Advanced Business Analytics
Prescriptive Analytics: Linear
Optimization
Advanced Business Analytics– Majid Karimi
Optimization
The pursuit of BEST
Optimization problems maximize or minimize some function (called an objective
function) of decision variables, often with a set of restrictions known as constraints.
Prescriptive Analytics: Linear Optimization 2 – 14
Advanced Business Analytics– Majid Karimi
Optimization Example 1
Manufacturing
A manufacturer wants to develop a production schedule and an inventory policy
that will satisfy demand in future periods. Ideally, the schedule and policy will
enable the company to satisfy demand and at the same time minimize the total
production and inventory costs.
Prescriptive Analytics: Linear Optimization 3 – 14
Advanced Business Analytics– Majid Karimi
Optimization Example 1
Manufacturing
A manufacturer wants to develop a production schedule and an inventory policy
that will satisfy demand in future periods. Ideally, the schedule and policy will
enable the company to satisfy demand and at the same time minimize the total
production and inventory costs.
What are the objective, decision variables, and constraints?
Prescriptive Analytics: Linear Optimization 3 – 14
Advanced Business Analytics– Majid Karimi
Optimization Example 2
Finance
A financial analyst must select an investment portfolio from a variety of stock and
bond investment alternatives. The analyst would like to establish the portfolio that
maximizes the return on investment.
Prescriptive Analytics: Linear Optimization 4 – 14
Advanced Business Analytics– Majid Karimi
Optimization Example 2
Finance
A financial analyst must select an investment portfolio from a variety of stock and
bond investment alternatives. The analyst would like to establish the portfolio that
maximizes the return on investment.
What are the objective, decision variables, and constraints?
Prescriptive Analytics: Linear Optimization 4 – 14
Advanced Business Analytics– Majid Karimi
Optimization Example 3
Marketing
A marketing manager wants to determine how best to allocate a fixed advertising
budget among alternative advertising media such as web, radio, television, newspaper, and magazine. The manager would like to determine the media mix that
maximizes advertising effectiveness.
Prescriptive Analytics: Linear Optimization 5 – 14
Advanced Business Analytics– Majid Karimi
Optimization Example 3
Marketing
A marketing manager wants to determine how best to allocate a fixed advertising
budget among alternative advertising media such as web, radio, television, newspaper, and magazine. The manager would like to determine the media mix that
maximizes advertising effectiveness.
What are the objective, decision variables, and constraints?
Prescriptive Analytics: Linear Optimization 5 – 14
Advanced Business Analytics– Majid Karimi
Optimization Example 4
Transportation
A company has warehouses in a number of locations. Given specific customer demands, the company would like to determine how much each warehouse should
ship to each customer so that total transportation costs are minimized.
Prescriptive Analytics: Linear Optimization 6 – 14
Advanced Business Analytics– Majid Karimi
Optimization Example 4
Transportation
A company has warehouses in a number of locations. Given specific customer demands, the company would like to determine how much each warehouse should
ship to each customer so that total transportation costs are minimized.
What are the objective, decision variables, and constraints?
Prescriptive Analytics: Linear Optimization 6 – 14
Advanced Business Analytics– Majid Karimi
Par Inc. Example — A Simple Maximization Problem
Par, Inc. is a small manufacturer of golf equipment and supplies. Its management has
decided to move into the market for medium- and high-priced golf bags. Par’s distributor
is enthusiastic about the new product line and has agreed to buy all the golf bags Par
produces over the next three months.
Management determined that each golf bag produced will require the following
operations:
1. Cutting and dyeing the material
2. Sewing
3. Finishing (inserting umbrella holder, club separators, etc.)
4. Inspection and packaging
Prescriptive Analytics: Linear Optimization 7 – 14
Advanced Business Analytics– Majid Karimi
Par Inc. Example (Continued) — Cost Parameters
The accounting department analyzed the production data, and arrived at prices for both
bags that will result in a profit contribution of $10 for every standard bag and $9 for every
deluxe bag produced.
Prescriptive Analytics: Linear Optimization 8 – 14
Advanced Business Analytics– Majid Karimi
Par Inc. Example (Continued) — Formulation
Steps to formulate a mathematical (optimization) model:
1. Understand the problem thoroughly
2. Describe the objective
3. Describe each constraint
4. Define the decision variables
5. Write the objective in terms of the decision variables
6. Write the constraints in terms of the decision variables
7. Formulate everything together
Prescriptive Analytics: Linear Optimization 9 – 14
Advanced Business Analytics– Majid Karimi
Par Inc. Example (Continued) — Formulation — Steps 2 & 3
Step 2 Objective: maximize the total contribution to profit
Step 3
Constraints 1: The number of hours of cutting and dyeing time used must be less
than or equal to the number of hours of cutting and dyeing time available.
Constraints 2: The number of hours of sewing time used must be less than or equal
to the number of hours of sewing time available.
Constraints 3: The number of hours of finishing time used must be less than or equal
to the number of hours of finishing time available.
Constraints 4: The number of hours of inspection and packaging time used must be
less than or equal to the number of hours of inspection and packaging time available.
Prescriptive Analytics: Linear Optimization 10 – 14
Advanced Business Analytics– Majid Karimi
Par Inc. Example (Continued) — Formulation — Step 4
Step 4 Defining Decision Variables:
S = number of standard bags
D = number of deluxe bags
Prescriptive Analytics: Linear Optimization 11 – 14
Advanced Business Analytics– Majid Karimi
Par Inc. Example (Continued) — Formulation — Steps 5 & 6
Step 5 Objective:
max 9S + 10D.
Step 6
Constraints 1:
7
10
S + 1D ≤ 630.
Constraints 2:
1
2
S +
5
6
D ≤ 600.
Constraints 3:
1S +
2
3
D ≤ 708.
Constraints 4:
1
10
S +
1
4
D ≤ 135.
Prescriptive Analytics: Linear Optimization 12 – 14
Advanced Business Analytics– Majid Karimi
Par Inc. Example (Continued) — Formulation — Steps 7
max 9S + 10D.
Subject to
7
10 S + 1D ≤ 630,
1
2
S +
5
6
D ≤ 600,
1S +
2
3
D ≤ 708,
1
10 S +
1
4
D ≤ 135,
Prescriptive Analytics: Linear Optimization 13 – 14
Advanced Business Analytics– Majid Karimi
Par Inc. Example (Continued) — Formulation — Steps 7
max 9S + 10D.
Subject to
7
10 S + 1D ≤ 630,
1
2
S +
5
6
D ≤ 600,
1S +
2
3
D ≤ 708,
1
10 S +
1
4
D ≤ 135,
S ≥ 0,
D ≥ 0.
Prescriptive Analytics: Linear Optimization 13 – 14
Advanced Business Analytics– Majid Karimi
Par Inc. Example (Continued) — Finding Solution via Excel Solver
What is the optimal production schedule?
Using a blank Excel file, create a spreadsheet model of the Par. Inc. example and
find the optimal number of standard and deluxe bags.
Prescriptive Analytics: Linear Optimization 14 – 14
