Formulate on spreedsheet and solve using solver with step by step procedure. Coa
ID: 443051 • Letter: F
Question
Formulate on spreedsheet and solve using solver with step by step procedure.
Coalco produces coal at three mines and ships it to four customers. The cost per ton of producing coal, the ash and sulfur content (per ton) of the coal, and the production capacity (in tons) for each mine are given in the table below.
The number of tons of coal demanded by each customer are provided as:
Demand (Tons)
The cost (in dollars) of shipping a ton of coal from a mine to each customer is given below:
Customer
It is required that the total amount of coal shipped contain at most 6% ash and at most 3.5% sulfur. Formulate an LP that minimizes the cost of meeting customer demands.
Mine Production Cost ($) Capacity Ash Content (Tons) Sulfur Content (Tons) 1 50 120 0.08 0.05 2 55 100 0.06 0.04 3 62 140 0.04 0.03Explanation / Answer
Formulation of the problem:
Decision Variables:As we are required to find out the quantities of coal to be shipped from mines to customers, therefore let Xij be the quantity of coal to be shipped from ith mine to jth customers. There are three mines so i can take the values from 1, 2, 3. Number of customers are four so j can have values 1, 2, 3, 4. Thus decision variables Xij takes the form of a matrix of order 3 by 4.
Objective function: As cost of production at mines and costs of transportation from mines to customers are given, therefore objective function has to be minimization of total cost s of production and transpotation.
Constraints: Just like any other transportation problem, there are supply and demand constraints. In addition, we are required to have constraint with respect to limits (at most) on average ash content and sulfur.
Supply constraints on account of capacities of mines: Sigma X1j <= 120; Sigma X2j <= 100 and Sigma X3j <= 140
Demand constraints on account of demand of customers: Sigma Xi1 =80; Sigma Xi2 = 70; Sigma Xi3 =60; and Sigma Xi4 =40
Constraint about ash content of at most 6%
.08 * Sigma X1j + .06 * Sigma X2j + .04 * Sigma X3j <= .06 * Sigma Sigma Xij
Similarly Constraint about sulfur of at most 3.5%
.05 * SigmaX1j + .04 * SigmaX2j + .03 * SigmaX3j <= .035 * Sigma sigma Xij
Lastly Xij >= 0 being quantities in tons of coal
Details of the solver solution are as follows
The answer sheet is as follows:
Mine Production Cost ($) Capacity Ash Content (Tons) Sulfur Content (Tons) 1 50 120 0.08 0.05 2 55 100 0.06 0.04 3 62 140 0.04 0.03 Transportation cost Mine Customer 1 2 3 4 1 4 6 8 12 2 9 6 7 11 3 8 12 3 5 Total cost Mine Customer 1 2 3 4 1 54 56 58 62 2 64 61 62 66 3 70 74 65 67 Decision Variables Constraints Objective 16060 Mine Customer Total 1 2 3 4 1 15 0 0 -1E-06 15 120 0.08 1.2 0.05 0.75 2 25 70 0 0 95 100 0.06 5.7 0.04 3.8 3 40 0 60 40 140 140 0.04 5.6 0.03 4.2 Total 80 70 60 40 250 12.5 8.75 80 70 60 40 15 8.75