IEOR160 HW5

Department of Industrial Engineering&Operations Research

IEOR160:Operations Research I(Fall2014)

Homework5

Due:Friday,Oct10th

Question1.Sheldon and Amy are trying to divide a pile of items into two portions.The values of the items are $4,$14,$24,$34,$21,$41,$25,$3,$5,$32,and$14.They want to divide items in a fair manner,i.e.they wish to minimize the di?erence between the total values of the two portions.Formulate an integer programming to help them solve this problem.

Hint:minimize|x|is equivalent to

minimize t

subject to t≥x

t≥?x

Question 2.Solve the following problem by branch and https://www.360docs.net/doc/df7704911.html,e the graphical method to solve the LP relaxation at each node.

max?x1+4x2

s.t.?10x1+20x2≤22

5x1+10x2≤49

x1≤5

x1,x2≥0and integer

Question3.Consider the following0?1knapsack problem:

maximize12x1+11x2+8x3+6x4

subject to8x1+8x2+6x3+5x4≤20

x1,x2,x3,x4∈{0,1}

a)Solve the problem using the branch and bound method.Note that you will have to solve the linear programming

relaxation at each node by solving a continuous knapsack problem.

b)Identify as many cover inequalities as you can for this problem.

c)Note that the following constraint is implied in the knapsack problem:if x1and x2are both equal to1,then

both x3and x4must be0.Write the corresponding constraint(s).

Question4.Consider a?rm that produces a single product at two di?erent plants.At each plant,the marginal production cost decreases as the production quantity increases.Speci?cally,the?rst300units cost$5per unit to produce,the next200units cost$3per unit,and all units beyond the?rst500cost$2per unit(note that the economies of scale occur only within each plant–i.e.producing200units in Plant1and200units in Plant2 means the?rm still pays$5per unit for all400units.)

The?rm must meet demand at two customer locations.The demand at Customer1is750units and Customer2 is400.The per unit shipping costs from each plant to each location are given in the table below.

Customer1Customer2

Plant1 1.52

Plant2 3.51

Formulate an integer program for minimizing total cost.