Chapter 6—Integer Linear Programming

PROJECT

74. Project 6.1- Air Operations Planning

As part of a joint airpower operations staff, you have been tasked to develop a new set of contingency plans for one squadron of F-1s, one squadron of F-2s, and one squadron of F-3s. The first mission you are given to plan involves four targets, two airfields (AF-1 and AF-2), and two command and control bunkers (CCB-1 and CCB-2). Each target must be attacked and destroyed to a particular level and these damage requirements are stated in terms of lbs of bombs on target. These levels have been defined as a minimum of 20,000 lbs on CCB-1, a minimum of 22,000 lbs on CCB-2, but no more than 40,000 lbs for either. The airfields must be damaged to a minimum of 10,000 lbs each, no more than 25,000 lbs each, and damage of up to 40,000 lbs on the two airfields combined.

There are certain constraints you have to take into account such as fuel requirements and bomb loads carried by each aircraft. Naturally, an aircraft burns fuel to and from the target as well as during time over the target. The following table summarizes these requirements:

AIRCRAFT	BOMBS (lbs)	Fuel mission to AF-1	Fuel mission to AF-2	Fuel mission to CCB-1	Fuel mission to CCB-2
F-1	1800	3500	3200	3900	3100
F-2	2200	3900	3400	3300	4200
F-3	2000	4600	4200	3000	3200

Planners have indicated you can plan on having 180,000 lbs of fuel on hand for the mission and an essentially unlimited supply of bombs, though your plan should indicate how many bombs are dropped in total and by each squadron. All fuel levels in the above table are provided in terms of lbs of fuel. Assume the fuel requirements indicated are for the complete trip.

There are 16 aircraft in each of the three squadrons. The F-1s and F-2s can fly up to two sorties each while the F-3s can support up to a total of 1.5 sorties each.

Additionally, there are two Unmanned Aerial Vehicles (UAVs) available for battle damage assessment, UAV-1 and UAV-2. Your plan will task these UAV assets to cover particular missions over the targets though you do not need to include details of the coverage (e.g., you will not worry about fuel, routing, etc.)

There are a total of 60 UAV missions available, 30 for each of UAV-1 and UAV-2. Due to differing capabilities, the "utility" of each UAV differs by target. The utility table is

	AF1	AF2	CCB1	CCB2
UAV-1	4	6	8	10
UAV-2	8	8	4	6

Additional constraints have been added (for a variety of reasons). These must be included in your analysis. These are:

(1)	F-3s can only hit 1 of the 4 targets; you decide which target that is and how many missions the F-3s fly against that target.
(2)	Any targets hit by F-1s are ONLY hit by F-1s.
(3)	All F-1 strikes must be photographed (by the UAVs).
(4)	Every strike, if photographed, is covered by UAV-1 or UAV-2, but there is to be no overlap of target coverage by the two assets (e.g., either UAV-1 or UAV-2 will cover AF-1, but not both).
(5)	All three aircraft types must be employed within the mission plan.

a.	Formulate this problem as a mixed integer programming problem.
b.	Implement the model in Excel and interpret the resulting solution to include the following items:
	1.	Total bombs required?
	2.	How many bombs are placed on each target?
	3.	Which target does the F-3 squadron service?
	4.	Which target does the F-1 squadron service?
	5.	How busy is each squadron in terms of both missions and bombs dropped?

75. Project 6.2- Dayton Electronics Manufacturing Inc. (DEMI)

The Dayton Electronic Manufacturing, Inc (DEMI) company manufactures two styles of remote keyless entry systems (the X30 and the X40) that various auto dealers supply to customers when a new (or used) automobile is purchased. DEMI currently operates four production facilities located in Springfield OH, Hartford, New Orleans, and Orlando. The manufactured items are shipped from the plants to regional distribution centers located in Trenton, Chicago and Seattle. It is from these regional locations that the product is distributed nationwide.

As more automobile manufactures include keyless entry as a standard option, and DEMI finds itself locked out of the manufacturer market, demand for DEMI's products have decreased. As a result, management is contemplating closing one or more of its production facilities. Distribution facilities are not currently being considered for closing.

Each production facility carries a fixed operating cost and a variable cost associated with building each of the products. Data has been compiled on production costs, resource availability, and resource usage at each of the production plants. That information is summarized in the table below.

Plant	Fixed Cost Per Month ($1000)	Production Cost (per 100) X30- X40	Production Time (hr/100) X30- X40	Available Hours per Month
Springfield	53	1100 1300	6 6	720
Hartford	38	1100 1250	7 8	780
New Orleans	25	1000 1000	5 5	530
Orlando	28	1200 1500	5 9	680

The entry systems are sold nationwide at the same prices: $24 for the X30 and $30 for the X40.

Current monthly demand projections at each distribution center for both products are given in the following table.

	Demand
	Trenton	Chicago	Seattle
X30	2200	3100	4000
X40	4500	5800	6000

The transportation costs between each plant and each distribution center, which are the same for either product, are shown in the following table:

Cost per 100	To
From	Trenton	Chicago	Seattle
Springfield	$200	$270	$450
Hartford	$100	$200	$700
New Orleans	$250	$240	$300
Orlando	$180	$220	$350

DEMI's problem is to

·	Determine which of the plants to close and which to keep open.
·	Determine the number of X30 and X40 to be produced at each plant.
·	Determine a shipping pattern from the plants to the distribution centers.
·	Maximize the net total monthly profit. If any plants were closed, what was the impact of the closing on profits?
·	Do not exceed the production capacities at any plant.

Formulate DEMI's problem as a fixed charge, integer program. Implement your model in Excel and solve the model to answer DEMI's questions.