This case covers the issue faced by a lumber company. Alabama Atlantic ships wood by train. However because of increasing shipping costs, the company is trying to figure out ways to reduce these costs by investigating other alternatives. The alternatives include: continue shipping by rail, switch to shipping by water and shipping by either rail or water depending which is less expensive for the particular route. We have used linear programming (Transportation Model) using excel solver to find the optimized solution by studying the given data. The results that we found suggest that we should use the option of shipping by rail or water however there are other factors that need to be considered also such as time sensitivity of goods, geographic location of the market, O.S & D (over, short, damage), weather and so on.
Every company wants to make profit and there are two ways to make profit: by selling most of their product at a profitable cost and by reducing their expenses. Logistics account for the major expenses of a company so our case study focuses on minimizing the transportation costs by using linear programming.
Allocating limited amount of resources among different projects or activities is a key issue that managers face. However this problem can be solved using linear programming which is a method of assigning resources in an optimized manner. Linear programming is a vital operations research tool that has been used to handle issues in financial, manufacturing and service organizations.
The Transportation problem is an application of the linear programming problem which deals with the transportation of goods from a specific source to a destination. It minimizes the total shipping cost by determining the amount of commodity to be transported from each source to each destination while keeping in mind the constraints on the supply and the demand of the commodity. The Transportation model can be applied to other areas as well other than just the delivery of a product, like employment scheduling, inventory control, job scheduling and so on.
Option 1: Continue shipping by rail.
Unit Cost by Rail ($1000s) to Market
= Quantity of wood moved by rail from Source i to Market j ,where i=1,2,3 and j=1,2,3,4,5.
Minimize Cost = 61+72+ 45+55+66+69+78+60+49+56+
Demand x11 + x12 + x13 + x14 + x15 = 15 x11 + x21 + x31 = 11 x21 + x22 + x23 + x24 +x25 = 20 x12 + x22 + x32 = 12 x31 + x32 + x33 + x34 + x35 = 15 x13 + x23 + x33 = 9
x14 + x24 + x34 = 10
x15 + x25 + x35 = 8
Xij>0 (where i=1,2,3 and j=1,2,3,4,5) Table 2
Excel Spreadsheet: Table 11 in appendix.
Minimize Cost = 61(6) + 45(9) + 69(2) + 49(10) + 56(8) + 59(3) + 66(12) = $2816 (thousands)
Network Distribution Model:
Option 2: Shipping by water
In Option 2, we consider to switch to shipping exclusively by water. And here we have the variables as followed:
Xij =Amount of woods shipped from source I to market j (i= 1,2,3; j= 1,2,3,4,5)
Cij = Unit Cost Needed from Source I to Market j (I = 1,2,3; j = 1,2,3,4,5)
Iij = Unit Cost Needed for investment from Source I to Market j (I = 1,2,3; j = 1,2,3,4,5)
Minimize Cost= (31+27.5) X11 + (38+30.3) X12+(24+23.8) X13+55X14+(35+28.5) X15+(36+29.3) X21+(43+31.8)X22+(28+27) X23+(24+25) X24+(31+26.5) X25+59X31+(33+28.3) X32+(36+27.5) X33+(32+26.8) X34+(26+24) X35
Supply Demand x11 + x12 + x13 + x14 + x15 = 15 x11 + x21 + x31 = 11 x21 + x22 + x23 + x24 +x25 = 20 x12 + x22 + x32 = 12 x31 + x32 + x33 + x34 + x35 = 15 x13 + x23 + x33 = 9
x14 + x24 + x34 = 10
x15 + x25 + x35 = 8 Table 3
Because shipping is unavailable from source