Definition 1.3.5 In conjunction with the four statistics ¯o, ¯u, ¯l+ and ¯l−, a facility’s cost of operation is determined by four constants:
• citem — the (unit) cost of a new item;
• csetup — the setup cost associated with placing an order;
• chold — the cost to hold one item for one time interval;
• cshort — the cost of being short one item for one time interval.
Case Study
Consider a hypothetical automobile dealership that uses a weekly periodic inventory review policy. The facility is the dealer’s showroom, service area and surrounding storage lot and the items that flow into and out of the facility are new cars. The supplier is the manufacturer of the cars and the customers are people convinced by clever advertising that their lives will be improved significantly if they purchase a new car from this dealer.
Example 1.3.5 Suppose space in the facility is limited to a maximum of, say S = 80, cars. (This is a small dealership.) Every Monday morning the dealer’s inventory of cars is reviewed and if the inventory level at that time falls below a threshold, say s = 20, then enough new cars are ordered from the supplier to restock the inventory to level S.*
• The (unit) cost to the dealer for each new car ordered is citem = $8000.
• The setup cost associated with deciding what cars to order (color, model, options, etc.) and arranging for additional bank financing (this is not a rich automobile dealer) is csetup = $1000, independent of the number ordered.
• The holding cost (interest charges primarily) to the dealer, per week, to have a car sit unsold in his facility is chold = $25.
• The shortage cost to the dealer, per week, to not have a car in inventory is hard to determine because, in our model, we have assumed that all demand will ultimately be satisfied. Therefore, any customer who wants to buy a new car, even if none are available, will agree to wait until next Monday when new cars arrive. Thus the shortage cost to the dealer is primarily in goodwill. Our dealer realizes, however, that in this situation customers may buy from another dealer and so, when a shortage occurs, he sweetens his deals by agreeing to pay “shorted” customers $100 cash per day when they come back on Monday to pick up their new car. This means that the cost of being short one car for one week is cshort = $700.
* There will be some, perhaps quite significant, delivery lag but that is ignored, for now, in our model. In effect, we are assuming that this dealer is located adjacent to the supplier and that the supplier responds immediately to each order.
Definition 1.3.6 A simple inventory system’s average operating costs per time interval are defined as follows:
• item cost: citem·o;¯
• setup cost: csetup ·u;¯
• holding cost: chold· ¯l+;
• shortage cost: cshort· ¯l−.
The average total cost of operation per time interval is the sum of these four costs. This sum multiplied by the number of time intervals is the total cost of operation.
Example 1.3.6 From the statistics in Example 1.3.4 and the constants in Example 1.3.5, for our auto dealership the average costs are:
• the item cost is $8000 · 29.29 = $234, 320;
• the setup cost is $1000 · 0.39 = $390;
• the holding cost is $25 · 42.40 = $1060;
• the shortage cost is $700 · 0.25 = $175.
Each of these costs is a per week average.
Cost Minimization
Although the inventory system statistic of primary interest is the average total cost per time interval, it is important to know the four components of this total cost. By varying the value of s (and possibly S) it seems reasonable to expect that an optimal (minimal average cost) periodic inventory review policy can be determined for which these components are properly balanced.
In a search for optimal (s, S) values, because ¯o= ¯dand ¯ddepends only on the demand sequence, it is important to note that the item cost is independent of (s, S). Therefore, the only cost that can be controlled by adjusting the inventory policy parameters is the sum of the average setup, holding, and shortage costs. In Example 1.3.7, this sum is called the average dependent cost. For reference, in Example 1.3.6 the average dependent cost is
$390 + $1060 + $175 = $1625 per week.
If S and the demand sequence is fixed, and if s is systematically increased, say from 0 to some large value less than S, then we expect to see the following.
• Generally, the average setup cost and holding cost will increase with increasing s.
• Generally, the average shortage cost will decrease with increasing s.
• Generally, the average total cost will have a ‘U’ shape indicating the presence of one (or more) optimal value(s) of s.
Example 1.3.7 is an illustration.
Example 1.3.7 With S fixed at 80, a modified version of program sis1 was used to study how the total cost relates to the value of s. That is, the cost constants in Exam-ple 1.3.5 were used to compute the average setup, holding, shortage, and dependent cost for a range of s values from 1 to 40. As illustrated in Figure 1.3.9, the minimum average dependent cost is approximately $1550 at s = 22.
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Figure 1.3.9.
Costs for an (s, 80) simple inventory system for s= 1, 2, . . . , 40.
As in the case study concerning the ice cream shop, the “raw” simulation output data is presented. In this case, however, because the parameter s is inherently integer-valued (and so there is no “missing” output data at, say, s = 22.5) no interpolating or approximating curve is superimposed. [For a more general treatment of issues surrounding simulation optimization, see Andrad´ottir (1998).]
1.3.6 EXERCISES
Exercise 1.3.1 Verify that the results in Example 1.3.1 and the averages in Exam-ples 1.3.2 and 1.3.3 are correct.
Exercise 1.3.2 (a) Using the cost constants in Example 1.3.5, modify program sis1 to compute all four components of the total average cost per week. (b) These four costs may differ somewhat from the numbers in Example 1.3.6. Why? (c) By constructing a graph like that in Example 1.3.7, explain the trade-offs involved in concluding that s = 22 is the optimum value (when S = 80). (d) Comment on how well-defined this optimum is.
Exercise 1.3.3 Suppose that the inventory level l(t) has a constant rate of change over the time interval a ≤ t ≤ b and both l(a) and l(b) are integers. (a) Prove that
Z b
a
l(t) dt = Z b
a
bl(t) + 0.5c dt = 1
2(b − a)¡l(a) + l(b)¢.
(b) What is the value of this integral if l(t) is truncated rather than rounded (i.e., if the 0.5 is omitted in the second integral)?
Exercise 1.3.4 (a) Construct a table or figure similar to Figure 1.3.7 but for S = 100 and S = 60. (b) How does the minimum cost value of s seem to depend on S? (See Exercise 1.3.2.)
Exercise 1.3.5 Provided there is no delivery lag, prove that if di ≤ sfor i = 1, 2, . . . , n, then ¯l− = 0.
Exercise 1.3.6 (a) Provided there is no delivery lag, prove that if S − s < di ≤ S for i= 1, 2, . . . , n then ¯l+ = S − ¯d/2. (b) What is the value of ¯l− and ¯u in this case?
Exercise 1.3.7 Use Definitions 1.3.3 and 1.3.4 to prove that the average inventory level equation
¯l= 1 n
Z n
0
l(t) dt = ¯l+− ¯l−
is correct. Hint: use the (·)+ and (·)− functions defined for any integer (or real number) x as
x+ = |x|+ x
2 and x− = |x| − x
2 and recognize that x = x+− x−.
Exercise 1.3.8 (a) Modify program sis1 so that the demands are first read into a circular array, then read out of that array, as needed, during program execution. (b) By experimenting with different starting locations for reading the demands from the circular array, explore how sensitive the program’s statistical output is to the order in which the demands occur.
Exercise 1.3.9a (a) Consider a variant of Exercise 1.3.8, where you use a conventional (non-circular) array and randomly shuffle the demands within this array before the de-mands are then read out of the array, as needed, during program execution. (b) Repeat for at least 10 different random shuffles and explore how sensitive the program’s statistical output is to the order in which the demands occur.