UNIVERSITY OF ECONOMIC OF HO CHI MINH CITY
School of International Business and Marketing
ENTERPRISE RESOURCE PLANNING (ERP)
APPLIED TO LOGISTICS & SUPPLY CHAIN
MANAGEMENT
This textbook is for internal circulation only | 2020
1
COURSE INTRODUCTION
Learning Objectives
• Understand the overview of ERP
• Recognize common problems in SCM
• Understand theoretical supply chains and common types of
problems
• Know how to use Excel Solver and QM for Windows software
applications to solve transportation and inventory problems in
SCM
2
COURSE INTRODUCTION
WEEK
LEARNING OBJECTIVES AND TOPICS
Week 1
TRANSPORTATION PROBLEMS
• Supply = Demand (P&T Co. case study)
• Supply ≠ Demand (Metro Water, Better Products Co., Energetic)
• Combination cannot be used for distributing units (Case study Energetic)
Homework 1 (Texago)
Week 2
VARIATION OF TRANSPORTATION PROBLEMS
• Unstable Demand between Min and Max Range (Case Study Middletown)
• The objective is to maximize profit (Case Study Nifty Co.)
Homework 2 (Case Study Better Products Co.)
3
COURSE INTRODUCTION
Week 3 INVENTORY MANAGEMENT
• The Basic Economic Order Quantity (EOQ) Model (Case study ACT)
• The EOQ model with Planned Shortages (ACT Co.)
• The EOQ Model with Quantity Discounts (ACT Co.)
Homework 3 (EOQ model for Computer Center)
Week 4 VARIATION OF EOQ MODEL IN INVENTORY MANAGEMENT
• The EOQ Model with Gradual Replenishment (SOCA)
Homework 4 (EOQ model for Speedy Wheels)
Homework 5 (EOQ model for Color View)
4
Common problems in SCM
• Supply chain refers to processes that move information and
material to and from the manufacturing and service processes of
the firm. These include the logistics processes that physically move
product and the warehousing and the storage processes that
position products for quick delivery to the customers.
(Jacobs & Chase, 2013).
5
6
Transportation problems
• Transportation problems received this name because many of their
applications involve determining how to transport goods optimally (
Hillier, F., & Hillier, M., 2013).
• In mathematics and economics, transportation problems refer to
the study of optimal transportation and allocation of resources.
7
Inventory problems
• The inventory control problems are problems faced by a firm that
must decide how much to order in each time period to meet
demand for its products.
• Typical questions include:
oHow much to store/order?
oWhen to place order?
oSize of each order?
oHow to classify inventory?
8
Software applications for transportation and
inventory problems in SCM
• Managers usually search best solutions to optimize the enterprise’s
resources usage by implementing software applications in order to
integrate all business activities.
• Some of used software applications for transportation and
inventory problems in SCM are Excel Solver and QM for Windows.
• QM for Windows application assist managers to make decisions on
how to use resources best.
9
Software applications in this course
• Excel Solver is a Microsoft Excel add-in program you can use for
what-if analysis (Microsoft, 2020).
• POM-QM for Windows (also known as POM for Windows and QM
for Windows) is a Decision Science software package developed
by Prentice Hall (Howard J. Weiss, 2010).
Free download at:
• https://qm-for-windows.software.informer.com
• https://wps.prenhall.com/bp_weiss_software_1/1/358/91664.cw/ind
ex.html
10
SOFTWARE APPLICATIONS IN
TRANSPORTATION
Learning Objectives:
• Understand transportation problem characteristics
• Apply Solver and QM for Windows to solve transportation problems
• Understand the variation of transportation problems
11
Characteristics of Transportation Problems
• Transportation problems in general are concerned (literally or
figuratively) with distributing any commodity from any group of
supply centers, called sources, to any group of receiving centers,
called destinations, in such a way as to minimize the total
distribution cost (Hillier, F. & Hillier, M., 2013).
12
Characteristics of Transportation Problems
The Requirements Assumption
Each source has a fixed supply of units, where this entire
supply must be distributed to the destinations.
Each destination has a fixed demand for units, where this
entire demand must be received from the sources.
The Feasible Solutions Property
A transportation problem will have feasible solutions if and only
if the sum of its supplies equals the sum of its demands.
13
Characteristics of Transportation Problems
The Cost Assumption
• The cost of distributing units from any particular source to any
particular destination is directly proportional to the number of
units distributed.
• This cost is just the unit cost of distribution times the number of
units distributed.
14
Characteristics of Transportation Problems
Terminology for a General Model in Transportation Problem
• Units of a commodity
• Sources
• Destinations
• Supply from a source
• Demand at a destination
• Cost per unit distributed from a source to a destination
15
Characteristics of Transportation Problems
Variation of transportation problems
• Supply ≠ Demand
• Combination cannot be used for distributing units
• Unstable Demand between Min and Max Range
• The objective is to maximize profit
16
Transportation problems (P&T case study)
• The P&T Company is a small family-owned business. It receives
raw vegetables, processes and cans them at its canneries, and
then distributes the canned goods for eventual sale. One of the
company’s main products is canned peas.
• The peas are prepared at three canneries (near Bellingham,
Washington; Eugene, Oregon; and Albert Lea, Minnesota) and then
shipped by truck to four distributing warehouses in the western
United States (Sacramento, California; Salt Lake City, Utah; Rapid
City, South Dakota; and Albuquerque, New Mexico).
17
Transportation problems (P&T case study)
• The company’s current approach for many years, the company has used
the following strategy for determining how much output should be
shipped from each of the canneries to meet the needs of each of the
warehouses. Current shipping strategy are:
• Since the cannery in Bellingham is furthest from the warehouses, ship its output to
its nearest warehouse, namely, the one in Sacramento, with any surplus going to
the warehouse in Salt Lake City.
• Since the warehouse in Albuquerque is furthest from the canneries, have its
nearest cannery (the one in Albert Lea) ship its output to Albuquerque, with any
surplus going to the warehouse in Rapid City.
• Use the cannery in Eugene to supply the remaining needs of the
warehouses. For the upcoming harvest season, an estimate has been
made of the output from each cannery, and each warehouse has been
allocated a certain amount from the total supply of peas. This
information is given in Table 2.1.
18
P&T current approach
19
P&T CURRENT APPROACH
Table 2.1 – Shipping data for the P&T Co.
Cannery
Output
Warehouse
Allocation
Bellingham
75 truckloads
Sacramento
80 truckloads
Eugene
125 truckloads
Salt Lake City
65 truckloads
Albert Lea
100 truckloads
Rapid City
70 truckloads
Total
300 truckloads
Albuquerque
85 truckloads
Total
300 truckloads
From \ To
Current
Shipping
Plan
Cannery
Warehouse
Sacramento
Salt Lake City
Rapid City
Albuquerque
Bellingham
75
0
0
0
Eugene
5
65
55
0
Albert Lea
0
0
15
85
From \ To
Cannery
Warehouse
Sacramento
Salt Lake City
Rapid City
Albuquerque
Bellingham
$464
$513
$654
$867
Eugene
352
416
690
791
Albert Lea
995
682
388
685
Shipping
Cost per
Truckload
Total shipping cost = 75($464) + 5($352) + 65($416) + 55($690) + 15($388) + 85($685)
= $165,595
QUESTIONS
They now are reexamining the current shipping strategy to
see if P&T Co. can develop a new shipping plan that would
reduce the total shipping cost to an absolute minimum.
1. If you were the CEO of the P&T Co., what do you concern
in this case study?
2. How do you solve this problem?
22
P&T case study
The Requirements Assumption
• Each source has a fixed supply of units, where this entire supply must
be distributed to the destinations. Similarly, each destination has a fixed
demand for units, where this entire demand must be received from the
sources.
• This assumption that there is no leeway in the amounts to be sent or
received means that there needs to be a balance between the total supply
from all sources and the total demand at all destinations.
P&T case study
The Feasible Solutions Property
• A transportation problem will have feasible solutions if and only if
the sum of its supplies equals the sum of its demands.
P&T case study
The Model
• Any problem (whether involving transportation or not) fits the model
for a transportation problem if it
(a) can be described completely in terms of a table like Table
2.4 that identifies all the sources, destinations, supplies, demands,
and unit costs, and
(b) satisfies both the requirements assumption and the cost
assumption. The objective is to minimize the total cost of distributing
the units.
25
P&T case study
The Unit cost Data for the P&T Co. Problem Formulated as a
Transportation Problem
From \ To
Cannery
Warehouse
Sacramento
Salt Lake City
Rapid City
Albuquerque
Bellingham
$464
$513
$654
$867
Eugene
352
416
690
791
Albert Lea
995
682
388
685
The Network Representation of a Transportation
Problems
distributing a product from
several sources or origins to
several destinations
Figure 2.1 – Network Representation of a P&T Problem
27
P&T case study
• The Transportation Problem is a linear programming problem to
demonstrate that the P&T Co. problem (or any other transportation
problem) is, in fact, a linear programming problem, let us formulate
its mathematical model in algebraic form.
• Let xij be the number of truckloads to be shipped from Cannery i to
Warehouse j for each i = 1, 2, 3 and j = 1, 2, 3, 4.
• The objective is to choose the values of these 12 decision variables
(the xij) so as to
28
x11
Minimize cost = 464x11 + 513x12 + 654x13 + 867x14 + 352x21 + 416x22 + 690x23 + 791x24 +
995x31 + 682x32 + 388x33 + 685x34,
subject to the constraints
+ x12 + x13 + x14
=
75
x21
+ x22
+ x23
+ x24
x31
x11
+ x21
x12
+ x34
+ x32
+ x23
x14
+ x33
x31
+ x22
x13
+ x32
+ x33
+ x24
+ x34
=
125
=
100
=
80
=
65
=
70
=
85
and xij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4), xij is integer number
29
Applying Excel Solver to Formulate and Solve
Transportation Problems
• The decisions to be made are the number of truckloads of peas to
ship from each cannery to each warehouse.
• The constraints on these decisions are that the total amount
shipped from each cannery must equal its output (the supply) and
the total amount received at each warehouse must equal its
allocation (the demand).
• The overall measure of performance is the total shipping cost, so
the objective is to minimize this quantity.
30
31
Applying Excel Solver and QM for Windows for
the P&T Co. problem
Applying QM for Windows
• Practice directly in QM for Windows
Step 1: Open QM  Modules  Transportation
Step 2: Define all sources and destinations (Figure 2.3)
Step 3: Input data (Figure 2.4)
Step 4: Click ‘Solve’
32
Figure 2.3 – Create data set for the P&T Co. Transportation
problem
33
Figure 2.4 – Input data for the P&T Co. Transportation problem
34
VARIATION OF TRANSPORTATION PROBLEMS
Learning objectives:
• Understand variation of transportation problems
• Enable to solve the transportation problems in Solver and QM for
windows
Variation of transportation problems
• Supply ≠ Demand
• Combination cannot be used for distributing units
• Unstable Demand between Min and Max Range
• The objective is to maximize profit
35
VARIATION OF TRANSPORTATION PROBLEMS
Supply ≠ Demand (Metro Water)
• Metro Water District is an agency that administers water distribution
in a large geographic region. The region is fairly arid, so the district
must purchase and bring in water from outside the region.
• The sources of this imported water are the Colombo, Sacron, and
Calorie rivers. The district then resells the water to users in its
region. Its main customers are the water departments of the cities
of Berdoo, Los Devils, San Go, and Hollyglass.
36
Variant 1: Supply ≠ Demand
Metro Water
• It is possible to supply any of these cities with water brought in from any
of the three rivers, with the exception that no provision has been made
to supply Hollyglass with Calorie River water. However, because of the
geographic layouts of the aqueducts and the cities in the region, the cost
to the district of supplying water depends upon both the source of the
water and the city being supplied.
• The variable cost per acre foot of water for each combination of river and
city is given in Table 3.1 Using units of 1 million-acre feet, the bottom
row of the table shows the amount of water needed by each city in the
coming year (a total of 12.5). The rightmost column shows the amount
available from each river (a total of 16).
37
Variant 1: Supply ≠ Demand
Metro Water (cont.)
• Since the total amount available exceeds the total amount needed,
management wants to determine how much water to take from
each river, and then how much to send from each river to each city.
The objective is to minimize the total cost of meeting the needs of
the four cities.
38
Variant 1: Supply ≠ Demand
Metro Water (cont.)
• Table 3.1 – Water Resources Data for Metro Water District
Cost Per Acre Foot
To
Berdoo
Los Devils
San Go
Hollyglass
Available
Colombo
River
Sacron River
$160
$130
$220
170
5
$140
$130
190
150
6
Calorie River
$190
$200
230
-
5
2
5
4
1.5
(million acre feet)
From
Needed
39
Variant 1: Supply ≠ Demand
Job Shop - Assigning Machines to Locations
• The Job Shop Company has purchased three new machines of
different types. There are five available locations in the shop
where a machine could be installed. Some of these locations
are more desirable than others for particular machines because
of their proximity to work centers that will have a heavy
workflow to and from these machines. (There will be no
workflow between the new machines.) Therefore, the objective
is to assign the new machines to the available locations to
minimize the total cost of materials handling. The estimated
cost per hour of materials handling involving each of the
machines is given in Table 3.2 for the respective locations.
Location 2 is not considered suitable for machine 2, so no cost
is given for this case.
40
Variant 1: Supply ≠ Demand
Job Shop
• Table 3.2 – Materials-Handling cost data for the Job Shop Co.
problem
Cost per Hour
Location
Machine
1
2
3
4
5
Total
Assignments
1
$13
$16
$12
$14
$15
1
2
$15
-
$13
$20
$16
1
3
$4
$7
$10
$6
$7
1
Total assigned
1
0
1
1
0
41
Variant 2: Combination cannot be used for
distributing units
Energetic
• The Energetic Company needs to make plans for the energy systems for
a new building.
• The energy needs in the building fall into three categories: (1) electricity,
(2) heating water, and (3) heating space in the building. The daily
requirements for these three categories (all measured in the same units)
are 20 units, 10 units, and 30 units, respectively.
• The three possible sources of energy to meet these needs are electricity,
natural gas, and a solar heating unit that can be installed on the roof.
The size of the roof limits the largest possible solar heater to providing
30 units per day. However, there is no limit to the amount of electricity
and natural gas available.
42
Variant 2: Combination cannot be used for
distributing units
Energetic (cont.)
• Electricity needs can be met only by purchasing electricity. Both
other energy needs (water heating and space heating) can be met
by any of the three sources of energy or a combination thereof.
• The unit costs for meeting these energy needs from these sources
of energy are shown in Table 3.2 below. The objectives of
management are to minimize the total cost of meeting all the
energy needs.
43
Variant 2: Combination cannot be used for
distributing units
Energetic (cont.)
• Table 3.3 – Cost data for the Energetic Co. Problem
Unit Cost
Need
Source
Electricity
Natural Gas
Solar Heater
Electricity
$400
⁃
⁃
Water Heating
$500
$600
$300
Space heating
$600
$500
$400
44
Different variation of transportation problems
• Supply ≥ Demand (Case Study Metro Water, Better Products Co.)
• Supply ≤ Demand (Case Study Job Shop)
• Combination cannot be used for distributing units (Case study
Energetic)
• Unstable Demand between Min and Max Range (Case Study
Middletown)
• The objective is to maximize profit (Case Study Nifty Co.)
45
Variant 3: Unstable Demand between Min and
Max Range
Middletown
• The Middletown School District is opening a third high school
and thus needs to redraw the boundaries for the areas of the
city that will be assigned to the respective schools.
• For the preliminary planning, the city has been divided into nine
tracts with approximately equal populations. (Subsequent
detailed planning will divide the city further into over 100 smaller
tracts.)
46
Variant 3: Unstable Demand between Min and
Max Range
Middletown (cont.)
• The school district management has decided that the
appropriate objective in setting school attendance zone
boundaries is to minimize the average distance that students
must travel to school.
• The unit costs for meeting these energy needs from these
sources of energy are shown in Table 3.3. The objective of
management is to minimize the total cost of meeting all the
energy needs.
47
Variant 3: Unstable Demand between Min and
Max Range
Middletown (cont.)
• Table 3.3 – Data for the Middletown school district problem
Tract
1
2
3
4
5
6
7
8
9
Minimum enrollment
Maximum enrollment
1
2.2
1.4
0.5
1.2
0.9
1.1
2.7
1.8
1.5
1,200
1,800
Distance (Miles) to School
2
1.9
1.3
1.8
0.3
0.7
1.6
0.7
1.2
1.7
1,100
1,700
3
2.5
1.7
1.1
2.0
1.0
0.6
1.5
0.8
0.7
1,000
1,500
Number of High
School students
500
400
450
400
500
450
450
400
500
48
Variant 4: The objective is to maximize
profit
Nifty
• Read the case on LMS.
49
Variant 4: The objective is to maximize
profit
Nifty
• Table 3.4 – Data for the Nifty Co. Problem
Unit profit
Customer
Plant
1
2
3
4
Production
quantity
1
55
42
46
53
8,000
2
37
18
32
48
5,000
3
29
59
51
35
7,000
Minimum purchase
7,000
3,000
2,000
0
Requested purchase
7,000
9,000
6,000
8,000
50
Glossary of Transportation
• Demand at a destination: The number of units that need to be
received by this destination from the sources.
• Destinations: The receiving centers for a transportation problem.
• Network simplex method: A streamlined version of the simplex
method for solving distribution network problems, including
transportation and assignment problems, very efficiently.
• Sources: The supply centers for a transportation problem.
51
Glossary of Transportation
• Supply from a source: The number of units to be distributed from
this source to the destinations.
• Tasks: The jobs to be performed by the assignees when
formulating a problem as an assignment problem.
• Transportation simplex method: A streamlined version of the
simplex method for solving transportation problems very efficiently.
52
INVENTORY MANAGEMENT
• SOFTWARE APPLICATIONS IN INVENTORY MANAGEMENT
• Introduction to Inventory
• Cost Components of Inventory Models
• The Basic Economic Order Quantity (EOQ) Model
• Case study: The Atlantic coast tire corp. (ACT) problem
53
SOFTWARE APPLICATIONS IN INVENTORY
MANAGEMENT
Learning Objectives:
After studying this topic, you are being able to:
• Identify the cost components of inventory models
• Describe the basic economic order quantity (EOQ) model
• Use a square root formula to obtain the optimal order quantity for
this model
• Use Excel Solver and QM for windows software to solve different
inventory problems
54
Introduction to Inventory Management
Jacobs & Chase (2017)
55
Introduction to Inventory Management
• Inventories pervade the
business world, including
manufacturers,
wholesalers,
and
retailers.
• For ex.: The average
cost of inventory in the
United States is 30 to 35
percent of its value.
• The purpose of inventory
management
is
to
determine how much and
when to order
Jacobs & Chase (2017)
56
Introduction to Inventory Management
• Managers use scientific inventory management comprising the
following steps:
a. Formulate a mathematical model describing the behavior of
the inventory system.
b. Seek an optimal inventory policy with respect to this model.
c. Use a computerized information processing system to
maintain a record of the current inventory levels.
d. Using this record of current inventory levels, apply the
optimal inventory policy to signal when and how much to replenish
inventory.
57
Cost Components of Inventory Models
• Acquisition cost: The direct cost of replenishing inventory,
whether through purchasing or manufacturing of the product.
Notation: c = unit acquisition cost.
• Setup cost: The setup cost to initiate the replenishing of inventory,
whether through purchasing or manufacturing of the product.
Notation: K = setup cost.
58
Cost Components of Inventory Models
• Holding cost: The cost of holding units in inventory
Notation: h = annual holding cost per unit held = unit holding
cost.
• Shortage cost: The cost of having a shortage of units, i.e., of
needing units from inventory when there are none there.
Notation: p = annual shortage cost per unit short = unit
shortage cost.
59
Cost Components of Inventory Models
• Combining of these cost components:
Annual acquisition cost = c times number of units added to
inventory per year.
Annual setup cost = K times number of setups per year.
Annual holding cost = h times average number of units in inventory
throughout a year.
Annual shortage cost = p times average number of units short
throughout a year.
TC = total inventory cost per year = sum of the above four annual
costs.
TVC = total variable inventory cost per year = sum of the variable
annual costs.
60
The Basic Economic Order Quantity (EOQ) Model
Where the Model is applicable
• A constant demand rate.
• The order quantity to replenish inventory arrives all at once just
when desired.
• Planned shortages are not allowed.
Reorder point = (daily demand) x (lead time).
Daily demand =
𝐴𝑛𝑛𝑢𝑎𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 𝑟𝑎𝑡𝑒
𝑡𝑜𝑡𝑎𝑙 𝑤𝑜𝑟𝑘𝑖𝑛𝑔 𝑑𝑎𝑦𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟
61
The Basic Economic Order Quantity (EOQ) Model
• Annual demand rate (denoted as D): number of units being
withdrawn from inventory per year
• Lead time: The amount of time between the placement of an order
and its receipt
• Reorder point: inventory level when placing an order
62
The Basic Economic Order Quantity (EOQ) Model
63
The Basic Economic Order Quantity (EOQ) Model
The Objective of the Model
• Since the model assumes that the order arrives at the same
moment that the inventory level drops to 0, this delivery
immediately jumps the inventory level up from 0 to Q. With the
constant demand rate, the inventory level then gradually drops
down over time at this rate until the level reaches 0 again, at which
point the process is repeated. This saw-toothed pattern is depicted
in Figure 4.2. The pattern is the same as in Figure 4.1, where Q =
1,000, but now we want to choose the best value of Q.
64
The Basic Economic Order Quantity (EOQ) Model
65
The Basic Economic Order Quantity (EOQ) Model
• The specific objective in choosing Q is to
Minimize TVC = total variable inventory cost per year.
• TVC excludes the cost of the product, since this is a fixed cost. TVC also does
not include any shortage costs, since the model assumes that shortages never
occur. Therefore,
TVC = annual setup cost + annual holding cost,
where
Annual setup cost = K times number of setups per year,
Annual holding cost = h times average inventory level.
As described in the preceding section,
K = setup cost each time an order occurs,
h = unit holding cost.
66
The Basic Economic Order Quantity (EOQ) Model
• For any inventory system fitting the basic EOQ model, here are
some key formulas.
𝑎𝑛𝑛𝑢𝑎𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 𝑟𝑎𝑡𝑒
𝐷
Number of setups per year =
= .
𝑜𝑟𝑑𝑒𝑟 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦
𝑄
𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑙𝑒𝑣𝑒𝑙 + 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑙𝑒𝑣𝑒𝑙
Average inventory level =
2
𝑄 + 0 𝑄
=
=
2
2
Total variable cost (TVC) = annual setup cost + annual holding cost
=
𝐷
𝑄
𝐾 +ℎ
𝑄
2
67
The Basic Economic Order Quantity (EOQ) Model
• The value of Q which gives the minimum value on the TVC curve is
the optimal order quantity Q*, when annual holding cost is equal to
annual setup cost
Annual holding cost = Annual setup cost.
𝑄
𝐷
ℎ =𝐾
2
𝑄
68
The Basic Economic Order Quantity (EOQ) Model
• This yields the following formula for Q* (The Square Root Formula
for the Optimal Order Quantity)
𝑄∗
=
2𝐾𝐷
ℎ
Where
D = annual demand rate,
K = setup cost,
h = unit holding cost.
69
Case study: The Atlantic coast tire corp. (ACT)
problem
Figure 4.3 – The pattern of inventory levels over time for the
185/70 R13 Eversafe tire under ACT’s current inventory policy
70
Case study: The Atlantic coast tire corp. (ACT)
problem
Read the case study on LMS
Questions:
• When a wholesaler (like ACT) places an order for goods, what
can cause the cost to exceed the purchase price?
• What are cost components of ACT inventory model?
71
The optimal inventory policy for the basic EOQ
model of ACT
The Square Root Formula for the Optimal Order Quantity
Number of setups per year = annual demand rate / order quantity =
D/Q
Average inventory level = (maximum level + minimum level)/2
TVC (Total Variable Cost) = annual setup cost + annual holding cost
𝑄 ∗=
2𝐾𝐷
ℎ
Q*: the optimal order quantity
D = annual demand rate, K = setup cost,
h = unit holding cost.
72
The current inventory policy of ACT
Figure 4.4 – A spreadsheet formulation of the basic EOQ model for the ACT problem when
using the current order quantity of Q = 1,000
73
Applying Excel Solver to formulate and solve the
basic EOQ model
• Figure 4.5 – Excel Solver solution
74
Applying QM for Windows to formulate and solve
the basic EOQ model
• Step 1: Create data set for ACT (Figure 4.6)
• Step 2: Input data for ACT on QM (Figure 4.7)
• Step 3: Click ‘Solve’
75
Applying QM for Windows to formulate and solve
the basic EOQ model
Figure 4.6 – Data settings in QM for Windows
76
Applying QM for Windows to formulate and solve
the basic EOQ model
Figure 4.7 – Input data in QM for Windows
77
VARIATION OF EOQ MODEL IN INVENTORY
MANAGEMENT
VARIATION OF EOQ MODEL IN INVENTORY MANAGEMENT
• The EOQ model with Planned Shortages (ACT)
• The EOQ Model with Quantity Discounts (ACT)
• The EOQ Model with Gradual Replenishment (SOCA)
78
VARIATION OF EOQ MODEL IN INVENTORY
MANAGEMENT
Learning objectives
• Understand variation of inventory problems
• Enable to solve the inventory problems in Solver and QM for
windows
79
Variant 1: The EOQ model with Planned Shortages
This model is a variation of the basic EOQ model described in the
preceding two sections. The difference arises in the third of its key
assumptions (Planned shortages are allowed):
Assumptions
• A constant demand rate.
• The order quantity to replenish inventory arrives all at once just when
desired.
• Planned shortages are allowed. When a shortage occurs, the affected
customers will wait for the product to become available again. Their
backorders are filled immediately when the order quantity arrives to
replenish inventory.
80
Variant 1: The EOQ model with Planned Shortages
Assumptions
• A constant demand rate.
• The order quantity to replenish inventory arrives all at once just
when desired.
• Planned shortages are allowed. When a shortage occurs, the
affected customers will wait for the product to become available
again. Their backorders are filled immediately when the order
quantity arrives to replenish inventory.
81
Variant 1: The EOQ model with Planned Shortages
Figure 5.1 – The pattern of inventory levels over time assumed by the EOQ model with
planned shortages, where both the order quantity Q and the maximum shortage S are the
decision variables.
82
Variant 1: The EOQ model with Planned Shortages
The Objective of the Model
• This model has two decision variables — the order quantity Q
and the maximum shortage S. The objective in choosing Q and
S is to
Minimize TVC = total variable inventory cost per year.
• This TVC needs to include the same kinds of costs as for the
basic EOQ model plus the cost of incurring the shortages. Thus,
TVC = annual setup cost + annual holding cost + annual
shortage cost.
83
Variant 1: The EOQ model with Planned Shortages
As for the basic EOQ model,
Annual Setup cost
=𝐾
𝐷
𝑄
Annual holding cost = h times (average inventory level when positive)
times (fraction of time inventory level is positive)
=ℎ
𝑄−𝑆
2
𝑄−𝑆
𝑄
=
(𝑄−𝑆)2
ℎ
2𝑄
84
Variant 1: The EOQ model with Planned Shortages
• To obtain a similar expression for the shortage costs, recall that
p = annual shortage cost per unit short
where the symbol p is used to indicate that this is the penalty for
incurring the shortage of a unit. Since this unit shortage cost only is
incurred during the fraction of the year when a shortage is occurring,
85
Variant 1: The EOQ model with Planned Shortages
Since this unit shortage cost only is incurred during the fraction of
the year when a shortage is occurring,
Annual shortage cost = p times (average shortage level when a
shortage occurs) times (fraction of time shortage
is occurring)
=𝑝
𝑆
2
𝑆
𝑄
=
𝑆2
𝑝
2𝑄
Combining these expressions gives
2
2
 D  (Q  S)
S
TVC  K    h
p
 Q
2Q
2Q
86
Variant 1: The EOQ model with Planned Shortages
The optimal inventory policy
•
Calculus now can be used to find the values of Q and S
that minimize TVC. This leads to the following formulas for their
optimal values, Q* and S*.
Q* 
h  p 2KD
p
h
where
D = annual demand rate,
K = setup cost,
h = unit holding cost,
p = unit shortage cost
 h 
S*  
Q*

 h  p
87
Variant 1: The EOQ model with Planned Shortages
After some algebra, these two formulas also yield
Maximum inventory level = Q* – S*
=
𝑝
2𝐾𝐷
ℎ+𝑝
ℎ
88
Variant 1: The EOQ model with Planned Shortages
Since the first square root is less than 1 and the second
square root is the value of Q* when planned shortages are not
allowed, the maximum inventory level for this model always will
be less than for the basic EOQ model. This level can be
considerably less if h is fairly large compared to p. This is good,
since we want the inventory levels to come down when the unit
holding cost goes up. Having shortages, a significant fraction of
the time also helps to drive down the annual holding cost.
Therefore, this model does a good job of reducing the
annual holding cost well below that for the basic EOQ model
when h is fairly large compared to p. When p is considerably
larger than h instead, the trade-offs between the cost factors will
lead to an optimal inventory policy that is not much different than
for the basic EOQ model.
89
Variant 1: The EOQ model with Planned Shortages
Application to the ACT Case Study
Table 5.1 –Data of the ACT problem
D=
6000
(demand/year)
K=
$115
(setup cost)
h=
$4.20
(unit holding cost)
p=
$7.50
(unit shortage cost)
90
Variant 1: The EOQ model with Planned Shortages
Application to the ACT Case Study
Table 5.1 –Data of the ACT problem
D=
6000
(demand/year)
K=
$115
(setup cost)
h=
$4.20
(unit holding cost)
p=
$7.50
(unit shortage cost)
91
Variant 1: The EOQ model with Planned Shortages
Applying Excel Solver to formulate and solve ACT’s planned
shortage problem.
Figure 5.2 – The results obtained for the ACT problem by applying either of the
Excel templates (Solver version or analytical version) for the EOQ model with
planned shortages
92
Variant 1: The EOQ model with Planned Shortages
Applying QM for Windows to formulate and solve ACT’s planned
shortage problem.
Step 1: Data settings for ACT – EOQ model with planned
shortage (Figure 5.3)
Step 2: Input data in QM for ACT – EOQ model with planned
shortage (Figure 5.4)
Step 3: Click ‘Solve’
93
Variant 1: The EOQ model with Planned Shortages
Figure 5.3 – Data settings in QM for ACT problem with planned shortage
94
Variant 1: The EOQ model with Planned Shortages
Figure 5.4 – Input data in QM for Windows
95
Variant 2: The EOQ Model with Quantity
Discounts
• Commonly, suppliers always wish to increase their sales by offering
quantity discounts for large orders (Refer to the Table 5.3 in ACT
discount case as an example).
• The drawback of placing larger orders is that this increases the
average inventory level and thereby increases the holding cost.
Therefore, we need to do a careful cost analysis to determine
whether it is worthwhile to take advantage of these quantity
discounts.
96
Variant 2: The EOQ Model with Quantity
Discounts
Assumptions
• Annual acquisition cost becomes a variable cost.
• Holding cost varies upon purchasing price.
• TVC = annual acquisition cost + annual setup cost + annual holding
cost.
97
Variant 2: The EOQ Model with Quantity
Discounts
The Objective of the Model
•
For the basic EOQ model, the only components of the total
variable inventory cost per year (TVC) are the annual setup cost
and the annual holding cost, since the annual cost of purchasing
the product is a fixed cost. Now, with quantity discounts, this annual
acquisition cost becomes a variable cost.
TVC = annual acquisition cost + annual setup cost + annual holding
cost.
98
Variant 2: The EOQ Model with Quantity
Discounts
TVC = annual acquisition cost + annual setup cost + annual
holding cost.
where
c = unit acquisition cost (as given in Table 5.3)
D = annual demand rate
K = setup cost
Q = order quantity (the decision variable),
h = unit holding cost.
I = inventory holding cost rate
h = Ic
99
Variant 2: The EOQ Model with Quantity
Discounts
The optimal inventory policy
The decision variable of this model is the order quantity Q. The
objective in choosing a feasible Q is to get a minimum total variable
cost. This TVC needs to include the same kinds of costs as for the
basic EOQ model plus the annual acquisition cost.
Thus,
Minimize TVC = total variable inventory cost per year.
Minimize TVC = 𝑐𝐷 + 𝐾
𝐷
𝑄∗
+ℎ
𝑄∗
2
100
Variant 2: The EOQ Model with Quantity
Discounts
Application to the ACT Case Study
Discount
quantity
1
Order quantity
Discount
Unit cost
0 – 749
0
$20.00
2
750 – 1,999
1%
$19.80
3
2,000 or more
2%
$19.60
101
Variant 2: The EOQ Model with Quantity
Discounts
Cost Analysis
• Even though ACT will continue to purchase a fixed total of 6,000
tires of the 185/70 R13 size per year, the annual acquisition
cost now depends on the size of the individual order quantities.
Therefore, to adapt the basic EOQ model to incorporate
quantity discounts, the total variable cost is calculated as shown
in the following Figure 5.6.
• The decision-making process for Inventory policy in Excel.
102
Figure 5.6 – The application of the Excel template (analytical) for the
EOQ model with quantity discounts to the ACT problem
103
Variant 2: The EOQ Model with Quantity
Discounts
The decision-making process for Inventory policy in QM for
Windows.
Step 1: Data settings in QM for Windows for EOQ model
with quantity discounts (Figure 5.7)
Step 2: Input data in QM for Windows for EOQ model with
quantity discounts (Figure 5.8)
Step 3: Click ‘solve’
The illustration is given below:
104
Variant 2: The EOQ Model with Quantity
Discounts
Figure 5.7 – Data settings in QM for Windows for EOQ model with quantity discounts
105
Variant 2: The EOQ Model with Quantity
Discounts
Figure 5.8 – Input data in QM for Windows for EOQ model with quantity discounts
106
Variant 3: The EOQ Model with Gradual
Replenishment
• One of the assumptions of the basic EOQ model is that the order
quantity to replenish inventory arrives all at once just when desired.
Having the order delivered all at once is common for retailers or
wholesalers (such as ACT), or even for manufacturers receiving
raw materials from their vendors. However, the situation often is
different with manufacturers when they replenish their finishedgoods and intermediate-goods inventories internally by conducting
intermittent production runs. The EOQ model with gradual
replenishment is designed to fit this situation.
• This model assumes that the pattern of inventory levels over time is
the one shown in Figure 5.10.
107
Variant 3: The EOQ Model with Gradual
Replenishment
Figure 5.10 – The pattern of inventory levels over time — rising during a production run and
dropping afterward — for the EOQ model with gradual replenishment
108
Variant 3: The EOQ Model with Gradual
Replenishment
Assumptions
• A constant demand rate.
• A production run is scheduled to begin each time the inventory
level drops to 0, and this production replenishes inventory at a
constant rate throughout the duration of the run.
• Planned shortages are not allowed.
109
Variant 3: The EOQ Model with Gradual
Replenishment
The objective of the model
•
The decision variable of this model is the production lot size Q. The
objective is choosing Q to
Minimize TVC = total variable inventory cost per year.
•
This TVC needs to include the same kinds of costs as for the basic
EOQ model
TVC = annual setup cost + annual holding cost
•
As for the basic EOQ model,
𝐷
Annual Setup cost = 𝐾
𝑄
Annual holding cost = h (average inventory level)
1
Average inventory level = (maximum inventory level)
2
Maximum inventory level = production lot size – demand during
production run
110
Variant 3: The EOQ Model with Gradual
Replenishment
The objective of the model
•
The goal is to find the value of Q as the optimal production lot size that
gives the overall minimum cost. This requires the annual setup cost equals to
the annual holding cost.
Minimize TVC = annual setup cost + annual holding cost
𝐷
𝑄∗
𝐷
=𝐾
+ℎ
(1 − )
𝑄∗
2
𝑅
• Where
D
=
annual
demand
rate
R
=
annual
production
rate
if
producing
continuously
K
=
setup
cost,
h = unit holding cost.
R = (daily production rate) (number of working days per year)
111
Variant 3: The EOQ Model with Gradual
Replenishment
The optimal inventory policy
•
The new square root formula is derived in the same way as
described for the basic EOQ model. The only reason the new
formula differs from the one for the basic EOQ model is that the
annual holding cost for the basic EOQ model now is being
multiplied by the factor, (1 - D/R). The reason for this factor is that
the maximum inventory level has changed from Q to
Maximum inventory level = production lot size – demand during
production run
112
Variant 3: The EOQ Model with Gradual
Replenishment
The optimal inventory policy
Maximum inventory level = production lot size – demand during
production run
=𝑄−
𝐷
𝑄
𝑅
= (1 −
𝐷
)𝑄
𝑅
The optimal production lot size can be obtained directly from a
square root formula that is similar to the one for the basic EOQ
model. The new formula is
2𝐾𝐷
𝑄∗=
𝐷
ℎ(1 − )
𝑅
113
Variant 3: The EOQ Model with Gradual
Replenishment
The optimal inventory policy
• The corresponding total variable inventory cost per year is
calculated from the following formula:
TVC = annual setup cost + annual holding cost
𝐷
𝑄∗
𝐷
=𝐾
+ℎ
1−
𝑄∗
2
𝑅
114
Variant 3: The EOQ Model with Gradual
Replenishment
The application to the SOCA case study
•
Read the case on LMS
Applying Excel Solver to formulate and solve SOCA’s gradual
replenishment problem (Figure 5.11).
115
Variant 3: The EOQ Model with Gradual
Replenishment
Figure 5.11 – The results obtained for the SOCA problem by applying the
Excel Solver for the EOQ model with gradual replenishment
116
Variant 3: The EOQ Model with Gradual
Replenishment
Applying QM for Windows to formulate and solve SOCA’s gradual
replenishment problem.
• Step 1: Data settings for SOCA – EOQ model with gradual
replenishment (Figure 5.12)
• Step 2: Input data for SOCA – EOQ model with gradual
replenishment (Figure 5.13)
• Step 3: Click ‘Solve’
The illustration is displayed below:
117
Variant 3: The EOQ Model with Gradual
Replenishment
Figure 5.12 – Data settings for SOCA – EOQ model with gradual replenishment
118
Variant 3: The EOQ Model with Gradual
Replenishment
Figure 5.13 – Input data for SOCA – EOQ model with gradual replenishment
119
Glossary of Inventory management
• Acquisition cost: The direct cost of acquiring units of a product, either
through purchasing or manufacturing, to replenish inventory.
• Backorder: An order that cannot be filled currently because the
inventory is depleted, but will be filled later when the inventory is
replenished.
• Constant demand rate: A fixed rate at which units need to be withdrawn
from inventory.
• Continuous-review system: An inventory system whose current
inventory level is monitored on a continuous basis.
• Cost of capital tied up in inventory: The rate of return from capital that
is foregone because that capital has been invested in the materials
being held in inventory.
120
Glossary of Inventory management
• Demand: The number of units of a product that will need to be
withdrawn from inventory during a specific period.
• Dependent demand: Demand for a product that is dependent upon the
demand for another product, generally because the former product is a
component of the latter product.
• Fixed cost: A cost that remains the same regardless of the decisions
made.
• Holding cost: The cost associated with holding units of a product in
inventory.
• Independent demand: Demand for a product that is independent of the
demand for all products.
121
Glossary of Inventory management
• Inventory system: the set of policies and controls that monitor
levels of inventory.
• Inventory: Goods being stored for future use or sale.
• Inventory policy: A rule that specifies when to replenish inventory
and by how much.
• Just-in-time (JIT) inventory system: A system that places great
emphasis on reducing inventory levels to a bare minimum, as well
as eliminating other forms of waste in the production process.
• Lead time: The amount of time between the placement of an order
and the delivery of the order quantity.
122
Glossary of Inventory management
• Material requirements planning (MRP): A computer-based system for
planning, scheduling, and controlling the production of all the
components of a final product.
• Manufacturing inventory: refers to items that contribute to or become
part of a firm’s product.
• Opportunity cost: When capital is used in a certain way, its opportunity
cost is the lost return because alternate opportunities for using this
capital must be foregone.
• Order quantity: The number of units of a product being acquired, either
through purchasing or manufacturing, to replenish inventory.
• Periodic-review system: An inventory system whose inventory level is
only checked periodically.
123
Glossary of Inventory management
• Production lot size: The number of units of a product being
produced during a production run.
• Quantity discounts: Reductions in the unit acquisition cost of a
product that are offered for ordering a relatively large quantity.
• Reorder point: The inventory level at which an order is placed.
• Safety stock: Extra inventory being carried to safeguard against
delivery delays.
• Scientific inventory management: A management science
approach to inventory management that involves using a
mathematical model to seek and implement an optimal inventory
policy.
124
Glossary of Inventory management
• Setup cost: The fixed cost associated with initiating the
replenishment of inventory, whether the administrative cost of
purchasing the product or the cost of setting up a production run to
manufacture the product.
• Shortage cost: The cost incurred when there is a need to withdraw
units from inventory and there are none available.
• Square root formula: The formula for calculating the optimal order
quantity for the basic EOQ model.
• Variable cost: A cost that is affected by the decisions made.
125