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Production and Operation Managements Professor JIANG Zhibin Department of Industrial Engineering & Management Shanghai Jiao Tong University Inventory Control Subject to Unknown Demand
In the economic recession of the early 1990s, some business that relied on direct consumer spending, suffered severe losses. Sears and Macy’s department store, long standing successes in American retail market made poor earning in 1991.
Several retailers enjoyed dramatic successes. Both The Gap and Limited in the fashion business did very well.
Wal-Mart Stores continues its ascendancy and surpassed Sear as the largest retailer in the United State.
In many situations the random component of the demand is too important to ignore.
As long as the expected demand per unit times is relatively constant, and the problem structure is not too complex, explicit treatment of demand uncertainty is desirable.
Two basic inventory control models subject to uncertainty:
Periodic review-the inventory level is known at discrete points in time only;
For one planning period-the objective is to balance the costs of overage and underage; useful for planning for determining run sizes for items with short useful lifetimes (Fashions, foods,newspaper)- newsboy model .
For multiple planning period-Complex, topics of research, and rarely implemented.
Continuous review-the inventory level is known at all times.
Example 5.1-Mac wishes to determine the number of copies of the Computer Journal he should purchased each Sunday. The demand during any week is a random variable that is approximately normally distributed, with mean 11.73 and standard deviation 4.74. Each copy is purchased for 25 cents and sold for 75 cents, and he is paid for 10 cents for each unsold copy by his supplier.
Discussion:
One obvious solution is to buy enough copies to meet the demand, which is 12 copies.
Wrong: If he purchase a copy that does not sell, his out-of-pocket expense is only 25-10-15 cents. However, if he is unable to meet the demand of a customer, he loses 75-25=50cents.
Suggestion: He should buy more than the mean. How many?
A general outline for analyzing most stochastic inventory problems is as follows:
Develop an expression for cost incurred as a function of both the random variable D and the decision variable Q.
Determine the expected value of this expression with respect to the density function or probability function of demand.
Determine the value of Q such that the expected cost function is to be minimized.
Development of Cost Function
Define G(Q, D) as the total overage and underage cost incurred at the end of the period when Q units are ordered at the start of the period and D is the demand.
Q-D is the demand units left at the end of the period as long as Q D;
If Q<D, then Q-D is negative and the number of units remaining on hand at the end of the period is 0.
Determine the value of Q that minimizes the expected cost G(Q).
G(Q) is convex such that Q(Q) has minimal value Since the slope is negative at Q=0, G(Q) is decreasing at Q=0.
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The newsboy model Optimal solution, Q * , such that The critical ratio is strictly between 0 and 1, meaning that for a continuous demand, this equation is always solvable. Fig5-3 Expected Cost Function for Newsboy Model The critical ratio.
Since F(Q*) is defined as the probability that the demand does not exceed Q*, the critical ratio is the probability of satisfying all the demand during the period if Q* units are purchased at the beginning of the period.
Example 5.1- Mac’s newsstand
Suppose that the demand for the Journal is approximately normally distributed with mean =11.73 and standard deviation =4.74. c 0 =25-10=15, and c u =75-25=50 cents. The critical ratio is c u /(c o +c u )=0.50/(0.15 +0.5)=0.77. Hence, he ought to purchase enough copies to satisfy all of the weekly demand with probability 0.77. The optimal Q* is the 77 th percentile of the demand distribution.
Q*= z+ =4.74 0.74+11.73=15.24 15 F(Q*) Fig. 5-4 Determination of the Optimal Order Quantity for Newsboy Example
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The newsboy model- Optimal Policy for Discrete Demand
In some cases, accurate representation of the observed pattern of demand in term of continuous distribution is difficult or impossible.
In the discrete case, the critical ratio will generally fall between two values of F(Q).
The optimal solution procedure is locate the critical ratio between two values of F(Q) and choose the Q corresponding to the higher value.
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The newsboy model- Optimal Policy for Discrete Demand
Example 5.2- Mac’s newsstand
f(4) =3/52 is obtained by dividing frequencies 4 (the numbers of times 3 that a given weekly demand 4 occur during a year, i.e. 52 weeks) by 52;
The critical ratio is 0.77, which corresponds to a value of F(Q) between Q=14 and Q=15 .
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The newsboy model- Extension to Include Starting Inventory
Suppose that the starting inventory is some value u and u>0 .
The optimal policy is simply to modify that for u=0.
The same ideal is that we still want to be at Q* after ordering.
If u<Q*, order Q*-u; If u>Q*, do not order.
Note that Q* should be understood as order-up-to point rather than the order quantity when u>0.
Example 5.2 (Cont.)-Suppose that Mac has received 6 copies of the Journal at the beginning of the week from other supplier. The optimal policy still calls for having 15 copies on hand after ordering, thus he would order the difference 15-6=9 copies.
For random demand, Q and R are regarded as independent decision variables;
Assumptions
Continuous review-demands are recorded as they occur;
Random and stationary demand-the expected value of demand over any time interval of fixed length is constant; the expected demand rate is unite/year.
Fixed positive lead time for placing an order;
Assume the following costs
Setup cost $K per order;
Holding cost at $h per unit held per year;
Proportional order cost of $c per item;
Stock-out cost $p per unit of unsatisfied demand, or shortage cost or penalty cost;
The response time is the amount of time required to effect a change in the on-hand inventory level.
The response time is the reorder lead time
The demand during the lead time is the random variable of interest.
It is assumed that demand during lead time is continuous random variable D with probability density function (pdf) f(x) and cumulative distribution function (cdf) F(x).
is the mean of the demand during the lead time. is the standard deviation.
Derivation of the Expected Cost Function- develop an expression for the expected average annual cost in terms of the decision variables (Q, R) and search for the optimal values of (Q, R) to minimize this cost.
Holding cost
Assume that the mean rate of demand is units per year;
The expected inventory level varies between s and Q+s , where s is the safety stock , defined as the expected level of on-hand inventory just before an order arrives , s=R- .
The average inventory is s+Q/2=R- +Q/2.
The holding cost should not be charged against the inventory level when it is negative.
Occurs only when the system is subject to shortage.
The number of units of excess demand is simply the amount by which the demand over the lead time, D, exceeds the reorder level, R.
The expected number of shortages that occurs in one cycle is determined by
n(R) As n(R) represents the expected number of stock-outs incurred in a cycle, the expected number of stock-outs incurred per unit time is n(R)/T= n(R)/Q .
The expected proportional order cost per unit of time is c;
Since this item is independent of variables Q and R, it does not affect the optimization, and thus may be ignored.
The Cost Function:
G(Q, R)=h(Q/2+R- )+K /Q+p n(R)/Q. The objective is to choose Q and R to minimize G(Q, R). The solution procedure requires iterating between (1) and (2) until the two successive values of Q and R are the same.
Example 5.4 Harvey’s Specialty Shop sells a popular mustard that purchased from English company. The mustard costs $10 a jar and requites a six-month lead time for replenishment stock. The holding cost is computed on basis 20% annual interest rate; the lost-of-goodwill cost is $25 a jar; and bookkeeping expenses for placing an order amount to about $50. During the six-month lead time, average 100 jars are sold , but with substantial variation from one six-month period to the next. The demand follows normal distribution and the standard deviation of demand during each six-month period is 25. How should Harvey control the replenishment of the mustard?
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Lot Size-Reorder Point System Q 1 =110 1-F(R 1 )=Q 1 h/p =110 2/(25 200)=0.044 (from Table A-4) z=1.70, and L(z)=0.0183 Solving R 1 = z+ =25 1.70+100=143; n(R 1 )= L(z)=25 0.0183=0.4575, and Q 1 =110 is not closed to Q 0 =100 enough to stop Q 2 =111 1-F(R 2 )=Q 2 h/p =111 2/(25 200)=0.0444 (from Table A-4) z=1.70, and L(z)=0.0183 Solving R 2 = z+ =25 1.70+100=143=R 1 ; Since both Q 2 and R 2 are only different from Q 1 and R 1 by at most 1, stop!
Results for Example 5.4: The optimal values of (Q, R)=(111, 143), that is, when Harvey’s inventory of this type mustard hits 143 jars, he should place an order for 111 jars.
Example 5.4 (Cont.): determine the following
Safety stock;
The average annual holding, setup, and penalty costs associated with the inventory control of the mustard;
The average time between placement of orders;
The proportion of order cycles in which no stock-outs occur>Among given number of order cycles, how many order cycles do not have stock-outs?
The holding cost is h(Q/2+s) =2(111/2+43)=$197/jar;
The setup cost is K /Q =50 200/111=$90.09/jar;
The penalty cost is p n(R)/Q =25 200 0.4575/111=$20.61/jar
Hence, the total average cost under optimal inventory control policy is $307.70/jar.
T=Q/ =111/200=0.556 yr=6.7months;
Compute the probability that no stock-out occurs in the lead time, which is the same as that the probability that the lead time demand does not exceeds the reorder point: P(D R)=F(R)=1-Qh/p =1-0.044=0.956;
The expected demand per cycle must be Q; the expected number of stock-outs per cycle is n(R). Hence, the proportion of demand that stock out is n(R)/Q=0.4575/111=0.004.
is interpreted as the proportion of cycles in which no stock-out occurs;
A Type 1 service objective is suitable when a shortage occurrence has the same consequence independent of its time or amount. For example, a production line is stopped whether 1 unit or 100 units are short.
However, Type 1 service does not illustrate how does the shortage occur.
Usually, when we say we would like provide 95% service, we mean that we would like to be able to fill 95% of the demand when they occur, rather than fill all of the demands in 95% of the order cycles. – not be specified by Type 1 Service .
In addition, different items have different cycle lengths, this measure will not be consistent among different products, making the proper choice of difficult.
Harvey feels uncomfortable with assumption that the stock-out cost is $25 and decide to use a service level criterion instead. Suppose that he chooses to use 98%.
Type 1 service: =0.98, find R to solve F(R)=0.98. From Table A-4, z=2.05, R= z+ =25 2.05+100=151.
Type 2 service: =0.98, n(R)=EOQ(1- ), which corresponds to L(z)= EOQ(1- )/ =100(1-0.98)/25=0.08. From Table A-4, z=1.02, then R= z+ =25 1.02+100=126.
The same values of and gives considerably different values of R.
n(R 1 )= (1- )Q 1 =(1-0.98)114=2.28; L(z)=n(R 1 )/ =2.28/25=0.0912 z=0.95 1-F(R 1 )=0.171; R 1 = z+ =124 R 2 =124; The optimal values of Q and R satisfying a 98 percent fill rate constraint are (Q, R)=(114, 124). The cost is $252, only $2 higher than for (100, 126)>>good approximation.
Type 1 service objective- find the order-up-to point Q so that all of the demand is satisfied in a given percentage of the periods, which can be determined by F(Q)= , where F(Q) is the probability that the demand during the period does not exceed Q.
Type 2 service objective
To find the Q to satisfy the Type 2 service objective , it is necessary to obtain an expression for the fraction of demand that stock out each period.
Define n(Q), the expected number of demands that stock out at the end of period.
Since the demand per period is , then the proportion of demand that stock out each period is n(Q)/ =1- , giving n(Q) =(1- ) .
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