Chapter 7 Transportation, Assignment, and Transshipment Problems Math 305 2008

Special Types of LPs Have used linear programming to formulate and solve two types of problem:   1. Determining optimal production processes (choosing a mix of products which maximize profits given limitations in resources) 2. Meeting minimum specifications (diet example: selecting foods which meet nutritional requirements at minimum cost. Others - allocating health care resources) Amazing feature of this model is its versatility: many different, seemingly unrelated, problems can be modeled as LPs

Transportation Problem A given (homogeneous) commodity is to be shipped from a set of different supply centers to another set of different destinations E.g. you make Honda Accords at 5 factories around the US and distribute the cars to 100 dealers this month E.g. tanks from 10 military bases in the US are to be shipped to Iraq, Afghanistan, and Bosnia

Transportation Problem Typical data format:   m supply centers n destinations si = amount of commodity at supply center i dj = amount of commodity required by destination j cij = cost to ship one unit from supply center i to destination j 1 2 ... n supply c11 c12 c1n s1 c21 c22 c2n s2 m cm1 cm2 cmn sm demand d1 d2 dn

Example Peas are canned in Washington, Oregon, and Minnesota and shipped to warehouses in California, Utah, South Dakota, and New Mexico warehouse cannery What are the decision variables? Objective function? Constraints? 1 2 3 4 supply 465 513 654 867 75 352 416 690 791 1 25 995 682 388 685 100 demand 80 65 70 85

AN LP! xij = amount shipped from source i to destination j minimize  cij xij (total shipping costs) subject to xij = si (use up all the supply)  xij = dj (meet all the demand) note: this assumes si =  dj  Special form:  x11 + x12 + ...+x1n = s1 x21 + x22 +...+ x2n = s2 ... xm1 + xm2... + xmn = sm   x11 + x21 + xm1 = d1 x12 + x22 + xm2 = d2

Cannery MINIMIZE 465 x11+513x12+654x13+867x14+352x21+416x22+690x23+791x24 +995x31+682x32+388x33+685x34 SUBJECT TO x11+x12+x13+x14 = 75 x21+x22+x23+x24 = 125 x31+x32+x33+x34 = 100 x11 +x21 +x31 = 80 x12 +x22 +x32 = 65 x13 +x23 +x33 = 70 x14 +x24 +x34 = 85.

Observations Solutions must be integers Special format can be exploited in solving the lp si >  dj can be handled by using a dummy destination Suppose warehouse 4 only wants 65 units warehouse cannery The special form of the problem leads to efficiencies in solving Won't cover 1 2 3 4 dummy supply 465 513 654 867 75 352 416 690 791 1 25 995 682 388 685 100 demand 80 65 70 20

Assignment Problem How do you assign n people to n jobs? Assume there is a cost, cij, associated with person i doing job j Job Person How is this a transportation problem? 1 2 ... n c11 c12 c1n c21 c22 c2n cn1 cn2 cnn

Assignment Problem Each person can do one job, si = 1 Each job is done by one person , dj = 1 cij = cost or benefit when person i does job j Job Person E.g. 5 students in a class divide a 5 problem exam 1 2 ... n supply c11 c12 c1n c21 c22 c2n cn1 cn2 cnn demand

Transshipment Problems Shipments are allowed to go through intermediate destinations Can be solved as a transportation problem Won't cover