निम्नलिखित में से कौन-से कथन सत्य हैं और कौन-से असत्य हैं ? अपने उत्तर के पक्ष में एक संक्षिप्त उपपत्ति या प्रति-उदाहरण दीजिए :

(i) यदि एक उत्पाद बार-बार ऑर्डर किया जाता है, तो स्टॉक की पुनरावृत्ति लागत का जोखिम कम होता है।

(ii) एक पंक्ति निर्देश के हल के लिए, आगमन दर, सेवा दर से अधिक नहीं होनी चाहिए।

(iii) एक परियोजना नेटवर्क का क्रांतिक पथ उस परियोजना को पूरा करने में लगने न्यूनतम काल को दर्शाता है।

(iv) एक अधिकतमकरण LPP के आधार सुसंगत हल के लिए आवश्यक और अनिवार्य प्रतिबंध है कि सभी `Z_j - C_j >= 0`

(v) यदि द्वैती अपरिबद्ध है, तो आद्य का सुसंगत हल होता है।

Solve the following cost minimizing assignment problem.

In the inventory model with finite replenishment rate, if the replenishment rate is equal to consumption rate, the holding cost is 0.

An assignment problem can be considered as a special case of transportation problem.

An oil engine manufacture purchase lubricants at the rate of  50 per unit from a vendor. The requirement of these lubricants is 1800 units per year. The cost of placing an order is  40 and inventory carrying cost per rupee per year is only 20 paise. Find the Economic Order Quantity (EOQ). Also find the cycle time.

Customers come to a coffee shop at the average rate of 32 per day (8 hours a day) in Poisson pattern. The waiter employed to serve the customers has left the job. The owner of the shop wants to hire a new the job. The owner of the shop wants to hire a new waiter. Two applicants, Abdul and Raju, have applied for the post. The service times of Abdul and Raju are exponentially distributed with mean 12 minutes and 10 minutes, respectively. Abdul and Raju demand salaries of ₹ 135 and ₹ 165 per day, respectively. If no service is available, the average loss to the owner is ₹ 400 per day. Who among the two should be selected for the job?

Solve the following LPP by Simplex method:

such that

From the optimal table of the solution to the problem find the optimal solution of the dual of the problem. Verify complementary slackness property for the primal-dual pair.