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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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