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Inventory Control Models Uncertainity

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  • 1. Inventory Control Models Ch 5 (Uncertainty of Demand) R. R. Lindeke IE 3265, Production And Operations Management
  • 2. Lets do a ‘QUICK’ Exploration of Stochastic Inventory Control (Ch 5)
    • We will examine underlying ideas –
    • We base our approaches on Probability Density Functions (means & std. deviations)
    • We are concerned with two competing ideas: Q and R
    • Q (as earlier) an order quantity and R a stochastic estimate of reordering time and level
    • Finally we are concerned with Servicing ideas – how often can we supply vs. not supply a demand (adds stockout costs to simple EOQ models)
  • 3. The Nature of Uncertainty
    • Suppose that we represent demand as:
        • D = D deterministic + D random
    • If the random component is small compared to the deterministic component, the models of chapter 4 will be accurate. If not, randomness must be explicitly accounted for in the model.
    • In this chapter, assume that demand is a random variable with cumulative probability distribution F(t) and probability density function f(t).
  • 4. Single Period Stochastic Inventory Models
    • These models have the objective of properly balancing the cost of Underage – having not ordered enough products vs. Overage – having ordered more than we can sell
    • These models apply to problems like:
      • Planning initial shipments of ‘High-Fashion’ items
      • Amount of perishable food products
      • Item with short shelf life (like the daily newspaper)
    • Because of this last problem type, this class of problems is typically called the “Newsboy” problem
  • 5. The Newsboy Model
    • At the start of each day, a newsboy must decide on the number of papers to purchase. Daily sales cannot be predicted exactly , and are represented by the random variable, D.
    • The newsboy must carefully consider these costs: c o = unit cost of overage
    • c u = unit cost of underage
    • It can be shown that the optimal number of papers to purchase is the fractile of the demand distribution given by F(Q*) = c u / (c u + c o ).
  • 6. Determination of the Optimal Order Quantity for Newsboy Example
  • 7. Computing the Critical Fractile:
    • We wish to minimize competing costs (Co & Cu):
      • G(Q,D) = Co*MAX(0, Q-D) + Cu*MAX(0, D-Q)
        • D is actual (potential) Demand
      • G(Q) = E(G(Q,D)) (an expected value)
      • Therefore:
  • 8. Applying Leibniz’s Rule:
    • d(G(Q))/dQ = C o F(Q) – C u (1 – F(Q))
    • F(Q) is a cumulative Prob. Density Function (as earlier – of the quantity ordered)
    • Thus: G’(Q*) = (Cu)/(Co + Cu)
    • This is the critical fractile for the order variable as stated earlier
  • 9. Lets see about this: Prob 5 pg 241
    • Observed sales given as a number purchased during a week (grouped)
    • Lets assume some data was supplied:
      • Make Cost: $1.25
      • Selling Price: $3.50
      • Salvageable Parts: $0.80
    • Co = overage cost = $1.25 - $0.80 = $0.45
    • Cu = underage cost = $3.50 - $1.25 = $2.25
  • 10. Continuing:
    • Compute Critical Ratio:
      • CR = Cu/(Co + Cu) = 2.25/(.45 + 2.25) = .8333
    • If we assume a continuous Probability Density Function (lets choose a normal distribution):
      • Z(CR)  0.967 when F(Z) = .8333 (from Std. Normal Tables!)
      • Z = (Q* -  )/  )
      • From the problem data set, we compute
        • Mean = 9856
        • St.Dev. = 4813.5
  • 11. Continuing:
    • Q* =  Z +  = 4813.5*.967 + 9856 = 14511
    • Our best guess economic order quantity is 14511
    • (We really should have done it as a Discrete problem -- Taking this approach we would find that Q* is only 12898)
  • 12. Newsboy’s Extensions
    • Assuming we have a certain number of parts on hand, u > 0
        • This extends the problem compared to our initial u = 0 assumption for the single period case
    • This is true only if the product under study has a shelf life that extends beyond one period
    • Here we still compute Q * will order only Q * - u (or 0 if u > Q * )
  • 13. Try one (in your Engineering Teams) :
    • Do Problem 11a & 11b (pg 249)
  • 14. Lot Size Reorder Point Systems
    • Earlier we considered reorder points (number of parts on hand when we placed an order) they were dependent on lead times as a dependent variable on Q, now we will consider R as an independent variable just like Q
    • Assumptions:
      • Inventory levels are reviewed continuously (the level of on-hand inventory is known at all times)
      • Demand is random but the mean and variance of demand are constant. (stationary demand)
  • 15. Lot Size Reorder Point Systems
      • There is a positive leadtime, τ. This is the time that elapses from the time an order is placed until it arrives.
      • The costs are:
        • Set-up cost each time an order is placed at $K per order
        • Unit order cost at $C for each unit ordered
        • Holding at $H per unit held per unit time (i.e., per year)
        • Penalty cost of $P per unit of unsatisfied demand
    Additional Assumptions:
  • 16. Describing Demand
    • The response time of the system (in this case) is the time that elapses from the point an order is placed until it arrives. Hence,
    • The uncertainty that must be protected against is the uncertainty of demand during the lead time.
    • We assume that D represents the demand during the lead time and has probability distribution F(t). Although the theory applies to any form of F(t), we assume that it follows a normal distribution for calculation purposes.
  • 17. Decision Variables
    • For the basic EOQ model discussed in Chapter 4, there was only the single decision variable Q .
    • The value of the reorder level, R , was determined by Q.
    • Now we treat Q and R as independent decision variables.
    • Essentially, R is chosen to protect against uncertainty of demand during the lead time, and Q is chosen to balance the holding and set-up costs. (Refer to Figure 5-5)
  • 18. Changes in Inventory Over Time for Continuous-Review (Q, R) System
  • 19. The Cost Function
    • The average annual cost is given by:
    • Interpret n(R) as the expected number of stockouts per cycle given by the loss integral formula (see Table A-4 (std. values)). And note, the last term is this cost model is a shortage cost term
    • The optimal values of (Q,R) that minimizes G(Q,R) can be shown to be:
  • 20. Solution Procedure
    • The optimal solution procedure requires iterating between the two equations for Q and R until convergence occurs (which is generally quite fast)
        • We consider that the problem has converged if 2 consecutive calculation of Q and R are within 1 unit
    • A cost effective approximation is to set Q=EOQ and find R from the second equation.
    • A slightly better approximation is to set Q = max(EOQ, σ)
      • where σ is the standard deviation of lead time demand when demand variance is high.
  • 21. Ready to Try one? Lets!
    • Try Problem 13a & 13b (pg 261)
    • Start by computing EOQ and then begin iterative solution for optimal Q and R values
  • 22. Service Levels in (Q,R) Systems
    • In many circumstances, the penalty cost, p , is difficult to estimate. For this reason, it is common business practice to set inventory levels to meet a specified service objective instead. The two most common service objectives are:
      • Type 1 service: Choose R so that the probability of not stocking out in the lead time is equal to a specified value.
      • Type 2 service. Choose both Q and R so that the proportion of demands satisfied from stock equals a specified value.
  • 23. Computations
    • For type 1 service, if the desired service level is α then one finds R from F(R)= α and Q=EOQ.
    • Type 2 service requires a complex interative solution procedure to find the best Q and R . However, setting Q=EOQ and finding R to satisfy n(R) = (1-β)Q (which requires Table A-4) will generally give good results.
  • 24. Comparison of Service Objectives
    • Although the calculations are far easier for type 1 service, type 2 service is generally the accepted definition of service.
    • Note that type 1 service might be referred to as lead time service, and type 2 service is generally referred to as the fill rate.
    • Refer to the example in section 5-5 to see the difference between these objectives in practice (on the next slide).
  • 25. Comparison (continued)
    • Order Cycle Demand Stock-Outs
    • 1 180 0
    • 2 75 0
    • 3 235 45
    • 4 140 0
    • 5 180 0
    • 6 200 10
    • 7 150 0
    • 8 90 0
    • 9 160 0
    • 10 40 0
    • For a type 1 service objective there are two cycles out of ten in which a stockout occurs, so the type 1 service level is 80%. For type 2 service, there are a total of 1,450 units demand and 55 stockouts (which means that 1,395 demand are satisfied). This translates to a 96% fill rate.
  • 26. Example: Type 1 Service Pr 5-16
    • Desire 95% Type I service Level
    • F(R) = .95  Z is 1.645 (Table A4)
    • From Problem 13:  was found to be 172.8 and  was 1400
    • Therefore: R =  Z +  = 172.8*1.645 + 1400 R = 1684.256  1685
    • Use Q = EOQ = 1265
  • 27. Example: Type 2 Service Pr 5-17
    • Require Iterative Solution:
  • 28. Example: Type 2 Service Pr 5-17 (cont.)
  • 29. (s, S) Policies
    • The (Q,R) policy is appropriate when inventory levels are reviewed continuously. In the case of periodic review, a slight alteration of this policy is required. Define two levels, s < S, and let u be the starting inventory at the beginning of a period. Then
    • (In general, computing the optimal values of s and S is much more difficult than computing Q and R.)