Background

In the classic inventory model presented in Lecture 7, it is assumed that the quality of all products is satisfactory, so that they can be sold to the market eventually. In practice however, product quality is an issue of not only economical, but also social, importance. For instance, during the launch period of PS4 console, Sony discovered that this new product has a 0.4% defect rate (LNS Research, 2013). In 2014, Toyota recalled 380,000 automobiles worldwide due to various defects and malfunctions (Fortune, 2014).

Despite the large impact of production defects on companies business performance, little is known about how it affects production and inventory dynamics. In this assignment, you are expected to

- Based on the linear inventory model, systematically investigate the relationship be- tween the level of defect and inventory system performance;

- Conceptually and mathematically extend the linear model, to incorporate more defect- related issues/characteristics;

- Through observation of output graphs and numerical assessments, compare and con- trast the difference in system performance, and analyse the effect of these extensions.

2 Linear inventory model with constant defect rate

The model is established for a single-product, single-echelon, periodic-review inventory sys- tem. For simplicity, all linear assumptions (free returns, backorder, etc.) are made in this model. Production lead-time is assumed to be constant and equal to 4 periods.
The following notations should be used in this assignment:

D_t Demand (consumption) in period t

Dˆ t Demand forecast for period t

O_t Order quantity in period t

I_t Inventory level at the end of period t

W_t Work-in-progress (WIP) level at the end of period t

β Coefficient of inventory feedback

λ Defect rate

ss Safety stock

L Production lead-time, L = 4

It is assumed that the defect rate in a single batch is constant. That is, for any period t, the number of defected products is λO_t. These products are to be discarded and should not be used to replenish the inventory.

The company uses naive forecasting to generate demand forecast, which is

Dˆ_t = D_t-1

A proportional order-up-to policy is used to determine the order quantity:

O_t = Dˆ_t + β (ss + 3Dˆt - I_t - W_t)

The inspection of defected product can only be made when the production is complete. So the balance equation of WIP is not affected by defects; only inventory balance equation is.

W_t = W_t-1 + O_t-1 - O_t-4

I_t = I_t-1 + (1 - λ)O_t-4 - D_t

Simulation analysis

Based on the mathematical model above (and the extended model later), establish a simu- lation model with spreadsheet to investigate the following problem:

- (Q₁) How do the feedback coefficient β and the defect rate λ affect the bullwhip ratio σ²_O/σ²_D and the NSAmp ratio σ²_I /σ²_D?

The demand follows a normal distribution with the expectation of 100 and standard deviation of 5.

Model evaluation and extension

It is essential to realize that any model is faulty from reality. More complex models are generally better at representing the real case. On the other hand, good models achieve better balance between model complexity and tractability.

For the model presented in §2, think about the following:

- (Q2) What assumptions are made in this model with regard to defects?
- (Q3) Are these assumptions valid? How do you judge validity?
- (Q4) How do you extend the model to relax the assumption(s) which is/are deemed invalid?

Comparative analysis

Conduct the simulation analysis again based on the extended model. Compare the simulation results and answer the following question:

- (Q5) Does the removal of assumptions change the answer to Q1? If so, to what degree?

Report

The following parts should be included in the final report.

1. Simulation analysis

You should provide evidence (e.g. sample screen-shots) of the simulation model and the procedure of analysis. Results in numerical and graphical forms (e.g. tables and figures) will greatly enhance the readability. You should also provide a summary to answer the question.

2. Model evaluation and extension

Proper justification should be given when evaluating the assumptions in the basic model. Reliable sources of information include (but are not limited to) published literature and case studies. Extended model needs to be presented in both conceptual and mathematical forms. New variables and parameters need to be properly defined before use.

3. Comparative analysis

Here you should carefully design the method of analysis and presentation of results. Again, numerical results and graphics (if designed properly) with the assistance of textual explanation is helpful.