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Solutions to Exam #1

John H. Vande Vate

Fall, 2002

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Scores

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12

9

2

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2

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< 40 < 50 <60 <70 <80 <90 <100

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Stats

• Range: 37-89

• Mean: 61

• Std. Dev: 11

• Curve: 11

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Concerns?

• Submit concerns about grading in writing

• Identify what you think is mis-graded and why.

• Will return regraded exams by the end of the semester.

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Question #1• 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.

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Question 1.A

What is the optimal frequency for direct shipments to our customers?

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.

Short Answer: Every unit incurs $0.10Long Answer: $0.10*Q is how much we pay per

shipmentThere will be D/Q shipments per yearWe will pay $0.10D per year regardless of the frequency

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Question 1.A

• Use the version of the EOQ model for several destinations:

• Q* = sqrt(2AD/hC)sqrt(P/(D+P)) • 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

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Question 1.A

• So, Q* = sqrt(2*1500D/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.

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Question 1.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*/2Which customers should be served with each

frequency? Be specific and be sure to show how you obtained your answer.

 

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Question 1.B

Convention: 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 unless we change mode.

 

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Question 1.BThe 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/2AD+ 450*225hC = 2AD + 225*225hC225*225hC = AD

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

So, customers closer than about 400 miles we can ship to twice as frequently.

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Question 2

Our carrier’s trucks average about 50 miles per hour and, because of Federal Transportation Regulations, about 10 hours per 24 hour day.

 

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

 

  

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Question 2.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

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Question 2.B

Factor the pipeline inventory costs into your model for calculating the optimal frequency. How does this change your answer from Question 1, Part A, Part B? 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 – 2 days worth of production

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Question 3

1 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.

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Question 3

 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)};

 

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Question 3

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)};

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Project TopicsProvide a linear description in terms of the Flow variables of:

The total quantity of goods shipped from Detroit annually:

Sum{path in paths that start at Detroit, dest in destinations} Flow[Detroit, path, dest];

Flow[Detroit, Path1, Los Angeles] + Flow[Detroit, Path2, Los Angeles] + Flow[Detroit, Path3, Los Angeles] + Flow[Detroit, Path4, Los Angeles] +Flow[Detroit, Path5, Seattle] + Flow[Detroit, Path6, Seattle] +Flow[Detroit, Path7, Seattle] + Flow[Detroit, Path8, Seattle]

B    

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AMPL Version

• Set CITIES;• Set ORIGINS;• Set DESTINS;• Set PATHS;• Set MODES;• Set EDGES within CITIES cross CITIES cross

MODES;• Set PATHEDGES{PATHS} within EDGES;• Param PathOrigin{PATHS};• Param PathDestin{PATHS};

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AMPL Version

• Var Flow{orig in ORIGINS, path in PATHS, dest in DESTINS: PathOrigin[path] = orig and PathDestin[path] = dest} >= 0;

• Sum{path in PATHS, dest in DESTINS: PathOrigin[path] = ‘Detroit’} Flow[‘Detroit’, path, destin]

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Question 3  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];

   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.    

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More Options

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]

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];

 

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Question 4

• 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.

 

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Question 4Model 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.

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Question 4

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  

The two models are identical!

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Resolution

Inventory per shipment: Q*Q/2P

Shipments per year: D/Q to each customer

Total Inventory contribution per customer: QD/2P.

With n identical customers P = nD

So 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.

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