INDU 6121: Assignment 2 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.

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 ???? = {1, … , |????|} must be produced using processes ???? = {1, … , |????|}. The following parameters and variable are used by an operations research analyst to formulate the problem:

Parameters: ???????? : The market limit for product ????. ???????? : The per-unit profit for selling product ????. ???????? : The total available time for process ???? (in hour) ???????????? : The required time of process ???? for one unit of product ???? (in hour)

Variables: ???????? : The number of product ???? produced and sold in the market.

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

???????????? ????????????????? ?????????

(1)

????????????????????? ?????????

? ???????? ???? ? ???? (2)

0 ? ???????? ? ???????? ???? ? ???? (3) ???????? integer ???? ? ???? (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 ???? = {1, … , |????|} must be produced to cover the demands over a planning horizon ???? = {1, … , |????|} . The following parameters and variable are used by an operations research analyst to formulate the problem:

Parameters: ???????????? : The demand of product ???? in period ????. ???????????? : The per-unit production cost of product ???? in period ????. ???????????? : The setup cost of producing product ???? in period ????. ????????? : The per-unit holding cost of product ???? in period ????. ???????? : The initial inventory of product k at the beginning of planning horizon. ???? : The limited capacity of the warehouse (in terms of m³). ???????? : The required space for one unit of product ???? (in terms of m³).

Variables: ???????????? : The amount of product ???? to be produced in period ????. ???????????? : The inventory level of product ???? at the end of period ????. ???????????? : 1 if product k is produced in period t, 0 otherwise.

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

?????????????????????????????????????? ??????????????????

+ ?????????????????????????? ??????????????????

+ ??????????????????????? ??????????????????

(1)

????????(?????1) + ???????????? = ???????????? + ???????????? ???? ? ????,???? ? ???? (2)

????????0 = ???????? ???? ? ???? (3)

???????????? ? ???????????????????????? ???? ? ????,???? ? ???? (4) ????????????????????? ?????????

? ???? ???? ? ???? (5)

???????????? ? {0,1}

???? ? ????,???? ? ???? (6)

???????????? , ???????????? ? 0

???? ? ????,???? ? ???? (7)

In the above model, ???????????? is the big-M value (9999999999).

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

Question 2-Part a) Considering ???????????? = 9999999999, report the Solution time, the obtained objective value, and the obtained solution.

Question 2-Part b) Considering ???????????? = ? ???????????????????????:?????????? , report the Solution time, the obtained objective value, and the obtained solution. Note that ???????????? = ? ???????????????????????:?????????? is not a constraint in the model. In fact, here we are tuning the values of ???????????? 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 1.1????.

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 ???? = {1, … , |????|} must be produced in facilities ???? = {1, … , |????|} and shipped to customers ???? = {1, … , |????|}. The following parameters and variable are used by an operations research analyst to formulate the problem:

Parameters: ???????? : The fixed cost of opening a facility at location ???? ? ????. ???????????????? The per-unit transportation cost of product ???? from facility ???? to customer ????. ???????????? : The demand of product ???? by customer ????. ???????? : The production capacity of facility ????. ???????????? : Amount of production capacity usage for one unit of product k in facility ????.

Variables: ???????? : 1 if a facility is open at location ????, 0 otherwise. ???????????????? : The amount of product ???? shipped from facility ???? to customer ????.

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

??????????????????????????????????????????????? ???????????????????????????

+ ????????????????? ?????????

(1)

????????????????? ?????????

= ???????????? ???? ? ????,???? ? ????

(2)

?????????????????????????????? ??????????????????

? ???????? ???? ? ???? (3)

???????????????? ? ???????????????????????? ???? ? ????,???? ? ???? (4) 0 ? ???????????????? ? ???????????? ???? ? ????, ???? ? ????, ???? ? ???? (5) ???????? ? {0,1}

???? ? ????,???? ? ???? (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 ???? for product ???? 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, ???????????????? is the big-M value (9999999999).

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

Question 3-Part a) Considering ???????????????? = 9999999999, report the Solution time, the obtained objective value, and the obtained solution.

Question 3-Part b) Considering ???????????????? = min ( ???????? ????????????

,????????????), report the Solution time, the obtained objective

value, and the obtained solution. Note that ???????????????? = min ( ???????? ????????????

,????????????) is not a constraint in the model. In

fact, here we are tuning the values of ???????????????? 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: ???? : The set of fire stations. ???? : The set of communities.

Parameters: ???????? : The cost of opening fire station ????.

???? : The total available budget for opening fire stations. ???????????? : 1 if fire station ???? covers community ????.

Variables: ???????? : 1 if fire station ???? is opened; 0 otherwise. ???????? : 1 if community ???? is covered, 0 otherwise.

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

????????????????????? ?????????

(1)

????????????????? ?????????

? ???? (2)

???????? ?????????????????????? ?????????

???? ? ????

(3)

???????? ? {0,1} ???? ? ???? (4) ???????? ? {0,1} ???? ? ???? (5)

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

What are the optimal solution and the optimal objective value?