Spring 2003 / AMBA / A. A. Elimam

“BUS886: STATISTCS AND OPERATIONS ANALYSIS”

Assignment 2: Linear Programming and Transportation

1. Consider the following problem [solve this problem on graph paper]

Max 4 X1 + 5 X2

Subject to: 2 X1 + 3 X2 <= 16

X1 + X2 = 6

X2 <= 4

X1>=0,X2 >= 0

(a) Solve this LP problem graphically (find optimum solution and its objective function value)

(b) Compute the slack and indicate whether the shadow price for each constraint = zero?

2. Two advertising media are being considered for promotion of a product. Radio ads cost $400 each while newspaper ads cost $600 each. The total budget is $ 7400 / week. The total number of ads should be at least 15, with at least 4 ad of each type. Each newspaper ad reaches 6,000 people while each radio ad reaches 2,000 people. The company wishes to reach the maximum number of people while meeting all constraints stated. How many ads of each type should be placed?

3.Consider the following LP model and Solver output.

Max. 5 Y + 8 Z
Subject to: / 2 Y + Z ≤ 10
Y + 3 Z ≤ 15
3 Y + 2 Z ≤ 24
Y, Z ≥ 0

Changing Cells

Final / Reduced / Objective / Allowable / Allowable
Cell / Name / Value / Cost / Coefficient / Increase / Decrease
$C$8 / Number Y / 3 / 0 / 5 / 11 / 2.333333333
$D$8 / Number Z / 4 / 0 / 8 / 7 / 5.5

Constraints

Final / Shadow / Constraint / Allowable / Allowable
Cell / Name / Value / Price / R.H. Side / Increase / Decrease
$E$10 / Labor LHS / 10 / 1.4 / 10 / 5 / 5
$E$11 / Machine LHS / 15 / 2.2 / 15 / 15 / 10
$E$12 / Material LHS / 17 / 0 / 24 / 1E+30 / 7
  1. What is the optimal objective function (OF) value?
  2. What is the largest value that the OF coefficient of Y can assume without changing the optimal solution?
  3. What is the smallest value that the OF coefficient of Z can assume without changing the optimal solution?
  4. What is the optimal OF value if the RHS of the second constraint decreases to 8?
  5. If an extra hour of labor costs $1, should we acquire this hour? Why?
  6. Would the solution change if
  • the OF coefficient of Z decreases to 6?Why?

or

  • the OF coefficient Y increases to 10? Why?

4. A farm family owns 80 acres of land & has $ 7,000 available for investment every year. Its members can produce a total of 3600 worker-hr. of labor during winter months & 4100 worker-hr. during summer. The family members can also work on a neighboring farm for $ 5/hr during the winter months and $ 7/hr during the summer. In addition to the salary they can make from working at the neighbors farms, income may also be obtained from three crops and two types of livestock: dairy cows and laying hens. Land, investment funds & worker-hours required as well as the annual income for the crops and livestock are as follows:

Type of Crop or Livestock /

Requirements

/ Income
$/year
Acres per year / Worker-hours / Cost
$/year
Winter / Summer
Cows / 1.5 / 50 / 30 / 1000 / 1800
Hens / 0 / 0.6 / 0.3 / 1 / 4
Soybeans/acre / 1 / 15 / 25 / 60 / 760
Corn/acre / 1 / 25 / 50 / 90 / 990
Oats/acre / 1 / 10 / 20 / 50 / 850

The chicken house can accommodate a maximum of 200 hens, and the size of the barn limits the herd to a maximum of 20 cows. Formulate and solve an to determine how much acreage should be planted in each of the crops and how many cows and hens should be kept, as well as the hours worked at the neighbors farms, to maximize the net annual income.

LP Formulation

W= Number of Cows to be kept in the farm

H= Number of Hens to be kept in the farm

Y= Number of acres planted with Soybeans

R= Number of acres planted with Corn

T= Number of acres planted with Oats

WH= Number of hours worked in neighboring farm in winter

SH= Number of hours worked in neighboring farm in summer

Maximize 800 W + 3 H + 700 Y + 900 R + 800 T +5 WH + 7 SH
Subject to
1.5 W + Y + R + T / <= 80
50 W + 0.6 H + 15 Y + 25 R + 10 T + WH / <= 3600
30 W + 0.3 H + 25 Y + 50 R + 20 T + + SH / <= 4100
1000 W + 1 H + 60 Y + 90 R + 50 T / <= 7000
W / <= 20
H / <= 200
W, H, Y, R, T, WH, SH / >= 0

5. A Company is considering Houston or Miami for a new plant. The existing plants are located in Boston City & San Francisco; they ship to 4 national warehouses in Chicago, Atlanta, Tulsa and San Diego. The unit costs of transportation, availability of goods, and the requirements are shown below:

City of
Origin /

Shipping costs/unit to warehouse in

/ Available units
Chicago / Atlanta / Tulsa / San Diego
Boston / $ 9 / $ 3 / $ 5 / $ 12 / 200
San Franc. / 6 / 9 / 8 / 2 / 300
Houston / 4 / 5 / 3 / 4 / 300
Miami / 8 / 5 / 7 / 10 / 300
Units Req. / 250 / 180 / 170 / 200

Please formulate the two transportation models required to locate the new plant in either Houston or Miami ? – [Hint: please make sure to define the variables, state the objective function, the capacity, demand and the non-negativity constraints].