Graduate Program in Business Information Systems Inventory Decisions with Uncertain Factors Aslı Sencer

BIS 517- Aslı Sencer2 Uncertainties in real life Demand is usually uncertain. Probability distributions are used to represent uncertain factors. Ex: Demand is normally distributed OR Demand is either 20,30,40 with respective probabilities 0.2, 0.5, 0.3.

BIS 517- Aslı Sencer3 Stochastic versus Deterministic Models Mathematical models involving probability are referred to as stochastic models. Deterministic models are limited in scope since they do not involve uncertain factors. But they are used to develop insight! Stochastic models are based on expected values, i.e. the long run average of all possible outcomes!

BIS 517- Aslı Sencer4 Example: Drugstore A drugstore stocks Fortunes.They sell each for $3 and unit cost is $2.10. Unsold copies are returned for $.70 credit. There are four levels of demand possible. How many copies of Fortune should be stocked in October? Payoff Table: Demand EventProbability ACTS Q = 20Q = 21Q = 22Q = 23 D = 20.2$18.00$16.60$15.20$13.80 D = D = D =

BIS 517- Aslı Sencer5 Solution: The expected payoffs are computed for each possible order quantity: Q = 20Q = 21Q = 22Q = 23 $18.00$18.44$17.90$16.79 Optimal stocking level, Q*=21 at an optimal expected profit of $18.44 If the probabilities were long-run frequencies, then doing so would maximize long-run profit.

BIS 517- Aslı Sencer6 Example: Drugstore Payoff Table (Figure 16-1)

BIS 517- Aslı Sencer7 Single-Period Inventory Decision: The Newsvendor Problem Single period problem (periodic review) Demand is uncertain (stochastic) No fixed ordering cost Instead of h ($/$/period) we have h E ($/unit/period=ch) Instead of p ($/unit) we have p S and p R -c Q: Order Quantity (decision variable) D: Demand Quantity Costs: c = Unit procurement cost h E = Additional cost of each item held at end of inventory cycle = unit inventory holding cost-salvage value to the supplier p S = Penalty for each item short (loss of customer goodwill) p R = Selling price

BIS 517- Aslı Sencer8 Modeling the Newsvendor Problem The objective is to minimize total expected cost, which can be simplified as: where is the expected demand.

BIS 517- Aslı Sencer9 Optimal order quantity of the Newsboy Problem Q* is the smallest possible demand such that

BIS 517- Aslı Sencer10 Example: Newsboy Problem A newsvendor sells Wall Street Journals. She loses p S = $.02 in future profits each time a customer wants to buy a paper when out of stock. They sell for p R = $.23 and cost c = $.20. Unsold copies cost h E = $.01 to dispose. Demands between 45 and 55 are equally likely. How many should she stock?

BIS 517- Aslı Sencer11 Example: Solution Discrete Uniform Distribution Demand is either 45,46,47,..., 55 each with a probability of 1/11. P(D<=Q*)=0.2Q*=47 units.

12 Newsvendor Problem (Figure 16-3) BIS 517- Aslı Sencer

13 Multiperiod Inventory Policies When demand is uncertain, multiperiod inventory might look like this over time.

BIS 517- Aslı Sencer14 Multiperiod Inventory Policies The multiperiod decisions involve two variables: Order quantity Q Reorder point r The following parameters apply: A = mean annual demand rate k = ordering cost c = unit procurement cost p s = cost of short item (no matter how long) h = annual holding cost per dollar value = mean lead-time demand

BIS 517- Aslı Sencer15 Multiperiod Inventory Policies: Discrete Lead-Time Demand The following is used to compute the expected shortage per inventory cycle: The following is used to compute the total annual expected cost:

BIS 517- Aslı Sencer16 Multiperiod Inventory Policies: Discrete Lead-Time Demand Solution Algorithm. Calculate the starting order quantity: Determine the reorder point r*: Determine optimal order quantity: This procedure continues –using the last Q to obtain r and r to obtain the next Q- until no values change.

BIS 517- Aslı Sencer17 Example: Annual demand for printer cartridges costing c = $1.50 is A = 1,500. Ordering cost is k = $5 and holding cost is $.12 per dollar per year. Shortage cost is p S = $.12, no matter how long. Lead-time demand has the following distribution. Find the optimal inventory policy.

BIS 517- Aslı Sencer18 Example: Solution The starting order quantity is: r* = 7 cartridges. B(7) = (8–7)(.03) + (9–7)(.01) + (10–7)(.01)=.08 and the optimal order quantity is:

BIS 517- Aslı Sencer19 Example: Solution (contd.) Q=290 leads to r=7, so the solution is optimal. The optimal inventory policy is: r* = 7 Q* = 290 Optimal annual expected cost is:

20 Multiperiod Discrete Backordering Iteration 1 BIS 517- Aslı Sencer

21 Multiperiod Discrete Backordering Iteration 10 BIS 517- Aslı Sencer

22 Multiperiod Discrete Backordering Summary BIS 517- Aslı Sencer