Use branch and bound method to solve the following LPP:
Minimize: Z = 4x1 + 3x2
Subject to the constraints:
and are integers.
Minimize: ( Z = 4x_1 + 3x_2 ) Subject to: ( 5x_1 + 3x_2 \geq 30 ), ( x_1 \leq 4 ), ( x_2 \leq 6 ), ( x_1, x_2 \geq 0 ) and integers. Step 1: Solve the relaxed LPP (ignore integer constraints). Constraints: 1. ( 5x_1 + 3x_2 \geq 30 ) → ( x_2 \geq \frac{30 - 5x_1}{3} ) 2. ( x_1 \leq 4 ) 3. ( x_2 \leq 6 ) 4. ( x_1, x_2 \geq 0 ) Step 2: Find feasible region vertices. * Intersection of ( 5x_1 + 3x_2 = 30 ) and ( x_1 = 4 ): ( 5(4) + 3x_2 = 30 ) → ( 20 + 3x_2 = 30 ) → _________ ____ ____ _______ _____ ____ __________ ___ _____.
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निम्नलिखित LPP को हल करने के लिए एकधा विधि का प्रयोग कीजिए : का अधिकतमीकरण कीजिए
जबकि :
तथा ।
ख) निम्नलिखित LPP की द्वैती लिखिए: का न्यूनतमीकरण कीजिए
जबकि : और x3 अप्रतिबंधित है।
On an average 96 patients per 24-hour a day require the services of an emergency clinic. Also on an average, a patient requires 10 minutes of active attention. Assume that the facility can handle only one emergency at a time. Suppose that it costs the clinic ₹ 100 per patient treated to obtain an average servicing time of 10 minutes, and that each minute of decrease in this average time would cost ₹ 10 per patient treated. How much would have to budgetted by the clinic to decrease the average size of the queue from 1x1/3 patients to 1/2 a patient?
| Activity | Time |
|---|---|
| 1 – 2 | 4 |
| 1 – 3 | 1 |
| 2 – 4 | 1 |
| 3 – 4 | 1 |
| 3 – 5 | 6 |
| 4 – 9 | 5 |
| 5 – 6 | 4 |
| 5 – 7 | 8 |
| 6 – 8 | 1 |
| 7 – 8 | 2 |
| 8 – 10 | 5 |
| 9 – 10 | 7 |
निम्नलिखित LPP को ग्राफीय विधि से हल कीजिए : का अधिकतमीकरण कीजिए जबकि :
तथा ।
एक फर्म तीन उत्पाद A, B और C बनाती है। यह दो प्रकार I और II का कच्चा माल प्रयोग करती है जिसकी क्रमशः 5000 और 7500 इकाइयाँ उपलब्ध हैं। प्रत्येक उत्पाद की इकाइयों के लिए आवश्यक कच्चा माल नीचे दिया गया है :
| कच्चा माल | उत्पाद की प्रति इकाई आवश्यकता | ||
|---|---|---|---|
| A | B | C | |
| I | 3 | 4 | 5 |
| II | 5 | 3 | 5 |
तीनों उत्पादों की न्यूनतम माँग क्रमशः 600, 650 और 500 इकाइयाँ है। मान लीजिए कि A, B और C की प्रत्येक इकाई पर लाभ क्रमशः ₹ 50, ₹ 50 और ₹ 80 है। लाभ का अधिकतमीकरण करने वाली LPP सूत्रित कीजिए।
ख) निम्नलिखित LPP को ग्राफीय विधि से हल कीजिए : का अधिकतमीकरण कीजिए जबकि :
तथा ।
A manufacturing company needs 2500 units of a particular item every year. The company buys it at the rate of ₹ 30 per unit. The order processing cost for this item is estimated at ₹ 15 and the cost of carrying a item in stock comes to about ₹ 4 per year. The company can manufacture this item internally. In that case it saves 20% of the price of the product. However, it estimates a set-up cost of ₹ 250 per production run. The annual production rate would be 4800 units. However, the inventory holding costs remain unchanged.
एक परियोजना की विभिन्न गतिविधियाँ और उनकी अवधि निम्नलिखित सारणी में दी गई है
| फैक्ट्री | गोदाम | क्षमता | |||
|---|---|---|---|---|---|
| D | E | F | G | ||
| A | 42 | 48 | 38 | 37 | 160 |
| B | 40 | 49 | 52 | 51 | 150 |
| C | 39 | 38 | 40 | 43 | 190 |
| माँग | 80 | 90 | 110 | 160 |
Given units,
units,
units,
units,
units and
units. Is it an optimal solution to the transportation problem:
If not, modify it to obtain a better feasible solution.