Transshipment Problem STRATEGIC RESOURCE ALLOCATION and PLANNING MGMT E-5050 Bamm Mining Company Transshipment Problem

Bamm Mining Company Bamm Mining Company is currently extracting rock from two mines. Once it is taken from the ground and loaded on a truck, it is sent to one of two plants for processing. The processed rock is then shipped to one of three builders’ supply stores, where it is sold for landscaping purposes. The cost of transportation, the supply available at each mine, and the processing capacity of each plant are given on the following table:

Daily Process Capacity Bamm Mining Company Cost per Ton for Shipping To Plant From Mine #1 #2 Daily Supply A $6.00 $8.00 320 tons B $7.00 $10.00 450 tons Daily Process Capacity 500 tons

Bamm Mining Company To From Plant #1 $13.00 $17.00 $20.00 #2 $19.00 The cost of shipping from each processing plant to each store and the daily demand are as follows: Cost per Ton for Shipping To From Plant Supply Store A Supply Store B Supply Store C #1 $13.00 $17.00 $20.00 #2 $19.00 $22.00 $21.00 Daily Demand 200 240 330

Bamm Mining Company REQUIREMENT: Formulate a linear program that can be used to determine how to meet the demands of the 3 supply stores at the least cost. Solve this by computer. How many tons should be shipped from each mine to each plant? How many tons should go from each plant to each store?

Bamm Mining Company Assume that the cost of processing the rock is $22.00 per ton at Plant 1 and $18.00 per ton at Plant 2. REQUIREMENT: Formulate a linear program to minimize the total processing and transportation cost. Solve this by computer. What is the optimal solution?

Solution Let : A1 = tons of ore from mine A to plant 1 B1 = tons of ore from mine B to plant 1 B2 = tons of ore from mine B to plant 2

Solution Let : X1 = tons shipped to Builder’s Home from plant 1 Y1 = tons shipped to Homeowners’ Hdqtrs from plant 1 Y2 = tons shipped to Homeowners’ Hdqtrs from plant 2 Z1 = tons shipped to Hardware City from plant 1 Z2 = tons shipped to Hardware City from plant 2

Solution Minimize cost = 6A1 + 8A2 + 7B1 + 10B2 + 13X1 + 19X2 + 17Y1 + 22Y2 + 20Z1 + 21Z2 subject to : A1 + A2 = 320 ( supply at mine A ) B1 + B2 = 450 ( supply at mine B ) A1 + B1 =< 500 ( capacity at plant 1 ) A2 + B2 =< 500 ( capacity at plant 2 )

Solution X1 + X2 = 200 ( demand at Builder’s Home ) Y1 + Y2 = 240 ( demand at Homeowners’ Hdqtrs ) Z1 + Z2 = 330 ( demand at Hardware City ) A1 + B1 = X1 + Y1 + Z1 Units shipped into plant 1 must equal units shipped out of plant 1 A2 + B2 = X2 + Y2 + Z2 Units shipped into plant 2 must equal units shipped out of plant 2 All variables => 0

Solution A1 = 50 tons of ore from mine A to plant 1 B1 = 450 tons of ore from mine B to plant 1 X1 = 200 tons shipped to Builder’s Home from plant 1 Y1 = 240 tons shipped to Homeowners’ Hdqtrs from plant 1 Z1 = 60 tons shipped to Hardware City from plant 1 Z2 = 270 tons shipped to Hardware City from plant 2 All other variables = 0 Minimum total cost = $19,160.00