Question 1) (20 points) Consider a pottery workshop that produces bowls, mugs, and plates. The number of labor hours and amount of clay needed to produce a unit of these products is presented in table below:

Optimization Assignment (100 points) This is a group assignment. Write the answers to each question in this word file. For the mathematical models either use Word Equations, or write down the formulas on a paper, take a photo, and copy the photo in this word file. When submitting the assignment, you MUST upload the word file and supporting Excel werkbooks that are named by each question. (You can alternatively submit only one Excel file in which each question is solved in a different worksheet.) Only one member of the group should submit the assignment. Question 1) (20 points) Consider a pottery workshop that produces bowls, mugs, and plates. The number of labor hours and amount of clay needed to produce a unit of these products is presented in table below: Cunit. Lab Clay Plane 3 2 Mag There are 80 hours of labor and 240 pounds of clay available on each day The revenue generated by selling a unit of plate, bowl, and mug is $35, $40 and $50, respectively 1- Write a linear optimization mathematical model to determine the number of plates, bowls, and mugs that needs to be produced in order to maximize the total revenue. The model must consider the restrictions of total labor hours and clay available during a day. Clearly define decision variables, objective function, and constraints for the optimization model 2- Use Excel to solve the optimization model. How many of cach product should be produced to maximize the total revenue? Question 2) (25 points) Consider a box with a limited weight of 50 and a limited volume of 8. We want to select a subset of the following items and put them into the box to maximize the total value. We cannot violate the weight and volume capacity of the box ItemA Watch Vamd Vale CIC G 1. Write a mathematical optimization model for this problem. Clearly define decision variables, objective function, and constraints for the optimization model. 2- Use Excel to solve the optimization model. Which items should be selected? What is the optimal value of the objective function? 3- (Bonus question) (5 points) Can you design a simple algorithm that simultaneously incorporates weights, volumes, and values of the items to solve this problem? volumes, and values of the items to solve this problem? Question 3) (25 points) Consider a machine in a manufacturing center that can process multiple types of products. During a day. the machine is available for 8 hours. We have a set of potential items that can be processed on this machine and after the process, cach item can be sold to a customer. Assume that we only can sell one copy of each item. Table below shows the processing times and the potential profit of each item. ADGDLOG Presentes del Ram 15 SEPTES 1- Which items the manufacturer should select to process and sell to the customer in order to maximize his profit? Write a mathematical optimization model for this problem Clearly define decision variables, objective function, and constraints for the optimization model. Use Excel to solve the optimization model 2- How long (in minutes) the machine remains idle during a day? 3. What is the solution of the problem if we assume that we can sell multiple copies of each item? (Hint: this means what happens if we consider integer values for decision variables instead of binary values when setting up the variable range constraints in Excel) How the objective function value changes if we consider this new assumption? Use excel to solve the problem with this new assumption Question 4) (25 points) Consider a transportation problem There are a set of suppliers and a set of customers in a region The customers have their known demand of a specific product Suppliers have supply capacities for the product. An operations manager wants to make shipment decisions, i.e., how many items to ship from each supplier to each customer, subject to the demand requirements and supply capacities. The demand and supply information are presented below. Spel SSCIA w 250 ES A shipping cost is charged for shipping cach unit of item from a supplier to a customer. The cost information is summarized below. According to this cost matrix, for example to ship 100 units of the item from supplier 2 to customer B, a shipping cost of is charged The goal is to minimize the total transportation costs. Costa From ABCBEHO 1. Which items the manufacturer should select to process and sell to the customer in order to maximize his profit? Write a mathematical optimization model for this problem. Clearly define decision variables, objective function, and constraints for the optimization model. Use Excel to solve the optimization model. 2. How long (in minutes) the machine remains idle during a day? 3. What is the solution of the problem if we assume that we can sell multiple copies of each item? (Hint: this means what happens if we consider integer values for decision variables instead of binary values when setting up the variable range constraints in Excel) How the objective function value changes if we consider this new assumption? Use excel to solve the problem with this new assumption Question 4) (25 points) Consider a transportation problem There are a set of suppliers and a set of customers in a region The customers have their known demand of a specific product Suppliers have supply capacities for the product. An operations manager wants to make shipment decisions, i.e., how many items to ship from each supplier to cach customer, subject to the demand requirements and supply capacities. The demand and supply information are presented below. Supermend SIA IN SIE ID 450 ESCI A shipping cost is charged for shipping each unit of item from a supplier to a customer. The cost information is summarized below. According to this cost matrix, for example to ship 100 units of the item from supplier 2 to customer 13, a shipping cost of is charged. The goal is to minimize the total transportation costs. KM Prow ACOLO 1 - How many items should be shipped from each supplier to each customer to minimize the total transportation costs? Formulate a mathematical model for this problem and use Excel to solve the problem. 2- What is the excess capacity of supplier in the optimal solution