The Exam Is to Be Done Individually, Without Collaboration with Anyone Else s1

ISyE 6203 Fall, 2002 Exam 1 Vande Vate

Instructions

You have until 10:55 am to complete the exam. Watch your time. If you are having difficulty with a question, you might wish to pass over it and return to it later if you have enough time at the end of the exam.

Be sure to put your name on your exam. Show your work – that will allow me to award partial credit. Make sure your answers are legible and clearly marked.

The exam is open book: you may use any notes or text material. The exam is to be done individually, without collaboration with anyone else. 1. (25 points) Our plant makes two products and ships them to customers across the country. Demand for the products is relatively constant at 10 units of each product each day at each customer. We have 1,000 customers across the country.

Product Selling Price Weight Product 1: $100 99 lbs Product 2: $100 1 lbs

The capacity of our trucks is 45,000 lbs. The average distance to customers is 1,000 miles and we pay our carrier $1.50/mile (one way, regardless of the load) plus an additional $0.10/lb (regardless of the distance). Our inventory carrying cost is 15%/year. Treat a year as 250 days.

A. What is the optimal frequency for direct shipments to our customers? Be sure to show your work and explain any differences between your calculations and those we used in class.

First, we can ignore the $0.10/lb as the total amount we pay under this heading will be the same regardless of our shipping frequency.

Second, we want to use the version of the EOQ model for several destinations: Q* = sqrt(2AD/hC)sqrt(P/(D+P)) Here, A is the cost per trip = $1,500 D is the annual demand for a single customer or 2,500 packages consisting of 1 unit of product 1 and one unit of product 2 h is 15% per year C is the value of a package or $200 P is the production rate which we take to be 1,000D So, Q* = sqrt(3000D/hC)sqrt(1,000D/1,001D) ≈ sqrt(3,000D/30) = sqrt(100D) = 10sqrt(2,500) = 500. Total weight of 500 packages is 50,000 lbs which exceeds the capacity of the truck. So, we should ship in full truckloads of 450 units each. That means the answer is to ship every 45 working days. B. You calculated the optimal frequency f* in Part A using the average distance to customers. Some of our customers are closer than average and others are further away. We have decided that to keep our operations simple, we will ship to each customer either

i. The optimal frequency f*

ii. Twice the optimal frequency 2f*:

iii. Half the optimal frequency f*/2 Which customers should be served with each frequency? Be specific and be sure to show how you obtained your answer.

We adopt the convention that if the optimal frequency is f* = once every 45 days then 2f* is twice every 45 days and f*/2 is once every 90 days.

Since shipping every 45 days requires full truckload shipments, reducing the shipping frequency will never be advantageous regardless of where the customer is.

The decision of whether to ship at the frequency f* or at the frequency 2f* will depend on the distance to the customer. We can find the breakpoint by setting the two cost formulas equal and finding the appropriate distance:

AD/450 + 450hC/2 = AD/225 + 225hC/2

2252hC/D = A = $1.5*breakpoint distance

So, customers closer than about 400 miles we can ship to twice as frequently. 2. (25 points) Our carrier’s trucks average about 50 miles per hour and, because of Federal Transportation Regulations, about 10 hours per 24 hour day.

C. Calculate the pipeline inventory costs inherent in the distribution system you designed in answer to Question 1, Part A.

It takes two days to reach the average customer. So, every item we make is on the road for an average of two days. Consequently, there is approximately two days production in the pipeline. That’s 2days * 10 packages per day/customer * 1000 customers = 20,000 packages each worth $200. So the value of the pipeline inventory is $4,000,000 and the inventory costs associated with that are 15% of $4 million = $600,000/year

D. Factor the pipeline inventory costs into your model for calculating the optimal frequency.

i. How does this change your answer from Question 1, Part A?

It doesn’t. Goods still take just as long to get there. If we increase the shipping frequency, there will just be more smaller shipments, but they will still add up to the same amount of goods on the road at any given time.

ii. How does this change your answer from Question 1, Part B?

It doesn’t. Changing the shipment quantity does not change pipeline inventory unless it gets the goods there more quickly.

3. (25 points) We want to develop a flows-on-paths formulation of a multi- modal logistics problem. You have been asked to provide examples to help the engineers in building the model. You have the following list of variables and are asked to use them in formulating each of the following constraints.

The variable Flow[origin, Path, destination] is the annual quantity of goods shipped from origin to destination via the path Path, where Path is a set of Edges from the origin to the destination via a number of intermediate stops. Each edge also indicates the mode used. So, for example, the edge (Detroit, Kansas City, Unit Train) indicates a Unit Train move from Detroit to Kansas City with no intermediate stops.

The paths from Detroit to Los Angeles are: Path1 := {(Detroit, Kansas City, Unit Train), (Kansas City, Denver, Loose Car), (Denver, Los Angeles, Loose Car)}

Path2 := {(Detroit, Los Angeles, Unit Train)};

Path3 := {(Detroit, Los Angeles, Loose Car)};

Path4 := {(Detroit, Kansas City, Loose Car), (Kansas City, Los Angeles, Loose Car)};

The paths from Detroit to Seattle are: Path5 := {(Detroit, Kansas City, Unit Train), (Kansas City, Denver, Loose Car), (Denver, Seattle, Loose Car)}

Path6 := {(Detroit, Seattle, Unit Train)};

Path7 := {(Detroit, Seattle, Loose Car)};

Path8 := {(Detroit, Kansas City, Loose Car), (Kansas City, Seattle, Loose Car)}; Provide a linear description in terms of the Flow variables of:

A. The total quantity of goods shipped from Detroit annually Sum{path in paths that start at Detroit, dest in destinations} Flow[Detroit, path, dest];

B. The total quantity of goods loaded onto Unit Trains in Detroit annually

Sum{path in the set of paths that start in Detroit with unit train shipments, dest in destinations} Flow[Detroit, path, dest];

C. The total quantity of goods passing through Kansas City annually

Sum{orig in Origins, path in paths that include Kansas City, dest in Destinations} Flow[orig, path, dest};

D. The total quantity of goods that passed through Denver arriving in Seattle annually

Sum{orig in Origins, path in paths that end in Seattle and include Kansas City}Flow[orig, path, Seattle]

E. The total quantity of goods that pass through both Kansas City and Denver each year.

Sum{orig in Origins, path in paths that include both Kansas City and Denver, dest in Destinations} Flow[orig, path, dest]; 4. (25 points) In class we discussed two models for the inventory at a production plant serving many identical customers with staggered shipments via a single loading dock.

Model 1: In this analysis, we calculated the inventory impact of each customer and showed that, when the number of customers is large (say 100), any single customer’s contribution to inventory at the plant is small – much smaller than Q/2, where Q is the quantity delivered in each shipment.

Model 2: In this analysis, we argued that, whatever the contribution of a single customer, the operations at a production facility serving several customers with shipments of size Q will be indistinguishable from those of a facility serving a single customer with shipments of shipments of size Q: The inventory at the plant will grow from 0 to Q, at which point the vehicle will depart and the inventory will return to 0. This process repeats and so the average inventory at the plant should be Q/2 in both cases.

What will the average inventory at the plant be? Assume the plant serves 100 identical customers each with annual demand D. It sends each customer shipments of size Q. To answer this question completely, you must

A. Select a correct model: Model 1 or Model 2.

Both are correct.

B. Explain what is wrong with the other model.

The two models are identical. The annual inventory at the plant from shipments to a single customer is Q*Q/2P per shipment times D/Q shipments or QD/2P. We assume the customers are identical and that production matches demand, so P = nD where n is the number of customers. Thus each customer contributes Q/2n in annual inventory at the plant. But there are n customers, so average annual inventory at the plant is … Q/2.