Question

. Problem Two: Chelsea Furniture produces three lines of office chairs for corporate clients, model A, model B and model C. Given demand studies, the company expects the demand for model A to be at least 400 units per day. In contrast, the demand for model B is expected to be at least 250 units per day. The demand for model C is expected to be at least 175 units per day. Due to limitations in their manufacturing facility, the company cannot exceed 750 units for model A, 475 units for model B, and 315 units for model C. Based on pre-existing contracts, the company has a contractual obligation to send at least 1,135 units every day. In addition, to prevent brand dilution, the marketing department asks to produce at most 1,400 units in total. If the sale of each model A yields a $125 profit, the sale of each model B results in a $88 profit, and each sold model C unit contributes $78 in profits, (a) how many units of each model should the company produce to maximize daily profits? (b) Create a Sensitivity Report and state, both the Reduced Cost, and the Shadow Price, (c) interpret the significance of both, Reduced Cost and the Shadow Price, (d) suppose the company cannot exceed 751 units for model A, how does that impact your optimal profit? (e) Suppose the marketing department decides to produce at most 1401 total units, how does that impact your optimal profit in your original model?

Answer 1

Part A. number of units to be produced to get the maximum profit

Creating the equation in linear programming:

x=number of units of model A

y=number of units of model B

z=number of units of model C

Maximize : z(Profit)= 125x+88y+78z

with the following constraints: x+y+z >= 1135

x+y+z <=1400

x<=750

y<=475

z<=315

x>=400

y>=250

z>=175

Solving through the innear programming we get

x=750 units: y= 475 units: z= 175 units

Total max profit =149200

Part B: Sensitivity analysis using the linear programming

Iteration-7 MinRatio C 125 88 78 oooooooo * x3 x3 s S2 S3 S4 S. So S, Sp 225 0 0 0 1 0 -1 0 0 0 1 1 350 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 -1 0 1 1 0 0 0 -1 1400 oooooo 1 0 0 1 750 0 0 0 0 1 0 0 0 0 0 475 1 0 1 0 - 100001 1 75 0 0 1 0 0 0 0 0 0 0 -1 2 125 88 78 880 370 0010 Z-C,000 88037000010 125 78 Z = 149200

Reduced cost or oppurtunity cost

For model A( 125$)= next best alternative profitwise is model B(88$)

For model B( 88$)= next best alternative profitwise is model A(125$)

For model C( 178$)= next best alternative profitwise is model A(125$)

Shawdow cost : for every unit lost in Model A we will loose contribution of $37 that is determined by Cj-Zj in the final optimal solution.

Part C.

Reduced cost or opputunity cost is determined to know the cost of loosing the next best alternative .

whereas shadow cost tell the lose of contribution for every production unit lost and thats how it gonna impact the profit stucture of the company.

Part D.

Now the profit is determined by keeping the Model A at maximum 750 units. so there wont be any change in the solution but when we use the resources used for model C for Model A its cost more by 10$ which is the Cjj-Zj of the S8 in the solution and over all profit would be 37$ for the model A - 10$ additinal cost . over all profit can be impacted by 27$

Note: As there are mulitple sub parts and there is no specific requirement given only the first 4 parts were answered.