ProblemsB$-25$

Problems

Game SitesOfficials

RaleighAtlantaDurhamClemsonDummy

A$210901801600$B$100701302000$C$1751051401700$D$80651051200$E$951151201000$

Table B$-39$

An Unbalanced AssignmentTableau with a Dummy Column

In solving this model,one team ofofficials would be assigned to the dummy column.If there were five games and only four teams ofofficials,a

dummy row

would be addedinstead ofa dummy column.The addition ofa dummy row or column does not affect thesolution method.

Prohibited assignments

are also possible in an assignment problem,just as prohibitedroutes can occur in a transportation model.In the transportation model,an

M

value wasassigned as a large cost for the cell representing the prohibited route.This same method isused for a prohibited assignment.A value of

M

is placed in the cell that represents the pro$-$hibited assignment.The steps ofthe assignment solution method are summarized here$\mathrm{.}1.$Perform row reductions by subtracting the minimum value in each row from all row values$\mathrm{.}2.$Perform column reductions by subtracting the minimum value in each column fromall column values$\mathrm{.}3.$In the completed opportunity cost table,cross out all zeros,using the minimumnumber ofhorizontal or vertical lines$\mathrm{.}4.$Iffewer than

m

lines are required $($where

m

$$

the number ofrows or columns$),$sub$-$tract the minimum uncrossed value from all other uncrossed values,and add thissame minimum value to all cells where two lines intersect.Leave all other valuesunchanged,and repeat step $3\mathrm{.}5.$If

m

lines are required,the tableau contains the optimal solution and

m

uniqueassignments can be made.Iffewer than

m

lines are required,repeat step $4.$

When demand exceeds supply,adummy row is added to theassignment tableau.

$1.$Green Valley Mills produces carpet at plants in St$.$Louis and Richmond.The carpet is then shippedto two outlets located in Chicago and Atlanta.The cost per ton ofshipping carpet from each ofthetwo plants to the two warehouses is as follows.

ToFrom

Chicago Atlanta

St$.$Louis$$40$$$65$Richmond$7030$

A

prohibited assignment

is givena large relative cost of

M

so that it will never be selected.

B$-26$Module BTransportation and Assignment Solution Methods

ToFrom A B C Supply

$12106$

$1$

$6006004153$

$2$

$40040097$

M

$3$

$3003001186$

$4$

$500300800000$

Dummy

$200200$

Demand

$9005009002,300$

The plant at St$.$Louis can supply $250$ tons ofcarpet per week;the plant at Richmond can supply $400$ tonsper week.The Chicago outlet has a demand of$300$ tons per week,and the outlet at Atlanta demands $350$tons per week.The company wants to know the number oftons ofcarpet to ship from each plant to eachoutlet in order to minimize the total shipping cost.Solve this tra