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O.J.'s Linear Programming Problem

Problem Description: O.J.'s Linear Programming Problem

In this problem we have to make a decision regarding the procurement of oranges from 10 suppliers. Each of the suppliers provides different grades of oranges. These different rates of oranges are used to make two kind of product the first kind of product is boxes of orange which is directly sold and second product is oranges which is first processed and then sold.

The revenue can be earned by selling both orange juice and boxes of oranges and revenue earned through per weight of orange juice is greater than the revenue earned by per weight of orange box.

The company has three processing plans the oranges are transported to these three processing plant from the suppliers and then the oranges are sorted and processed in this plants.

There are cost associated with the purchase of oranges, transport of oranges, sorting of oranges, processing of orange juice and recycling of oranges. The oranges which remain over and above the capacity of plant are then recycled.

In this problem our objective is to determine how much oranges we need to purchase from each of the given supplier so that they can be used for orange box orange juice. We also need to determine out of these oranges purchased, how many of the oranges we need to to transport to the given three plants.

Formulation of Problem

Assumptions

In this case our first assumption is that the total amount of oranges which are purchase from the supplier are either used to make Orange box or they are used to make orange juice.

The second assumption which we have assumed in this case is that all the oranges which are procured from the supplier reach to either plant a plant b or plant c. There are no losses of orange in the transportation

Decision Variable

In this case our decision variables are the number of oranges which are procured from suppliers. We also need to make decisions that how much of the oranges which are procured from supplier will be used to make Orange box and how much of those will be used to make orange juice. 

Second set of decision variable is how much of the oranges that are procured from a particular supplier will be transported to plant a, plant b and plant c.

Objective Function

In this case our objective function is to maximize the profit. The profit is calculated as revenue minus total cost. There are two components of total revenue the first component is is the revenue which is earned by selling Orange box and the second component is the revenue which is earned by selling orange juice. The total amount of orange which are procured for orange box are multiplied with the yield and then the resultant is further multiplied with the selling price in order to calculate the total revenue which is earned from the orange box. The total amount of orange which are procured for oranges are multiplied with the yield of orange juice and the resultant is further multiplied with the selling price in order to calculate the revenue that is earned from orange juice.

There are multiple cost associated with the production of orange box and orange juice. The first cost is cost of purchase of raw material, II cost is is the cost of transportation, the third cost is cost of salting oranges, IV cost is is cost of processing orange juice and fixed cost is recycling cost.

Constraints

There are multiple constraints in the processing of orange box and orange juice. The first constraint is that the total amount of oranges that are procured from the supplier cannot be greater than the capacity of the supplier. The second constraint is that the total oranges which are procured from a supplier are equal to the sum of oranges which are required for orange box and sum of oranges which are procured for oranges juice.

The third constant is that the total amount of oranges which are procured from supplier will be transported to either plant a, plant b or plant c.

 All the constraint is explained in detail below:

Constraints

     

Total weight of oranges purchased = total weight of oranges shipped

a1+b1+c1 = z1

   
 

a2+b2+c2=z2

   
 

…….

   
 

a3+b3+c3=z3

   

Orange bags + orange juice = total orange purchased

x1+y1=z1

   
 

x2+y2=z2

   
 

…….

   
 

x10+y10=z10

   

Orange purchased from supplier < capacity of supplier

Z1

<=

4000

 

Z2

<=

5000

 

Z3

<=

9000

 

Z4

<=

6000

 

Z5

<=

3000

 

Z6

<=

5500

 

Z7

<=

4500

 

Z8

<=

5000

 

Z9

<=

7000

 

Z10

<=

3500

Average quality of orange bag

6

 

Average quality of orange juice

7.5

 

Minimum purchase

∑Z>3000

   

Results: O.J.'s Linear Programming Problem

lb purchased total

Lb purchased for bag

lb purchased for juices

Lb purchased for bag

lb purchased for juices

Lb purchsed total

z1

x1

y1

3000

1000

4000

z2

x2

y2

2400

2400

4800

z3

x3

y3

5000

4000

9000

z4

x4

y4

3000

2500

5500

z5

x5

y5

1500

1500

3000

z6

x6

y6

2400

2800

5200

z7

x7

y7

2300

2200

4500

z8

x8

y8

2205

2290

4495

z9

x9

y9

3200

3800

7000

z10

x10

y10

1900

1600

3500

     

26905

24090

50995

Transported to A

Transported to B

Transported to C

Total

 

0

2000.000001

2000

4000

 

4800

0

0

4800

 

0

4500

4500

9000

 

0

0

5500

5500

 

0

1500

1500

3000

 

0

5200

0

5200

 

0

2250

2250

4500

 

0

4495

0

4495

 

0

3500

3500

7000

 

0

1750.000001

1750

3500

 

4800

25195

21000

   

 

Total cost

 

Purchase cost

11870

Sorting cost

5405.5

Transportation cost

4186.75

Juice transportation cost

1204.5

Processing cost

12326.5

Recycling cost

2250

Total cost

37243.25

Revenue

88928.625

Objective function

51685.375

It can be observed that the respected amounts which are mentioned in the table are required to be purchased from suppliers. These oranges should be use other to produce Orange box and orange juice. The respective amount of orange box and orange juice are also described.

 The total revenue earned is $ 88928, the total cost is $37243 and net profit is $51685.37

Summary of O.J.'s Linear Programming Problem

 Initially we have divided the product which is procured from the supplier in to the product which is used to to create Orange box and product which is used to create orange juice. In the second step it is decided how much of oranges is to be transported to each of given three plants.

In the second step all the necessary cost which are incurred in these processes are described and analyzed. In the third step revenue which is generated from the orange box and orange juice is computed on the basis of of their yield and selling prices.

In the third step all the necessary constituents are incorporated and the grades of oranges which are used to make Orange box and the grades of oranges which are used to make orange juice are calculated respectively.

In the final step all the values are fitted into to excel solver and the optimal solution is calculated from the excel solver.

References for O.J.'s Linear Programming Problem

Charnes, A., & Cooper, W. W. (1962). Programming with linear fractional functionals. Naval Research logistics quarterly9(3‐4), 181-186.

Dantzig, G. B. (1998). Linear programming and extensions (Vol. 48). Princeton university press.

Bazaraa, M. S., Jarvis, J. J., & Sherali, H. D. (2011). Linear programming and network flows. John Wiley & Sons.

Remember, at the center of any academic work, lies clarity and evidence. Should you need further assistance, do look up to our Management Assignment Help

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