1 Assignment Problem Assignment problem is also known as a special case of LP problem or transportation problem; with which unit of demand and supply is “1” Its LP formulation Our objective here is to determine its solution using heuristic algorithm – similar to what we did in the transportation lecture. (to p2) (to p3)
2 LP formulation Total Total 1 LP: Min 210Xar + 90Xaa + 180Xad + …… Xdc s.t. Xar+Xaa+Xad+Xac = 1 ; Xar+Xbr+Xcr+Xdr = 1 Xbr+Xba+Xbd+Xbc = 1 ; Xaa+Xba+Xca+Xda = 1 Xcr+Xca+Xcd+Xcc = 1 ; Xad+Xbd+Xcd+Xdd = 1 Xdr+Xda+Xdd+Xdc = 1 ; Xac+Xbc+Xcc+Xdc = 1 all Xij = 0 or 1 for i=a,b,c,d & j=r,a,d,c (to p1)
3 Heuristic algorithm Its logical flow: –We make use of the “opportunity cost” concept –It is defined as follows: How it works? (to p4)
4 Steps Step 1: F or each column/row, find its minimum cost and subtract from its respective column/row Step 2: D etermine its feasible solution by crossing rows/columns with most “0” values Step 3: S olution is obtained if total crossed lines = total numbers of rows/column Otherwise, select min cost of uncrossed cells and subtracting it from all uncrossed and add it to double crossed cells Step 4: R epeat step 4 until solution is obtained. Example (to p5)
5 Example Consider the following example: Step 1 : For row, select its min and subtract from them (to p6)
6 Step 1 Step 1: for column, select min cost and subtract from them Step 2: Determine its feasible solution (to p7)
7 Step 2 Step 3: Only 3 lines. No good since we need four lines Thus, we select the min cost for uncrossed = 15 We subtract them from uncrossed cells and add to it double crossed Which resulting as ………. (to p8)
8 Steps 3 & 4 Step 4: We have four line above, Stop. Optimal solution is obtained Solution is: or Important notes (to p9)
9 Important Note Note 1: It is a (nxn) matrix i.e. total supply= total demand If not, we add row/column to them Note 2: We assign a big value M to a route that is not feasible one How computer package works? Tutorial (to p10)
10 Tutorial Appendix B –37, 38, 40, 46
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