Linear Programming Ollies Oil Company in Texas has three oil

Linear Programming

Ollie\'s Oil Company in Texas has three oil wells with capacities of 93, 88 and 95 thousand barrels per day respectively. Ollie\'s Oil also owns five refineries along the Gulf Coast, all of which have been operating at stable demand levels. Three pump stations have been built to move the oil from the wells to the refineries. Oil can flow from any one of the wells to any of the pump stations and from any one of the pump stations to any of the refineries. The company is looking to develop a minimum cost schedule for their systems. The refineries\' requirements are as given the table 1. The company\'s cost accounting system recognizes charges by the segment of pipeline that is used. These daily costs are given in tables 2 in thousands of dollars per thousand barrels. Questions: a. Formulate a linear model to minimize the company\'s cost and put it in standard format. Clearly define your decisions variables, objective function and constraints

Solution

Formulation of the given problem is similiar to the standard Transportation problem (special type of LPP) but it has two phases. Firstly oil moves from three wells to any of the three pumps, second phase is movement from three pumps to any of the five Refineries. Therefore we need to find out how much quantity of oil from which well to which refinery through which pump should be moved so that the total cost of these movements is minimum subject to availability and requirement constraints mentioned herein.

Let me denote Qijk is the quantity of oil in thousand barrels (because quantities and cost are given in terms of thousand barrels) that moves from ith well to kth refinery through jth pump. i takes the values 1,2,3 for well1, well2 and well3. j takes the values 1,2,3 for pump1, pump2 and pump3. Similarly k takes the values 1,2,3,4,5 for the corresponding refinaries numbered 1,2,3,4 and 5 respectively. Therefore in all there are 3*3*5 =45 decision variables represented by Qijk (quantity in ,000 barrels from ith well to kth refinery through jth pump) all >= 0

The Objective function is minimization of total cost of movements between wells and pumps and between pumps and Refineries. Similar to decision variables, cost coefficients can be defined as Cijk as costs in thousands of dollor per thousand barrels movement/transfer from ith well to kth refinery through jth pump. Cijk = Cij(table1) + Cjk(table2) where Cij is cost from ith well to jth pump and Cjk is cost from jth pump to kth refinery ( data given in two tables uses the same units thousands of dollors per thousand barrels)

For example C111 = 1.52 + 5.15 = 6.67    C123 = 1.60 + 6.05 =7.65    C335 =1.30 + 5.87 = 7.17

Therefore Objective function is defined as Minimization of Total sum of Cijk * Qijk for i=1,2,3 ; j=1,2,3 and k=1,2,3,4,5

Now let us define the constraints about capacities of wells

Total sum of oil moved from well i will be sum of Qijk when i is kept constant and values of j and k varies.

Therefore Capacities constraints are Total sum of Q1jk <= 93 ; Total sum of Q2jk <=88 and Total sum of Q3jk <=95

Similarly requirement constraints of refineries are given as follows by keeping k constant and i and j varies

Total sum of Qij1 >=30 Total sum of Qij2 >= 57 Total sum of Qij3 >= 48 Total sum of Qij4 >=91

Total sum of Qij5 >=48