AGEC 622 (Spring 2012) ANSWERS for LP Homework 2 · (70 pts) The ABC lumber company has three sources of wood and five markets where wood is demanded. The annual quantity of wood

AGEC 622 (Spring 2012)

ANSWERS for LP Homework 2

Due Date: February 4th by noon

Turn in to: Fabian or TA (Alexander M.)

Late Assignments: -10 points per day late

Total Points Possible: 200

1. (70 pts) The ABC lumber company has three sources of wood and five markets where wood is demanded. The annual quantity of wood available in

the supply source 1, 2, and 3 are 15, 20, and 15 million board feet, respectively. The amount that can be sold at market 1, 2, 3, 4, and 5 is 11, 12, 9,

10, and 8 million board feet, respectively. The company currently transports all of the wood by train. It wishes to evaluate its transportation schedule,

possibly shifting some or all of its transportation to ships. The unit cost of shipment (in $10,000 along the various routes using both methods is

described in the table below.

Table A. Cost per unit of truck transportation

Supply source Market 1 Market 2 Market 3 Market 4 Market 5

Supply 1 51 62 35 45 56

Supply 2 59 68 50 39 46

Supply 3 49 56 53 51 37

Table B. Cost per unit of train transportation

Supply source Market 1 Market 2 Market 3 Market 4 Market 5

Supply 1 48 68 48 none 54

Supply 2 66 75 55 49 57

Supply 3 none 61 64 59 50

The management needs to decide to what extent to continue to rely on truck transportation.

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Wood Source 1

Supply = 1.5E7 board feet Demand = 1.1E7 board feet

Demand = 1.2E7 board feet

Demand = 9E6 board feet



Wood Source 2

Supply = 2.0E7 board feet

Wood Source 3


Truck Transportation



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a) (10 pts) Formulate a linear programming model for the ABC lumber company for the case that this company uses only rail transport including the

math formulation and the corresponding tableau.

Answer:

4

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b) (10 pts) Solve the problem in (a) using Excel. Print out the excel spreadsheet that contains your LP model, the answer report and the sensitivity

report. If you make any additional assumptions in setting up your model, please state them explicitly.

Answer:

Answer Report:

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Sensitivity Report:

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c) (10 pts) Write down and interpret the optimal values for the objective function, shadow prices, decision variables, and reduced costs.

Answer:

The minimum cost to distribute the wood from the three sources by truck transportation is $23,160 Billions of Dollars

The reduced cost of the variable (Amount supplied from source 1 to market 2) is telling that if we force to move 1 unit of into the

optimal system, then we will increase the total cost by $40,000.

The shadow price of constraint 1 (source 1 annual supply) is telling that at the optimal solution, if it is increased by one unit the supply at this

point (source 1), the total cost (objective function) will have a decrement of $80,000.

The shadow price of constraint 8 (Required demand from market 5) is telling that at the optimal solution, if it is increased by one unit the RHS of

the constraint, then the total cost (objective function) will have an increment of $460,000. Variable Optimal Value (board feet)

X11 6,000,000 X12 0 X13 9,000,000 X14 0 X15 0 X21 2,000,000 X22 0 X23 0 X24 10,000,000 X25 8,000,000 X31 3,000,000 X32 12,000,000 X33 0 X34 0 X35 0

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d) (10 pts) How much does it cost if the company use only train transport? Also write down the optimal values of decision variables. Hint: plugging the

large number in the objective function for the routes that have no ship transport. You do not need to turn in the associated Sensitivity Reports for this

question.

Answer:

The total cost to the company using only train transport is $26,520 Billions of Dollars, so it looks a better deal to use truck transport instead of only train

transport. Variable Optimal Value (board feet)

Y11 11,000,000

Y12 0

Y13 4,000,000

Y14 0

Y15 0

Y21 0

Y22 0

Y23 5,000,000

Y24 10,000,000

Y25 5,000,000

Y31 0

Y32 12,000,000

Y33 0

Y34 0

Y35 3,000,000

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e) (10 pts) How much does it cost to use the cheapest available mode of transportation on each route? Also write down the optimal values of decision

variables. You do not need to turn in the associated Sensitivity Reports for this question.

Wood Source 1

Supply = 1.5E7 board feet Demand = 1.1E7 board feet


Demand = 9E6 board feet



Wood Source 2


Wood Source 3


Train Transportation






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Answer:

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Thus, Mixing Train and Truck Transportation, the total cost can be reduced to $22,980 Billions of Dollars Variable Optimal Value (board feet)

X11 0 X12 0 X13 9,000,000 X14 0 X15 0 X21 2,000,000 X22 0 X23 0 X24 10,000,000 X25 8,000,000 X31 3,000,000 X32 12,000,000 X33 0 X34 0 X35 0 Y11 6,000,000 Y12 0 Y13 0 Y14 0 Y15 0 Y21 0 Y22 0 Y23 0 Y24 0 Y25 0 Y31 0 Y32 0 Y33 0 Y34 0 Y35 0

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f) (10 pts) How would your answer in part (a) change if you learned that the supply source 2 and the demand at market 3 were both expected to increase

by 10 million board feet? You do not need to turn in the associated Sensitivity Reports for this question.

Answer:

At f) the objective function (total cost) is minimized at $27,740 Billions of Dollars.

However at a) the total cost was minimized at $23,160 Billions of Dollars.

SOLUTION f) Variable Optimal Value (board feet)

SOLUTION a) Variable Optimal Value (board feet)

Supply1 to Market1 X11 0

Supply1 to Market1 X11 6,000,000


Supply1 to Market2 X12 0 Supply1 to Market3 X13 15,000,000



Supply1 to Market4 X14 0 Supply1 to Market5 X15 0.00E+00

Supply1 to Market5 X15 0.00E+00


Supply2 to Market1 X21 2,000,000 Supply2 to Market2 X22 0



Supply2 to Market3 X23 0 Supply2 to Market4 X24 10,000,000





Supply3 to Market1 X31 3,000,000 Supply3 to Market2 X32 12,000,000



Supply3 to Market3 X33 0 Supply3 to Market4 X34 0




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2. (60 points) A dairy processes milk using capacity, labor, and refrigerator space and sells low fat milk, high fat milk, and butter milk. In addition, the

dairy has the option of using butterfat to make butter, which is then sold. The process of converting butterfat to butter is constrained by churn time.

Essentially, the dairy must choice between 3 different production processes. The first process uses raw milk to produce low fat milk, the second

process uses raw milk to produce high fat milk, and the third process uses butterfat from the first two processes to produce butter.

Process 1 Process 2 Process 3

Use raw milk

to make low fat

milk

Use raw milk

to make high

fat milk

Use butterfat

to make

butter

500 500

Low fat milk yield (gal) 450

High fat milk yield (gal)

491

Butterfat yield (gal) 50 9

Butterfat used (gal)

3000

Butter yield (pounds)

8000

Labor used (hours) 3.2 2.2 2.4

Refrigerator space (cu ft) 2700 2700 1000

Churn use (hours)

1.5

Processing cost ($) .15 per gal .16 per gal .45 per gal

Packaging cost ($) .15 per gal .15 per gal .30 per gal

The dairy has raw milk costs of $0.50 per gallon. The company has a contract to sell the high fat milk to one of its customers equal to 200,000

gallons and the company is able to sell the dairy products for the following prices:

Sales price

Low fat milk ($/gal) 2.00

High fat milk ($/gal) 1.20

Butterfat ($/gal) 2.20

Butter ($/pound) 2.50

The resources used in production are constrained to the following:

Availability

Labor available (hours) 2,300

Raw milk capacity (gal) 1,500,000

Refrigerator space (cu ft) 2,000,000

Churn time (hours) 12

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a) (10 pts) Should this problem be set up as a profit maximization model or a cost minimization model? Why?

Answer:

In this case the amount of product you make it unknown and the volume you make influences your profits so profit maximization is appropriate, we

would only minimize cost if you specified the volume of products to make.

b) (20 pts) Set up a linear programming model to determine the optimal production pattern for this dairy. Provide a discussion verifying that the

model you have set up satisfies the homogeneity of units criteria.

Answer: “ ” means units

Initial Input:

Input Processes:

Final Outputs:

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Process 1

Process 3

Process 2

(Input: Raw Milk for Low

Fat Milk)

(Low Fat Milk to be sold)

(Butter to be sold)

A Dairy Problem

(Input: Raw

Milk for High Fat Milk)

(High Fat Milk to be sold)

(Butterfat Milk from

Process 1& 2 to be sold)

Blender of Butterfat

(Input: Butterfat

Milk to be used on process 3)

(Raw Material)

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Thus the problem can be stated as:

Raw Capacity

Labor available

Refrigeration Space

Churn Space

High Fat Milk Contract

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Balances:

Low Fat Milk Balance

High Fat Milk Balance

Butterfat Milk Balance

Butter Balance

Raw-Milk Balance

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Non-negativity constraints.

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c) (20 pts) Provide a solved version of the LP model you have set up and interpret the results.

Answer:

Thus, it looks that the global optimum is achieved with a profit of $472,620 dollars.

The optimal values of the variables are:

The constraint results:

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d) (10 pts) You should have used at least 3 different types of constraints in your model. Describe the different types of constraints and their purpose

in the model.

Answer:

As mentioned, this is a disassembly problem, so, it can be identified:

1 upper bound on sales:

1 upper bound on disassembly:

Non-negativity constraints: , , etc.

3 Resource limitation (availability) constraints:

Labor available

Refrigeration Space

Churn Space

Balance constraints.

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3. (40 pts) Suppose you are consulting with an investor to determine how much of 4 stocks to buy. From previous years' experience, the investor has

observed the following data on returns per five hundred dollars invested from each of the four stocks:

Obs Stock 1 Stock 2 Stock 3 Stock 4

1 55 45 35 20

2 60 70 28 20

3 18 28 30 22

4 -30 42 12 19

5 50 15 40 21

6 60 40 22 18

7 41 30 35 20

8 16 19 36 20

The investor has 500,000 to invest.

a. Formulate the investor's problem using the E-V criterion.

Answer:

Where:

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Check that the problem can be explicitly stated as:

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b. Solve it for the following five risk aversion coefficients and explain in a basic fashion why the solutions change. 0.01, 0.001, 0.0001, 0.00001,

0.00

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4. (20 points) Suppose you have the mixed integer model

Max 32121 35823 YYYXX

..ts 1X 19Y 39Y 0

2X 25Y 0

1Y 3Y = 2

1X , 2X 0

1Y , 2Y , 3Y = 0 or 1

Explain the coefficients and relationship of the variable ,, 21 YY and 3Y in this problem along with constraints relating 1X , 2X and 1Y , 2Y , 3Y .

Answer:

Check that the constraint implies that these variables have always the value of one: and

The constraint thus, if and

The constraint implies that

So, the maximum is achieved with value of 48

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*************************Thank You************************

Documents

AGEC 622 (Spring 2012) ANSWERS for LP Homework 2 · (70 pts) The ABC lumber company has three sources of wood and five markets where wood is demanded. The annual quantity of wood