A manufacturing cell may have multiple sites in series reserved for a single machine type; the sites are created for a machine that is used to negate the intra-cell backtracking movements of parts, as demonstrated in section 5.4.5. The model presented in this section is used to assign a
1 4 2 5 3
Flow Direction 4 First Site Second Site
86 | P a g e single machine to sites within the cell when two possible sites are available. The problem of assigning multiple mobile machines to multiple sites in parallel is a topic for future research.
The following notation is applicable to the intra-cell machine assignment model:
Sets:
K= {1, 2, …,T} is the index set of time periods; k denotes a single time period and k ϵ K.
Decision variables:
xk is a zero-one variable indicating the assignment of the machine to the first site.
yk is a zero-one variable indicating the assignment of the machine to the second site.
ak is a zero-one variable indicating that the machine in the first site has been gained by reconfiguration in period k.
bk is a zero-one variable indicating that the machine in the first site has been removed by reconfiguration in period k.
pk is a zero-one variable indicating that the machine in the second site has been gained by reconfiguration in period k.
qk is a zero-one variable indicating that the machine in the second site has been removed by reconfiguration in period k.
Parameters:
αk is the volume of parts that will perform an intra-cell backtracking movement if a machine is not available in the first site in period k.
βk is the volume of parts that will perform an intra-cell backtracking movement if a machine is not available in the second site in period k.
ck is the cost per part, per intra-cell backtracking movement in period k.
rk is the cost of relocating a single machine within the cell in period k.
Nk is the maximum number of machines of the specified type that have been made available to the cell in period k; Nk ϵ {0, 1, 2}.
The objective of the model is to minimize the cost of backtracking flow and the cost of relocating a machine to negate the undesirable flow. The assignment models of sections 6.3 and 6.4 will propose the movements of machines in and out of cells. The number of machines of a given type that have been allocated to the cell in period k is indicated by the variable Nk. The challenge in representing the objective function is to impose a reconfiguration cost when a machine is relocated internally to the cell and not to impose a reconfiguration cost when sites xk or yk gain a machine from another cell. The cost of inter-cell machine relocation has already been taken into consideration by the previous inter-cell assignment models. The necessity to impose a reconfiguration cost only when a machine is relocated within the cell has resulted in the creation of additional variables ak, bk, pk and qk.
87 | P a g e The first summation of the objective function (equation 90) represents the backtracking flow cost if the machine in question is not made available in the first site, i.e. if xk = 0 the cost ckαkis imposed. The second summation of the objective function represents the backtracking flow cost if the machine in question is not made available in the second site. Note that if xk = 0 and yk = 0, no backtracking cost is incurred at all since the parts will be performing an inter-cell part movement. This cost is already accounted for by the inter-cell machine assignment models. The third summation represents a reconfiguration cost if a machine is removed from the second site and placed in the first site. The fourth summation represents a reconfiguration cost if a machine is removed from the first site and placed in the second site. A reconfiguration cost is therefore imposed if a machine is moved internally. If a machine is lost/gained in either site to/from another cell, no reconfiguration cost is incurred in this model since inter-cell machine relocation costs are already accounted for.
𝑚𝑖𝑛 Cost = ∑ 𝑐𝑘𝛼𝑘𝑦𝑘(1 − 𝑥𝑘)
𝑘∈𝐾
+ ∑ 𝑐𝑘𝛽𝑘𝑥𝑘(1 − 𝑦𝑘)
𝑘∈𝐾
+ ∑ 𝑟𝑘(𝑎𝑘𝑞𝑘)
𝑘∈𝐾
+ ∑ 𝑟𝑘(𝑝𝑘𝑏𝑘)
𝑘∈𝐾
(90)
Subject to:
𝑥𝑘− 𝑥(𝑘−1)− 𝑎𝑘+ 𝑏𝑘 = 0, ∀ 𝑘 ∈ 𝐾 (91) 𝑦𝑘 − 𝑦(𝑘−1)− 𝑝𝑘+ 𝑞𝑘= 0, ∀ 𝑘 ∈ 𝐾 (92)
𝑎𝑘+ 𝑏𝑘 ≤ 1, ∀ 𝑘 ∈ 𝐾 (93)
𝑝𝑘+ 𝑞𝑘 ≤ 1, ∀ 𝑘 ∈ 𝐾 (94)
𝑥𝑘+ 𝑦𝑘 = 𝑁𝑘, ∀ 𝑘 ∈ 𝐾 (95)
𝑥𝑘, 𝑦𝑘, 𝑎𝑘, 𝑏𝑘, 𝑝𝑘, 𝑞𝑘∈ {0,1} (96) The objective function does not add a charge for both a machine loss and gain at sites. In the inter- cell machine assignment models the assumption is that the cost of restarting a cell after reconfiguration is included in the cost per machine loss/gain. When an inter-cell machine relocation occurs, two cells must start-up under a new configuration, therefore a cost is charged to the cell that gains a machine and the cell that loses a machine. With respect to an intra-cell machine swap between sites in series, a cost is charged to the objective function once since only one cell is affected.
The first constraint of the model is a balance equation which will enforce ak =1 if a machine is gained in the first site and bk = 1 if a machine is removed from the first site. Likewise, the second constraint is also a balance equation that will cause pk and qk to indicate the gain or loss of a machine at the second site. The third and fourth constraints allow either a loss or gain to be
88 | P a g e reflected in each site in each period k. The third and fourth constraints ensure the correct functioning of the first and second constraints. The fifth constraint ensures that all machines that have been made available to the cell are allocated to the positions available. The final constraint restricts all variables to zero-one values.