCU(Q*, n*)] = $545,344 (see Table 1 for details).

#### 5.2. Sensitivity Analysis with Respect to the Scrap Rate x

#### 5.3. Sensitivity Analysis with Respect to Outsourcing Proportion π

_{2}exists (see Figure 7) at β

_{2}= 0.188, in the numerical example with a uniformly distributed scrap rate range of [0, 0.2]. This represents that if the outsourcing cost factor β

_{2}is less than the critical point (i.e., C

_{π}< $118.80), then the completely outsourcing policy (i.e., π = 1) turns into the better choice than a partial or no outsourcing option, in terms of cost savings.

_{1π}/E[T

_{π}]). It demonstrates that as the outsourcing proportion π increases, the in-house machine utilization ratio declines accordingly. This information can assist production managers in scheduling the multi-product fabrication plan.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

Q | n | δ_{i} | δ_{n}_{−1} − δ_{n} | δ_{n}_{+1} − δ_{n} | (δ_{n}_{−1} + δ_{n}_{+1}) − 2(δ_{n}) |
---|---|---|---|---|---|

600 | 1 | δ_{n}_{−1} = $558,725 | $476 > 0 | $3,624 > 0 | $4,100 > 0 |

2 * | δ_{n} = $558,248 | ||||

3 | δ_{n}_{+1} = $561,872 | ||||

1000 | 2 | δ_{n}_{−1} = $546,669 | $12 > 0 | $1,696 > 0 | $1,708 > 0 |

3 * | δ_{n} = $546,657 | ||||

4 | δ_{n}_{+1} = $548,353 | ||||

1400 | 2 | δ_{n}_{−1} = $548,221 | $2,352 > 0 | $40 > 0 | $2,392 > 0 |

3 * | δ_{n} = $545,869 | ||||

4 | δ_{n}_{+1} = $545,909 | ||||

1800 | 3 | δ_{n}_{−1} = $549,890 | $1,184 > 0 | $46 > 0 | $1,230 > 0 |

4 | δ_{n} = $548,707 | ||||

5 | δ_{n}_{+1} = $548,753 | ||||

2200 | 4 | δ_{n}_{−1} = $553,887 | $708 > 0 | $44 > 0 | $752 > 0 |

5 * | δ_{n} = $553,179 | ||||

6 | δ_{n}_{+1} = $553,223 | ||||

2600 | 5 | δ_{n}_{−1} = $558,994 | $467 > 0 | $40 > 0 | $508 > 0 |

6 * | δ_{n} = $558,527 | ||||

7 | δ_{n}_{+1} = $558,567 | ||||

3000 | 6 | δ_{n}_{−1} = $564,727 | $330 > 0 | $37 > 0 | $366 > 0 |

7 * | δ_{n} = $564,398 | ||||

8 | δ_{n}_{+1} = $564,434 | ||||

3400 | 7 | δ_{n}_{−1} = $570,850 | $243 > 0 | $33 > 0 | $277 > 0 |

8 * | δ_{n} = $570,606 | ||||

9 | δ_{n}_{+1} = $570,639 | ||||

3800 | 8 | δ_{n}_{−1} = $577,231 | $186 > 0 | $30 > 0 | $216 > 0 |

9 * | δ_{n} = $577,045 | ||||

10 | δ_{n}_{+1} = $577,075 | ||||

4200 | 9 | δ_{n}_{−1} = $583,795 | $146 > 0 | $28 > 0 | $174 > 0 |

10 * | δ_{n} = $583,649 | ||||

11 | δ_{n}_{+1} = $583,676 | ||||

4600 | 10 | δ_{n}_{−1} = $590,491 | $117 > 0 | $26 > 0 | $143 > 0 |

11 * | δ_{n} = $590,374 | ||||

12 | δ_{n}_{+1} = $590,399 | ||||

5000 | 11 | δ_{n}_{−1} = $597,286 | $96 > 0 | $24 > 0 | $119 > 0 |

12 * | δ_{n} = $597,191 | ||||

13 | δ_{n}_{+1} = $597,214 | ||||

5400 | 12 | δ_{n}_{−1} = $604,159 | $79 > 0 | $22 > 0 | $101 > 0 |

13 * | δ_{n} = $604,079 | ||||

14 | δ_{n}_{+1} = $604,102 |

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**Figure 1.**The on-hand inventory level of perfect quality items in the proposed model (in green) as compared to that in the same model without outsourcing (in black).

**Figure 3.**The on-hand inventory of products stored at the customer’s side I

_{c}(t) in the proposed model (where D is the quantity per shipment, and I is the left-over quantity after demand λt

_{n}is satisfied [18]).

**Figure 5.**The joint effects of different lot size Q and scrap rate x on the expected system cost E[TCU(Q, n)].

**Figure 8.**The effect of variation in outsourcing proportion π on the machine utilization ratio (t

_{1π}/E[T

_{π}]).

**Table 1.**Iterations of the proposed searching procedure for optimal n* (please refer to Section 4.1 for definition for δ

_{n}, δ

_{n}

_{−1}, and δ

_{n}

_{+1}).

n | Q | δ_{n} | δ_{n−1} | δ_{n+1} | δ_{n−1} − δ_{n} | δ_{n+1} − δ_{n} |
---|---|---|---|---|---|---|

1 | 895 | $553,091 | – | $546,386 | – | −$6,705 ≤ 0 |

2 | 1100 | $546,386 | $553,091 | $545,344 | $6,705 ≥ 0 | −$1,042 ≤ 0 |

3 | 1229 | $545,344 | $546,386 | $545,824 | $1,042 ≥ 0 | $481 ≤ 0 |

4 | 1323 | $545,824 | $545,344 | $546,902 | −$481 ≥ 0 | $1,078 ≥ 0 |

**Table 2.**The effects of variations in outsourcing ratio π on optimal production–shipment policy and on major system cost components (where Q* and n* denote the optimal replenishment batch size and number of shipments, respectively).

π | Q* | n* | Total Outsourcing Cost | Total In-House Production Cost | E[TCU(Q*, n*)] | |||
---|---|---|---|---|---|---|---|---|

Amount | % to Total System Costs | Amount | % to Total System Costs | Amount | Increase % | |||

0.00 | 979 | 2.0 | $0 | 0.0% | $515,237 | 100.0% | $515,237 | – |

0.05 | 1201 | 3.0 | $34,250 | 6.5% | $490,278 | 93.5% | $524,527 | 1.8% |

0.10 | 1206 | 3.0 | $62,611 | 11.9% | $464,933 | 88.1% | $527,544 | 2.4% |

0.15 | 1210 | 3.0 | $90,663 | 17.1% | $439,882 | 82.9% | $530,545 | 3.0% |

0.20 | 1215 | 3.0 | $118,412 | 22.2% | $415,120 | 77.8% | $533,532 | 3.6% |

0.25 | 1219 | 3.0 | $145,862 | 27.2% | $390,642 | 72.8% | $536,505 | 4.1% |

0.30 | 1222 | 3.0 | $173,019 | 32.1% | $366,445 | 67.9% | $539,464 | 4.7% |

0.35 | 1226 | 3.0 | $199,887 | 36.9% | $342,523 | 63.1% | $542,410 | 5.3% |

0.40 | 1229 | 3.0 | $226,471 | 41.5% | $318,873 | 58.5% | $545,344 | 5.8% |

0.45 | 1231 | 3.0 | $252,775 | 46.1% | $295,490 | 53.9% | $548,265 | 6.4% |

0.50 | 1234 | 3.0 | $278,804 | 50.6% | $272,369 | 49.4% | $551,173 | 7.0% |

0.55 | 1236 | 3.0 | $304,561 | 55.0% | $249,509 | 45.0% | $554,070 | 7.5% |

0.60 | 1237 | 3.0 | $330,052 | 59.3% | $226,903 | 40.7% | $556,955 | 8.1% |

0.65 | 1238 | 3.0 | $355,281 | 63.5% | $204,548 | 36.5% | $559,829 | 8.7% |

0.70 | 1239 | 3.0 | $380,251 | 67.6% | $182,441 | 32.4% | $562,691 | 9.2% |

0.75 | 1239 | 3.0 | $404,966 | 71.6% | $160,577 | 28.4% | $565,543 | 9.8% |

0.80 | 1239 | 3.0 | $429,430 | 75.6% | $138,953 | 24.4% | $568,384 | 10.3% |

0.85 | 1352 | 4.0 | $453,238 | 79.4% | $117,913 | 20.6% | $571,150 | 10.9% |

0.90 | 1352 | 4.0 | $477,210 | 83.1% | $96,708 | 16.9% | $573,918 | 11.4% |

0.95 | 1352 | 4.0 | $500,943 | 86.9% | $75,734 | 13.1% | $576,677 | 11.9% |

1.00 | 646 | 2.0 | $529,294 | 94.4% | $31,120 | 5.6% | $560,414 | 8.8% |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chiu, Y.-S.P.; Liang, G.-M.; Chiu, S.W.
Solving a Fabrication Lot-Size and Shipping Frequency Problem with an Outsourcing Policy and Random Scrap. *Math. Comput. Appl.* **2016**, *21*, 45.
https://doi.org/10.3390/mca21040045

**AMA Style**

Chiu Y-SP, Liang G-M, Chiu SW.
Solving a Fabrication Lot-Size and Shipping Frequency Problem with an Outsourcing Policy and Random Scrap. *Mathematical and Computational Applications*. 2016; 21(4):45.
https://doi.org/10.3390/mca21040045

**Chicago/Turabian Style**

Chiu, Yuan-Shyi Peter, Gang-Ming Liang, and Singa Wang Chiu.
2016. "Solving a Fabrication Lot-Size and Shipping Frequency Problem with an Outsourcing Policy and Random Scrap" *Mathematical and Computational Applications* 21, no. 4: 45.
https://doi.org/10.3390/mca21040045