Submission: Please upload a single doc/docx or pdf document using Blackboard before the deadline, stated below. Late submissions will not be graded.
This midterm is an individual exam, i.e., you must solve the problems individually
Please carefully read the problems
There are five problems and the points for each problem are given next to each problem
Your solutions must be legible otherwise you can lose credits
Correct answers with wrong reasoning will not get partial credits. For instance
Question: ‘Is it true that Dincer Konur is a teacher? Explain why?’
Answer: Yes, he works at a university.
The answer is Yes, but the reasoning is wrong as not everyone working at a university is teacher. The correct answer would be ‘Yes, he teaches EMGT 365.’
Suggestions: Start with the problems that you think to be easy.
Due: October 7, 2013 Monday 6:00pm (cst)
Define the meaning of any notation that you use
Constraints and objective functions should be functions of your decision variables!
You are not asked to solve any problem, you are asked to mathematically formulate problems, so excel is not needed in this midterm
Do not make assumptions on the values of the parameters given. For instance, "A is cheaper than B, so I will not use B and I am not defining it as a decision variable" should not be done for this midterm. Parameter values are given for easier construction of mathematical models and notational simplicity.
DistanceTransportation CostPurchase CostWeekly Supply(in miles)Per Unit Per MilePer UnitCapacitySupplier 1150$0.10$62000 unitsSupplier 2200$0.15$52500 unitsSupplier 3125$0.20$41500 unitsSupplier 4100$0.25$71750 units
Problem 1 (20 Points): Multi-Sourcing Shipping Plan
Suppose that you are the Manager of the Purchasing Department in a manufacturing company. Currently, you are trying to outsource one of the raw materials (a steel pipe) required for manufacturing your product. Specifically, you know that you need to purchase and ship exactly 5,000 units of steel pipe each week to the manufacturing plant. There are four possible suppliers that you can purchase and ship the steel pipes from. Each supplier has different purchase costs per unit of steel pipe and different shipping costs due to their distances. Furthermore, each supplier has limited weekly supply of steel pipes. The table below summarizes the following data:
The distance of each supplier to the manufacturing plant (in miles)
The purchase cost charged by each supplier per steel pipe ($ per unit of steel pipe)
The weekly supply capacity of each supplier (units of steel pipe)
The transportation cost charged by the supplier for shipping one unit of steel pipe per mile ($ per steel pipe per mile)
As the purchasing manager, you want to minimize the total cost of weekly steel pipe supply to the manufacturing plant, which includes weekly purchase costs and weekly transportation costs. To do so, you need to determine how much steel pipe to ship weekly from each supplier (assume that you can purchase and ship fractional number of steel pipes from the suppliers). While doing so, you need to consider the following:
Weekly transportation cost should not exceed 80% of the total weekly cost
Since suppliers 3 and 4 are closer to the manufacturing plant, you want the total steel pipes purchased and shipped weekly from suppliers 3 and 4 to be more than or equal the total steel pipes purchased and supplied weekly from suppliers 1 and 2
Since you want to build good relations with your suppliers, you want to purchase and ship at least 250 units of steel pipe from each supplier each week
You want to purchase and ship at least 2 units of steel pipe from supplier 1 for each unit of steel pipe you purchase and ship from supplier 4
In this problem, you are asked to formulate the above outsourcing problem mathematically as a linear model. To do so, define your decision variables, objective and express your objective function and constraints in terms of your decision variables, and combine everything to get the final formulation. 3
Problem 2 (30 points): Teller Management
Phelps County Bank in Rolla, MO is open Monday to Friday from 9am to 5pm. From past experience, the bank knows that it needs the number of tellers shown for each time period seen in the Table below for any day of the week. The number of tellers available in each time period should be greater than or equal to the number of tellers needed.
The bank hires two types of tellers: full-time tellers and part-time tellers.
Full-time tellers work 9am–5pm five days a week, except for 1 hour off for lunch. The bank determines when a full-time employee takes lunch hour, but each full-time teller must go between noon and 1pm or between 1pm and 2pm. Full-time employees are paid (including fringe benefits) $8 per hour (this includes payment for lunch hour).
Each part-time teller must work exactly 3 consecutive hours each day. Each part-time teller can start working at beginning of hours, i.e., a part-time teller will not start working at 10:30am, he/she can either start at 10am or 11am. A part-time teller is paid $5 per hour (and receives no fringe benefits).
To maintain adequate quality of service, the bank has decided that at most five part-time tellers can be hired. Assume that fractional number of tellers can be hired.
Mathematically formulate a linear programming model to meet the teller requirements to minimize costs. To do so, define your decision variables, objective and express your objective function and constraints in terms of your decision variables, and combine everything to get the final formulation. 4
Problem 3 (10 points + 2 points bonus): Linear Programming Properties
Please answer the following questions as correct or wrong and explain your reasoning briefly.
a) (4 points) Suppose that the feasible region of a linear model is bounded. Can this linear model be unbounded?
b) (4 points) Suppose that a linear model has alternative optima. Is it possible that there are more than three corner solutions which are optimum?
c) (4 points) Is it true that any linear model, which is not unbounded, will always have at least one corner point as the optimum solution?
Problem 4 (10 points): Sending Orcs from Isengard to Helm’s Deep
Suppose that you are Saruman and you want to send as many orcs as you can from Isengard to Helm’s Deep to destroy Rohan. The orcs should get across the Isen River and at most 5,000 orcs can pass Isen River towards the East and at most 5,000 orcs can pass Isen River towards the West. The orcs passing towards the West will make it to Helm’s Deep. The orcs passing towards the East need to get across the Adorn River towards the West. At most 3,000 orcs can pass Adorn River towards the West and orcs passing towards the West of Adorn River will make it to Helm’s Deep. The map below is the map Saruman have.
Mathematically formulate Saruman’s problem as a maximum flow problem by defining your decision variables, objective function, and the constraints. 5
WarehouseStore 1Store 2Warehouse-100150Store 1100-50Store 215050-Distance (in miles)
Problem 5 (30 Points, 15 points each): Distribution Management with Lateral Shipments
Suppose that you are the distribution manager of a warehouse. You need to determine how to ship 100 units of a specific product to two retailer stores. Each retailer store requires 50 units of the product. There are three carriers that you can use to ship the product from the warehouse to the retailer stores and there is another carrier that can ship the product between the retailer stores. The specifications of the carriers are as follows:
Carrier 1: It charges $15 to pick up one unit of the product from the warehouse. It charges $0.25 to ship one unit of the product one mile. It can ship at most 40 units of the product. It can only ship to retailer store 1 from the warehouse.
Carrier 2: It charges $8 to pick up one unit of the product from the warehouse. It charges $0.30 to ship one unit of the product one mile. It can ship at most 30 units of the product. It can ship to both retailer stores from the warehouse.
Carrier 3: It charges $12 to pick up one unit of the product from the warehouse. It charges $0.20 to ship one unit of the product one mile. It can ship at most 50 units of the product. It can only ship to retailer store 2 from the warehouse.
Carrier 4: It charges $10 to pick up one unit of the product from any retailer store. It charges $0.40 to ship one unit of the product one mile. It can ship as much as you want. It can only ship between the retailer stores.
The distances from the warehouse to each retailer store and between the retailer stores are given in the table below:
As the distribution manager, you want to determine the minimum cost shipment plan from the warehouse to the retailers.
Express the above distribution management problem as a network optimization problem by defining and drawing the network. That is,
Define the nodes and explain what they represent, explain whether they are supply nodes, demand nodes, or transshipment nodes and determine their node values, if any needed.
Define the arcs and explain what the flow on each arc represents, determine the arc costs and arc capacities, if any needed.
Explain what kind of a network optimization problem should be solved on the network you defined to find the best distribution plan.
Mathematically formulate the network optimization problem you defined above by defining your decision variables, objective function, and the constraints. Combine everything to achieve you final mathematical formulation.