Example Rainbow Colors. Impact of R on the costs and Q

Transcription

1 Example ainbow Colors,=80,5 eorder level is larger than expected demand during lead time. Why? Optimal order quantity is larger than EO. Why? Safety stock s=-=5-90=5 units Impact of on the costs and inventory, therefore Total cost As Holding cost Fixed cost Shortage cost 5 5

2 Optimal as a function of h pd As the order quantity increases, the reorder level decreases holding cost and setup cost, therefore so that we can bring the holding cost, although a lower means shortage cost The Impact of Holding Cost on the Optimal, As h goes up, both and go down, but drops at a faster rate! i

3 The Impact of Stockout Cost on the Optimal, As p goes up, goes??? and goes??? The Impact of Stockout Cost on the Optimal, As p goes up, goes down and goes up! p

4 Summary:, Models Balance between holding cost, setup/fixed cost, and shortage cost To save on the shortage cost, we want large To save on the holding cost, we want small and small To save on the fixed cost, we want large Choose and to strike a good balance among these three costs!!! Service Levels in, Models Esma Gel, ınar Keskinocak, 007 ISYE 0 Fall 0

5 Service objectives Type I service level The proportion of cycles in which no stockouts occur Example: 90% Type I service level There are no stockouts in 9 out of 0 cycles on average Type II service level fill rate, Fraction of demand satisfied on time Service objectives - Example Order cycle Demand Stock-outs TOTAL: Fraction of periods with no stock-outs = 8/0 Type I service = 80% = 0.8 Fraction of demand satisfied on time = 50-55/50=0.96 Type II service = 96% = 0.96 In general, is it easier to achieve an x% Type I service or Type II service level? 5

6 6 Type I service level, : Long-run average proportion of cycles with no stock-outs : robability of having no stock-outs in a cycle : robability of having no stock-outs during lead time : robability that demand during lead time is less than!!! ecap : z z Z D D D Set =EO Find z that satisfies z= Set =z+ safety stock + expected demand during lead time Type I service level, : Long-run average proportion of cycles with no stock-outs : robability of having no stock-outs in a cycle : robability of having no stock-outs during lead time : robability that demand during lead time is less than!!! ecap : z z Z D D D Set =EO Find z that satisfies z= Set =z+ safety stock + expected demand during lead time Why is =EO optimal in this case?

7 Example ainbow Colors ainbow Colors paint store uses a, inventory system to control its stock levels. For a popular eggshell latex paint, historical data show that the distribution of monthly demand is approximately Normal, with mean 8 and standard deviation 8. eplenishment lead time for this paint is about weeks. Each can of paint costs the store $6. Although excess demands are backordered, each unit of stockout costs about $0 due to bookkeeping and loss-of-goodwill. Fixed cost of replenishment is $5 per order and holding costs are based on a 0% annual interest rate. What is the optimal lot size order quantity and reorder level? What is the expected inventory level safety stock just before an order arrives? Example ainbow Colors ainbow Colors is not sure whether the $0 estimate for the shortage cost is accurate. Hence, they decided to use a service level approach. What are the optimal, values if they want to achieve no stockouts in 90% of the order cycles? satisfy 90% of the demand on time? 7

8 Example ainbow Colors Input Monthly demand Normal mean=8 std.dev.=8 = weeks, c=$6, K=$5 h=ic=0.6=$.8/unit/year = 0.9 or = 0.9 Computed input d=8=6 units/year Expected annual demand Expected demand during lead time 8 units / year 5 weeks / year Variance of demand during lead time Annual variance weeks 90 units Variance of lead time demand ainbow Colors Type I service Find, to have 90% Type I service level =EO=75 z= = 0.9 z=.8 = z+ = =08 For 90% Type I service level,=75,08 emember: With unit penalty cost of $0, we found,=80,5. What is the Type I service level that corresponds to,=80,5? = z+ 5=.8z+90 z= = % Type I service level when,=80,5 8

9 Type II service level : Fraction of demand met on time - : Fraction of demand not met on time stock-outs ecap: Expected # of stockouts d per unit time T since T d Expected # of stockouts per unit time Expected demand per unit time With this information, for a given,, we can compute. ainbow Colors For 90% Type I service level we found,=75,08 What is the Type II service level which corresponds to this policy? The same policy results in 90% Type I service and 99% Type II service!! z.8 L z L

10 Finding the optimal, for a desired Type II service level emember : stock - out cost p : Optimal solution when we have d[ K p ] h h pd Finding the optimal, for a desired Type II service level Optimal solution when we have stock - out cost From Substitute d[ K p ] h : p into h p d : h pd p : 5 Imputed shortage cost Kd h To be solved simultaneously with 0

11 Impact of service level on For a given As =- i.e., As the service level, the reorder level as well Finding the optimal, for a desired Type II service level Kd h 0 =EO Start with a 0 value and iterate until the values or the values converge

12 Example ainbow Colors Iteration 0: Compute EO Kd h Example ainbow Colors Iteration : Compute given 0 and then compute given L z L z 0.56 z 0. z To find weneed -.Look at thenormaltable =75 =89 =87

13 Example ainbow Colors Iteration : Compute given and then compute given Example ainbow Colors Iteration : Compute given and then compute given L z L z 0.69 z 0.8 z To find weneed -.Look at thenormaltable. 0 =75 =89 =90 =87 =85

14 Example ainbow Colors Iteration : Compute given and then compute given L z L z 0.59 z 0. z =75 =89 =90 STO! values converged, optimal,=90,85 =87 =85 =85