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INDU6121: Assignment 2

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Submission Deadline: Friday, November 20th.
In this assignment, you need to implement and solve the given models by IBM CPLEX Optimization
Studio (OPL). You need to consider a maximum time limit of 30 minutes in solving the given problems.
For each problem, some data are given in an Excel file. After solving the problem, you need to report
these details:

1- Solution time
2- The obtained objective value
3- The obtained solution

Note that for this assignment, you are not allowed to you the Excel Solver.

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You need submit the hard and soft copy including this information:

1- Hard/Soft copy:
a. OPL codes for each part of each question separately. On the top of each page, specify

the question and its corresponding part (a, b, c, d, etc).
b. The outputs (Solution time, The obtained objective value, The obtained solution)

For the soft copy, you need to submit the original OPL files in addition to the report file. Do not just copy
and paste the code in word and make it pdf. The marker will check your code. You can submit only one
.rar or .zip file including OPl codes and pdf of your report. The name of this file must
Your_Student_ID.rar or Your_Student_ID.zip.

Question 1- This question is about the Factory Planning problem that is explained in the
slides of the course. In the general case of this problem, products 𝐼𝐼 = {1, … , |𝐼𝐼|} must be produced using
processes 𝐽𝐽 = {1, … , |𝐽𝐽|}. The following parameters and variable are used by an operations research
analyst to formulate the problem:

Parameters:
𝑑𝑑𝑖𝑖 : The market limit for product 𝑖𝑖.
𝑝𝑝𝑖𝑖 : The per-unit profit for selling product 𝑖𝑖.
𝑏𝑏𝑗𝑗 : The total available time for process 𝑗𝑗 (in hour)
π‘Žπ‘Žπ‘–π‘–π‘—π‘— : The required time of process 𝑗𝑗 for one unit of product 𝑖𝑖 (in hour)

Variables:
π‘₯π‘₯𝑖𝑖 : The number of product 𝑖𝑖 produced and sold in the market.

Using this notation, the following model is proposed by the operations research analyst.

π‘šπ‘šπ‘Žπ‘Žπ‘₯π‘₯ �𝑝𝑝𝑖𝑖π‘₯π‘₯𝑖𝑖
π‘–π‘–βˆˆπΌπΌ

(1)

οΏ½π‘Žπ‘Žπ‘–π‘–π‘—π‘—π‘₯π‘₯𝑖𝑖
π‘–π‘–βˆˆπΌπΌ

≀ 𝑏𝑏𝑗𝑗 𝑗𝑗 ∈ 𝐽𝐽 (2)

0 ≀ π‘₯π‘₯𝑖𝑖 ≀ 𝑑𝑑𝑖𝑖 𝑖𝑖 ∈ 𝐼𝐼

(3)

π‘₯π‘₯𝑖𝑖 integer 𝑖𝑖 ∈ 𝐼𝐼 (4)

The data for this problem are given in Factory_Planning.xlsx.

Question 1-Part a) Report the Solution time, the obtained objective value, and the obtained solution.

Question 1-Part b) The business owner has realized that she has two options;

1) pay $10,000 to have 10% more available time for the available time of process 1.
2) Stick the available times for processes as in the previous part of the question and do not pay

anything.

Which option do you recommend to the business owner?

Question 2- This question is about the Production planning problem with setup cost that is
already discussed in the slides of the course. In the general case of this problem that we discussed in
Quiz 1, different types of products 𝐾𝐾 = {1, … , |𝐾𝐾|} must be produced to cover the demands over a
planning horizon 𝑇𝑇 = {1, … , |𝑇𝑇|} . The following parameters and variable are used by an operations
research analyst to formulate the problem:

Parameters:
π‘‘π‘‘π‘˜π‘˜π‘˜π‘˜ : The demand of product π‘˜π‘˜ in period 𝑑𝑑.
π‘π‘π‘˜π‘˜π‘˜π‘˜ : The per-unit production cost of product π‘˜π‘˜ in period 𝑑𝑑.
π‘žπ‘žπ‘˜π‘˜π‘˜π‘˜ : The setup cost of producing product π‘˜π‘˜ in period 𝑑𝑑.
β„Žπ‘˜π‘˜π‘˜π‘˜ : The per-unit holding cost of product π‘˜π‘˜ in period 𝑑𝑑.
π‘Žπ‘Žπ‘˜π‘˜ : The initial inventory of product k at the beginning of planning horizon.
𝐢𝐢 : The limited capacity of the warehouse (in terms of m³).
π‘’π‘’π‘˜π‘˜ : The required space for one unit of product π‘˜π‘˜ (in terms of mΒ³).

Variables:
π‘₯π‘₯π‘˜π‘˜π‘˜π‘˜ : The amount of product π‘˜π‘˜ to be produced in period 𝑑𝑑.
π‘ π‘ π‘˜π‘˜π‘˜π‘˜ : The inventory level of product π‘˜π‘˜ at the end of period 𝑑𝑑.
π‘¦π‘¦π‘˜π‘˜π‘˜π‘˜ : 1 if product k is produced in period t, 0 otherwise.

Using this notation, the following model is proposed by the operations research analyst.

π‘šπ‘šπ‘–π‘–π‘šπ‘šοΏ½οΏ½π‘π‘π‘˜π‘˜π‘˜π‘˜π‘₯π‘₯π‘˜π‘˜π‘˜π‘˜
π‘˜π‘˜βˆˆπ‘‡π‘‡π‘˜π‘˜βˆˆπΎπΎ

+ οΏ½οΏ½π‘žπ‘žπ‘˜π‘˜π‘˜π‘˜π‘¦π‘¦π‘˜π‘˜π‘˜π‘˜
π‘˜π‘˜βˆˆπ‘‡π‘‡π‘˜π‘˜βˆˆπΎπΎ

+ οΏ½οΏ½β„Žπ‘˜π‘˜π‘˜π‘˜π‘ π‘ π‘˜π‘˜π‘˜π‘˜
π‘˜π‘˜βˆˆπ‘‡π‘‡π‘˜π‘˜βˆˆπΎπΎ

(1)

π‘ π‘ π‘˜π‘˜(π‘˜π‘˜βˆ’1) + π‘₯π‘₯π‘˜π‘˜π‘˜π‘˜ = π‘‘π‘‘π‘˜π‘˜π‘˜π‘˜ + π‘ π‘ π‘˜π‘˜π‘˜π‘˜
𝑑𝑑 ∈ 𝑇𝑇,π‘˜π‘˜ ∈ 𝐾𝐾 (2)

π‘ π‘ π‘˜π‘˜0 = π‘Žπ‘Žπ‘˜π‘˜ π‘˜π‘˜ ∈ 𝐾𝐾 (3)

π‘₯π‘₯π‘˜π‘˜π‘˜π‘˜ ≀ π‘€π‘€π‘˜π‘˜π‘˜π‘˜π‘¦π‘¦π‘˜π‘˜π‘˜π‘˜ 𝑑𝑑 ∈ 𝑇𝑇,π‘˜π‘˜ ∈ 𝐾𝐾 (4)
οΏ½π‘’π‘’π‘˜π‘˜π‘ π‘ π‘˜π‘˜π‘˜π‘˜
π‘˜π‘˜βˆˆπΎπΎ

≀ 𝐢𝐢 𝑑𝑑 ∈ 𝑇𝑇 (5)

π‘¦π‘¦π‘˜π‘˜π‘˜π‘˜ ∈ {0,1}

𝑑𝑑 ∈ 𝑇𝑇,π‘˜π‘˜ ∈ 𝐾𝐾 (6)

π‘₯π‘₯π‘˜π‘˜π‘˜π‘˜ , π‘ π‘ π‘˜π‘˜π‘˜π‘˜ β‰₯ 0

𝑑𝑑 ∈ 𝑇𝑇,π‘˜π‘˜ ∈ 𝐾𝐾 (7)

In the above model, π‘€π‘€π‘˜π‘˜π‘˜π‘˜ is the big-M value (9999999999).

The data for this problem are given in Inventory_problem.xlsx.

Question 2-Part a) Considering π‘€π‘€π‘˜π‘˜π‘˜π‘˜ = 9999999999, report the Solution time, the obtained objective
value, and the obtained solution.

Question 2-Part b) Considering π‘€π‘€π‘˜π‘˜π‘˜π‘˜ = βˆ‘ π‘‘π‘‘π‘˜π‘˜π‘˜π‘˜β€²π‘˜π‘˜β€²βˆˆπ‘‡π‘‡:π‘˜π‘˜β€²β‰₯π‘˜π‘˜ , report the Solution time, the obtained objective
value, and the obtained solution. Note that π‘€π‘€π‘˜π‘˜π‘˜π‘˜ = βˆ‘ π‘‘π‘‘π‘˜π‘˜π‘˜π‘˜β€²π‘˜π‘˜β€²βˆˆπ‘‡π‘‡:π‘˜π‘˜β€²β‰₯π‘˜π‘˜ is not a constraint in the model. In
fact, here we are tuning the values of π‘€π‘€π‘˜π‘˜π‘˜π‘˜ with the hope that the model finds the solution faster.
Question 2-Part c) The business owner has realized that he has two options;

1) pay $100,000 to increase the capacity of the warehouse by 10%. So, if he chooses this
option the new capacity will be 1.1𝐢𝐢.

2) Does not pay this extra cost and stick the current warehouse and the obtained solution.

Which option do you recommend to the business owner?

Question 3- This question is about the Capacitated Facility Location Problem that is explained
in the slides of the course. In a general case of this problem, different types of products 𝐾𝐾 = {1, … , |𝐾𝐾|}
must be produced in facilities 𝐼𝐼 = {1, … , |𝐼𝐼|} and shipped to customers 𝐽𝐽 = {1, … , |𝐽𝐽|}. The following
parameters and variable are used by an operations research analyst to formulate the problem:

Parameters:
𝑓𝑓𝑖𝑖 : The fixed cost of opening a facility at location 𝑖𝑖 ∈ 𝐼𝐼.
π‘π‘π‘˜π‘˜π‘–π‘–π‘—π‘— The per-unit transportation cost of product π‘˜π‘˜ from facility 𝑖𝑖 to customer 𝑗𝑗.
π‘‘π‘‘π‘˜π‘˜π‘—π‘— : The demand of product π‘˜π‘˜ by customer 𝑗𝑗.
𝑒𝑒𝑖𝑖 : The production capacity of facility 𝑖𝑖.
π‘Žπ‘Žπ‘˜π‘˜π‘–π‘– : Amount of production capacity usage for one unit of product k in facility 𝑖𝑖.

Variables:
𝑧𝑧𝑖𝑖 : 1 if a facility is open at location 𝑖𝑖, 0 otherwise.
π‘₯π‘₯π‘˜π‘˜π‘–π‘–π‘—π‘— : The amount of product π‘˜π‘˜ shipped from facility 𝑖𝑖 to customer 𝑗𝑗.

Using this notation, the following model is proposed by the operations research analyst.

π‘šπ‘šπ‘–π‘–π‘šπ‘šοΏ½οΏ½οΏ½π‘π‘π‘–π‘–π‘—π‘—π‘˜π‘˜π‘₯π‘₯π‘˜π‘˜π‘–π‘–π‘—π‘—
π‘—π‘—βˆˆπ½π½π‘–π‘–βˆˆπΌπΌπ‘˜π‘˜βˆˆπΎπΎ

+ �𝑓𝑓𝑖𝑖𝑧𝑧𝑖𝑖
π‘–π‘–βˆˆπΌπΌ

(1)

οΏ½π‘₯π‘₯π‘˜π‘˜π‘–π‘–π‘—π‘—
π‘–π‘–βˆˆπΌπΌ

= π‘‘π‘‘π‘˜π‘˜π‘—π‘—
𝑗𝑗 ∈ 𝐽𝐽,π‘˜π‘˜ ∈ 𝐾𝐾

(2)

οΏ½οΏ½π‘Žπ‘Žπ‘˜π‘˜π‘–π‘–π‘₯π‘₯π‘˜π‘˜π‘–π‘–π‘—π‘—
π‘—π‘—βˆˆπ½π½π‘˜π‘˜βˆˆπΎπΎ

≀ 𝑒𝑒𝑖𝑖 𝑖𝑖 ∈ 𝐼𝐼 (3)

π‘₯π‘₯π‘˜π‘˜π‘–π‘–π‘—π‘— ≀ π‘€π‘€π‘˜π‘˜π‘–π‘–π‘—π‘—π‘§π‘§π‘–π‘– 𝑑𝑑 ∈ 𝑇𝑇,π‘˜π‘˜ ∈ 𝐾𝐾 (4)
0 ≀ π‘₯π‘₯π‘˜π‘˜π‘–π‘–π‘—π‘— ≀ π‘‘π‘‘π‘˜π‘˜π‘—π‘— π‘˜π‘˜ ∈ 𝐾𝐾, 𝑖𝑖 ∈ 𝐼𝐼, 𝑗𝑗 ∈ 𝐽𝐽 (5)
𝑧𝑧𝑖𝑖 ∈ {0,1}

𝑑𝑑 ∈ 𝑇𝑇,π‘˜π‘˜ ∈ 𝐾𝐾 (6)

In (1), the objective function minimizes the total transportation of products plus the opening cost of
facilities. Constraint (2) implies that demand of customer 𝑖𝑖 for product π‘˜π‘˜ must be satisfied by the
shipments from different facilities. Constraint (3) implies that the total production capacity in each
facility is limited. Constraint (4) ensures that we can have production in a facility if that facility is open.

In the above model, π‘€π‘€π‘˜π‘˜π‘–π‘–π‘—π‘— is the big-M value (9999999999).

The data for this problem are given in Extended_CFLP.xlsx.

Question 3-Part a) Considering π‘€π‘€π‘˜π‘˜π‘–π‘–π‘—π‘— = 9999999999, report the Solution time, the obtained objective
value, and the obtained solution.

Question 3-Part b) Considering π‘€π‘€π‘˜π‘˜π‘–π‘–π‘—π‘— = min (
𝑒𝑒𝑖𝑖
π‘Žπ‘Žπ‘˜π‘˜π‘–π‘–

,π‘‘π‘‘π‘˜π‘˜π‘—π‘—), report the Solution time, the obtained objective

value, and the obtained solution. Note that π‘€π‘€π‘˜π‘˜π‘–π‘–π‘—π‘— = min (
𝑒𝑒𝑖𝑖
π‘Žπ‘Žπ‘˜π‘˜π‘–π‘–

,π‘‘π‘‘π‘˜π‘˜π‘—π‘—) is not a constraint in the model. In

fact, here we are tuning the values of π‘€π‘€π‘˜π‘˜π‘–π‘–π‘—π‘— hoping that the model finds the solution faster.

Question 3-Part c) The business owner has realized that he has two options;

1) pay $300,000 to another company to satisfy the demands of all customers for product type
5. In this case, the business owner is not responsible to the transportation cost of that
product, but must still minimize the total transportation cost of other products and also the
opening of the facilities.

2) Does not pay this extra cost and stick the current plan and the obtained solution.

Which option do you recommend to the business owner?

Question 4- This question is about the Budgeted maximum coverage problem that is already
explained in the slides of the course. The following parameters and variable are used by an operations
research analyst to formulate the problem:

Sets:
𝐽𝐽 : The set of fire stations.
𝐼𝐼 : The set of communities.

Parameters:
𝑐𝑐𝑗𝑗 : The cost of opening fire station 𝑗𝑗.

𝐡𝐡 : The total available budget for opening fire stations.
π‘Žπ‘Žπ‘–π‘–π‘—π‘— : 1 if fire station 𝑗𝑗 covers community 𝑖𝑖.

Variables:
π‘₯π‘₯𝑗𝑗 : 1 if fire station 𝑗𝑗 is opened; 0 otherwise.
𝑦𝑦𝑖𝑖 : 1 if community 𝑖𝑖 is covered, 0 otherwise.

Using this notation, the following model is proposed by the operations research analyst.

π‘šπ‘šπ‘Žπ‘Žπ‘₯π‘₯�𝑦𝑦𝑖𝑖
π‘–π‘–βˆˆπΌπΌ

(1)

�𝑐𝑐𝑗𝑗π‘₯π‘₯𝑗𝑗
π‘—π‘—βˆˆπ½π½

≀ 𝐡𝐡
(2)

𝑦𝑦𝑖𝑖 β‰€οΏ½π‘Žπ‘Žπ‘–π‘–π‘—π‘—π‘₯π‘₯𝑗𝑗
π‘—π‘—βˆˆπ½π½

𝑖𝑖 ∈ 𝐼𝐼

(3)

π‘₯π‘₯𝑗𝑗 ∈ {0,1} 𝑗𝑗 ∈ 𝐽𝐽 (4)
𝑦𝑦𝑖𝑖 ∈ {0,1} 𝑖𝑖 ∈ 𝐼𝐼 (5)

The data for this problem are given in Budgeted_maximum_coverage_problem.xlsx.

What are the optimal solution and the optimal objective value?

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