Major Topics Already Covered

1. Major Topics Already Covered • Major roles of Warehouses in contemporary supply chains • Warehouse processes and the organization of the material flow • Warehouse equipment • Overview of the warehouse design and control problem(s) • Warehouse Activity Profiling

2. Next Issues • Storage configuration and storage policies • the forward/reserve problem • order-picking: batching, zoning, and routing • Pallet-building • Warehouse layout • Configuring and controlling automated storage and retrieval equipment • Cross-docking

3. Warehouse Storage Configuration and Storage Policies Bibliography Bartholdi & Hackman: Chapter 6 Francis, McGinnis, White: Chapter 5 Askin and Standridge: Sections 10.3 and 10.4

4. I/O Storage Policies • Main Issue: Decide how to allocate the various storage locations of a uniform storage medium to a number of SKU’s.

5. Types of Storage Policies • Dedicated storage: Every SKU i gets a number of storage locations, N_i, exclusively allocated to it. The number of storage locations allocated to it, N_i, reflects its maximum storage needs and it must be determined through inventory activity profiling. • Randomized storage: Each unit from any SKU can by stored in any available location • Class-based storage: SKU’s are grouped into classes. Each class is assigned a dedicated storage area, but SKU’s within a class are stored according to randomized storage logic.

6. Location Assignment under dedicated storage policy • Major Criterion driving the decision-making process:Enhance the throughput of your storage and retrieval operations by reducing the travel time <=> reducing the travel distance • How? By allocating the most “active” units to the most “convenient” locations...

7. “Convenient” Locations • Locations with the smallest distanced_j to the I/O point! • In case that the material transfer is performed through a forklift truck (or a similar type of material handling equipment), a proper distance metric is the, so-called, rectilinear or Manhattan metric (or L1 norm): d_j = |x(j)-x(I/O)| + |y(j)-y(I/O)| • For an AS/RS type of storage mode, where the S/R unit can move simultaneously in both axes, with uniform speed, the most appropriate distance metric is the, so-called Tchebychev metric (or L norm): d_j = max (|x(j)-x(I/O)|,|y(j)-y(I/O)|)

8. “Active” SKU’s • SKU’s that cause a lot of traffic! • In steady state, the appropriate “activity” measure for a given SKU i: Average visits per storage location = (number of units handled per unit of time) / (number of allocated storage locations) = TH_i / N_i

9. Problem Representation Location SKU N_1 1 1 1 c_ij = (TH_i/N_i)*d_j N_i i 1 j N_S S L 1

10. Problem Formulation • Decision variables: x_ij = 1 if location j is allocated to SKU I; 0 otherwise. • Formulation: min S_i S_j [(TH_i/N_i) * d_j] * x_ij s.t.  i, S_j x_ij = N_i  j, S_i x_ij = 1  i, j, x_ij  {0,1} => x_ij  0

11. Remarks • The previous problem representation corresponds to a balanced transportation problem: Implicitly it has been assumed that: L = S_iN_i • For the problem to be feasible, in general, it must hold that: L  S_iN_i • If L - S_iN_i > 0, the previous balanced formulation is obtained by introducing a fictitious SKU 0, with N_0 = L - S_iN_i and TH_0 = 0

12. A fast solution algorithm • Rank all the available storage locations in increasing distance from the I/O point, d_j. • Rank all SKU’s in decreasing “turns”, TH_i/N_i. • Move down the two lists, assigning to the next most highly ranked SKU i, the next N_i locations.

13. 5 9 8 7 5 6 7 8 9 5 6 7 6 5 4 4 4 5 8 8 6 7 4 5 4 3 3 3 6 7 6 7 5 6 5 4 2 2 2 3 6 3 4 5 3 5 4 3 2 1 0 1 2 4 5 I/O I/O Example A: 20/10=2 B: 15/5 = 3 C: 10/2 = 5 D: 20/5 = 4 A A A A A B B A D D D A A B B A A C C D D B

14. Locating the I/O point • In many cases, this location is already predetermined by the building characteristics, its location/orientation with respect to the neighboring area/roads/railway tracks, etc. • Also, in the case of an AS/RS, this location is specified by the AS/RS technical/operational characteristics. • In case that the I/O point can be placed at will, the ultimate choice should seek to enhance its “proximity” to the storage locations.

15. 5 9 8 7 5 6 7 8 9 5 6 7 6 5 4 4 4 5 8 8 6 7 4 5 4 3 3 3 6 7 6 7 5 6 5 4 2 2 2 3 6 3 4 5 3 5 4 3 2 1 0 1 2 4 5 Option B 6 7 9 10 12 13 14 11 5 5 8 4 8 11 4 5 6 7 9 10 12 13 3 3 4 5 6 8 9 10 11 7 12 4 6 8 10 2 11 2 3 5 7 9 0 1 2 3 4 5 6 7 8 9 10 I/O I/O Locating the I/O point: Example 1 Option A

16. I/O Locating the I/O point: Example 2 Option A 12 10 10 10 18 16 14 12 14 16 18 16 14 12 10 8 8 8 10 16 12 14 8 14 12 6 10 10 8 6 6 12 14 12 6 4 8 10 10 8 4 4 6 12 10 6 8 6 4 2 0 2 4 8 10 O Option C 10 8 5 6 10 12 14 14 8 12 6 14 6 10 10 8 6 6 8 12 14 12 14 12 10 8 6 6 8 10 12 6 14 8 6 8 12 14 14 12 10 6 6 10 14 12 10 8 6 5 6 8 10 12 14 I

17. Example 2 (cont.) • Option A: U-shaped or cross-docking configuration • amplifies the convenience/inconvenience of close/distant locations • appropriate for product movement with strong ABC skew • provides flexibility for interchanging between shipping and receiving docking capacity • allows for “dual command” operation of forklifts, reducing, thus, the deadhead traveling • minimizes truck apron and roadway • Option C: Flow-through configuration • attenuates the convenience difference among storage locations • conservative design: more reasonably convenient storage locations but fewer very convenient • more appropriate for extremely high volume • preferable when the building is long and narrow • limits the opportunity for efficiencies for “dual command” operations

18. Storage Sizing • Dedicated storage: • How many storage locations, N_i, should be dedicated to each SKU i? • Randomized Storage: • How many storage locations, N, should be employed for the storage of the entire SKU set? • Class-based storage: • How should SKU’s be organized into classes? • How many storage locations, N_k, should be dedicated to each SKU class k?

19. Possible Approaches to Storage Sizing under Dedicated Storage • Issue resolved/predetermined through operational (e.g., replenishment) policies, contractual agreements, etc. • “Service-level” type of analysis: • Determine the number of storage locations, N_i to be assigned to each SKU i in a way that tends to control the probability that there will be a lack of storage space in any operational period (e.g., day). • Cost-based Analysis • Select N_i’s in a way that minimizes the total operational cost over a given horizon, taking into consideration the cost of owning and operating the storage space and equipment, and also any additional costs resulting from space shortage and/or the need to contract additional storage space.

20. Sizing dedicated storage based on “service level” requirements • F_i(Q_i) = Prob{stor. loc. requested by SKU i in a single period  Q_i} i.e., the cumulative distribution function of the storage locations requested by SKU i during any single operational period (day) • Assuming independence of daily storage requirements across SKU’s, for an assignment of N_i locations to each SKU i, Prob{no storage shortages in a single day} = _i F_i(N_i) and Prob{1 or more storage shortages} = 1 - _i F_i(N_i)

21. Sizing dedicated storage based on “service level” requirements (cont.) • Formulation I: Fixed service level, P min _i N_i s.t. _i F_i(N_i)  P N_i  0  i • Formulation II: Fixed storage space, S max _i F_i(N_i) s.t. _i N_I  S N_i  0  i

22. Sizing randomized storage based on “service level” requirements • F(Q) = Prob{stor. loc. requested by all SKU’s in a single period  Q} i.e., the cumulative distribution function of the storage locations requested by all SKU’s during any single operational period (day) • For a storage size of N locations: Prob{No storage shortage in a single period} = F(N) • Problem Formulations: • Formulation I: Fixed service level, P min N s.t. F(N)  P N  0 • Formulation II: Fixed storage space, S max F(N) s.t. 0  N  S

23. Class-Based Storage Sizing and Location Assignment • Divide SKU’s into classes, using ABC (Pareto) analysis, based on their number of turns TH_i/N_i. • Determine the required number of storage locations for each class C_k • ad-hoc adjustment of the total storage requirement of the class SKU’s N_k = p * _{iC_k } N_i • Class-based “service-level” type of analysis, e.g., • Assign to each class the requested storage locations, prioritizing them according to their number of turns, TH_k (=_{iC_k } TH_i)/N_k • min _k N_k • s.t. • _k F_k(N_k)  P • N_k  0  k

24. A simple cost-based model for (dedicated) storage sizing • Model-defining logic:Assuming that you know your storage needs d_ti, for each SKU i, over a planning horizon T, determine the optimal storage locations N_i for each SKU i, by establishing a trade-off between the • fixed and variable costs for developing this set of locations, and operating them over the planning horizon T, and • the costs resulting from any experienced storage shortage.

25. A simple cost-based model for (dedicated) storage sizing (cont.) • Model Parameters: • T = length of planning horizon in time periods • d_ti = storage space required for SKU i during period t • C_0 = discounted present worth cost per unit storage capacity owned during the planning horizon T • C_1t = discounted present worth cost per unit stored in owned space during period t • C_2t = discounted present worth cost per unit stored in leased space or per unit of space shortage during period t • Model Decision Variables: • N_i = “owned” storage capacity for SKU i • Model Objective: • min TC (N_1,N_2,…,N_n) = S_i [C_0 N_i + S_t {C_1t [min(d_ti, N_i)] + C_2t [max(d_ti - N_i, 0)]}]

26. A fast solution algorithm for the case of time-invariant costs • For each SKU i: • Sequence the storage demands appearing in the d_ti, t=1,…T, sequence in decreasing order. • Determine the frequency of the various values in the ordered sequence obtained in Step 1. • Sum the demand frequencies over the sequence. • When the obtained partial sum is first equal to or greater than C’ = C_0/ (C_2-C_1) stop; the optimum capacity for SKU i, N_i, equals the corresponding demand level.

27. Example • Problem Data: N=1; T=6; d = < 2, 3, 2, 3, 3, 4,>; C_0 = 10, C_1 = 3, C_2 = 5 • Solution: C’ = C_0/(C_2-C-1) = 10/(5-3) = 5 => N = 2

28. A probabilistic version • Additional data • p(d_ti) = probability mass function for the storage requirement of SKU i during period t • New objective function • min E [TC (N_1,N_2,…,N_n)] = S_i [C_0 N_i + S_t {C_1 E[min(d_ti, N_i)] + C_2 E[max(d_ti - N_i, 0)]}] = S_i [C_0 N_i + S_t {C_1[S_{d_ti  N_i} d_ti p(d_ti) + S_{d_ti > N_i} N_i p(d_ti) ]+ C_2 [S_{d_ti > N_i} (d_ti - N_i) p(d_ti) ]}]

29. A probabilistic version (cont.) • Model Properties: • Separable • Piece-wise linear • convex • Optimality Condition: N_i = Q_ij optimal <=> S_t Fc_ti(Q_ij)  C_0 / (C_2 - C_1)  S_t Fc_ti(Q_i,j+1) <=> S_t [1-F_ti (Q_ij) +p(Q_ij)]  C_0 / (C_2 - C_1)  S_t [1-F_ti (Q_i,j+1)+p(Q_i,j+1)] where F_ti( ) is the cdf for the storage requirements of SKU i in period t.

30. Storage Configuration and Policiesfor “Unit Load” warehouses: Topics covered • Storage Policies: Assigning storage locations of a uniform storage medium to the various SKU’s stored in that medium • Dedicated • Randomized • Class-based Criterion: Maximize productivity by reducing the traveling effort / cost • The placement of the I/O point(s) Criterion: Maximize productivity by reducing the traveling effort / cost

31. Storage Configuration and Policiesfor “Unit Load” warehouses: Topics covered (cont.) • Storage sizing for various SKU’s: Determine the number of storage locations to be assigned to each SKU / group of SKU’s. Criterion: • provide a certain (or a maximal) “service level” • minimize the total (space+equipment+labor+shortage) cost over a planning horizon • Next major theme: Storage Configuration for better space exploitation • floor versus rack-based storage for pallet-handling warehouses • determining the lane depth (mainly for randomized storage) (based on Bartholdi & Hackman, Section 6.3)

32. Determining the Employment (and Configuration) of Rack-based storage • Basic Logic: • For each SKU, • compute how many pallet locations would be created by moving it into rack of a given configuration; • compute the value of the created pallet locations; • move the sku into rack if the value it creates is sufficient to justify the rack. • Remark: In general, space utilization will be only one of the factors affecting the final decision on whether to move an SKU into rack or not. Other important factors can be • the protection that the rack might provide for the pallets of the considered SKU; • the ability to support certain operational schemes, e.g., FIFO retrieval; • etc.

33. Examples on evaluating the efficiencies from moving to rack-based storage • Case I: Utilizing 3-high pallet rack for an SKU of N=4 (pallets), which is not stackable at all. • Current footprint: 4 pallet positions • Introducing a 3-high rack in the same area creates 3x4=12 position, 8 of which are available to store other SKU’s. What are the gains of exploiting these new locations vs the cost of purchasing and installing the rack? • Case II: Utilizing a 3-high pallet rack for an SKU with N=30 (pallets), which are currently floor-stacked 3-high, to come within 4 ft from the ceiling. • Current footprint: 10 pallet positions • Introducing a 3-high rack does not create any new positions, and it will actually require more space in order to accommodate the rack structure (cross-beams and the space above the pallets, required for pallet handling)

34. Lane Height Lane Depth (3-deep) Lanes Determining an efficient lane depth(in case of randomized storage) • A conceptual characterization of the problem: • More shallow lanes imply more of them, and therefore, more space is lost in aisles (the size of which is typically determined by the maneuvering requirements of the warehouse vehicles) • On the other hand, assuming that a lane can be occupied only by loads of the same SKU, a deeper lane will have many of its locations utilized over a smaller fraction of time (“honeycombing”). • So, we need to compute an optimal lane depth, that balances out the two opposite effects identified above, and minimizes the average floor space required for storing all SKU’s. Aisle

35. Notation • w = pallet width • d = pallet depth • g = gap between adjacent lanes • a = aisle width • x = lane depth • n = number of SKU’s • N_i = max storage demand by SKU i • z_i = column height for SKU I • lane footprint = (g+w)(d*x+a/2)

36. Key results • Assuming that the same lane depth is employed across all n SKU’s, under floor storage, the average space consumed per pallet is minimized by a lane depth computed approximately through the following formula: x_opt = [(a/2dn)*_i (N_i /z_i)] • The optimal lane depth for any single SKU i, which is stackable z_i pallets high, is x_opt = [(a/2d)*(N_i /z_i)] • Assuming that the same lane depth is employed across all n SKU’s, under rack storage, the average space consumed per pallet is minimized by a lane depth computed approximately through the following formula: x_opt = [(a/2dn)*_i N_i ]