END395 Operations Research II Term Project Spring 2023 Part I Deadline: March, 17, 2023 - Friday 18:30 In this project, you will work on a production scheduling problem of a chemical company that produces batches of chemical products on demand. Equipment1, Equipment2, and Equipment3 are the three primary types of equipment used in the production facility. There are currently 17 equipment, where 8 of them are of type Equipment1, 5 of them are of type Equipment2, and the remaining 4 are of type Equipment3. Demand from customers arrives in the form of orders. Orders consist of batches of one, two, or three custom chemical products, and they must be supplied at the exact time of order due date. For each product in an order, a sort of special pilot plant consisting of specific quantities of these equipment should be constructed. Some products may need to be processed by more than one of these pilot plants. In such cases, one or more batches of intermediate products are produced for further processing. Typically, the equipment on hand is not sufficient to meet the current demand. The production scheduling department of the company prepares the production schedule in a weekly manner. Once a new order is received, the industrial engineer in charge of production scheduling prepares a precedence tree for each product of the corresponding order. Through the precedence tree, each product of the order is divided into a number of jobs, where each job requires the use of a specific pilot plant (that includes specific quantities of equipment) for a specific duration. As an example, order O197886 includes two custom chemical products X1A31 and X7B71. X1A31 requires an intermediate stage and should be broken into three jobs and X7B71 should be broken into two jobs. Precedence trees of corresponding chemical products are given in the Figure 1. The equipment requirements, duration, and precedence relations for each job of each product of order O197886 are provided in Table 1. As the table indicates, for job1 of X1A31, a pilot plant that connects 2 Equipment1 and 2 Equipment3 should be established and this pilot plant should be used for four weeks. Figure 1: Precedence chart of the sample order O197886 1 Table 1: Detailed information for the jobs of order O197886 Currently, the production scheduling department has to schedule four orders, each of which contains a single specific chemical product. The precedence tree of the products of these orders are constructed and the detailed information for the jobs of orders is prepared as provided in Table 2. Each order is placed at the beginning of its arrival week and is due to the beginning of the week specified in the Week due column. As an example, order O3951 arrives at the beginning of the first week of the current plan and all of its jobs should be completed by the beginning of week 9. The last row of the table provides the total quantities of equipment available in each week for these orders. Table 2: Current orders of the company Part I Formulate the problem as a mixed integer problem (MIP) where the objective is to minimize the total throughput time of orders. Note that, throughput time for an order is the elapsed time between order arrival and order completion, where order completion occurs when all jobs of an order are completed. In this part, an order can be delivered late. The project report should provide a clear description of the decision variables, 2 the objective function, and constraints After constructing the main model, analyze the alternative objective functions. Alternative objectives are given as: Minimizing makespan Minimizing total lateness Minimizing maximum lateness Describe how the mathematical model you constructed should be updated for each alternative objective and solve the corresponding model for the same instance with each of these alternative objectives. Compare the optimal solution and the run time for each objective you analysed. Time schedule and submission requirements: A small problem instance: The data for a small problem instance will be provided on February 27th, 2023, Monday for you to use for your model validation. Real problem instances: The data for a set of real problem instances will be provided on March 6 th, 2023, Monday. In the project report, for each objective, provide the following results of each instance in a Results table the objective function value of the best found solution, the solution time (CPU time) of your model the percentage gap attained at the end of the run and write a discussion on the difficulty of the instances and difficulty of the objectives based on the results in your Results table for each objective. For the most difficult model, adjust the settings file as described in each of the following ways separately and report the run results with a discussion on the effect of each of the following changes in the settings file. In Mixed Integer Programming tab of the settings file open the Cuts (Kesiler) part and remove all cuts and solve the model. Open Mixed Integer ProgrammingStrategy part and change branching direction to deepest first and solve the model. Also change to uppest first and solve again. In Mathematical Programming tab, open Emphasis part. Change it to emphasize optimality over feasibility (olurluluktan ok eniyilii vurgula) and solve. Also change it to emphasize integer feasibility over optimality (eniyilikten ok tamsay olurluluu vurgula) and solve. Deadline: March 17th 2023 Friday 18:30 (No extension and late submissions are not accepted) Project report should be prepared in English and should include all the necessary information specified above. Report format will be announced later and you should prepare your reports in the correct format to get full credit. You must submit all your CPLEX project files (including data sets you used) in a described format. The format will be announced and CPLEX files that are not sent with the described format will not be evaluated.

Variant 4: Equipment Substitutability In the first part of the project, it was not possible to use an equipment different than the specified type for a job. In this variant, it may be possible to use alternative equipment to accomplish some jobs. For example, the resource requirements of Job1 of product X1A31 of order O1977886 are four weeks use of three Equipment1 and one Equipment3 (see the first data row of Table 2.3). In this variant, due to equipment substitutability, the four weeks use of three Equipment1 and one Equipment3 (see the second row of Table 2.3) can be substituted for the five weeks use of one Equipment1 and two Equipment2 and one Equipment3 , Job3 of product X1A31 and Job2 of product X7B71 of order O1977886 have no substitution alternative. The objective is to minimize the total throughput time of orders. Table 2.3: Sample order with equipment substitutability

Variant 5: Equipment Rental In the first part of the project, the company did not have the option of weekly rental of equipment. In this variant, the orders have strict deadlines (i.e., an order cannot be delivered late) and the company has the option of renting equipment in order to meet the deadlines. The rental rate for 2 equipment i is ri per week and we assume that the equipment rental company has unlimited source of equipment for each week. The company wants to determine the number of equipment to be rented for each week with the objective of minimizing the total equipment rental cost.