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Chapter 7 Transportation, Assignment, and Transshipment Problems

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:

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Chapter 7 Transportation, Assignment, and Transshipment Problems

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

  2. 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

  3. 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

  4. 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

  5. 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?

  6. 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 ...

  7. 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.

  8. 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

  9. 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?

  10. 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

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

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