#517 Tri-County Utilities, Inc. supplies natural gas to

Tri-County Utilities, Inc. supplies natural gas to - Math

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3. Tri-County Utilities, Inc., supplies natural gas to customers in a three-county area. The company purchases natural gas from two companies: Southern Gas and Northwest Gas. Demand forecasts for the coming winter season are as follows: Hamilton County, 400 units; Butler County, 200 Units; and Clermont County, 300 units. Contracts to provide the following quantities have been written: Southern Gas, 500 units; and Northwest Gas, 400 units. Distribution costs for the counties vary, depending upon the location of the suppliers. The distribution costs per unit (in thousands of dollars) are as follows:

To

From	Hamilton	Butler	Clermont
Southern Gas	10	20	15
Northwest Gas	12	15	18

A) Develop a network representation of this problem.

B) Develop a linear programming model that can be used to determine the plan that will minimize total distribution costs.

C) Describe the distribution plan and show the total distribution cost.

D)Recent residential and industrial growth in Butler County has the potential for increasing demand by as much as 100 units. Which supplier should Tri-County contract with to supply the additional capacity?

 

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General guidance

Concept and reason

Linear programming (LP) uses linear algebraic relationships to represent a firm’s decisions, given a business objective and resource constraints. The main objective of business decisions involves maximizing profit or minimizing costs.

Decision variables are mathematical symbols representing levels of activity of a firm.

The objective function is a linear mathematical relationship describing an objective of the firm, in terms of decision variables. It can be maximized or minimized.

Constraints are the requirements or restrictions placed on the firm by the operating environment, stated in linear relationships of the decision variables.

Parameters are numerical coefficients and constants used in the objective function and constraints.

Fundamentals

Graphical method can be used when there are two decision variables.

The formulated linear programming model is,

Here,

Step 1 of 4

(a)

Following table represents the given information as follows:

Part a

Following network diagram represents the network representation of this problem.

Explanation | Hint for next step

For the given information,

Following network diagram represents the network representation of this problem.

Step 2 of 4

(b)

The objective is to determine the plan that will minimize total distribution costs.

The variables X, Y, Z represents the amount of X units shipped from supply node Y to demand node Z.

From the given information,

Let represents that the units shipped from supply node “southern” to demand node “Hamilton”.

Let represents that the units shipped from supply node “southern” to demand node “Butler”.

Let represents that the units shipped from supply node “southern” to demand node “Clermont”.

Let represents that the units shipped from supply node “Northwest” to demand node “Hamilton”.

Let represents that the units shipped from supply node “Northwest” to demand node “Butler”.

Let represents that the units shipped from supply node “Northwest” to demand node “Clermont”.

Hence, the LPP formulated with respect to the given information is as follows:

The objective function is,

The subject to constraints are,

The non-negative constraints are,

Explanation | Hint for next step

For the given information, the LPP is,

The objective function is,

The subject to constraints are,

The non-negative constraints are,

Step 3 of 4

(c)

The optimum solution for the given LPP is 11900.

Hence, the optimum solution for the given LPP is calculated as follows:

Part c

The optimum solution for the obtained LPP is 11900.

Explanation | Hint for next step

The optimum solution for the obtained LPP is 11900.

Step 4 of 4

(D)

Butler county demand is increasing by 100 units.

So, we change our Butler demand in LINGO from 200 to 300.

To, decide what company (Southern Gas or Northwest Gas) is best suited to provide this increase, we increase the supply at each company in LINGO by 100.

Part D

Butler county demand is increasing by 100 units and increase the supply at each company in LINGO by 100.

Explanation

Butler county demand is increasing by 100 units and increase the supply at each company in LINGO by 100.

Answer

Part a

Following network diagram represents the network representation of this problem.

Part c

The optimum solution for the obtained LPP is 11900.

Part D

Butler county demand is increasing by 100 units and increase the supply at each company in LINGO by 100.