 A company can make two products P1 & P2. Profit: P1 â€“ 15, P2 â€“ 5 per unit.
Requirements per unit â€“
P1  P2  Maximum  
Process 1  20 min.  10 min.  1200 min. 
Process 2  40 min.  10 min.  1600 min. 
Solve by simplex method.
 An electronics company wants to decide most profitable product mix. Profit per unit of Transistors â€“ Rs. 120, Resistors â€“ Rs. 60, CRT â€“ Rs. 40.
Resource requirements per unit â€“
ProductÂ Â Â Â Â Â Â Â Â Â Engg. Services (Hrs.)Â Â Â Â Â Â Â Â Â Â Â Â Â Direct Labour (Hrs.)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Admin (Hrs.)
TransistorÂ Â Â Â Â Â Â 1.0Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 10.0Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 2.0
ResistorÂ Â Â Â Â Â Â Â Â Â 1.0Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 4.0Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 2.0
CRTÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â 1.0Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 5.0Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 6.0
TotalÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â 100.0Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 600.0Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 300.0
Solve by simplex.
 ABC Ltd. Can make two products P1 & P2.
Product  Casting Hrs. / unit  Finishing Hrs. / unit  Profit / unit  Max. Sales (units) 
P1  2  3  7  – 
P2  1  3  4  200 
Max. 390  Max. 810 
Find optimal product mix to maximize profit by simplex method.
 Max. Z = 20 x1 + 6 x2 + 8 x3
Subject to constraints â€“
8 x1 + 2 x2 +3 x3 <= 250
4 x1 + 3 x2 <= 150
2×1 + x3 <= 50
x1, x2, x3 >= 0
 Max. Z = 10 x1 + 6 x2 + 6 x3
Subject to constraints â€“
3 x1 + 2 x2 +2 x3 <= 250
2 x1 + 3 x2 +3 x3 <= 270
x1 <= 60
x1, x2, x3 >= 0
 Max. Z = 30 x1 + 25 x2
Subject to constraints â€“
x1 + 2 x2 <= 80
3 x1 + 2 x2 <= 120
x1, x2 >= 0
 Minimum requirements of diet are 4000 units of vitamins, 50 units of minerals & 1400 units of calories per day. Two products A & B are available at a price of Rs. 4 & Rs. 2 per unit. 1 unit of product A contains 200 units of vitamins, 1 unit of minerals & 40 calories. 1 unit of product B contains 100 units of vitamins, 2 units of minerals & 40 calories. Find optimal product mix to minimize total cost by simplex method.
 Minimize Z = 27 x1 + 30 x2
Subject to constraints â€“
(2/3) x1 + (1/2) x2 >= 10
(1/3) x1 + (1/2) x2 >= 6
x1, x2 >= 0
 Product A offers a profit of Rs. 25/ per unit and product B yields a profit of Rs. 40/ per unit. To manufacture the products â€“ leather, wood and glue are required in the amount shown below:
Product  Resources for one unit  
Leather
(in Kg.) 
Wood
(in sq. Mts.) 
Glue
(in lts.) 

A  0.50  4  0.2 
B  0.25  7  0.2 
Available resources include 2100kgs. Of leathers, 28000 sq. metres of wood and 1,400 litres of glue;
 State the objective function and constraints in mathematical form.
 Find the optimum solution.
 Which resources are fully consumed? How much of each resource remains unutilized?
 What are the shadow prices of resources?
 Using simplex method solve the following LPP & explain the solution.
Max. Z = 6 x1 â€“ 3 x2
Subject to constraints â€“
2 x1 â€“ x2 <= 2
x1 <= 4
x1, x2 >= 0
 The Simplex Tableau for a maximization problem of linear programming is given below:
Product Mix  X_{1}  X_{2}  S_{1}  S_{2}  Quantity
(b_{1})


C_{i}  X_{i}  
5

X_{2}  1  1  1  0  10 
0

S_{2}  1  0  1  1  2 
C_{i}  4  5  0  0  
Z_{i}  5  5  5  0  
C_{i} – Z_{i}  1  0  5  0 
Answer the following questions, giving reasons in brief:
 Is there solution optimal?
 Are there more than one optimal solution?
 Is the solution degenerate?
 Is the solution feasible?
 If S_{1 }is slack in machine A (in hours/ week) and S_{2} is slack in machine B (in hours/week), which of these machines is being used to the full capacity when producing according to this solution?
 How many units of two products X_{1} and X_{2} are being produced and what is the total profit?
 Machine A has to be shut down for repairs for 2 hours next week. What will be the effect on profits?
 How much would you prepare to pay for another hour (per week) of capacity each on machine A and machine B?
 A new product is proposed to be introduced which would require processing time of 45 minutes on machine A and 30 minutes on machine B. It would yield a profit of Rs. 3/ per unit.Â Do you think it is advisable to introduce this product?
 Zigma Electronics produces two models of electronic products using Resistors, Capacitors and Chips. The following table gives the entire Technological and other details in this regard:
Resource  Unit resource requirement  Maximum Availability  
Model 1  Model 2  
Resistor
Capacitor Chips 
2
2 0 
3
1 4 
1200
1000 800 
Unit profit (Rs).  3  4 
After formulating the above problem as a Linear Programming Problem the following optimal Simplex Solution table is obtained.
Profit
Coefficient 
Basis
Variables 
Solution
Values 
C:  3  4  0  0  0  
X:  _{X1}  _{X2}  _{S1}  _{S2}  _{S3}  
C  X  B  
3
0 4 
X_{1}
S_{3} X_{2} 
450
400 100 
1
0 0 
0
0 1 
1/4
2 _{1/3} 
Â¾
2 1/2 
0
1 0 

Z=Rs.1750  Z  3  4  5/4  0.25  0  
= CZ  0  0  5/4  0.25  0 
 Determine the value of each resource.
 In terms of optimal profit, determine the worth of one Resistor, one Capacitor and one Chip.
 Determine the range of the applicability of the shadow prices (duel prices) for each resource.
 If the available number of chips is reduced to 350 units, will you be able to determine the new optimum solution directly from the given information? Explain.
 Standard Manufacturers produce three products P, Q, and R which generate profits of Rs. 20/, Rs. 12/, and Rs. 8/ per unit. Three operations are needed for each product on three machines M_{1}, M_{2}, and M_{3}. The maximum working hours available for each of these three machines are 1300, 900 and 400 respectively. One of the Simplex Method Solution is given in the following table
c  x  b  20  12  8  0  0  0 
X_{1}  X_{2}  X_{3}  S_{1}  S_{2}  S_{3}  
0  S1  160  0  0  4/5  1  4/5  4/5 
12  X2  120  0  1  3/5  0  2/5  3/5 
20  X1  140  1  0  1/5  0  1/5  4/5 
Z  20  12  56/5  0  4/5  44/5  
= CZ  0  0  16/5  0  4/5  44/5 
On the basis of above table, answer the following questions:
 Which Machine is not fully utilized? If the balance working hrs. of this machine are shifted to M_{2}, what will be the effect on the solution?
 Retaining the optimality, find the range of working hrs. of the third Machine..
 Within what range of profit of each product, the solution will remain optimal?
 Keeping the Shadow Prices intact, find the range for the working hours of M_{2}.
 Without altering the optimality, is it possible to reduce the availability of the working hours of the M_{2} to 200 hours?
 If it is decided to increase the capacities of all three machines by 25% of their respective present capacities, what will be the new product mix?
 An engineering company BMS Ltd. Produces three products A, B, and C using three machines M1, M2 and M3. The resource constraints on M1, M2, and M3 are 100, 40, and 60 hours respectively. The profits earned by the products. A, B, and C are Rs. 2, Rs.5 and Rs.8 per unit respectively.Â A simplex optimal solution to maximize the profit is given below where x_{1}, x_{2} and x_{3} are quantities of products A,B, and C produced by the company and s_{1}, s_{2} and s_{3} represent the slack in the resources M1, M2, and M3.Â Study the solution given below and answer the following questions.
C  X
Variables in the basis 
x_{1}  x_{2}  x_{3}  s_{1}  s_{2}  s_{3}  b
Â 
5  X2  1/3  1  0  1/6  1/3  0  8/3 
8  X3  5/6  0  1  –  2/3  0  56/3 
0  S3  7/3  0  0  1/2  1/3  1  44/3 
= CZ  19/3  0  0  1/6  1/3  0 
 Indicate the shadow price of each resource. Which of the resources are abundant and which are scare?
 What profit margin for product A do you expect the marketing department to secure if it is to be produced, and justify your advice?
 Within what range, the profit of product B can change for the above solution to remain optimal?
 How would an increase of 10 hours in the resource M2 affect the optimality?
 If the company BMS Ltd. Wishes to raise production which of the three resources should be given priority for enhancement?
 A business problem is formulated and expressed below as an LPP. (Profit is in Rs. and Resources are in units).
Objective Function: Maximize Z = 80 X_{1} + 100 X_{2}
X_{1} + 2X_{2} < 750Â Â Â Â Â Â Â Â Â â€¦â€¦â€¦ (Resource 1)
5X_{1} + 4X_{2} < 1800Â Â Â Â Â Â â€¦â€¦â€¦ (Resource 2)
3X_{1} + X_{2} < 900Â Â Â Â Â Â Â Â Â Â â€¦â€¦â€¦ (Resource 3)
X_{1} X_{2} > 0
Simplex algorithm of LPP, applied to the above problem yielded following solutions.
Basis  Bi  
C_{b}  X_{b}  X1  X2  S1  S2  S3  
100  X_{2}  0  1  5/6  1/6  0  300 
80  X_{1}  1  0  2/3  1/3  0  120 
0  S_{3}  0  0  7/6  5/6  1  240 
C_{j}  80  100  0  0  0  
(= C_{j }– Z_{j} )  0  0  30  0  0 
 Answer the following questions with justification.
 Is the solution optimal and unique?
 Is the above solution infeasible?
 What is the maximum profit as per optimal solution?
 Which resources are abundant and which are scare as per optimal solution?
 Find out the range of coefficient of X1 in the objective function for which the above solution remains optimal.
Can you obtain the solution values of basic variables from the optimal solution when resource constraint (1) Changes to 800 units? If yes, find the new values of the basic variables.
Source: Vipin Saboo Tutorials
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