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The Objective Function
The objective of this problem is different from the objectives in the previous examples in which only profit was Maximized (or cost minimized). In this problem profit is not Maximized, but rather the audience exposure is Maximized.
This objective function demonstrates that although a linear programming model must either Maximize or Minimize some objective, the objective itself can be in terms of any type of activity or valuation.
For this problem the objective of audience exposure is determined by summing the audience exposure gained from each type of advertising
Maximize Z = 20, 000 X 1 + 12, 000 X 2 + 9, 000 X 3
Where Z = the total number of audience exposures
20, 000 X 1 = the estimated number of exposures from television commercials
12, 000 X 2 = the estimated number of exposures from radio commercials
9, 000 X3 = the estimated number of exposures from newspaper ads
Model Constraints
The first constraint in this model reflects the limited budget of Rs. 100, 000 allocated for advertisement,
Rs. 15, 000 X 1 + 6, 000 X 2 + 4, 000 X 3 < 100, 000
where
Rs. 15, 000 X 1 = the amount spent for television advertising
6, 000 X 2 = the amount spent for radio advertising
4, 000 X 3 = the amount spent for newspaper advertising
The next three constraints represent the fact that television and radio commercials are limited to four and ten, respectively, while newspaper ads are limited to seven.
X 1 < 4 commercial
X 2 < 10 commercials
X 3 < 7 ads
The final constraint specifies that the total number of commercials and ads cannot exceed fifteen due to the limitations of the advertising firm:
X 1 + X 2 + X 3 < 15 commercials and ads
The complete linear programming model for this problem is summarized as
Maximize Z = 20, 000 X 1 + 12, 000 X 2 + 9, 000 X 3
Subject to
Rs. 15, 000 X 1 + 6, 000 X 2 + 4, 000 X 3 < Rs. 100, 000
X 1 < 4
X 2 < 10
X 3 < 7
X 1 + X 2 + X 3 < 15
X 1 , X 2 , X 3 > 0
Example 6 Transportation
The Philips Television Company produces and ships televisions from three warehouses to three retail stores on a monthly basis. Each warehouse has a fixed demand per month. The manufacturer wants to know the number of television sets to ship from each warehouse to each store in order to minimize the total cost of transportation.
Each warehouse has the following supply of televisions available for shipment each month.
Warehouse Supply (sets)
- Karachi 300
- Lahore 100
Islamabad 200
600
Each retail store has the following monthly demand for television sets:
Store Demand (sets)
A. Faisalabad 150 B. Peshawar 250 C. Hyderabad 200
600
The costs for transporting television sets from each warehouse to each retail store are different as a result of different modes of transportation and distances. The shiping costs per television set for each route are,
The three demand constraints are developed in the same way except that television sets can be supplied from any of the three warehouses. Thus, the amount shipped to one store is the sum of the shipments from the three warehouses:
X1A + X2A + X3A = 150
X1B + X2B + X3B = 250
X1C + X2C + X3C = 200
The complete linear programming model for this problem is summarized as:
Minimize Z = Rs. 6X1A+8X1B+1X1C + 4X2A + 2X2B + 3X2C + 3X3A + 5X3B + 7X3C
subject to X1A + X1B + X1C = 300
X2A + X2B + X2C = 100
X3A + X3B + X3C = 200
X1A + X2A + X3A = 150
X1B + X2B + X3B = 250
X1C + X2C + X3C = 200
Xij > 0
