Slides14_15 - Lecture 14/15 Transportation Problems...

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1 Lecture 14/15 Transportation Problems Application Consider the following problem: A product is produced at 3 supply points 1,2,3, and is available in amounts 50,30,60, respectively. The product is demanded at 4 demand points 1,2,3,4 in amounts 40,50,10,40. Shipping costs c ij from ith supply point to jth demand point are given by the following array: 3 6 9 9 7 6 10 11 5 7 9 12 What is a minimum cost shipping plan for meeting all demands with available supply? 50 30 60 supplies demands 40 50 10 40 Example A
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2 General model We have: m supply centers i=1,2. ..,m and supplies s i , 1 i m; n demand points j=1,2,. ..,n and demands d j , 1 j n; unit shipping costs c ij , 1 i m, 1 j n. Task: Find min cost “feasible” shipment. (“feasible” means here that we do not undership or overship for each i,j-pair.) Linear programming formulation Variables: x ij for each i,j-pair. They denote how many units are shipped from i to j. ∑∑ == m i n j ij ij x c 11 min = m i j ij d x 1 , = n j i ij s x 1 , , 0 ij x s.t. for all j for all i for all i,j. (min total shipping cost) (meet demands) (no overshipping) (no negative shipping)
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3 Graphical representation 12 6 3 9 9 6 7 11 10 9 7 5 50 30 60 40 50 10 40 supply, i demand, j bipartite graph Graphical representation 12 6 3 9 9 6 7 11 10 9 7 5 50 30 60 40 50 10 40 supply, i demand, j 30 20 20 20 10 40 Feasible solution 1: Total cost: 3*20+6*30+6*20+10*10+5*20+12*40 = 1040
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4 Graphical representation 12 6 3 9 9 6 7 11 10 9 7 5 50 30 60 40 50 10 40 supply, i demand, j 10 40 10 30 10 40 Feasible solution 2: Total cost: 3*40+6*10+6*30+7*10+9*10+12*40 = 1000 Balanced problem ∑∑ == + = m i n j j i n d s d 11 1 > m i n j j i d s (The previous example is balanced.) If the problem is not balanced, then we convert it to a balanced problem in the following way: If Our transportation problem is balanced if: = m i n j j i d s then we add a fictitious demand node n+1 with Now, the problem is balanced. c i,n+1 is interpreted as storage cost at site i (0 in text). Example: 50 30 100 40 50 10 40 d 5 =40
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5 Balanced problem ∑∑ == + = n j m i i j m s d s 11 1 < m i n j j i d s (The previous example is balanced.) If the problem is not balanced, then we convert it to a balanced problem in the following way: If Our transportation problem is balanced if: = m i n j j i d s then we add a fictitious supply node m+1 with Now, the problem is balanced.
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Slides14_15 - Lecture 14/15 Transportation Problems...

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