After the Van earthquake, ABC Foundation decided to send tent, blanket, sleeping

Question

After the Van earthquake, ABC Foundation decided to send tent, blanket, sleeping bag and camping mat to aggrieved citizens. Three different cargo planes are available to send the materials. Each cargo plane has three compartments for storing cargo: front, center, and back. These compartments have capacity limits on both weight and space. Weights capacities of the compartments of the planes are shown in Table 1. Volume capacities are given proportional to the volume of one tent in Table 2. For example; front part of Plane 1 can carry at most 50 tents. It is assumed that the volume of one tent is equivalent to the total volume of five blankets. Additionally, one blanket and two sleeping bags have the same volume. The volume of one mat is equivalent to the volume of one sleeping bag.

Table 1: Weight capacities of planes (kg)
                            Plane 1            Plane 2        Plane 3
Front                          100                 90            150
Center                         160                 80             90
Back                              80                  70            70

Table 2: Total volume of planes (tent)
                                    Plane 1        Plane 2         Plane 3
Front                                 50              50                80
Center                                30             20                40
Back                                   40             30                50

Furthermore, the weight of the cargo in the respective compartments must be the same proportion of that compartment’s weight capacity to maintain the balance of the airplane. Moreover, since the sticks of the tents can damage, tents cannot be put on the center of Plane 1 and Plane 2. As mat and sleeping bag are used together, their amount in numbers should be the equal.

As much as tents, blankets, sleeping bags and camping mats are needed at the earthquake area. The number of products that ABC owns is shown in Table 3. The weight and benefit coefficient of each product are also given in Table 3. Product sequence according to benefit is “tent, blanket, sleeping bag and mat”.

Table 3: Unit benefit coefficients, weights and total number of available materials
                       benefit(per unit)         weight(kg/unit)        total amount
Blanket                    5                                  1                            500
Tent                       10                                 3                             100
Sleeping bag            4                                0.8                            200
Mat                         3                                0.5                            150

ABC wants to maximize the total benefit of the sent materials. Formulate a LP model for this problem (closed form with summation symbol,   ).

Explanation / Answer

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