Singher Cola is one of the most successful cola companies in Asia. The first Sin
ID: 458114 • Letter: S
Question
Singher Cola is one of the most successful cola companies in Asia. The first Singher Cola was produced in 1933, and since then, it has been exported to 50 countries worldwide. The cola is still a top seller in Thailand, and in one of its main distribution centers, a manager has been presented with a linear programming problem. One of the Singher factories in the north- ern part of Bangkok supplies four different small villages from three of its plants, and the unitary cost of each box for shipment (in US$) is presented in the table below.
There are other constraints that need to be considered when formulating the linear programming problem. Not all plants are able to supply the same quantity of cola required. According to the last estimation, Plant A can supply a maximum of 35 per day, Plant B can supply 50, and Plant C can supply 40. However, city A needs at least 45 every day, City B needs 20, and Cities C and D need 30 to fulfill demand. Therefore, the manager has to combine the production from the three plants to satisfy the demand while still keeping a reasonable shipping price.
Determine a suitable linear programming model that can address the issue and create its parameter table. (Hint: Formulating the problem as a transportation problem will help with the solution.)
Require Decison variable, Objecctive, Constraints
City A City B City C City D Plant A 8 6 10 9 Plant B 4 8 12 9 Plant C 14 9 16 5Explanation / Answer
The problem can be best modeled as Transportation problem. Let A, B, and C represents plants A, B, and C. Let 1, 2, 3, and 4 represents cities A, B, C, and D. The given problem of Singher cola can be represented in tabular form as shown below:
From/To
City A (1)
City B (2)
City A
(3)
City A
(4)
Supply
Plant A
8
6
10
9
35
Plant B
4
8
12
9
50
Plant C
14
9
16
5
40
Requirements
45
20
30
30
First check whether the total supply matches the total requirements for balance TP.
Total supply = 35 + 50 + 40 = 125
Total requirements = 45 + 20 + 20 + 30 = 125
Since total supply = total requirements = 125 boxes, the transportation problem is balanced.
Decision variable:
Manager of Singher Cola has to take decision regarding how much box should be shipped from particular plant to the particular city.
Let, xij be number of boxes to be shipped from Plant i (i = A, B, C) to the cities j (j = 1, 2, 3, 4)
xA1 = number of boxes shipped from plant A to city 1(A)
xA2 = number of boxes shipped from plant A to city 2(B)
xA3 = number of boxes shipped from plant A to city 3(C)
xA4 = number of boxes shipped from plant A to city 4(D)
xB1 = number of boxes shipped from plant B to city 1(A)
xB2 = number of boxes shipped from plant B to city 2(B)
xB3 = number of boxes shipped from plant B to city 3(C)
xB4 = number of boxes shipped from plant B to city 4(D)
xC1 = number of boxes shipped from plant C to city 1(A)
xC2 = number of boxes shipped from plant C to city 2(B)
xC3 = number of boxes shipped from plant C to city 3(C)
xC4 = number of boxes shipped from plant C to city 4(D)
Objective:
While shipping the boxes between the plant and cities the manager wants to keep shipping price at reasonable rate, in other words, managers objective is to minimize the shipping cost. The objective function for manger is formulated as:
Minimize Z = $8xA1 + $6xA2 + $10xA3 + $9xB4 + $4xB1 + $8xB2 + $12xB3 + $9xB4 + $14xC1 + $9xC2 + $16xC3 + $5xC4
Constraints:
For Singhar Cola, two physical constraints has to be considered. First, there is a limit on the number of boxes that can be shipped from processing plants. Singhar Cola can ship no more than 35, 50, and 40 boxes from Plants A, B, and C. These restrictions can be formulated as follows:
xA1 + xA2 + xA3 + xA4 35 Capacity Restriction for Plant A
xB1 + xB2 + xB3 + xB4 50 Capacity Restriction for Plant B
xC1 + xC2 + xC3 + xC4 40 Capacity Restriction for Plant C
Second limitation is to satisfy all the requirements of Cities A, B, C, and D. That is, all the 45, 20, 30 and 30 boxes requirement of Cities A, B, C, and D are to be satisfied. These restrictions can be formulated as follows:
xA1 + xB1 + xC1 = 45 Minimum Requirement at City A (1)
xA2 + xB2 + xC2 = 20 Minimum Requirement at City B (2)
xA3 + xB3 + xC3 = 30 Minimum Requirement at City C (3)
xA4 + xB4 + xC4 = 30 Minimum Requirement at City D (4)
To these limitation one more common restriction is that all xij should be non-negativity, that is, number of boxes cannot be negative.
xij 0
The model can be summarized as follows:
Minimize Z = $8xA1 + $6xA2 + $10xA3 + $9xB4 + $4xB1 + $8xB2 + $12xB3 + $9xB4 + $14xC1 + $9xC2 + $16xC3 + $5xC4
Subject to:
xA1 + xA2 + xA3 + xA4 35 Capacity Restriction for Plant A
xB1 + xB2 + xB3 + xB4 50 Capacity Restriction for Plant B
xC1 + xC2 + xC3 + xC4 40 Capacity Restriction for Plant C
xA1 + xB1 + xC1 = 45 Minimum Requirement at City A (1)
xA2 + xB2 + xC2 = 20 Minimum Requirement at City B (2)
xA3 + xB3 + xC3 = 30 Minimum Requirement at City C (3)
xA4 + xB4 + xC4 = 30 Minimum Requirement at City D (4)
xij 0, for all I and j
From/To
City A (1)
City B (2)
City A
(3)
City A
(4)
Supply
Plant A
8
6
10
9
35
Plant B
4
8
12
9
50
Plant C
14
9
16
5
40
Requirements
45
20
30
30