Quantitative approaches to facilities planning: Deterministic models, single and multi facility models, Conventional layout model: Block stacking, location-allocation models, Layout Models: Warehouse layout models, waiting line models, Storage models.

1. Rashmi S
Dept. of IE&M
JSS Academy of Technical Education

2.  Quantitative approaches to facilities planning:
 Deterministic models
 Single and multi facility models
 Conventional layout model: Block stacking,
location allocation models,
 Layout Models: Warehouse layout models,
waiting line models, Storage models.

13.  The single facility location problems is to determine the location of a
new facility, say X ∗, that minimizes f(X).
 In many applications the cost per unit distance is a constant thus the
minimization problem often reduces to a determination of
the location that minimizes distance.
 The problem of single facility location can be stated
as follows:
Determine the location of a single new facility with respect to a number of
existing facilities that minimizes an appropriate defined total cost function
which is chosen to be proportional to distance.

16.  The objective of SFLP : to locate a single new facility in relation to a
set of existing facilities such that total TRANSPORTATION between
the new and set of existing facilities is minimized .
 If given set of existing facilities with their co-ordinates on XY plane and
movement of materials from new facility to all these existing facilities, then
the objective is to find the optimal location for new facility.
 In such case rectilinear distance is followed for such decision.
 The rectilinear distance between any two points whose co-ordinates are (X1,
Y1) and (X2, Y2) is given by formula:
d12 = |X1-X2|+|Y1-Y2|

29. Single –Facility Minisum Location Problem

33. SINGLE FACILITY MINIMAX LOCATION
PROBLEM
Objective :
Minimizing maximum distance between new and
any existing facility
• Generally, transformation of terms with absolute values the n-facilities rectilinear
distance location problem can be solved using linear programming methods.
• The Euclidean distance problem is more difficult to solve since the function to be
minimized is continuous but non-differentiable over the entire range of both x and y
coordinates

34. Note:
• Generally, transformation of terms with absolute values
the n-facilities rectilinear distance location problem can
be solved using linear programming methods.
• The Euclidean distance problem is more difficult to
solve since the function to be minimized is continuous
but non-differentiable over the entire range of both x
and y coordinates

35.  Let us assume m facilities which are located at (a1,b1), (a2,b2)….(am, bm) in XY plane.
 (ai,bi) is the location ith existing facility on the XY plane.
 Now idea is locate the new emergency facility in the XY plane to serve the existing facilities.
 Hence objective function is to locate a new emergency facility at (x,y) such that the maximum distance
from new emergency facility to any of existing facilities is minimized.
 This type of problem is called as MINIMAX Location problem.
fi(x,y) = Distance between the new facility and existing facility i.
fi(x,y) = |x-ai|+|y-bi|
fmax(x,y) = max(1<=i<=m) {|x-ai|+|y-bi|}
The distance between the new facilities and the existing facilities may be rectilinear or Euclidean.

36.  Assume that there are m different shops existing in an industry.
 In the event of fire in any one of these shops, a costly fire fighting equipment should reach
the spot as soon as possible from its base location.
 Generally , movements within any industry are rectilinear in nature.
 If fire fighting equipment is not reaching in time in the event of fire in any of the shops, the
damage to the industry would be very heavy.
 So the objective is to locate the new fire fighting equipment within the industry such that
maximum distance it has to travel from its base location to any of the existing shops is
minimized.
 The methods of finding the optimal location is presented below.
 Step 1: Find C1, C2, C3, C4 and C5 using following formulas:

43.  Block Stacking
 Location allocation models

45.  Block stacking refers to unit loads stacked on top of each other and stored on the warehouse floor in lanes or blocks.
 Block stacking involves storage of unit loads in stacks within storage rows. It is frequently used when large
quantities of few products are to be stored and product is stackable to reasonable height without load crushing.
 Used in food, beverages, appliances, product products and others
 Important design question is how deep the storage rows must be ?
 Block stacking are used to achieve high space utilization at low cost. It is often designed with row depths of 15, 20,
30, or more.
 During storage and retrieval of lot, vacancies can occur in storage rack/row
 To achieve FIFO (First In First Out) lot rotation these vacant storage positions cant be used to store other
products/lots until all loads have been withdrawn from the row.
 The space loses resulting from unusable storage positions are referred to as “Honeycomb loss”.
 It is important to define the storage parameters (stack height, load width, load depth, aisle allowances, etc.)

47.  The pallets are stacked to a specific height based on some criteria such as pallet condition, the weight of
the load, height clearance and the capability of the warehouse forklifts.
 The pallets are retrieved from the block in a last in, first out (LIFO) manner.
 This does not allow for removing stock based on date basis or first in (FIFO).
 Removal of stock can cause honeycombing to occur where empty spaces occur that cannot be filled until
the whole lane is empty.
 This method is cheap to implement as it involves no racking and can be operated in any warehouse with
open floor space.
 Design of block stacking storage system is characterized by
i) depth of the storage row (x)
ii) No. of storage rows required for given product lot (y)
iii) Height of stack (z)
If height of the stack is fixed then the key decision variable is the depth of the storage row.

48. Lanes
Lane Depth
(3-deep)
Lane Height An efficient storage mode when
• there are multiple pallets per SKU;
• inventory is turned in large increments,
I.e., several loads of the same SKU are
received or withdrawn at one time.
Main problems:
• Loss of space due to “honeycombing”
• not effective utilization of the vertical
dimension of the facility

49. Advantages of floor stacking:
 Very low setup costs
 Flexible
Disadvantages of floor stacking:
 Low density storage (Requires a large storage facility to store only a small amount of stock).
 Poor ventilation of products
 Storage height depends on a number of variable factors
 Only one SKU can be effectively stored in a lane, empty pallet spaces are created that cannot be utilized
effectively until an entire lane is emptied.
 You have to move the top pallet to get to the pallet underneath (LIFO)

50.  Single product: factors that may influence the
optimum row depth include lot sizes, load
dimensions, aisles widths, row clearances,
allowable stacking heights, storage /retrieval
times and storage /retrieval distribution.
 Basic Block Stacking Model

51.  Another important concept of facilities location
problem is not just identifying where the new
facilities needs to located but also find which
customers (Existing facilities) will be served by
each of the new facilities.
 Location allocation problems also involves
determining optimum number of new facilities.

53.  The decision variable in the location allocation
problem are
 “n” i.e. no. of new facilities
 Z Allocation matrix
 Xj (j= 1,2, …n) the location of new facilities
Note: Constraints of equation ensure that each existing
facility interacts with only one NEW facility.

54.  Due to difficulty in solving the general problem,
the location allocation problem is solved by using
Enumeration procedure.
 i.e. for given value of n, the resulting problem is
solved and value of the objective function ψn
recorded, then optimum value of n is obtained
by searching for minimum value of ψn

55.  Simply enumerate all location combinations for
each allocation combination and specify
minimum cost solution.
 The no. of possible allocations given “m”
customer and “n” new facilities is given by

56.  Focus on determining the location of products for storage in
a warehouse.
 Three types of storage policies can be used to select storage
locations (or slots):
1. Dedicated (or Fixed Slot) Storage: a particular set of storage slots or
location is assigned to a specific product, hence number of slots equal
to maximum inventory level for the product must be provided

57. 2. Randomized (or Open Slot or Floating Slot) Storage:
Each unit of a particular product is equally likely to be retrieved when a
retrieval operation is performed, likewise each empty storage slot is
equally likely to be selected for storage when storage operation is done
3. Class-based Storage—a combination of dedicated and randomized
storage, where each Stock keeping unit (SKU) is assigned to one of several
different storage classes.

58. Where

66. Probability

67. 1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
https://www.youtube.com/watch?v=EJGo_6uu
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71.  Three products A,B, C are to be stored in the warehouse with only one type of
product stored in a given storage bay.
 Product A requires 3600ft2 of storage space and enters and leaves at the rate of
750 loads per month.
 Product B requires 6400ft2 of storage space and enters and leaves at the rate of
900 loads per month.
 Product C requires 4000ft2 of storage space and enters and leaves at the rate of
800 loads per month.
 > Rectilinear travel is used and is measured between the centriods of storage bays
 > Objective function = Minimize the total expected distance travelled.

76.  Among many analytical approaches used in design of facilities, the Waiting
line model analysis or Queueing theory is one.
 Waiting Lines analysis includes study of waiting lines.
 It may be accumulation of parts on conveyors at workstation, pallet loads of
material at receiving or shipping, in-process inventory accumulation,
customers at tool crib, customers at checkout station in grocery store, lift
truck waiting for maintenance etc….
 Wait time is affected by the design of the waiting line system.

77.  A waiting line system (or queuing system) is defined by two elements: the
population source of its customers and the process or service system itself.
 Reason for including waiting line model in FPD is to treat the waiting lines in
facilities of all types.
 Failure to consider congestion is the major reason for facilities plans to fail.
 Elements of waiting system are
1. Customer
2. Servers
3. Queue discipline
4. Service discipline

78. 1.Customer- trucks @receiving dock/pallets loads of material arriving at a m/c, employees near tool
crib
1.Servers: they are entities or combination of entities that provide service to customers.
Ex: S/R machines, order pickers, industrial trucks, cranes, elevators, m/c tools, computer terminals
3. Queue discipline: refer to behavior of customers in the waiting line as well as design of waiting line
Ex: customer waiting in queue, some may refuse to wait, single waiting line or separate waiting
for each server.
4. Service discipline: manner in which customers are served
Ex: customers served singly, serve more than one customer @ a time, FCFS, based on priority
based or random basis

116. Model A – Single Channel

132. Reference reading:
https://wps.prenhall.com/wps/media/objects/2234/2288589/ModD.pdf

133. JSS Academy of Technical Education
Bangalore 560060
JSS Academy of Technical Education
Bangalore 560060

134. NUMERICAL

141. Probability of N customers

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