Fixed and Varied Parameters for the Simulation-Optimization Study

For the purposes of an experimental design for the simulation-optimization study, it is important to determine which parameters are to be fixed and which are to vary. The demand distribution, mean demand, coefficient of variation of demand, and variation of market share of each product will be held constant while the length of review period, lead time, number of products, shelf-life of products, coefficient of variation of demand, proportion of overall demand for each product, price, holding cost, and substitution factors will all vary. This factorial design will be applied to the case of two and three product variants within a product class. However, in the case of three product variants, it will be assumed that all products have similar properties in terms of demand and cost structures. Therefore, Si will also be assumed to be the same for all products. It is of interest to analyze the impact of an additional product variant keeping overall

product class demand constant. Finally, one major objective of this study is to develop a heuristic for the n=2 product case that can work well under varying operating conditions. Table 1 below summarizes both the fixed and varied input parameters for the simulation-optimization study

Fixed Input parameters Level

Demand distribution Negative Binomial

μd 22 CV 0.7 Factors Levels N 2,3 items R 2,3 days L 0,1 days mi 4,6 days

E(κi) Equal, Unbalanced

hi

$0.0083/unit/day, $0.0833/unit/day

pi $4,$5

vi $1,$2

Product Based Substitution High, Low Age Based Substitution High, Low

Overall, a total of 696 distinct combinations of the parameter values in Table 1 above will be evaluated out of a possible 210 = 1024. This is a fractional factorial design as some parameter combinations will be eliminated due to redundancy and necessity based on the intentions of this study. Since the n = 3 case for number of products introduces more complexity and takes longer to search for the optimal solution, only the special cases are evaluated where all products are assumed to have identical properties in market share, costs, and shelf-life. In this case, it can be easily assumed that the order-up-to level will be the same for all three products. Based on the parameters chosen, there are 96 combinations of parameters that will be evaluated for the n = 3 case and 600 combinations of parameters for the n = 2 case. Of the 600 for the n = 2 case, 96 of the combinations will be identical to the 96 chosen for the n = 3 case, and the remaining 504 combinations will involve two products that have at least one property that differs from the other. The ultimate goal of this study is to be able to examine the effects of factors such as product- based substitution, age-based substitution, lead time, shelf-life of products, cost structures, and number of products on the simulated optimal solutions.

The fixed parameters concern properties of product demand. The overall daily market demand will follows a negative binomial distribution with a mean of 22 and standard deviation of 0.7. The expected market share for each product will vary between 0.5 and 0.7; however, there is uncertainty in this parameter. A uniform distribution will be assumed for the market share of a product. In other words, the realized market share of a product can fall anywhere within 0.2 of the expectation in this two product case. For example, if the expected market share is 0.7 for a particular variant, then the realized market share can be anywhere from 0.5 to 0.9. This equates to a standard deviation of 0.033. This standard deviation of market share will remain constant throughout the study.

As for the varied parameters, the review period will be either 2 or 3 days and lead time will be zero or 1 day. However, only the following review period/lead time pairs are to be

evaluated: (2,0), (2,1), and (3,0). Having (2,1) vs. (3,0) enables a comparative study on the effect of lead time on optimal stocking levels. Having (2,0) offers a smaller cycle length and allows for the examination of how optimal inventory policy is affected by increased shelf-life relative to the coverage period. Thus, it is decided that (3,1) is not necessary for inclusion in the simulation study as it will only yield marginal additional information.

The shelf-life of the products will vary between 4 and 6 days. It is possible for two product variants to have different shelf-lives. In all scenarios examined, the shelf-life of each product will exceed the cycle length. The maximum differential between product shelf-life and cycle length, therefore, is 4 days.

In terms of cost structure, there will be two scenarios for the marginal holding cost, hi,

applied to average inventory based on the length of the simulation run. One scenario is that the holding cost is relatively low, $0.0083 per item per day and the other scenario is that the marginal holding cost is relatively high, $0.0833 per item per day. These values are arbitrarily chosen and represent a stark contrast between what can be perceived as low versus high holding cost. The retail price will vary between $4 and $5 while the unit purchase cost will vary between $1 and $2. One requirement for the price and unit purchase cost is that the margin, pi-vi, must be the same for each product. Thus, the profit margin for each product will be either $2, $3, or $4.

In terms of substitution, there are two levels for both product-based and age-based substitution. A high level of product-based substitution with high level of age-based substitution will translate to (α = 0.4 β = 0.5, ω = 0.8, γ = 0.8, and δ = 0.7), high level of product-based

substitution with low age-based substitution will translate to (α = 0.2 β = 0.7, ω = 0.3, γ = 0.8, and δ = 0.2), low level of product-based substitution with high age-based substitution will translate to (α = 0.7, β = 0.2, ω = 0.8, γ = 0.3, and δ = 0.2), and a low level of product-based substitution with low age based substitution will translate to (α = 0.2, β = 0.2, ω = 0.3, γ = 0.3, and δ = 0.2).