1 Inventory Theory – Part II Quantity Discounts Constrained Inventory Management Single Facility EOQ A quantity of commodity held for some time to satisfy some future demand.

2 Check it out! The Engineering Manager’s and Management Scientist’s Handy-Dandy Cheat Sheet for Deterministic Continuous Review Inventory Models This assignment is easy with the cheat sheet

3 Quantity Discounts All Units Discounts: the discount is applied to ALL of the units in the order. Incremental Discounts: the discount is applied only to the number of units above the breakpoint.

4 All Units Discount

5 A Really nice property of the all unit discount solution For all units discounts, the optimal will occur at the bottom of one of the cost curves or at a breakpoint. (It is generally at a breakpoint). Compare the cost at the largest (Q) realizable EOQ and all of the breakpoints succeeding it.

6 An Example Assume D = 600 per year, K = $8, and I =.20

7 ALL Units Discount Q TC(Q) TC 0 (Q) TC 1 (Q) TC 2 (Q) 400

8 The Example Continued I choose Q = 500.

9 Incremental Discount

10 Not so nice properties of the incremental discounting solution Minimum cost point will never occur at a breakpoint If the EOQ for an interval is in the interval, it still may not be optimal Must compute the EOQ for each price break –if it falls within the interval, compute the average cost –pick the best

11 Incremental Discounting Average annual inventory cost C(Q) is the cumulative cost to purchase Q units

12 Incremental Discounting Example Minimize average annual cost C(Q) / Q is average unit cost

13 Incremental Discounting Example

14 Incremental Discounting Example Not realizable

15 Incremental Discounting

16 Incremental Discounting Example Not realizable Minimize average annual cost G 0 (400) = $ G 1 (519) = $204.58

17 Resource Constrained Multi-Item Inventories Consider an inventory system of n items in which the total amount available to spend is C and items cost respectively c 1, c 2,..., c n. Then this imposes the following constraint on the system:

18 The Model Obviously this can be solved using the generalized Lagrangian approach! An EMS graduate

19 Form the Lagrangian  is called the Lagrangian multiplier

20 Solve the Lagrangian The approach: 1. Pick a value for  2. Compute Q j and 3. Repeat 1. and 2. until

21 What about other constraints? Max warehouse space Holding costs Average Stock on-hand I bet you can’t work an example of this so called generalized Lagrangian approach.

22 Warehouse space constraint

23 Solving…

24 Example of the so-called generalized Lagrangian Approach

25 Find EOQ for each item independently Q x sq. ft., / unit 1,000 sq. ft. limitation

26 Optimal Constrained Solution let Excel show the way…

27 Single Facility Production Model 1.n items to produce 2.D j = constant demand rate for product j 3.P j = constant production rate for product j 4.h j = holding cost per unit per unit time for product j 5.K j = fixed setup cost for product j 6.s j = setup time for product j (not sequence dependent) 7.no stock outs permitted Determine production lot sizes (Q j ) that will minimize relevant costs.

28 Single Facility Production Model Let T = cycle time in which one lot of each product is produced. Then Q j = D j T in order to meet demands And G(Q j ) = K j D j / Q j + (1 - D j /P j ) h j Q j /2 Note that

29 Total relevant costs: P-D -D Q/P (P-D) Q/P = (1-D/P) Q Q j = D j T

30 T* must include production time + setup time Solution Set T = Maximum{T*,T min }

31 The Inevitable Example (1-D/P) h to Excel and beyond…

32 Single Facility Production Model Some final thoughts is the fraction of time the facility is not producing 1. 2.If products are made on different facilities Then use independent lot models. 3.If setup times or setup costs depend upon the rotation order, then have a more difficult sequencing problem.

33 EOQ Homework I always work extra problems – not only the ones that are assigned! Text Chapter 4: 4,8,10,12,13,14,17,20,21,22 24,26,27,28,29,30,33,35,40 + Handout