Integer Programming
In this table, Xij is the number of units transported from ith warehouse to jth demand city.
TABLE 2: Unit Transportation Costs
The annual demand is:
Hence, it can be expressed as:
x11+x21+x31+x41+x51+x61=85x1,10+x2,10+x3,10+x4,10+x5,10+x6,10=7
The warehouses’ maximum capacity and fixed cost are:
The capacity constraint can be expressed as:
x11+x12++x1,10≤80x61+x62++x6,10≤70
Each city should be supplied by a single warehouse for no more than 60%. Hence:
x11,x21,,x61≤0.6*85x1,10, x2,10,,x6,10≤0.6*7
At least one of the following three warehouses should be opened: Phoenix, Austin, and/or Dallas). Hence, there should be at least 1 unit transported from at least one of these warehouses:
x11+x12++x1,10+x31+x32++x3,10+x41+x42++x4,10≥1
At most one of two (Austin or Dallas) may be opened. Hence:
x31+x32++x3,10*x41+x42++x4,10=0
Finally, all Xij should be non-negative.
The objective function is the total cost. The mathematical expression would be quite huge. This cost consists of the sum of two parts. The first part is the sum of the products of the corresponding elements in TABLE 1 and TABLE 2. The second part is fixed cost for all opened warehouses. This objective function should be minimized.
Executive Summary
In this project we have been asked by IMSE LogSmart management to develop a facility location and transportation plan for the year. In order to create an optimal plan, the total cost of transportation should be minimized, given to the constraints on demand, warehouses’ maximum capability and special restrictions on the optimal cost and design. The linear optimization problem has been developed. After solving this problem in software, we have obtained the following solution (Solver): The total cost of transportation is $27,104, when Phoenix, Indianapolis, Austin, Boston and Atlanta warehouses are opened and Dallas is closed. The numbers of items transported (from warehouses to demand cities) are given below in the table
This table should be interpreted in the following way: the company should transport 23 units from Phoenix to Los Angeles, 8 units – from Phoenix to San Diego and 4 units from Phoenix to Seattle. From Indianapolis, 30 units are supplied to New York, 16 – to Chicago and 4 – to Detroit. From Austin: 16 units are transported to Los Angeles. 13 units are transported to Houston, 6 units are transported to San Diego, 7 units are transported to Dallas, 4 units are transported to Jacksonville and 3 units are transported to Seattle. The Dallas warehouse is not opened and it does not provide transportation to any city. From Boston, 51 units are transported to New York and 9 units are transported to Philadelphia. Finally, from Atlanta, 4 units are transported to New York, 11 units are transported to Chicago, 9 units are transported to Houston, 7 units are transported to Philadelphia, 6 units are transported to Dallas, 5 units are transported to Jacksonville and 3 units are transported to Detroit.
This plan is the optimal transportation plan that provides the lowest possible total transportation cost. Since the optimal cost is less than $30,000, this solution will help the company’s management to receive a substantial benefit.
Appendix
Objective function:
241x11+37x12++142*x1,10++87x61+218x62++264x6,10++1990y1+2164y2+2690y3+2620y4+3396y5+3566y6→to min
Constraints:
x11+x12++x1,10≤80x61+x62++x6,10≤70
x11+x21+x31+x41+x51+x61=85x1,10+x2,10+x3,10+x4,10+x5,10+x6,10=7
x11,x21,,x61≤0.6*85x1,10, x2,10,,x6,10≤0.6*7
x11+x12++x1,10+x31+x32++x3,10+x41+x42++x4,10≥1
x31+x32++x3,10*x41+x42++x4,10=0
x11,x6,10≥0, integery1,y2=0, 1depending on opened warehouses
Works Cited
Solver. "Solver". N.p., 2016. Web. 23 Feb. 2016. http://www.solver.com/
