In this document all the knowledge clips and extra information given in the lectures are summarized for the course Decision making in Supply Chains. The examples are also taken into account.
Lecture 1 Introduction 10-1-2022
linear programming and integer programming.
Quantitative methods are useful to make decisions in companies. You can develop methods and
models on costs or demand. Business problems tend to be very complex. To improve and analyse
them, quantitative methods are required.
Example logistics network
There is a certain supply chain. The
distribution network needs to deliver
products to the customers. You want to
ensure that the distribution network is used
in an ideal way.
Example package delivery
Given the customer demand and depot, you need to figure out
the best route. Many companies are solving these problems on a
daily basis, such as DHL, PostNL and picnic. The changing order
can be of influence on the route in a Picnic order.
The models can be extended with more variables and time
constraints.
the basic point is that when you have a situation where you have
scarce resources, you need to make decisions. And the models we
are going to analyse will contribute to the decision making.
Major course topic:
- Develop models ((integer) linear programs) for decision making problems
- Spreadsheet-based optimization models
- Report and communication model assumptions, findings and recommendations.
The modelling process
It is important to ask the right questions to be able to identify the problem. You also need to discuss
whether the company is really willing to implement the new model. Capturing the right data is also
important. The data and problem should be aligned, otherwise you cannot make an improved model.
Steps of the modelling process
The modelling process can be characterized as a seven-step process:
1. Problem definition
2. Data collection
3. Model development
4. Model verification
5. Optimization and decision making
6. Model communication to management
7. Model implementation
It is not always the case that the steps are followed in the detailed order. It could be the case that
some steps have to be repeated. These are represented in the feedback loops.
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,Step 1: Problem definition
• A management science model is initiated when an organization believes it has a problem, e.g.,
losing money, market share declining, etc -> something has to be done better and/ or improved
• The analyst investigates and properly defines the organization’s problem; it is crucial to identify the
real problem
• Defining the problem includes specifying the organization’s objectives, problem’s restrictions, and
the parts of the organization to study before the problem can be solved
Step 2: Data collection
• The analyst collects data to estimate the value of parameters that affect the organization’s problem
• The goal is to have the right data in a consistent format
• A key issue is that data is often not in the form the analyst requires and is stored in different places
and formats
• These estimates are the input of a mathematical model (Step 3)
Step 3: Model development
• The analyst uses quantitative skills to develop a model of the problem, e.g., analytical,
optimization, or simulation models
• The model should represent the real problems accurately, but it should also be as simple as
possible: a balance between accuracy and complexity is crucial
• We will study many mathematical models for a variety of case studies
• The main tools we will use are Linear Programming (LP) and Integer Linear Programming (ILP)
Step 4: Model verification
• The analyst tries to determine whether the model developed in the previous step is an accurate
representation of reality
• A first step in determining how well the model fits reality is to check whether the model is valid for
the current situation
• Ideally, the model should be verified on as many test cases (instances) as possible, also on corner
cases
• A proper verification of the model validates its correctness and increases the chance that the
organization will accept recommendations
• If the model’s outputs are not as expected, the model may need a refinement (feedback loop with
previous phases)
Step 5: Optimization and decision making
• Given a model and a set of possible decisions, the analyst must now choose the decision that best
meets the organization’s objectives within the problem’s restrictions
• This often requires the optimization of an objective (i.e., maximizing profit or minimizing cost)
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,• The optimization phase is the most difficult phase from a mathematical standpoint; commercial
software packages can often be used to solve the mathematical model
• There are two main classes of solution methods: exact methods (provide an optimal solution, but
may take long computing times) and heuristic methods (may not be able to provide an optimal
solution, but take short computing times)
Step 6: Model communication to management
• The analyst presents the model and the recommendations to the organization
• The analyst might present several alternatives and let the organization choose the best one
• The gap between analysts and managers represents a major challenge; it is up to the analyst to
present the model and its outcomes in a clear and convincing way
• A successful communication often requires to involve key people in the organization from the
beginning; Step 6 should occur throughout the modelling process, not just toward the end
• The model should be as intuitive and user-friendly as possible; the ability to ask what-if questions
often helps; the support of graphics, customized menus, toolbars, and data tables is vital
Step 7: Model implementation
• If the organization has accepted the validity of the study, the analyst helps implement its
recommendations
• The implemented system must be monitored and updated constantly to ensure that the model
enables the organization to meet its objectives
• People who will run the model every day must be trained at, e.g., understanding how to enter
inputs, run what-if analyses, and interpret model’s output
• Organizations will likely expand the model after the initial implementation, so models should be
designed keeping an eye also on future needs and developments
Remarks on the seven-step process
• The process does not always go smoothly and its logical chronological order is not always followed
• Even in successful applications, the analyst might considerably backtrack throughout the process
• Many models are never implemented even though they are technically perfectly correct
• The most frequent cause of failure is miscommunication, e.g., people in the organization do not
understand how the model, or the model suggests actions that top managers do not want to take
The model is a beginning, not an end
• This course places emphasis on developing (I)LP models, i.e., on Step 3
• We also address aspects of communication to management, i.e., Step 6
• Nevertheless, the complete model is not the end of the process: it is a starting point
• The model is a tool for gaining insights, e.g., by answering what-if questions and by conducting
sensitivity analysis.
3
, Knowledge clips Chapter 1
A linear program is a mathematical model that optimizes (minimizes or maximizes) a linear objective
function over a set of continuous decision variables subject to linear equality and inequality
constraints.
So there are 4 decision variables, objective function is minimize Z and
the constraints are the other functions stated.
A minimization problem can be converted to a maximization problem
(and vice versa) by multiplying the objective function with -1
Key assumptions
1. Proportionality. The contribution to the objective function and the amount of resources used in
each constraint are proportional to the value of each decision variable. If I need 3 hours for product A
and 5 hours for product B, the total time is 3 * number of product A + 5* number of product B
2. Additivity. The value of the objective function and the total resources used is the sum of the
objective function contribution and the resources used for all decision variables. We add the values
to get the total value of the objective function and the constraints.
3. Divisibility. The decision variables are continuous. It can only model situations that can be in
fractions of units. For example products, but staff planning would be hard, since you cannot have 3/5
person.
4. Certainty. All parameters are known.
How to derive LP formulations? Example
The Puck and Pawn Company produces hockey sticks and chess sets. Each hockey stick yields a profit
of $2, and each chess set of $4. A hockey stick requires 4 hours of processing at machine centre A
and 2 hours at machine centre B. A chess set requires 6 hours at machine centre A, 6 hours at
machine centre B, and 1 hour at machine centre C. Machine centre A has a maximum of 120 hours of
available capacity per day, machine centre B of 72 hours, and machine centre C of 10 hours.
The first thing to do is to write down the model properly. So first state the decision variables. in this
case these are the hockey sticks and the chess sets. Automatically you already get 2 constraints: both
should be at least 0. So:
• H ≥ 0 : number of produced hockey sticks
• C ≥ 0 : number of produced chess sets
Then we need to make an objective. We can assume that we want to maximize our
profit. So we get: Z= 2H + 4C, because the profit of H is 2 per product and 4 for C.
Then we get the constraints. We have 1 constraint for each machine center. For
centre A: 4 hours*H + 6*C and we have 120 available. The same is done for B and C.
so 2*H+ 6*C < 72. And for C we get 1*C < 10. We do not need to forget the non-
negative constraints.
How to derive LP formulations
1. Understand the problem
2. Describe the objective in own words
3. Describe each constraints in own words
4. Define and describe the decision variables
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