I have add a custom column in customer grid by copying core **Grid.php** file to local folder. I want to show **last order date** of customer in a column. So I have tried below code in collection.

    protected function _prepareCollection()
        {
            $collection = Mage::getResourceModel('customer/customer_collection')
                ->addNameToSelect()
                ->addAttributeToSelect('email')
                ->addAttributeToSelect('created_at')
                ->addAttributeToSelect('group_id')
    			->addAttributeToSelect('track_device')
    			->addAttributeToSelect('inchoo_socialconnect_fid')
                ->addAttributeToSelect('inchoo_socialconnect_gid')
                ->joinAttribute('billing_postcode', 'customer_address/postcode', 'default_billing', null, 'left')
                ->joinAttribute('billing_city', 'customer_address/city', 'default_billing', null, 'left')
                ->joinAttribute('billing_telephone', 'customer_address/telephone', 'default_billing', null, 'left')
                ->joinAttribute('billing_region', 'customer_address/region', 'default_billing', null, 'left')
                ->joinAttribute('billing_country_id', 'customer_address/country_id', 'default_billing', null, 'left');
    			
    
           // add last_order_date column     
    			$sql ='SELECT MAX(o.created_at)'
    				. ' FROM ' . Mage::getSingleton('core/resource')->getTableName('sales/order') . ' AS o'
    				. ' WHERE o.customer_id = e.entity_id ';
    			$expr = new Zend_Db_Expr('(' . $sql . ')'); 
    		
    			$collection->getSelect()->from(null, array('last_order_date'=>$expr));
         
    	 
            $this->setCollection($collection);
    
            return parent::_prepareCollection();
        }

here is the column code--

    $this->addColumn('last_order_date', array(
    			'header'    => Mage::helper('customer')->__('Last Order Date'),
    			'type'      => 'datetime',
    			'align'     => 'center',
    			'index'     => 'last_order_date',
                'filter'    => true,
                'sortable'  => true ,     			
    			'gmtoffset' => true,        
            ));

column showing correct values but **Sorting** and **Filtering** not working for this column. What is wrong there.