11

Exam #2

John H. Vande Vate

Fall, 2002

22

Question 1• Formulate a linear mixed integer programming model to solve the following

problem.

• The county is trying to site a landfill within its incorporated boundaries. All of the towns within the county are opposed to having the landfill located near them. So the county’s challenge is to find a location within its borders that maximizes the distance between the landfill and the nearest town. Since the county is in Kansas and all the roads run North-South or East-West, the county proposes to use rectilinear distance. Formulate a linear mixed integer programming model to help them.

• THIS IS A MAXMIN

33

Question 1Cont’d

City

MinLatitude

MaxLatitude

Min

Lon

gtit

ude

Max

Lon

gtit

ude

Comment: There is no guarantee that any of these numbers is non-negative.

44

Question 1Cont’d

param MinLatitude; /* The minimum latitude of the county in decimal form.*/

param MaxLatitude; /* The maximum latitude of the county in decimal form.*/

param MinLongitude; /* The minimum longitude of the county in decimal form.*/

param MaxLongitude; /*The maximum longitude of the county in decimal form*/

set TOWNS; /* The set of towns in the county */

param Latitude{TOWNS}; /* The latitude in decimal form of each town */

param Longitude{TOWNS}; /* The longitude in decimal form of each town */ 

var Lon >= MinLongitude, <= MaxLongitude; /* The longitude of the landfill */

var Lat >= MinLatitude, <= MaxLatitude; /* The latitude of the landfill */

You may not assume that the landfill’s coordinates will fall on some pre-defined grid or must be from some finite set of choices.

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Question 1Cont’d

param Latitude{TOWNS};

param Longitude{TOWNS};

param M = 2*(MaxLongtitude + MaxLatitude – MinLongtitude - MinLatitude) 

var Lon >= MinLongitude, <= MaxLongitude;

var Lat >= MinLatitude, <= MaxLatitude;

var xdist(TOWNS); /* horizontal distance between towns and landfill */

var ydist(TOWNS); /* vertical distance between towns and landfill */

var mindist; /* minimum of xdist + ydist */

Var XToggle{TOWNS} binary; /* Is the town East of the Land Fill? */

Var YToggle{TOWNS} binary; /* Is the town North of the Land fill? */

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Question 1Cont’d

Maximize Objective: mindist;s.t. EastDisplacement{t in TOWNS}:

xdist(t) <= Lon - Longtitude(t) + M*XToggle[t];s.t WestDisplacement{t in TOWNS}:

xdist(t) <= Longtitude(t) – Lon + M*(1-XToggle[t])s.t. NorthDisplacement{t in TOWNS}:

ydist(t) <= Lat – Latitude(t) + M*YToggle[t];s.t. SouthDisplacement{t in TOWNS}:

ydist(t) <= Latitude(t) – Lat + M*(1-YToggle[t]);s.t. DefineTheMin{t in TOWNS}:

mindist <=xdist(t) + ydist(t);

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Question 2• Given a set of customer locations, we wish to position a

single facility so as to minimize the maximum rectilinear distance between the facility and the customer locations. In class we speculated that one solution to this problem might be obtained by solving two 1-dimensional problems:

• Find the position X on the North-South axis that minimizes the maximum distance North or South to the customers.

• Find the position Y on the East-West axis that minimizes the maximum distance East and West to the customers.

• Locate the facility at the position (X, Y).

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A. Every such position (X, Y) is a solution to our 2-dimensional problem:

False.

Question 2

99

Question 2

1010

Question 3

• An on-line retailer like Amazon uses UPS and other parcel service companies to deliver its products to consumers. The retailer only charges its customers for freight if they use a premium service (e.g., Next Day Air or Second Day). These carriers charge based on the service level required (Next Day Air, Second Day, Ground) and distance. The distance charges are computed through delivery zones. These correspond roughly to concentric circles of increasing radius about the retailer’s location.

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Question 3 Cont’d

A. In locating its distribution centers, which is the most reasonable objective for the retailer to focus on? (Choose One. Be sure to justify your choice):

• Minimize the maximum distance to customer • Minimize the sum of the distances to

customers• Minimize the total gallon-miles

(distance*volume) to customers• Maximize the minimum distance to customers

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Question 3 Cont’dSeveral answers make sense:• Minimize the distance*volume where volume is calculated in terms of

the number of ground orders customers place. This reduced our retailer’s freight expense.

• Minimize the distance*volume where volume is calculated in terms of the number of premium and ground orders customers place. Even if the consumer pays for premium services, we would like the total cost of our goods to be as small as possible. To enhance our competitive position.

• Minimizing the total distance to customers. This answer penalizes our best customers to some extent though.

• Minimizing the maximum distance is defensible on the grounds we want the freight costs to be fair and don’t want to discriminate unduly against customers in any region or if we want to provide good service guarantees.

• Unless our product smells, there’s not much point in maximizing the minimum distance.

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Question 3 Cont’d

• B. How would your answer change if the retailer charged a fixed fee for basic freight (e.g., one basic freight charge for all customers regardless of location) as well as special fees for premium services?

• Minimize the distance*volume where volume is calculated in terms of the number of premium and ground orders customers place. We would like the total cost of our goods to be as small as possible. To enhance our competitive position. Other variations make sense.

• Minimizing the maximum distance is defensible on the grounds we want the freight costs to be fair and don’t want to discriminate unduly against customers in any region or if we want to provide good service guarantees.

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Question 4

• Indicate the most appropriate routing methodology in each of the following situations. The choices in each case are:

– Integer Programming– Nearest Neighbor– Spacefilling Curve– Minimum Spanning Tree Heuristic

• Be sure to justify your answer in each case.

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Question 4 Cont’d• A parcel delivery company is just opening its

operations in Urumqi, a town in Northwest China near Mongolia, Kazakhstan and Kyrgyzstan. Deliveries are made by local employees who peddle three-wheeled vehicles from customer to customer. This is a small operation and investments in the business are extremely limited.

• Since the investments are limited, we should not use expensive solutions like integer programming. Also the area is small, the labor is cheap, and the vehicle expenses are small (depreciation, fuel, etc.); so that any reasonable solution for the routing will be fine. But the local employees may not understand complex solutions, so it’s best to use Nearest Neighbor or perhaps spacefilling curves.

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Question 4 Cont’d

• An oil company operates drilling platforms in the Gulf of Mexico. It provides regular helicopter service that originates from its base in Mobile, visits all 20 platforms and returns to the base. The service runs every two hours and is intended to transport crews and supplies to the platforms.

• The transportation costs are really high here, and the service should be completed before 2 hours. So finding an optimum solution is needed. And 20 is a reasonable number, so integer programming will work well here.

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Question 4 Cont’d• An emergency medical team has arrived at the scene of a major

industrial accident. The area is now safe, but the injured are widely and randomly dispersed. The team moves as a unit from victim to victim providing immediate first aid and performing triage (assessing the urgency, medical requirements and chances of survival for each victim in an effort to sequence and allocate subsequent treatment capacity).

• It’s obvious that the time is very important here. We do not have time to compute as the information is only available at the last minute. Nearest neighbor algorithm is probably the only reasonable candidate. It would also be hard to justify anything else.

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Question 5• The rail ramp handles approximately 80,000 vehicles per year for

roughly 200 dealerships spread across an area of approximately 40,000 sq miles (200 miles X 200 miles). The car hauler serving the rail ramp uses rigs that can carry 10 vehicles at a time and travel an average of 35 miles per hour. The drivers work 40 hours per week on average though their schedules can vary to accommodate long hauls lasting several days. A typical route involves about two hours to load the rig and several stops at dealerships each lasting approximately one hour. On average, dealerships receive two vehicles each visit.

• Provide an estimate of the number of rigs and drivers the car hauler needs at the rail ramp. Be sure to clearly organize and explain your thinking including carefully indicating the units of each term in the calculations.

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Question 5 Cont’d

(rigs) drivers 59135

8000

driver oneby year / deliveries 13550*2.7 So

weeks50 yr 1

ek/driver/wedeliveries 7.214.8

40

hr 14.8 Total

hr 07.25*hr / miles 35

miles 14.4 sdealershipBetween

2hr timeLoad

hr 5 stops 5 sdealership 5About

hr 72.5miles/hr 35

miles 200 haul Line

r trips/yea000,8trip vehicles/10

vehicles80,000

customersbetween miles 14

2001

1

mileper sdealership 200

1

200*200

200 2